10. Digital Filters
12/20/20
Digital Signal Processing
Prof. Hesham Tolba
Alexandria University
Faculty of Engineering
Electrical Engineering Department
Alexandria
2020
Prof. Hesham Tolba
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Digital
Filters
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Digital Signal Processing
Prof. Hesham Tolba
Digital Signal Processing
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10. Digital Filters
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Table of contents
1.
General Considerations.
2.
Design of FIR Filters.
q Symmetric and Antisymmetric FIR Filters
q Design of Linear-Phase FIR Filters Using Windows
q Design of Linear-Phase FIR Filters by the Frequency-Sampling
Method.
q Design of Optimum Equiripple Linear-Phase FIR Filters.
3.
Design of IIR Filters from Analog Filters.
q IIR Filter Design by Approximation of Derivatives.
q IIR Filter Design by Impulse Invariance.
q IIR Filter Design by the Bilinear Transformation.
4.
Frequency Transformations.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
General Considerations
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Prof. Hesham Tolba
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Introduction
q In the filter design process
q The desired filter characteristics are specified in the
frequency domain in terms of the desired magnitude and phase
response of the filter.
q The coefficients of a causal FIR or IIR filter that closely
approximates the desired frequency response specifications
are determined.
q Depending on the nature of the problem and on the
specifications of the desired frequency response, FIR or IIR
filter is chosen.
q FIR filters are employed when there is a requirement for a
linear-phase characteristic within the passband of the
filter.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Introduction …
q If there is no requirement for a linear-phase characteristic,
either an IIR or an FIR filter may be employed.
q An IIR filter has lower sidelobes in the stopband than an FIR
filter having the same number of parameters.
q Thus, if some phase distortion is either tolerable, an IIR
filter is preferable.
q Its implementation involves fewer parameters,
q It requires less memory
q It has lower computational complexity.
q Today, FIR and IIR digital filter design is greatly facilitated
by the availability of numerous computer software programs.
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Prof. Hesham Tolba
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication
q The issue of causality is considered here by examining the
impulse response !(#) of an ideal LPF with frequency response
characteristic
%,
! " =$
(,
" ≤ "!
"! < " ≤ *
q The impulse response of this filter is
"!
,
*
+ , = " sin " ,
!
!
,
*
"!,
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Prof. Hesham Tolba
" ≤ "!
"! < " ≤ *
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
q A plot of
!(#) for %! = '/) is as shown.
Unit sample response of an ideal LPF.
q It is clear that the ideal LPF is noncausal and hence it cannot
be realized in practice.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
q
One possible solution is to introduce a large delay #" in !(#)
and arbitrarily to set ! # = * for # < #".
q
The resulting system no longer has an ideal frequency response
characteristic.
q
If we set ! # = * for # < #", the FS expansion of , % results in
the Gibbs phenomenon.
q
This is not limited to the realization of a LPF, but hold, for
all the other ideal filter characteristics.
q
Necessary and sufficient conditions that , % must satisfy for
the causality of the resulting filter is given by the PaleyWiener theorem.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
q
Paley-Wiener Theorem.
q If
! "
has finite energy and ! " = $ for " < $, then
"
) *+ & ' , ' < ∞
!"
if & ' is square integrable and if
the above integral is finite, then we can associate
with & ' a phase response Θ ' , so that the
resulting filter with frequency response
q Conversely,
& ' = & ' .#$
%
is causal.
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Prof. Hesham Tolba
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
q Causality imposes some tight constraints on an LTI system.
,# %
,
the
real
and
imaginary
components
of
the
frequency
,$ %
response , % .
q Causality also implies a strong relationship between
q This could be illustrated by decomposing
and
!(#) into an even and
an odd sequence, i.e.,
! # = !% # + !" #
where
!% # =
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.
! # + !(−#)
/
and
Prof. Hesham Tolba
!" # =
.
! # − !(−#)
/
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
+(,) is causal, it is possible to recover +(,) from its even
part +" , for ( ≤ , ≤ ∞ or from its odd component +# , for % ≤ , ≤ ∞.
q Now, if
q It can be easily seen that
and
! # = /!% # 1 # − !% * 2 # ,
#≥*
! # = /!" # 1 # + ! * 2 # ,
#≥.
+# ( = ( for , = (, we cannot recover + (
also must know + ( .
q Since
q In any case, it is apparent that
strong relationship between +# ,
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Digital Signal Processing
Prof. Hesham Tolba
+# , = +" ,
and +" , .
Prof. Hesham Tolba
Digital Signal Processing
from +# ,
and hence we
for , ≥ %, so there is a
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
!(#) is is absolutely summable (i.e., BIBO stable), the
frequency response , % exists, and
q If
, % = ,# % + 5,$ %
!(#) is real-valued and causal, the symmetry
properties of the FT imply that
q In addition, if
&
!% # ↔ ,# %
&
!" # ↔ ,$ %
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Causality & Its Implication …
q Since
, %
!(#) is completely specified by !% # , it follows that
is completely determined if we know ,# % .
q Alternatively,
q
, %
is completely determined from ,$ %
& !(*).
,# % and ,$ % are interdependent and cannot be specified
independently if the system is causal.
q Equivalently, the magnitude and phase responses of a causal
filter are interdependent and hence cannot be specified
independently.
,# % for a corresponding real, even, and absolutely
summable sequence !% # , we can determine , % .
q Given
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Digital Signal Processing
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Consider a stable LTI system with real and even impulse
response !(#). Determine , % if
. − 7 cos %
,# % =
,
. − /7 cos % + 7'
q Solution:
q The first step is to determine
noting that
7 < ..
!% # , which can be done by
,# % = ,# = <
()%!"
where
,# = =
. − 7 = + =*+ //
= − 7 =' + . //
=
. − 7 = + =*+ + 7' (= − 7)(. − 7=)
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The ROC has to be restricted by the poles at
should include the unit circle.
q Hence the ROC is
5$ = 6 and 5% = %/6 and
6 < 8 < %/ 6 .
+" , is a two-sided sequence, with 5$ = 6 contributing to the
causal part and 5% = %/6 contributing to the anticausal part.
q Thus
q By using a partial-fraction expansion, we obtain
q Substitution leads to
q The FT of
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Digital Signal Processing
Prof. Hesham Tolba
+(,) is
+" , =
% & %
6 + < (,)
:
:
+(,) = 6& 9(,)
! " =
%
% − 6>'()
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The relationship between the real and imaginary components of
the FT of an absolutely summable, causal, and real sequence can
be easily established from ! # = /!% # 1 # − !% * 2 # , # ≥ *.
q The FT relationship of this equation is
% ,
! " = !* " + ?!+ " = @ !* A B(" − A)CA − +" (
* ',
where >(%) is the FT of the unit step sequence 1(#).
q Although the unit step sequence is not absolutely summable, it
has an FT
B " = '2 " +
%
%
%
"
= '2 " + − ? cot
,
'()
%−>
:
:
:
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Prof. Hesham Tolba
−* ≤ " ≤ *
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Using the previous two equations and carrying out the
integration, we obtain the relation between ,# %
and ,$ %
as
. ,
%−@
,$ % =
? ,# @ cot
B@
/' *,
/
,$ % is uniquely determined from ,# %
integral relationship.
q Thus
through this
q The integral is called a discrete Hilbert transform.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Summary
q Implications of causality in the design of frequency-selective
filters are as follows
a) , %
cannot be zero, except at a finite set of points in
frequency;
b)
, % cannot be constant in any finite range of frequencies
and the transition from passband to stopband cannot be
infinitely sharp [this is a consequence of the Gibbs
phenomenon, which results from the truncation of !(#) to
achieve causality];
c) ,# %
and ,$ % are interdependent and are related by the
discrete Hilbert transform.
q As a consequence,
, %
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and C %
Prof. Hesham Tolba
cannot be chosen arbitrarily.
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Summary …
, % and the
fact that ideal filters are not achievable in practice,
attention will be limited to the class of LTI systems specified
by the difference equation
q Knowing the restrictions that causality imposes on
.
0
D # = − E 7- D # − F + E G- H # − F
-)+
-)/
which are causal and physically realizable.
q Such systems have a frequency response
*12∑0
-)/ G- J
, % =
*12. + ∑.
-)+ 7- J
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Digital Signal Processing
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10. Digital Filters
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Summary …
q The basic digital filter design problem is to approximate any
of the ideal frequency response characteristics with a system
that has the frequency response
, % =
*12∑0
-)/ G- J
*12. + ∑.
-)+ 7- J
by properly selecting the coefficients {7- } and {G- }.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Characteristics of Practical Frequency-Selective Filters
q Ideal filters are noncausal and hence physically unrealizable for
real-time signal processing applications.
! " of the filter cannot be zero, except at a
finite set of points in the frequency range.
q Causality implies that
! " cannot have an infinitely sharp cutoff from
passband to stopband.
q In addition,
q Although the frequency response characteristics possessed by ideal
filters may be desirable, they are not absolutely necessary in most
practical applications.
q If we relax these conditions, it is possible to realize causal
filters that approximate the ideal filters as closely as we desire.
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Digital Signal Processing
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10. Digital Filters
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Characteristics of Practical Frequency-Selective Filters …
q It is not necessary to insist that the magnitude
, %
be
constant in the entire passband of the filter.
q A small amount of ripple
in the passband, as
shown, is tolerable.
q It is not necessary for
, % to be zero in the
stopband.
q A small, nonzero value or
a small amount of ripple
in the stopband is also
tolerable.
Magnitude characteristics of physically realizable filters.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Characteristics of Practical Frequency-Selective Filters …
q The transition of the frequency response from passband to
stopband defines the transition band of the filter.
q The band-edge frequency
while the frequency %4
%3 defines the edge of the passband,
denotes the beginning of the stopband.
q The width of the transition band is
%4 − %3; it is usually
called the bandwidth of the filter.
q For example, if the filter is lowpass with a passband edge
frequency %3, its bandwidth is %3.
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Digital Signal Processing
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10. Digital Filters
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Characteristics of Practical Frequency-Selective Filters …
q If there is ripple in the passband, its value is denoted as
and the magnitude , %
2+ ,
varies between the limits . ± 2+ .
q The ripple in the stopband is denoted as
2' .
q To accommodate a large dynamic range in the graph of the
frequency response of any filter, it is common practice to use
a logarithmic scale for the magnitude , % .
q Consequently, the ripple in the passband is
/* log +/ 2+ dB, and
that in the stopband is /* log +/ 2' dB.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Characteristics of Practical Frequency-Selective Filters …
q In
1.
2.
3.
4.
any
the
the
the
the
filter design problem we can specify
maximum tolerable passband ripple 2+ 567 ,
maximum tolerable stopband ripple 2' 567 ,
passband edge frequency %3, and
stopband edge frequency %4 .
{7- }
and {G- } in the frequency response characteristic, which best
approximate the desired specification.
q Based on these specifications, we can select the parameters
, % approximates the specifications
depends on the criterion used in the selection of the filter
coefficients {7- } and {G- } as well as on the numbers (P, Q) of
coefficients.
q The degree to which
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Digital Signal Processing
Prof. Hesham Tolba
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Digital Signal Processing
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10. Digital Filters
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of FIR Filters
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters
G with input H(,) and output I(,) is
described by the difference equation
q An FIR filter of length
I , = K# H , + K$ H , − % + ⋯ + K. H , − G + %
.'$
= N K- H , − O
-/0
where {K- } is the set of filter coefficients.
q Alternatively, we can express the output sequence as the convolution
of the unit sample response +(,) of the system with the input signal,
thus we have
.'$
I , = N +(O)H , − O
-/0
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
q The last two equations are identical in form and hence it
follows that G- = ! F , F = *, ., … , P − ..
q The filter can also be characterized by its system function
.'$
! 8 = N +(O)8'-/0
q The roots of
, =
constitute the zeros of the filter.
q An FIR filter has linear phase if its unit sample response
satisfies the condition
+ , = ±+(G − % − ,),
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# = *, ., … , P − .
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
q Incorporating the above-defined symmetry and antisymmetry
*conditions in , = = ∑0*+
leads to
-)/ !(F)=
! 8 = + ( + + % 8'$ + + : 8'% + ⋯ + +(G − :)8'(.'%) + +(G − %)8'(.'$)
=
8'(.'$)/%
G−%
+
+
:
(.'4)/%
N
+ , 8(.'$'%-)/% ± 8'(.'$'%-)/%
,
G
QRR
G
STSU
&/0
.⁄% '$
=
8'(.'$)/%
N
+ , 8(.'$'%-)/% ± 8'(.'$'%-)/% ,
&/0
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Digital Signal Processing
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10. Digital Filters
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
*and multiplying both
=*+ for = in , = = ∑0*+
-)/ !(F)=
*(0*+)
sides of the resulting equation by =
, leads to
q substituting
8'
.'$
! =*+ = ±!(8)
q This result implies that the roots of
,(=) are identical to the
roots of , =*+ .
,(=) must occur in reciprocal pairs,
is a root or a zero of ,(=), then ./=+ is also a
q Consequently, the roots of
i.e., if =+
root.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
!(#) of the filter is real,
complex-valued roots must occur in complex-conjugate pairs.
q If the unit sample response
=+ is a complex-valued root, =∗+
is also a root.
q Hence, if
=* 0*+ , =*+ = ±,(=),
,(=) also has a zero at ./=∗+ .
q As a consequence of
q The shown figure illustrates the symmetry
that exists in the location of the zeros of
a linear-phase FIR filter.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
, = on the unit circle, yields the expression for
.
, %
q When ! # = !(P − . − #), , % can be expressed as
q Evaluating
, % = ,; % >'(2 .'$ /%
where ,; % is a real function of % and can be expressed as
G−%
!6 " = +
+:
:
.⁄% '$
!6 " = :
N
(.'4)/%
N
&/0
+ , VQW "
&/0
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+ , VQW "
G−%
−, ,
:
G−%
−, ,
:
Prof. Hesham Tolba
G
QRR
G
STSU
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
q The phase characteristic of the filter for both
G−%
,
:
X " =
G−%
−"
+ *,
:
−"
q When
G odd and G even is
YZ !6 " > (
YZ !6 " < (
+ , = −+(G − % − ,), the unit sample response is antisymmetric.
G odd, the center point of the antisymmetric + ,
consequently,
q For
!
q If
P−.
=*
/
G is even, each term in + ,
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Digital Signal Processing
Prof. Hesham Tolba
is , = (G − %)/:,
has a matching term of opposite sign.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
q The frequency response of an FIR filter with an antisymmetric
unit sample response can be expressed as
, % = ,; % J1 *12(0*+)/'=,/'
where
(0*?)/'
,; % = /
E
! # STU %
P−.
−# ,
/
P
VWW
! # STU %
P−.
−# ,
/
P
XYXU
>)/
0⁄' *+
,; % = /
E
>)/
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
q The phase characteristic of the filter for both
G odd and G even
is
'
P−.
−%
,
/
C % = /
\'
P−.
−%
+ ',
/
/
TZ ,; % > *
TZ ,; % < *
q These general frequency response formulas can be used to design
linear-phase FIR filters with symmetric and antisymmetric unit
sample responses.
+ , : the number of filter coefficients that
specify the frequency response is (G + %)/: when G is odd or G/:
when G is even.
q For a symmetric
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
q For a symmetric
! # ,
!
P−.
=*
/
antisymmetric
there are (P − .)// filter coefficients when P is odd and P//
coefficients when P is even to be specified.
q The choice of a symmetric or antisymmetric unit sample response
depends on the application.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Symmetric & Antisymmetric FIR Filters …
+ , = −+(G − % − ,) and G is odd ➞ !6 ( = ( and !6 * =
( ➞ not suitable as either an LPF or an HPF.
q For example, if
G even also
results in !6 ( = ( ➞ we would not use the antisymmetric condition in
the design of a LP linear-phase FIR filter.
q Similarly, the antisymmetric unit sample response with
+ , = +(G − % − ,) yields a linear-phase FIR
filter with a nonzero response at " = (, if desired, i.e.,
q The symmetry condition
G−%
!6 ( = +
+:
:
(.'4)/%
(.'4)/%
!6 ( = :
N
N
+ , ,
G
QRR
G
STSU
&/0
+ , ,
&/0
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Summary
q In summary, the problem of FIR filter design is simply to
determine the P coefficients ! # , # = *, ., … , P − ., from a
specification of the desired frequency response ,A % of the
FIR filter.
q The important parameters in
the specification of ,A %
as shown.
are
q In the following subsections,
design methods based on
specification of ,A % will be
described.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows
q We begin with the desired frequency response specification
,A %
q
!A #
and determine the corresponding !A # .
is related to ,A %
by the FT relation
8
!7 " = N +7 , >'()&
&/0
where
+7 , =
% ,
@ ! " >()& C"
:* ', 7
,A % , we can determine the unit sample response
by evaluating the above integral.
q Thus, given
!A #
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
+7 , is infinite in duration and must be truncated at
some point, say at , = G − %, to yield an FIR filter of length G .
q The obtained
+7 , to a length G − % is equivalent to multiplying
by a “rectangular window”, defined as
q Truncation of
+7 ,
%,
\ , =$
(,
, = (, %, … , G − %
Q^_S`aYWS
q Thus the unit sample response of the FIR filter becomes
+ , = !A # \ ,
* + ,
, = (, %, … , G − %
=$ #
(,
Q^_S`aYWS
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Digital Signal Processing
41
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
!7 " with b "
the (truncated) FIR filter, i.e.,
q The convolution of
!" =
where b "
is
yields the frequency response of
% ,
@ ! c b " − c Cc
:* ', 7
.'$
b " = N \ , >'()&
&/0
q The FT of the rectangular window is
.'$
b " = N >'()& =
&/0
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Digital Signal Processing
Prof. Hesham Tolba
% − >'().
sin "G/2
= >'()(.'$)/%
'()
%−>
sin "/2
Prof. Hesham Tolba
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q This window function has a magnitude response
] %
=
sin %P/2
,
sin %/2
−' ≤ % ≤ '
and a piecewise linear phase
P−.
,
/
C % =
P−.
−%
+ ',
/
−%
12/20/20
bcXU sin %P/2 ≥ *
bcXU sin %P⁄2 < *
Prof. Hesham Tolba
Digital Signal Processing
43
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The magnitude response of the window function is as shown for
P = \. and e..
q The width of the main lobe is
)'/P ➞ as P increases, the main
lobe becomes narrower.
] % are
relatively high and remain
unaffected by an increase in P.
q The side lobes of
P, the width
of each side lobe decreases & its
height increases in such a manner
that the area under each sidelobe
remains invariant to changes in P.
q With an increase in
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Digital Signal Processing
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q This characteristic behavior is not evident from observation of
the figure because ] % has been normalized by P such that the
normalized peak values of the sidelobes remain invariant to an
increase in P.
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Digital Signal Processing
45
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The characteristics of the rectangular window play a
significant role in determining the resulting frequency
response of the FIR filter obtained by truncating !A # to
length P.
q The convolution of
,A % .
,A %
with ] %
has the effect of smoothing
P is increased, ] % becomes narrower, and the smoothing
provided by ] % is reduced.
q As
] % result in some undesirable ringing
effects in the FIR filter frequency response , % , and also in
relatively larger sidelobes in , % .
q The large sidelobes of
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Prof. Hesham Tolba
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q These undesirable effects are best alleviated using windows
that
q do not contain abrupt discontinuities in their time-domain
characteristics, and
q have correspondingly low sidelobes in their frequency-domain
characteristics.
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Digital Signal Processing
47
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q Several window functions that possess desirable frequency
response characteristics are listed as shown.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The time-domain characteristics of the windows are shown below.
Rectangular
Hamming
Hanning
Blackman
T
K
B
L
Shapes of several window functions.
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Prof. Hesham Tolba
Digital Signal Processing
49
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The frequency response characteristics of the Hanning is as
shown below.
Frequency responses of Hanning window for (a) , = ./ and (b) , = 0/.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The frequency response characteristics of the Hamming is as
shown below.
Frequency responses of Hamming window for (a) , = ./ and (b) , = 0/.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The frequency response characteristics of the Blackman is as
shown below.
Frequency responses of Blackman window for (a) , = ./ and (b) , = 0/.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q All of these window functions have significantly lower side
lobes compared with the rectangular window.
P, the width of the main lobe is also
wider for these windows compared to the rectangular window.
q For the same value of
q Consequently, these window functions provide more smoothing
through the convolution operation in the frequency domain, and
as a result, the transition region in the FIR filter response
is wider.
q To reduce the width of this transition region, we can simply
increase the length of the window, which results in a larger
filter.
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Digital Signal Processing
53
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR filters Using Windows …
q The shown table summarizes these important frequency-domain
features of the various window functions.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Suppose that we want to design a symmetric LP linear-phase FIR filter
having a desired frequency response
%>'()
!7 " = $
(,
.'$ ⁄% ,
( ≤ " ≤ "!
Q^_S`aYWS
(G − %)/: units is incorporated into !7 "
of forcing the filter to be of length G.
q A delay of
in anticipation
q The corresponding unit sample response, obtained by evaluating the
$
,
integral in +7 , = %, ∫', !7 " >()& C", is
% )$ ()
+7 , =
@ >
:* ')$
&'.'$
% 9% C"
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=
,−/
5
,
,−/
" +− 5
sin %$ + −
Prof. Hesham Tolba
noncausal & infinite
in duration
+≠
,−/
5
Digital Signal Processing
55
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Clearly,
!A #
q Multiplying
is noncausal and infinite in duration.
!A #
by the rectangular window sequence,
f # =g
.,
*,
# = *, ., … , P − .
VhcXibTSX
leads to an FIR filter of length P having the unit sample
response
+ , =
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Digital Signal Processing
Prof. Hesham Tolba
,−/
5
,
,−/
" +− 5
89: %$ + −
Prof. Hesham Tolba
; ≤ + ≤ , − /,
Digital Signal Processing
+≠
,−/
5
56
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q If
P is selected to be odd, the value of ! #
P−.
%!
=
/
'
q The magnitude of the frequency response , %
as shown below for P = e. and P = .*..
at # = P − . ⁄/ is
!
of this filter is
LPF designed with a
rectangular window:
(a) , = 0/ and (b) , =
/;/.
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Digital Signal Processing
57
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations
q Relatively large oscillations (ripples) occur near the band
edge of the filter.
q The oscillations increase in frequency as
P increases, but
they do not diminish in amplitude.
q These large oscillations are the direct result of the large
side lobes existing in ] %
of the rectangular window.
,A % , the
oscillations occur as the large constant-area side lobes of
] % move across the discontinuity that exists in ,A % .
q As this window function is convolved with
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
*12>
is basically a FS representation of
,A % = ∑B
>)/ !A # J
,A % , the multiplication of !A # with a rectangular window is
identical to truncating the FS representation of the desired
filter characteristic ,A % .
q Since
, %
due to the nonuniform convergence of the FS at a discontinuity.
q The truncation of the FS is known to introduce ripples in
q The oscillatory behavior near the band edge of the filter is
called the Gibbs phenomenon.
q A window function that contains a taper and decays toward zero
gradually should be used to alleviate the presence of large
oscillations in both the passband and the stopband.
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Digital Signal Processing
59
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q The shown figures
illustrate , % of the
resulting filter when
some of the window
functions are used to
taper !A # .
LP FIR filter designed with
rectangular window (. = ;$).
q The window functions
eliminate the ringing
effects at the band edge
and result in lower
sidelobes at the expense
of an increase in the
width of the transition
band of the filter.
12/20/20
Digital Signal Processing
Prof. Hesham Tolba
LP FIR filter designed with
Blackman window (. = ;$).
Prof. Hesham Tolba
Digital Signal Processing
LP FIR filter designed with
Hamming window (. = ;$).
LP FIR filter designed with
< = = Kaiser window (. = ;$).
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method
q
In this method, we specify the desired frequency response ,A %
at a set of equally spaced frequencies
%- =
q
q
/'
F+j ,
P
P−.
,
/
P
F = *, . … , − .,
/
.
j=*
Vi
/
F = *, . … ,
P VWW
P XYXU
and solve for ! # of the FIR filter from these %-.
To reduce sidelobes, we optimize the frequency specification in
the transition band of the filter.
This optimization can be accomplished by linear programming.
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Digital Signal Processing
61
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
q The basic symmetry property of the sampled frequency response
function is exploited to simplify the computations.
q We begin with the desired frequency response of the FIR filter
0*+
, % = E ! # J*12>
>)/
q Suppose that we specify the frequency response of the filter at
the frequencies mentioned above (i.e., at %- ).
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
q Then from the previous equation we obtain
, F+j ≡,
/'
F+j
P
0*+
1', -=D >
0
,
, F+j ≡ E ! # J
F = *, . … , P − .
>)/
q It is a simple matter to invert the previous equation and
express ! # in terms of , F + j .
q Multiplying both sides of the previous equation by the
exponential J1(',-C)/0) , k = *, ., … , P − . and sum over F = *, ., … , P − .,
the RHS reduces to P l ! k J*1(',DC)/0) .
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Digital Signal Processing
63
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
q Thus we obtain
0*+
1', -=D >
.
0
! # =
E , F+j J
,
P
# = *, . … , P − .
-)/
! # from the
specification of the frequency samples , F + j , F = *, . … , P − ..
q This relationship allows computing the values of
j = *, the previous two equations reduce to the
DFT of the sequence {! # } and its IDFT, respectively.
q Note that when
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Digital Signal Processing
Prof. Hesham Tolba
Prof. Hesham Tolba
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
{! # } is real, the frequency samples , F + j
symmetry condition
q Since
satisfy the
, F + j = ,∗ k − F − j
q This symmetry condition, along with the symmetry conditions for
{! # }, can be used to reduce the frequency specifications from
P points to (P + .)// points for P odd and P// points for P
even.
q Thus the linear equations for determining
{! # } from {, F + j }
are considerably simplified.
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Digital Signal Processing
65
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
, % = ,; % J1 *12(0*+)/'=,/' is sampled at the
frequencies %- = /' F + j /P, F = *, . … , P − ., we obtain
q In particular, if
, F + j = ,;
/'
(F + j) J1 E,/'*',(-=D)12(0*+)/'0
P
where n = * when {! # } is symmetric and n = . when {! # } is
antisymmetric.
q A simplification occurs by defining a set of real frequency
samples {o F + j }
o F + j = −. - ,;
12/20/20
Digital Signal Processing
Prof. Hesham Tolba
/'
(F + j) ,
P
Prof. Hesham Tolba
F = *, ., … , P − .
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
q Using the two previous equations to eliminate
,; %- , we get
, F + j = o F + j J1,- J1 E,/'*',(-=D)(0*+)/'0
, F + j translates into a
corresponding symmetry condition for o F + j , which can be
exploited to simplify the expressions for the FIR filter
impulse response {! # } for the four cases j = *, j = .//, n = *,
and n = ..
q The symmetry condition for
q The results are summarized in the table shown in the next
slide.
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Digital Signal Processing
67
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
Unit Sample Response: * + = ±*(, − / − +)
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Prof. Hesham Tolba
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Linear-Phase FIR Filters by the FrequencySampling Method …
q The frequency sampling method provides us with another means
for designing linear-phase FIR filters.
q Its major advantage lies in the efficient frequency-sampling
structure, which is obtained when most of the frequency samples
are zero.
q The optimum values for the samples in the transition band are
obtained from the tables in Appendix B.
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Digital Signal Processing
69
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Determine the coefficients of a linear-phase FIR filter of
length P = .p which has a symmetric unit sample response and a
frequency response that satisfies the conditions
!6
%,
O = (, %, :, h
:*O
O=j
= g (. j,
%f
(,
O = f, k, l
q Solution:
q Since !(#) is symmetric and the frequencies are selected to
correspond to the case j = *, we use the corresponding
formula in the above table to evaluate !(#).
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q In this case,
/'F
,
.p
q The result of this computation is
o F = −. - ,;
!
!
!
!
!
!
!
!
*
.
/
\
)
p
e
q
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=!
=!
=!
=!
=!
=!
=!
.)
.\
./
..
.*
s
r
F = *, ., … , q
= −*. *.)../rs\
= −*. **.s)p\*s
= *. *)*****4
= *. *.//\)p)
= −*. *s.\rr*/
= −*. *.r*rsre
= *. \.\\.qe
= *. p/
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Digital Signal Processing
71
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The frequency response
characteristic of this
filter is as shown.
q We should emphasize that
,; % is exactly equal to
the values given by the
specifications above at
%- = /'F/.p.
Frequency response of linear-phase FIR filter.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Determine the coefficients of a linear-phase FIR filter of
length P = \/ which has a symmetric unit sample response and a
frequency response that satisfies the condition
!6
%,
:*(O + m)
= g n$,
h:
(,
O = (, %, :, h, j, f
O=k
O = l, o, … , %f
+
'
where u+ = *. \qrsqsp for j = *, and u+ = *. \pq*)se for j = .
u+ were obtained from the tables of optimum
transition parameters given in Appendix B.
q These values of
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Prof. Hesham Tolba
Digital Signal Processing
73
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Solution:
q The appropriate equations for this
computation are given in the
+
'
previous table for j = * & j = .
q These computations yield the unit
sample responses in the shown
table.
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Prof. Hesham Tolba
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The corresponding frequency response characteristics are
illustrated in the shown figures, respectively.
Frequency response of linear-phase
FIR filter (0 = ?' and D = /).
q Note that the filter BW for
12/20/20
Frequency response of linear-phase
FIR filter (0 = ?' and D = +/').
m = %/: is wider than that for m = (.
Prof. Hesham Tolba
Digital Signal Processing
75
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations
q The optimization of the frequency samples in the transition
region of the frequency response can be explained by evaluating
the system function , = , given by
0*+
. − =*0J1',D
, F+j
, = =
E
,
P
. − J1',(-=D)/0=*+
-)/
on the unit circle and using the relationship
, F + j = o F + j J1,- J1 E,/'*',(-=D)(0*+)/'0
to express , %
12/20/20
Digital Signal Processing
Prof. Hesham Tolba
in terms of o F + j .
Prof. Hesham Tolba
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q Thus for the symmetric filter we obtain
, % =
sin
%w
0*+
o F+j
2 − 'j E
%
'
P
-)/ sin 2 − P (F + j)
J*12(0*+)/'
where
−o P − F ,
o F+j =v
.
o P−F− ,
/
12/20/20
Prof. Hesham Tolba
j=*
.
j=
/
Digital Signal Processing
77
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q Similarly, for the antisymmetric linear-phase FIR filter,
obtain
, % =
sin
%w
0*+
o F+j
2 − 'j E
%
'
P
-)/ sin 2 − P (F + j)
we
J*12(0*+)/' J1,/'
where
o P−F ,
o F+j =v
.
−o P − F − ,
/
j=*
.
j=
/
o F + j in the passband are set to
in the stopband are set to zero.
q The values of
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Digital Signal Processing
Prof. Hesham Tolba
Prof. Hesham Tolba
Digital Signal Processing
−.
-
and those
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
p O + m in the transition band, the value of ! "
is computed at a dense set of frequencies (e.g., at "& = :*,/q, , =
(, %, … , q − %, where, for example, q = %(G).
q For any choice of
q The value of the maximum side lobe is determined, and the values of
the parameters {p O + m } in the transition band are changed in a
direction of steepest descent, which, in effect, reduces the maximum
sidelobe.
q The computation of
{p O + m }.
! "
is now repeated with the new choice of
! " is again determined and the values of
the parameters {p O + m } in the transition band are adjusted in a
direction of steepest descent, which, in turn, reduces the sidelobe.
q The maximum sidelobe of
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q This iterative process is performed until it converges to the
optimum choice of the parameters {o F + j } in the transition
band.
q There is a potential problem in the frequency-sampling
realization of the FIR linear-phase filter.
q The frequency-sampling realization of the FIR filter introduces
poles and zeros at equally spaced points on the unit circle.
q In the ideal situation, the zeros cancel the poles and,
consequently, the actual zeros of ,(=) are determined by the
selection of the frequency samples {, F + j }.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q In a practical implementation, quantization effects preclude a
perfect cancellation of the poles and zeros.
q The location of poles on the unit circle provide no damping of
the round-off noise that is introduced in the computations.
q As a result, such noise tends to increase with time and,
ultimately, may destroy the normal operation of the filter.
q To mitigate this problem, we can move both the poles and zeros
from the unit circle to a circle just inside the unit circle,
say at radius x = . − y, where y is a very small number.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q Thus the system function of the linear-phase FIR filter becomes
0*+
. − x0=*0J1',D
, F+j
, = =
E
P
. − xJ1'2,(-=D)/0=*+
-)/
q The corresponding two-pole filter realization given previously
can be modified accordingly.
x < . ensures that roundoff
noise will be bounded and thus instability is avoided.
q The damping provided by selecting
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters
q The window & the frequency-sampling method are relatively
simple techniques for designing linear-phase FIR filters.
q They possess some minor disadvantages, which may render them
undesirable for some applications.
q A major problem is the lack of precise control of the critical
frequencies such as %3 and %4 .
q The filter design method described here is formulated as a
Chebyshev approximation problem.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q It is viewed as an optimum design criterion.
q This criterion implies that
weighted approximation error between the desired
and the actual frequency responses is spread
evenly across both the passband and the stopband
of the filter minimizing the maximum error
q The resulting filter designs have ripples in both the passband
and the stopband.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Consider the design of an LPF with passband edge frequency
and stopband edge frequency %4 .
%3
q From the shown general specifications,
in the passband, the filter frequency
response satisfies the condition
. − 2+ ≤ ,; % ≤ . + 2+ ,
% ≤ %3
q Similarly, in the stopband, the filter
frequency response is specified to
fall between the limits ±2' , i.e.,
−2' ≤ ,; % ≤ 2' ,
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% > %4
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
+ , = +(G − % − ,) & G odd.
q In this case, the real-valued frequency response characteristic
! 6 " is
q Case 1: Symmetric unit sample response.
G−%
!6 " = +
+:
:
(.'4)/%
N
+(,) cos "
&/0
G−%
−,
:
O = G − % ⁄: − , and define a new set of filter
parameters {6(O)} as
q If we let
G−%
,
:
6(O) =
G−%
:+
−O ,
:
+
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Prof. Hesham Tolba
O=(
O = %, :, … ,
Digital Signal Processing
G−%
:
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Thus, we can write
,; %
to the compact form
(0*+)/'
,; % =
E
7(F) cos %F
>)/
q Case 2: Symmetric unit sample response.
even.
q In this case, ,; %
is expressed as
0⁄' *+
,; % = /
E
!(#) cos %
>)/
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! # = !(P − . − #) & P
Prof. Hesham Tolba
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−#
/
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
, to O = G⁄: − , and
define a new set of filter parameters {K(O)} as
q Again, we change the summation index from
G
G
−O ,
O = %, :, … ,
:
:
q With these substitutions !6 " becomes
K(O) = :+
./%
!6 " = N K(O) cos " O −
&/0
q It is convenient to rearrange !6 "
"
!6 " = cos
:
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Digital Signal Processing
Prof. Hesham Tolba
%
:
further into the form
.⁄% '$
N
s (O) cos "O
K
&/0
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
z F } are linearly related to the
{G
coefficients {G F } as follows
q where the coefficients
z * = .G . ,
G
/
z . = /G . − /G * ,
G
P
z F = /G F − /G F − . ,
G
F = ., /, … , − /
/
P
P
z
G
− . = /G
/
/
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Case 3: Antisymmetric unit sample response.
P odd.
q In this case, ,; %
! # = −!(P − . − #) &
is
(0*?)/'
,; % = /
E
!(#) sin %
>)/
P−.
−#
/
q Now, define a new set of filter parameters
{(F) = /!
12/20/20
Digital Signal Processing
Prof. Hesham Tolba
P−.
−F ,
/
Prof. Hesham Tolba
F = ., /, … ,
Digital Signal Processing
{{(F)} as
P−.
/
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
# to F = (P − .)⁄/ − # and
using the above-defined sequence {{(F)}, we get
q Changing the summation index from
(0*+)/'
,; % =
E
{(F) sin %F
-)+
,; %
q It is convenient to rearrange
further into the form
.'4 /%
!6 " = sin "
N
ut (O) cos "O
&/0
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q The coefficients
{tu O } are linearly related to the {u O } as follows
P−\
P−.
={
,
/
/
P−p
P−\
{|
= /{
/
/
⋮
⋮
{|
{| F − . − {| F + . = /{ F ,
.
{| * − {| / = { .
/
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Prof. Hesham Tolba
/≤F≤
Digital Signal Processing
P−p
/
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Case 4: Antisymmetric unit sample response.
even.
q In this case, !6 "
+ , = −+(G − % − ,) & G
is
.⁄%'$
!6 " = : N +(,) sin "
&/0
G−%
−,
:
q If we change the summation index from
, to O = G/: − , and define a
new set of filter parameters {C(O)} as
G
G
C(O) = :+
−O ,
O = %, :, … ,
:
:
then
./%
%
!6 " = N C(O) sin " O −
:
-/$
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q It is convenient to rearrange
,; %
further into the form
0⁄'*+
%
z (F) cos %F
,; % = sin
E B
2
>)/
z F } are linearly related to the
where the coefficients {B
{B F } as follows
z P − . = /B P ,
B
/
/
P
z F−. −B
z F = /B F ,
B
/≤F≤ −.
/
.
z * − B . =B .
B
/
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
!6 " in these
four cases are summarized as
shown.
q The expressions for
q The rearrangements in Cases 2, 3
& 4 allows us to express
as
!6 " = v " w "
!6 "
where
.,
%
cos / ,
~ % =
sin %,
%
sin / ,
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ÄSX .
ÄSX /
ÄSX \
ÄSX )
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
and
G
Ç % = E j(F) cos %F
-)/
with {j(F)} representing the parameters of the filter, which are
linearly related to the unit sample response !(#) of the FIR
filter.
Å in the sum is Å = (P − .)// for Case 1, Å = (P −
\)// for Case 3, and Å = P// − . for Case 2 and Case 4.
q The upper limit
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q In addition to the common framework given above for the
representation, the real-valued desired frequency
response !76 " & the weighting function b " on the
approximation error, are defined.
q
!76 " is simply defined to be unity in the passband and
zero in the stopband; some different types of !76 " are
as shown.
q
b " allows us to choose the relative size of the
errors in the different frequency bands.
q
b " is usually normalized to unity in the stopband and
set to b " = <% /<$ in the passband, i.e.,
] % =g
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2' ⁄2+ ,
.,
% TU hcX ÉÄSSÑÄUW
% TU hcX ShVÉÑÄUW
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q
] % is then simply selected in the passband to reflect our
emphasis on the relative size of the ripple in the stopband to
the ripple in the passband.
,A; % and ] % , the weighted
approximation error is defined as
q With the specification of
Ö % = ] % ,A; % − ,; %
= ] % ,A; % − ~ % Ç %
,A; %
=] % ~ %
− Ç %
~ %
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
Ü % and a modified desired
]
are defined for mathematical
q A modified weighting function
á A; %
frequency response ,
convenience, as
Ü % =] % ~ %
]
,A; %
~ %
q Then the weighted approximation error may be expressed as
á A; % =
,
Ü % ,
á A; % − Ç %
Ö % =]
for all four different types of linear-phase FIR filters.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
Given the error function Ö % , the Chebyshev approximation problem
is basically to determine the filter parameters {j F } that
minimize the maximum absolute value of Ö % over the frequency
bands in which the approximation is to be performed, i.e.,
G
àTU
HIJK{D - }
àÄâ Ö %
2∈O
=
àTU
HIJK{D - }
Ü % ,
á A; % − E j F {äã %F
àÄâ ]
2∈O
-)/
where x represents the set (disjoint union) of frequency bands
(passbands and stopbands of the desired filter) over
which the optimization is to be performed.
q The solution to this problem based on a theorem in the theory of
Chebyshev approximation, called the alternation theorem.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Alternation Theorem. Let
[(, *).
x be a compact subset of the interval
w " = ∑ >-/0 m O VQW "O to be
the unique, best weighted Chebyshev approximation to ! 76 " in x,
is that the error function { " exhibit at least | + : extremal
frequencies in x.
q A necessary and sufficient condition for
| + : frequencies {" ? } in x such
that " $ < " % < ⋯ < " >@% , { " ? = { " ?@$ , and
q That is, there must exist at least
Ö %P
= àÄâ Ö % ,
2∈O
å = ., / … , Å + /
{ " alternates in sign between two successive
extremal frequencies ➞ the theorem is called the alternation
theorem.
q The error function
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Consider the design of a LPF with passband
stopband %4 ≤ % ≤ '.
* ≤ % ≤ %3 and
,A; % and the weighting
are piecewise constant, we have
q Since the desired frequency response
function ] %
BÖ %
B
=
] % ,A; % − ,; %
B%
B%
B,A; %
=
=*
B%
{%P } corresponding to the peaks of
Ö % also correspond to peaks at which ,; % meets the error
tolerance.
q Consequently, the frequencies
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
!6 " is a trigonometric polynomial of degree |, for Case 1,
for example,
q Since
>
>
!6 " = N m O VQW "O = N m O
-/0
>
-/0
= N mA O VQW "
-
N }&- VQW "
&
&/0
-
-/0
!6 " can have at most | − % local maxima and minima
in the open interval ( < " < *.
q It follows that
q In addition,
of { " .
q Therefore,
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" = ( and " = * are usually extrema of !6 "
!6 "
and, also,
has at most | + % extremal frequencies.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
%3 and %4 are also extrema of Ö % ,
is maximum at % = %3 and % = %4 .
q The band-edge frequencies
since Ö %
Å + \ extremal frequencies
for the unique, best approximation of the ideal LPF.
q As a consequence, there are at most
in Ö %
Å+/
extremal frequencies in Ö % ➞ the error function for the LPF
design has either Å + \ or Å + / extrema.
q The alternation theorem states that there are at least
Å+/
alternations or ripples are called extra ripple filters.
q In general, filter designs that contain more than
q When the filter design contains the maximum number of
alternations, it is called a maximal ripple filter.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q The alternation theorem guarantees a unique solution for the
Chebyshev optimization problem.
q At the desired extremal frequencies
equations
Ü %> ,
á A; %> − Ç %>
]
{%> }, we have the set of
= −. > 2,
# = *, ., … , Å + .
where 2 represents the maximum value of the error function
Ö % .
] % as unity in the stopband and 2' /2+ in the
passband, it follows that 2 = 2' .
q If we select
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
Ü %> ,
á A; %> − Ç %>
]
# = *, ., … , Å + . can be rearranged as
q The set of linear equations
Ç %>
−. > 2
á A; %> ,
+
=,
Ü %>
]
= −. > 2,
# = *, ., … , Å + .
or, equivalently, in the form
G
−. > 2 á
E j(F) cos %> F + Ü
= ,A; %> ,
] %>
# = *, ., … , Å + .
-)/
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Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
{j(F)} and 2 as the parameters to be determined,
the previous equation can be expressed in matrix form as
q Treating the
%
%
⋮
⋮
%
cos " #
cos " $
⋮
⋮
cos " >@$
cos :" #
cos :" $
⋮
⋮
cos :" >@$
…
…
⋮
⋮
…
cos |" #
cos |" $
⋮
⋮
cos |" >@$
 "#
%⁄b
 "$
−%⁄b
⋮
⋮
 " >@$
−% >@$ ⁄b
m(()
m(%)
=
⋮
m(|)
?
Ä 76 " #
!
Ä 76 " $
!
⋮
⋮
Ä 76 " >@$
!
q Initially, we know neither the set of extremal frequencies
nor the parameters {j(F)} and 2.
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{%> }
Digital Signal Processing
107
Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q To solve for the parameters, we use an iterative algorithm,
called the Remez exchange algorithm, in which we begin by
guessing at the set of extremal frequencies, determine Ç %
2, and then compute the error function Ö % .
and
Ö % we determine another set of Å + / extremal frequencies
and repeat the process iteratively until it converges to the
optimal set of extremal frequencies.
q From
q Although the above matrix equation can be used in the iterative
procedure, matrix inversion is time consuming and inefficient.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q A more efficient procedure is to compute
< analytically, according to
the formula
<=
Ä 76 "# + Å$ !
Ä 76 "$ + ⋯ + Å>@$ !
Ä 76 ">@$
Å# !
Å#
Å$
Å>@$
−
+ ⋯+



b "#
b "$
b ">@$
where
>@$
Å- = Ç
&/0
&B-
%
cos "- − cos "&
< follows immediately from the matrix
equation.
q Thus with an initial guess at the | + : extremal frequencies, we
compute <.
q The above expression for
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q since
Ç %
is a trigonometric polynomial of the form
G
Ç % = E j(F)H- ,
H = cos %
-)/
and since we know that the polynomial at the points H> ≡ cos %> ,
# = *, ., … , Å − ., has the corresponding values
Ç %>
á A; %> −
=,
−. > 2
,
Ü %>
]
# = *, ., … , Å + .
we can use the Lagrange interpolation formula for Ç % .
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Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Thus
w "
can be expressed as
w " =
q where
Ç %>
∑>-/0 w "- }- / H − H∑>-/0 }- / H − H-
is as given above, H = cos %, H- = VS %- , and
G
n- = ç
>)/
>Q12/20/20
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.
H- − H>
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q Having the solution for
function Ö %
Ç % , we can now compute the error
from
Ü % ,
á A; % − Ç %
Ö % =]
on a dense set of frequency points.
q Usually, a number of points equal to
.eP, where P is the
length of the filter, suffices.
Ö % ≥ 2 for some frequencies on the dense set, then a new
set of frequencies corresponding to the Å + / largest peaks of
Ö % are selected and the computational procedure beginning
with the repetition of computation of 2.
q If
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Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
Å + / extremal
frequencies is selected to correspond to the
peaks of Ö % , the algorithm forces 2 to
increase in each iteration until it
converges to the upper bound and hence to
the optimum solution for the Chebyshev
approximation problem.
q Since the new set of
Ö % ≤ 2 for all
frequencies on the dense set, the optimal
solution has been found in terms of the
polynomial , % .
q In other words, when
q A flowchart of the algorithm is shown.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
Ç % ,
the unit sample response ! # can be computed directly, without
having to compute the parameters {j(F)}.
q Once the optimal solution has been obtained in terms of
q In effect, we have determined
,; % = ~ % Ç %
which can be evaluated at % = /'F/P, F = *, ., … , (P − .)//, for P
odd, or P// for P even.
!(#) can
be determined from the formulas given in the table in the next
slide.
q Then, depending on the type of filter being designed,
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Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
Unit Sample Response: R > = ±R(0 − + − >)
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q A computer program written by Parks and McClellan is available
for designing linear-phase FIR filters based on the Chebyshev
approximation criterion.
q It is implemented with the Remez exchange algorithm.
q This program can be used to design LPFs, HPFs, or BPFs,
differentiators, and Hilbert transformers.
q A number of software packages for designing equiripple linear-
phase FIR filters are now available.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
q The Parks-McClellan program requires a number of input
parameters which determine the filter characteristics.
q The following parameters must be specified:
NFILT:
The filter length, denoted above as M.
JTYPE:
Type of
JTYPE =
JTYPE =
JTYPE =
filter:
1 results in a multiple passband/stopband filter.
2 results in a differentiator.
3 results in a Hilbert transformer.
NBANDS: The number of frequency bands from 2 (for a lowpass filter)
to a maximum of 10 (for a multiple-band filter).
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Optimum Equiripple Linear-Phase FIR
Filters …
LGRID:
The frequency bands specified by lower and upper
cutoff frequencies, up to a maximum of 10 bands (an
array of size 20, maximum). The frequencies are given
in terms of the variable @ = %/5", where @ = ;. C
corresponds to the folding frequency.
FX:
An array of maximum size 10 that specifies the desired
frequency response D#% % in each band.
WTX:
An array of maximum size 10 that specifies the weight
function in each band.
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Design of IIR Filters
Frequency Transformations
Example
P = e. with a passband edge
frequency é3 = *. . and a stopband edge frequency é4 = *. .p.
q Design a lowpass filter of length
q Solution:
q The lowpass filter is a two-band filter with passband edge
frequencies (*, *. .) and stopband edge frequencies (*. .p, *. p).
(., *) and the weight function is
arbitrarily selected as (., .).
q The desired response is
e.,
*. *,
.. *,
.. *,
.,
/
*. .,
*. .p,
*. *
.. *
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The impulse response and
frequency response of
P = e. FIR filter are as
shown.
q The resulting filter has
a stopband attenuation
of -56 dB and a passband
ripple of 0.0135 dB.
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Design of IIR Filters
Frequency Transformations
Example …
P to .*. while
maintaining all the other
parameters the same, the
resulting filter has the
impulse response and
frequency response shown.
q Increasing
q The stopband attenuation
is -85 dB and the passband
ripple is reduced to
0.00046 dB.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q It is possible to increase the attenuation in the stopband by
P = e.,
q decreasing the weighting function ] % = 2' /2+ in the
passband.
q Keeping the filter length fixed, say at
P = e. and a weighting function (*. ., .), we obtain a filter
that has
q a stopband attenuation of -65 dB,
q a passband ripple of 0.049 dB.
q With
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Design of IIR Filters
Frequency Transformations
Example
P = \/ with a passband edge
frequencies é3+ = *. / & é3' = *. \p and a stopband edge frequencies
é4+ = *. . & é4' = *. )/p.
q Design a bandpass filter of length
q Solution:
q This passband filter is a three-band filter with a stopband
range of (*, *. .),a passband range of (*. /, *. \p), and second
stopband range of (*. )/p, *. p).
(.*. *, .. *, .*. *), or as
(.*. *, *. ., .. *), and the desired response in the three bands is
(*. *, .. *, *. *).
q The weighting function is selected as
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Thus the input parameters to the program are
\/,
*. *,
*. *,
.*. *,
.,
*. .,
.. *,
.. *,
\
*. /,
*. *
.*. *
*. \p,
*. )/p,
*. p
2' is .* times smaller
than the ripple in the passband due to the fact that errors in
the stopband were given a weight of .* compared to the passband
weight of unity.
q Note that the ripple in the stopbands
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Design of IIR Filters
Frequency Transformations
Example …
q The impulse response and
frequency response of the
bandpass FIR filter of
P = \/ are illustrated as
shown.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of FIR Differentiators
q An ideal differentiator has a frequency response that is
linearly proportional to frequency.
q Similarly, an ideal digital differentiator is defined as one
that has the frequency response
,A % = 5%,
−' ≤ % ≤ '
q The unit sample response corresponding to
,A %
is
. ,
. ,
? ,A % J12> B% =
? 5%J12> B%
/' *,
/' *,
cos '#
=
,
−∞ < # < ∞,
#≠*
#
!A # =
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of FIR Differentiators …
q The ideal differentiator has an antisymmetric unit sample
response (!A # = −!A −# , !A * = *).
q The design of linear-phase FIR differentiators based on the
Chebyshev approximation criterion is considered here with focus
on FIR designs in which ! # = −!(P − . − #).
,; % of the FIR
has the characteristic that ,; * = *.
q Recall that in Case 3,
q Both cases 3 & 4 filter types satisfy the
condition “zero response at zero
frequency” that the differentiator should
satisfy.
Response Functions for Linear-Phase
FIR Filters
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of FIR Differentiators …
q A full-band differentiator is impossible to achieve with an FIR
filter having an odd number of coefficients, since ,; ' = * for
P odd.
q However, in practice, full-band differentiators are rarely
required.
q In most cases, the desired frequency response characteristic
need only be linear over the limited frequency range * ≤ % ≤
/'é3, where é3 is called the BW of the differentiator.
/'é3 ≤ % ≤ ', the desired response may be either
left unconstrained or constrained to be zero.
q In the range
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of FIR Differentiators …
q In the design of FIR differentiators based on the Chebyshev
approximation, the weighting function b(") is specified as
E % =
/
,
%
; ≤ % ≤ 5"@&
in order that the relative ripple in the passband be a constant.
" and the
increases as " varies from ( to :*ÉC .
q Thus the absolute error between the desired response
approximation !6 "
q However,
] %
?=
ensures that the relative error
FGH
'("()*+!
E % % − D% %
=
FGH
'("()*+!
/−
D% %
%
is fixed within the passband of the differentiator.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Use the Remez algorithm to design a linear-phase FIR
differentiator of length P = e*. The passband edge frequency is
*. . and the stopband edge frequency is *. .p.
q Solution:
q The input parameters to the program are
e*,
*. *,
.. *,
.. *,
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*. .,
*. *
.. *
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*. .p,
*. p
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The frequency response characteristic
is illustrated as shown.
q Also shown in the same figure is the
approximation error over the passband
a * ≤ é ≤ *. . of the filter.
Frequency response and approximation
error for . = ;0 FIR differentiator
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations
q The important parameters in a
differentiator are its length P,
its BW é3, and the peak relative
error 2 of the approximation.
/* log +/ 2 versus é3
with P as a parameter is as
shown for P even.
q The value of
%0 log "# G vs H! for . = =, J, $;, 4%, and ;=
FIR differentiator
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Design of IIR Filters
Frequency Transformations
Observations …
/* log +/ 2 versus é3 for P
odd is as shown.
q The value of
q These results are useful in the
selection of the filter length,
given specifications on the in-band
ripple and the cutoff frequency é3.
q These graphs reveals that even-
length differentiators result in a
significantly smaller approximation
error 2 than comparable odd-length
differentiators.
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FIR differentiator
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q Designs based on
é3 = *. )p.
P odd are particularly poor if the BW exceeds
q The problem is basically the zero in the frequency response at
% = ' (é = .⁄/).
é3 < *. )p, good designs are obtained for P odd, but
comparable-length differentiators with P even are always
better in the sense that the approximation error is smaller.
q When
q In view of the obvious advantage of even-length over odd-length
differentiators, a conclusion might be that even-length
differentiators are always preferable in practical systems.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q However, we should note that the signal delay introduced by any
linear-phase FIR filter is (P − .)//, which is not an integer
when P is even.
q In many practical applications, this is unimportant.
q In some applications where it is desirable to have an integer-
valued delay in the signal at the output of the differentiator,
we must select P to be odd.
q These numerical results are based on designs resulting from the
Chebyshev approximation criterion.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q It is also possible to design
linear-phase FIR differentiators
based on the frequency-sampling
method.
q For example, the frequency response
characteristics of a wideband (é3 =
*. p) differentiator, P = \*, designed
by frequency-sampling method is
shown.
q The graph of the absolute value of
the approximation error as a
function of frequency is also shown.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers
q An ideal Hilbert transformer is an all-pass filter that imparts
a 90° phase shift on the signal at its input.
q Hence the frequency response of the ideal Hilbert transformer
is specified as
,A % = g
−5,
5,
*≤%≤'
−' ≤ % ≤ *
q Hilbert transformers are frequently used in communication
systems and signal processing.
q For example, in the generation of single-sideband modulated
signals, radar signal processing, and speech signal
processing.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers
q The unit sample response of an ideal Hilbert transformer is
!A # =
/
,
. ,
.
? ,A % J12> B% =
? 5J12> B% − ? 5J12> B%
/' *,
/' *,
/
/ sin' '#//
,
= v'
#
*,
#≠*
#=*
q !A
#
is infinite in duration and noncausal.
q !A
#
is antisymmetric [i.e., !A # = −!A −# ].
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers …
q Focus is on the design of linear-phase FIR Hilbert transformers
with antisymmetric unit sample response [i.e., + , = + G − % − , ].
q This choice is consistent with having a purely imaginary frequency
response characteristic ! 7 " .
+7 , is antisymmetric, ! 6 " is zero at " = ( for both G odd
and even and at " = * when G is odd.
q When
q Clearly, then, it is impossible to design an all-pass digital
Hilbert transformer.
q In practical applications, an all-pass Hilbert transformer is
unnecessary; its BW need only cover the BW of the signal to be
phase shifted.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers …
q Consequently, we specify the desired
transform filter as
,A; % = .,
,; %
of a Hilbert
/'éU ≤ % ≤ /'éV
where éU and éV are the lower and upper cutoff frequencies,
respectively.
q Note that the ideal Hilbert transformer with unit sample
response !A #
as given above is zero for # even.
q This property is retained by the FIR Hilbert transformer under
some symmetry conditions.
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Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers …
,; % =
and suppose that éU = *. p − éV to ensures a
symmetric passband about the midpoint frequency é = *. /p.
q Having this symmetry in the frequency response, ,; % = ,; (' −
%), thus
q Consider the Case 3 filter type for which
∑(0*+)/'
{(F) sin %F
-)+
(01.)/)
I
(01.)/)
J(K) sin %K =
,-.
I
J(K) sin " − % K
,-.
(01.)/)
=
I
J(K) sin %K cos "K
,-.
(01.)/)
=
I
J(K) −/
,4.
sin %K
,-.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers …
q Equivalently, we have
(01.)/)
I
/ − −/
,4.
J(K) sin %K = ;
,-.
q Clearly,
{(F) must be equal to zero for F = *, /, ), ….
q The relationship between
{! # } is,
J(K) = 5*
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,−/
−K
5
{{ F } and the unit sample response
or, equivalently
Prof. Hesham Tolba
Digital Signal Processing
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,−/
/
− K = J(K)
5
5
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers …
q If
{(F) is zero for F = *, /, ), …, then
*(K) =
;,
K = ;, 5, M, … ,
;,,
q This holds only for
q
,−/
RSR:
5
,−/
OPQ
PTT
5
OPQ
K = /, ., C, … ,
P odd, but does not hold for P even.
P odd is preferable since the computational complexity is
roughly one half of that for P even.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of Hilbert Transformers …
q When the design is performed by the Chebyshev approximation
criterion using the Remez algorithm, the filter coefficients
are selected to minimize the peak approximation error
2=
=
àÄâ
,A; % − ,; %
àÄâ
. − ,; %
',W5 X2X',W6
',W5 X2X',W6
q Thus the weighting function is set to unity and the
optimization is performed over the single frequency band (i.e.,
the passband of the filter).
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Design a Hilbert transformer with parameters
and éV = *. )p.
P = \., éU = *. *p
q Solution:
q We observe that the frequency response is symmetric, since
éU = *. p − éV.
q The parameters for executing the Remez algorithm are
\.,
*. *p,
.. *
.. *
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.
*. )p
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The result of this design
is the unit sample
response and frequency
response as shown.
q We observe that, indeed,
every other value of !(#)
is essentially zero.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations
q If the filter design is restricted to a
symmetric frequency response, then there
are basically three parameters of interest,
G, <, and ÉN .
:( log $0 < versus Üá (the transition
width) with G as a parameter is shown.
q A plot of
G, there is no
performance advantage of using G odd over G
even, and vice versa.
q For comparable values of
q The computational complexity in implementing
a filter for G odd is less by a factor of 2
over G even as previously indicated.
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%0 log "# G vs OP for . =
4, =, M, J, $K, $;, 4$, 4%, ;4, ;=.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q Therefore,
P odd is preferable in practice.
q For design purposes, the graphs in the previous figure suggest
that, as a rule of thumb,
PéU ≈ −*. e. log +/ 2
q Hence this formula can be used to estimate the size of one of
the three basic filter parameters when the other two parameters
are specified.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters
q The method based on the use of windows to truncate the impulse
response !A(#) and obtain the desired spectral shaping was the
first method proposed for designing linear-phase FIR filters.
q The frequency-sampling method and the Chebyshev approximation
method were developed in the 1970s and have since become very
popular in the design of practical linear-phase FIR filters.
q The major disadvantage of the window design method is the lack
of precise control of the critical frequencies, such as %3 and
%4, in the design of a LP FIR filter.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters …
%3 and %4, in general, depend on the type of
window and the filter length P.
q The values of
q The frequency-sampling method provides an improvement over the
window design method, since ,;(%) is specified at the
frequencies %- = /'F/P or %- = '(/F + .)/P and the transition
band is a multiple of /'/P.
q This filter design method is particularly attractive when the
FIR filter is realized either in the frequency domain by means
of the DFT or in any of the frequency-sampling realizations.
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Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters …
,;(%- ) is
either zero or unity at all frequencies, except in the
transition band.
q The attractive feature of these realizations is that
q The Chebyshev approximation method provides total control of
the filter specifications, and, as a consequence, it is usually
preferable over the other two methods.
q For an LPF, the specifications are given in terms of the
parameters %3, %4 , 2+ , 2' and P.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters …
q We can specify the parameters
the filters relative to 2' .
%3, %4, P, and 2, and optimize
q By spreading the approximation error over the passband and the
stopband of the filter, this method results in an optimal
filter design (the maximum sidelobe level is minimized).
q The Chebyshev design procedure based on the Remez exchange
algorithm requires that we specify the length of the filter,
the critical frequencies %3 and %4 , and the ratio 2' /2+ .
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters …
q However, it is more natural in filter design to specify (%3 ,
%4, 2+ , and 2' and to determine the filter length that
satisfies the specifications.
q Although there is no simple formula to determine the filter
length from these specifications, a number of approximations
have been proposed for estimating P from %3, %4 , 2+ , and 2' .
q A particularly simple formula for approximating
á =
P
P is
−/* log +/ 2+ 2Y − .\
.). eíé
where íé is the transition band, defined as íé = (%4 − %3)//'.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters …
q A more accurate formula is
á =
P
ìB(2+ 2Y) − é(2+ 2Y)íé'
+.
íé
where, by definition,
ìB(2+ 2Y) = *. **p\*s log +/ 2+ ' + *. *q..) log +/ 2+ − *. )qe. log +/ 2Y
− *. **/ee log +/ 2+ ' + *. ps). log +/ 2+ + *. )/qr
é 2+ , 2Y = ... *./ + *. p./)) log +/ 2+ − log +/ 2Y
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Comparison of Design Methods for Linear-Phase FIR
Filters …
q These formulas are useful in obtaining a good estimate of the
filter length required to achieve the given specifications íé,
2+ & 2Y.
q The estimate is used to carry out the design, and if the
resulting 2 exceeds the specified 2Y, the length can be
increased until we obtain a sidelobe level that meets the
specifications.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters from Analog Filters
q Several methods can be used to design IIR digital filters.
q The techniques described here are all based on converting an
analog filter into a digital filter.
q The design of a digital filter begins in the analog domain, then
it is converted into the digital domain.
q An analog filter can be described by its system function,
!Q à =
âà
∑. }-à= -/0
äà
∑R
-/0 m-à
where {m - } and {} - } are the filter coefficients, or by its impulse
response
8
!Q à = @ +(ã)>'ST Cã
'8
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters from Analog Filters …
, ã can
be described by the linear constant-coefficient differential
equation
q The analog filter having the rational system function
.
E
-)/
j-
0
B- D(î)
B- H(î)
=
E
n
BîBî-)/
where H î denotes the input signal and D î
of the filter.
denotes the output
q Each of these three equivalent characterizations of an analog
filter leads to alternative methods for converting the filter
into the digital domain.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters from Analog Filters …
, ã is
stable if all its poles lie in the left half of the ã-plane.
q Recall that an analog LTI system with system function
q Consequently, if the conversion technique is to be effective,
it should possess the following desirable properties:
1.
The 5ï axis in the ã-plane should map into the unit circle
in the ñ-plane ➞ there will be a direct relationship
between the two frequency variables in the two domains.
2.
The LHP of the ã-plane should map into the inside of the
unit circle in the ñ-plane ➞ a stable analog filter will be
converted to a stable digital filter.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters from Analog Filters …
q Recall that physically realizable and stable IIR filters cannot
have linear phase.
q Recall that a linear-phase filter must have a system function
that satisfies the condition
, = = ±=*., =*+
where =*. represents a delay of Q units of time
q But if this were the case, the filter would have a mirror-image
pole outside the unit circle for every pole inside the unit
circle.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters from Analog Filters …
q Hence the filter would be unstable.
q Consequently, a causal and stable IIR filter cannot have linear
phase.
q If the restriction on physical realizability is removed, it is
possible to obtain a linear-phase IIR filter, at least in
principle.
q This approach involves performing a time reversal of the input
signal H # , passing H −# through a digital filter , = , timereversing the output of , = , and finally, passing the result
through , = again.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Design of IIR Filters from Analog Filters …
q This signal processing is computationally cumbersome and offer
no advantages over linear-phase FIR filters.
q Consequently, when an application requires a linear-phase
filter, it should be an FIR filter.
q In the design of IIR filters, we specify the desired filter
characteristics for the magnitude response only.
q This does not mean that the phase response unimportant.
q Since the magnitude and phase characteristics are related, we
specify the desired magnitude characteristics and accept the
obtained phase response.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q One of the simplest methods for converting an analog filter
into a digital filter is to approximate the differential
equation
.
0
B- D(î)
B- H(î)
E
j=
E
n
BîBî-)/
-)/
by an equivalent difference equation.
q This approach is often used to solve a linear constant-
coefficient differential equation numerically on a digital
computer.
BD(î)/Bî at time î = #u, we substitute the
backward difference D #u − D(#u − .) /u.
q For the derivative
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q Thus
BD î
D #u − D(#u − .) D # − D(# − .)
ó
=
=
Bî Z)>[
u
u
where u represents the sampling interval and D # ≡ D(#u).
q The analog differentiator with output
function , ã = ã.
BD(î)/Bî has the system
q The digital system that produces the output
has the system function , = = . − =*+ /u.
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Prof. Hesham Tolba
Digital Signal Processing
D #u − D(#u − .) /u
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q Consequently, as shown, the frequency-domain equivalent for the
previous relationship is
ã=
. − =*+
u
Substitution of the \(>) backward difference for
the derivative implies the mapping 4 = + − (1. ⁄[.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q The 2nd derivative
B' D î ⁄Bî' is replaced by the second
difference
B' D î
ò
Bî'
Z)>[
=
B BD î
Bî Bî
Z)>[
D #u − D #u − u /u − D #u − u − D #u − /u /u
u
D # − /D # − . + D(# − /)
=
u'
=
q In the frequency domain, this is equivalent to
ã'
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. − /=*+ + =*' / − U1.
=
=
u'
V
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Digital Signal Processing
)
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
Fth derivative D(î) results in the
equivalent frequency-domain relationship
q The substitution for the
-
ã =
/ − U1.
,
V
q Consequently, the system function for the digital IIR filter
obtained as a result of the approximation of the derivatives is
, = = ,](ã)<
4) +*(7. ⁄[
where ,](ã) is the system function of the analog filter
A, \(Z)
A, ^(Z)
∑0
characterized by the equation ∑.
-)/ j, =
-)/ n, .
AZ
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AZ
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q The mapping from the
à-plane to the ç-plane as given by à = % − 8'$ ⁄n
is equivalent to
==
.
. − ãu
à = ?å in the previous
equation leads to
q Substituting
==
.
.
ïu
=
+
5
. − 5ïu . + ï' u'
. + ï' u'
å varies from −∞ to ∞, the
corresponding locus of points in the
ç-plane is a circle of radius %/: and
with center at 8 = %/:, as shown.
q As
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Digital Signal Processing
The mapping from the S-plane to the
U-plane
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
ã-plane into
corresponding points inside this circle in the ñ-plane, and
points in the RHP of the ã-plane are mapped into points outside
this circle.
q The above mapping takes points in the LHP of the
q This mapping has the desirable property that a stable analog
filter is transformed into a stable digital filter.
q However, the possible location of the poles of the digital
filter are confined to relatively small frequencies.
q As a consequence, the mapping is restricted to the design of LP
and BP filters having relatively small resonant frequencies.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q It is not possible, for example, to transform a HP analog
filter into a corresponding HP digital filter.
q To overcome the limitations in the mapping given above, more
complex substitutions for the derivatives have been proposed.
q An
Åth order difference of the form
G
BD î
.
D #u + Fu − D(#u − Fu)
ó
= E jBî Z)>[ u
u
-)+
has been proposed, where {j- } are a set of parameters that can
be selected to optimize the approximation.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q The resulting mapping between the
ã-plane and the ñ-plane is
>
à=
%
N m- 8- − 8'n
-/$
q When
ô = J12 , we have
>
:
à = ? N m- sin "O
n
-/$
which is purely imaginary; thus
>
å=
:
N m- sin "O
n
-/$
is the resulting mapping between the two frequency variables.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Approximation of Derivatives …
q By proper choice of the coefficients
the 5ï-axis into the unit circle.
q Furthermore, points in the LHP in
inside the unit circle in ñ.
{j- } it is possible to map
ã can be mapped into points
q Despite achieving the two desirable characteristics with the
'
mapping of ï = ∑G-)+ j- sin %F, the problem of selecting the set of
[
coefficients {j- } remains.
q This is a difficult problem.
q Since simpler techniques exist for converting analog filters
into IIR digital filters, the use of the Åth-order difference as
a substitute for the derivative will not be emphasized here.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q
Convert the analog BPF with system function
%
à + (. % % + é
into a digital IIR filter by use of the backward difference for the
derivative.
Solution:
q Substitution for W from à = % − 8'$ ⁄n into !(à) yields
%
! 8 =
%
% − 8'$
+
(.
%
+é
n
n% / % + (. :n + é. (%n%
=
: % + (. %n
%
%−
8'$ +
8'%
% + (. :n + é. (%n%
% + (. :n + é. (%n%
!Q (à) =
q
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q We observe that this mapping has introduced two additional
poles in the conversion from ,](ã) to ,(=).
q As a consequence, the digital filter is significantly more
complex than the analog filter.
q This is a major drawback to the mapping given above.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance
q Objective: Design an IIR filter having a unit sample response
! # that is the sampled version of the impulse response of the
analog filter, i.e.,
+ , ≡ + ,n ,
, = (, %, :, …
where u is the sampling interval.
H](î) with spectrum ö](õ) is sampled
at a rate õ4 = ./u, the spectrum of the sampled signal is ö é =
õ 4 ∑B
-)*B ö] (é − F)õ4 , where é = õ/õ4 is the normalized frequency.
q Recall that when a CT signal
õ4 is
less than twice the highest frequency contained in ö](õ).
q Recall also that aliasing occurs if the sampling rate
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q Expressed in the context of sampling the impulse response of an
analog filter with frequency response ,](õ), the digital filter
with ! # ≡ !] #u has the frequency response
B
! é = õ4 E
,] (é − F)õ4
-)*B
or, equivalently,
B
! % = õ4 E
,] (% − /'F)õ4
-)*B
or
! ïu =
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B
.
/'F
,] ï −
E
u -)*B
u
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q The frequency response of an LP analog
filter and the frequency response of the
corresponding digital filter are shown.
% has the
frequency response of the corresponding
analog filter if u is selected
sufficiently small to completely avoid
or at least minimize the effects of
aliasing.
q The digital filter with !
q The impulse invariance method is inappropriate for designing HP
filters due to the spectrum aliasing that results from the
sampling process.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
ñ-plane and the ã-plane by
the sampling process, rely on the generalization of the
previous equation which relates the ñ-transform of !(#) to the
Laplace Transform of !](î), as
q The mapping of points between the
,(=)<
=
()%89
B
.
/'F
E
,] ã −
u -)*B
u
where
*>
! = = ∑B
>)/ !(#)=
B
,(=)<
()%89
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= E !(#)J*4[>
>)/
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q When
ã = 5ï, the relation
,(=)<
=
()%89
B
.
/'F
E
,] ã −
u -)*B
u
reduces to
! ïu =
B
.
/'F
E
,] ï −
u -)*B
u
ã = ú + 5ï, and expressing the complex variable in
polar form as = = xJ12, the mapping = = J4[ becomes
q Substituting
xJ12 = J_[J1`[
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q Clearly, we must have
x = J_[
and
% = ïu
ú < * implies that * < x < . and ú > * implies that
x > .; when ú = *, we have x = ..
q Consequently,
ã is mapped inside the unit circle in =
and the RHP in ã is mapped outside the unit circle in =.
q Therefore, the LHP in
5ï-axis is mapped into the unit circle in = as
indicated above.
q Also, the
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q However, the mapping of the
5ï-axis into the unit circle is not
one-to-one.
% is unique over the range (−', '), the mapping % = ïu
implies that the interval −'/u ≤ ï ≤ '/u maps into the
corresponding values of −' ≤ % ≤ '.
q Since
'/u ≤ ï ≤ \'/u also maps into the
interval −' ≤ % ≤ ' and, in general, so does the interval
(/F − .)'/u ≤ ï ≤ (/F + .)'/u, when F is an integer.
q The frequency interval
ï to % in the digital domain is many-toone, which simply reflects the effects of aliasing due to
sampling.
q Thus the mapping from
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
ã–plane to the
ñ-plane for the relation = = J4[ is
as shown.
q The mapping from the
( = %89 maps strips of width ',/[ (for _ <
/) in the 4-plane into points in the unit
circle in the b-plane.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q Expressing the system function of the analog filter in partial-
fraction form and assuming that the poles of the analog filter
are distinct, we can write
.
,] ã = E
-)+
{ã − ù-
where {ù- } are the poles of the analog filter and {{- } are the
coefficients in the partial-fraction expansion.
q Consequently,
.
!] î = E {- J3, Z ,
î≥*
-)+
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
q If we sample
!] î
periodically at î = #u, we have
.
! # = !] #u = E {- J3, [>
-)+
q The system function of the resulting digital IIR filter becomes
B
B
.
.
B
, = = E ! # =*> = E E {- J3, [> =*> = E {- E J3, [=*+
>)/
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>)/
-)+
Prof. Hesham Tolba
-)+
Digital Signal Processing
>
>)/
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
ù- < * and yields
q The inner sum converges because
B
E J3, [=*+
>
=
>)/
.
. − J3, [=*+
q Therefore, the system function of the digital filter is
.
, = =E
-)+
.
. − J3, [=*+
q Observe that the digital filter has poles at
=- = J3, [,
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F = ., /, … , Q
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Impulse Invariance …
ã–plane to the ñ-plane
by the relationship =- = J , F = ., /, … , Q, the zeros in the two
domains do not satisfy the same relationship.
q Although the poles are mapped from the
3, [
q Therefore, the impulse invariance method does not correspond to
the simple mapping of points given by
= = J4[
+
,(=) given by , = = ∑.
-)+ +*%&, 9 (7.
was based on a filter having distinct poles.
q The development that resulted in
q It can be generalized to include multiple-order poles.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Convert the analog filter with system function
ã + *. .
ã + *. . ' + s
,] ã =
into a digital IIR filter by means of
the impulse invariance method.
q Solution:
q We note that the analog filter has a
zero at ã = −*. . and a pair of complexconjugate poles at ù- = −*. . ± 5\ as
shown.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q No need to determine the impulse response
q Instead, we directly determine
.
, = =E
-)+
!] î .
, = , as given by
.
. − J3, [=*+
from the partial-fraction expansion of ,] ã .
q Thus we have
, ã =
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*. p
*. p
+
ã + *. . − 5\ ã + *. . + 5\
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Then,
, = =
*. p
*. p
+
. − J*/.+[J1?[=*+ . − J*/.+[J*1?[=*+
q Since the two poles are complex conjugates, we can combine
them to form a single two-pole filter with system function
. − J*/.+[ cos \u =*+
, = =
. − /J*/.+[ cos \u =*+ + J*'/.+[=*+
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The magnitude of the frequency response characteristic of
this filter is as shown for u = *. . and
u = *. p.
Frequency response of digital filter
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q For purpose of comparison,
the magnitude of the
frequency response of the
analog filter is plotted as
shown.
q We note that aliasing is
significantly more prevalent
when u = *. p than when u = *. ..
Frequency response of analog filter
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Also, note the shift of the
resonant frequency as u
changes.
q Selecting a small value for
u
is important to minimize the
effect of aliasing.
q Due to the presence of
aliasing, the impulse
invariance method is
appropriate for the design of
LP & BP filters only.
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Prof. Hesham Tolba
Frequency response of analog filter
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation
q The design techniques described above are appropriate only for
LPFs and a limited class of BPFs.
q The bilinear transformation overcomes the limitation of the two
design methods described previously.
5ï-axis into the
unit circle in the ñ-plane only once ➞ avoiding aliasing of
frequency components.
s plan to z
q It is a conformal mapping that transforms the
S are mapped inside the unit circle in
the ñ-plane and all points in the RHP of S are mapped into
corresponding points outside the unit circle in the ñ-plane.
q All points in the LHP of
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
q Consider an analog linear filter with system function
, ã =
G
ã+7
q This system is also characterized by the differential equation
BD î
+ 7D î = GH(î)
Bî
q Instead of substituting a finite difference for the derivative,
suppose that we integrate the derivative and approximate the
integral by the trapezoidal formula.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
q Thus
Z
D î = ? Dd û Bû + D(î" )
Z:
where Dd î
denotes the derivative of D î .
q The approximation of the above integral by the trapezoidal
formula at î = #u and î" = #u − u yields
D #u =
u d
D #u + Dd #u − u + D #u − u
/
î = #u yields
#u = −7D #u + GH(#u)
q Thus the differential equation at
Dd
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
q Using the two previous equations, we get
.+
q The
7u
7u
Gu
D # − .−
D #−. =
H # −H #−.
/
/
/
ñ -transform of this difference equation is
.+
7u
7u *+
Gu
ü = − .−
= ü = =
. + =*+ ö(=)
/
/
/
q Consequently, the system function of the equivalent digital
filter is
, = =
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Prof. Hesham Tolba
ü =
Gu// . + =*+
=
ö(=) . + 7u⁄/ − . − 7u⁄/ =*+
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Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
or, equivalently
, = =
G
/ . − =*+
u . + =*+ + 7
q Clearly, the mapping from the
ã=
ã-plane to the ñ-plane is
/ . − =*+
u . + =*+
q This is called the bilinear transformation.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
q The previous derivation of the bilinear transformation holds
for Qth-order differential equations.
q Recalling that
= = xJ12 and
ã = ú + †ï , thus ã =
/ . − =*+
u . + =*+
can
be expressed as
/ =−.
ã= l
u =+.
/ xJ12 − . /
x' − .
/x sin %
= l 12
=
+5
'
u xJ + . u . + x + /x cos %
. + x' + /x cos %
q Consequently,
/
x' − .
ú= l
u . + x' + /x cos %
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Prof. Hesham Tolba
and
ï=
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/x sin %
l
u . + x' + /x cos %
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Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
x < ., then ú < *, and if x > ., then ú > *.
q Consequently, the LHP in ã maps into the inside of the unit
circle in the ñ-plane and the RHP in ã maps into the outside of
the unit circle.
q When x = ., then ú = * and
q If
ï=
/
sin %
/
%
l
= l tan
u . + cos % u
2
q or, equivalently,
% = / hÄU*+
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ïu
/
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
q The relationship between the frequency variables in the two
domains % = / hÄU*+ [ïu⁄/] is as shown.
q We observe that the entire range
in ï is mapped only once into
the range −§ ≤ • ≤ '.
bilinear
q However, the mapping is highly
nonlinear.
q We observe a frequency
compression (frequency warping)
due to the nonlinearity of the
arctangent function.
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Prof. Hesham Tolba
Mapping between e and ` resulting
from the bilinear transformation.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
IIR Filter Design by Bilinear Transformation …
q Note that the bilinear transformation maps the point
the point = = −..
ã = ∞ into
for bilinear
q Consequently, the single-pole LPF described above by
, ã =
G
ã+7
which has a zero at ã = ∞, results in a digital filter that has
a zero at = = −. .
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Convert the analog filter with system function
!Q à =
à + (. %
à + (. % % + %k
into a digital IIR filter by means of the bilinear transformation.
The digital filter is to have a resonant frequency of ê6 = */:.
q Solution:
q The analog filter has a resonant frequency at at
q This frequency is to be mapped into
of the parameter n.
q From W = 5⁄V Y tan[%⁄2], we must select
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Prof. Hesham Tolba
å6 = j.
ê6 = */: by selecting the value
n = %/: in order to have å6 = */:.
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Design of IIR Filters
Frequency Transformations
Example …
q Thus the desired mapping is
à=j
% − 8'$
% + 8'$
q The resulting digital filter has the system function
, = =
*. ./r + *. **e=*+ − *. .//=*'
. + *. ***e=*+ + *. sqp=*'
=*+ term in the denominator of ,(=) is
extremely small and can be approximated by zero.
q The coefficient of the
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Thus we have the system function
*. ./r + *. **e=*+ − *. .//=*'
, = =
. + *. sqp=*'
q This filter has poles at
*. sp.
ù+,' = *. srq J±1,/' and zeros at =+,' = −.,
q Therefore, we have succeeded in designing a two-pole filter
that resonates near % = '//.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q In this example,
u was selected to map ï; into the desired •;.
q The design of the digital filter usually begins with
specifications in the digital domain, which involve the
frequency variable •.
q These specifications in frequency are converted to the analog
domain by means of the relation in ï = /⁄u l tan[%⁄2].
q The analog filter is then designed that meets these
specifications and converted to a digital filter by means of
the bilinear transformation in ã = /⁄u . − =*+ ¶ . + =*+ .
u is transparent and may be
set to any arbitrary value (e.g., u = .).
q In this procedure, the parameter
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
\-dB
bandwidth of *. /', using the bilinear transformation applied to
the analog filter
q Design a single-pole lowpass digital filter with a
!Q à =
åV
à + åV
where ïg is the \-dB bandwidth of the analog filter.
q Solution:
q The digital filter is specified to have its
ê! = (. :*.
−\-dB gain at
q In the frequency domain of the analog filter
corresponds to
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•! = *. /'
ïg = [/⁄u] l tan 0.1' = *. ep/u
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Thus the analog filter has the system function
!Q à =
(. kf/n
à + (. kf/n
q This represents our filter design in the analog domain.
ã = /⁄u . − =*+ ¶ . + =*+
to convert the analog filter into the desired digital filter.
q Thus we obtain
q Applying the bilinear transformation
, = =
*. /)p . + =*+
. − *. p*s=*+
where the parameter u has been divided out.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q The frequency response of the digital filter is
, = =
*. /)p . + J*1e
. − *. p*sJ*1e
• = *, , * = ., and at • = *. /', we have , *. /' = *. q*q,
which is the desired response.
q At
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Characteristics of Commonly Used Analog Filters
q IIR digital filters can be obtained by beginning with an analog
filter and then using a mapping to transform the ã-plane into
the ñ-plane.
q Such a mapping preserves, as much as possible, the desired
characteristics of the analog filter.
q In this section, the important characteristics of commonly used
analog filters will be described here.
q Discussion is limited to LPFs.
q Subsequently, several frequency transformations that convert a
LP prototype filter into either a BP, HP, or band-elimination
filter will be described.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Butterworth Filters
q LP Butterworth filters are all-pole filters characterized by
the magnitude-squared frequency response
! å
%
=
1
% + å/å!
%R
=
1
% + y' å/åC
%R
where Q is the order of the filter, ïg is its cutoff
frequency, ï3 is the passband edge frequency, and ./ . + y'
the band-edge value of , ï ' .
, S , −ã
follows that
q Since
evaluated at ã = 5ï is simply equal to , ï
, ã , −ã =
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Prof. Hesham Tolba
is
'
, it
1
% + −^) /W)$
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Butterworth Filters …
, S , −ã
spaced points.
q The poles of
occur on a circle of radius ïg at equally
q From the previous equation we find that
−/5
05J
= −.
+/.
= J1 '-=+ ,/. ,
F = *, ., … , Q − .
and hence
/K = 0J .#"/5 .#
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5K_/ "/5` ,
Prof. Hesham Tolba
2 = $, 3, … , 5 − 3
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Butterworth Filters …
q For example, the pole positions for
Q = ) and Q = p Butterworth
filters are illustrated as shown.
Pole positions for Butterworth filters.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Butterworth Filters …
q The frequency response
characteristics of the class of
Butterworth filters are as shown
for several values of Q.
butterworth monotonically
decrease
with freq
butterworth filter
'
is monotonic in
, ï
both the passband and stopband.
q Note that
q The order of the filter required to
meet an attenuation 2' at a
specified frequency ï4 is easily
determined from
D W
)
=
1
/ + W/W;
)<
=
1
/+
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c)
W/W=
Frequency response for Butterworth
filters.
)<
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Butterworth Filters …
q Thus at
ï = ï4 we have
1
. + y'
and hence
Q=
ï4/ï3
'.
= 2''
©V™ ./2'' − .
©V™ 2/y
=
/ ©V™ ï4/ï!
©V™ ï4/ï3
where, by definition,
2' = ./ . + 2'
q Thus the Butterworth filter is completely characterized by the
parameters Q, 2' , y, and the ratio ï4 /ï3.
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Design of IIR Filters
Frequency Transformations
Example
q Determine the order and the poles of a LP Butterworth filter
that has a −\-dB bandwidth of 500 Hz and an attenuation of )*
dB at 1000 Hz.
q Solution:
q The critical frequencies are the
−\-dB ï! and the stopband
frequency ï4 , which are ï! = .***' & ï4 = /***'.
q Recall that
, ï3
'
=
h
+= `& /`>
+
)<
q For an attenuation of 40 dB,
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Prof. Hesham Tolba
= +=i) and , ï4
−)* W¨ = .* log +/
'
= += `
h
8 /`>
+
j))
+
)<
= j) .
)
⟹ 2' = *. *..
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Hence
©V™ ./2'' − .
©V™ +/ .*k − .
Q=
=
= e. e)
/ ©V™ ï4/ï!
/ ©V™ +/ /
q To meet the desired specifications, we select
Q = q.
q The pole positions are
ã- = .***'J1 ,⁄'= '-=+ ,/+k ,
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Prof. Hesham Tolba
F = *, ., … , e
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters
q There are two types of Chebyshev filters.
q Type I Chebyshev filters are all-pole filters that exhibit
equiripple behavior in the passband and a monotonic
characteristic in the stopband.
q The family of type II Chebyshev filters contains both poles
and zeros and exhibits a monotonic behavior in the passband
and an equiripple behavior in the stopband.
q The zeros of this class of filters lie on the imaginary axis in
the ã-plane.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q The magnitude squared of the frequency response characteristic
of a type I Chebyshev filter is given as
! å
%
=
1
% + y'u'.
å/åC
Where y is a parameter of the filter related to the ripple in
the passband and u.(H) is the Qth-order Chebyshev polynomial
defined as
VQW í VQW '$ H ,
VQW_ í VQW_'$ H ,
u.(H) = $
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Prof. Hesham Tolba
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H ≤%
H >%
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q The Chebyshev polynomials can be generated by the recursive
equation
u.=+ H = /H u. H − u.*+ H ,
Q = ., /, …
where u" H = ., u+ H = H; from the recursive relation we obtain
u' H = /H' − ., u? H = )H? − \H, and so on.
q Some of the properties of these polynomials are as follows:
u. H ≤ . for all H ≤ ..
2. u. . = . for all Q.
3. All the roots of the polynomial u. H
− . ≤ H ≤ ..
1.
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occur in the interval
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
y is
related to the ripple in
the passband, as shown,
for Q odd and Q even.
q The filter parameter
Q odd, u. * = * and
hence , * ' = ..
q For
Type 1
Q even, u. * = . and
hence , * ' = ./ . + y' .
q For
Type I Chebyshev filter characteristic.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q At the band-edge frequency
ï = ï3, we have u. . = ., so that
.
. + y'
or, equivalently
y' =
= 1 − 2h
.
−1
. − 2'+
where 2h is the value of the passband ripple.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q The poles of a type I Chebyshev filter lie on an ellipse in the
ã-plane with major axis
x+ = ï3
n' + .
/n
x' = ï3
n' − .
/n
and minor axis
where n is related to y according to the equation
n=
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. + y' + .
y
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
íth-order filter by first
locating the poles for an equivalent í
Butterworth filter
that lie on circles of radius ì$ or radius ì% , as shown.
butterworth circles
q Denoting the angular positions of the poles
chebyshev ellipse
of the Butterworth filter as
q The pole locations are determined for an
th -order
*
:O + % *
+
,
O = (, %, … , í − %
:
:í
then the positions of the poles for the
Chebyshev filter lie on the ellipse at the
coordinates H- , I- , O = (, %, … , í − %, where
î- =
H- = ì% cos î- , O = (, %, … , í − %
ï- = ì$ sin î- , O = (, %, … , í − %
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Determination of the pole
locations for a Chebyshev filter.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q A type II Chebyshev filter contains zeros as well as poles.
q The magnitude squared of its
frequency response is given as
! å
%
=
Type 2
1
% + y'
75` ï4/ï3 /75` ï4/ï
where u.(H) is the Qth-order
Chebyshev polynomial and ï4
is the stopband frequency as
shown.
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Prof. Hesham Tolba
Type II Chebyshev filter
characteristic.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q
The zeros are located on the on the imaginary axis at the points
óåS
à- =
,
O = (, %, … , í − %
sin î-
q
The poles are located at the points (ñ-, \-), where
ñ- = åSH-/ H%- + I%-,
O = (, %, … , í − %
\- = åSI-/ H%- + I%-,
O = (, %, … , í − %
where {H-} & {I-} are as defined previously with } now related to the
ripple in the stopband through the equation
$/R
% + % − <%% /<W
}=
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Chebyshev Filters …
q Observe that the Chebyshev filters are characterized by the
parameters í, ò, <% & åS /åC .
ò, <% & åS /åC , we can determine
the order of the filter from the equation
q For a given set of specifications on
öQõ
í=
% − <%% +
öQõ åS /åC +
% − <%% % + ò%
åS /åC
%
/ò<%
−%
=
cosh'$ </ò
cosh'$ åS /åC
where, by definition,
<% =
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Prof. Hesham Tolba
%
% + <%
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Determine the order and the poles of a type I LP Chebyshev
filter that has a .-dB ripple in the passband, a cutoff
frequency ï3 = .***', a stopband frequency ï4 = /***', and an
attenuation of )* dB or more for ï ≥ ï4 .
q Solution:
q First, we determine the order of the filter.
q We have
.* log +/ . + y' = .
. + y' = .. /ps
y' = *. /ps
y = *. p*rr
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Also,
/* log +/ 2' = −)*
2' = *. *.
y, 2' & ï4/ï3, we get
©V™ +/ .se. p)
Q=
= ). *
©V™ +/ / + \
q Substituting the values of
q Thus a type I Chebyshev filter having four poles meets the
specifications.
q The pole positions are determined from the above-mentioned
relations.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q First, we compute
}, ì$ , and ì% .
q Hence
n = .. )/s,
q The angles
x+ = .. *e ï3,
x' = *. \ep ï3
{î- } are
î- =
*
:O + % *
+
,
:
o
O = (, %, :, h
q Therefore, the poles are located at
H+ + 5D+ = −*. .\sï3 ± 5*. sqsï3
H' + 5D' = −*. \\qï3 ± 5*. )*peï3
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q
The filter specifications are very similar to the specifications
given in the previous example (Butterworth filter).
q
In that case the number of poles required was seven.
q
The Chebyshev filter required only four poles.
q
In general, the Chebyshev filter meets the specifications with
fewer poles than the corresponding Butterworth filter.
q
Alternatively, if we compare a Butterworth filter to a Chebyshev
filter having the same number of poles and the same passband and
stopband specifications, the Chebyshev filter will have a smaller
transition bandwidth.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Elliptic Filters
q Elliptic (or Cauer) filters exhibit equiripple behavior in both
the passband and the stopband, as shown for Q odd and Q even.
q This class of filters contains both poles and zeros and is
characterized by the magnitude-squared frequency response
! å
%
=
1
% + y'>. å/åC
where >.(H) is the Jacobian
elliptic function of order Q
(tabulated by Zverev) and y
is a parameter related to the
passband ripple.
q The zeros lie on the
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elliptical filter
5ï-axis.
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Digital Signal Processing
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Elliptic Filters …
q Recall that the most efficient designs occur when we spread the
approximation error equally over the passband and the stopband.
q Elliptic filters accomplish this objective and, as a
consequence, are the most efficient.
q Such filters yield the smallest-order filter for a given set of
specifications.
q Equivalently, we can say that for a given order and a given set
of specifications, an elliptic filter has the smallest
transition bandwidth.
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Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Elliptic Filters …
q The filter order required to achieve a given set of
specifications in passband ripple 2+ , stopband ripple 2' , and
transition ratio ï3/ï4 is given as
í=
q ï3/ï4 q
q ò/< q
% − ò% /<%
% − W& /W8
%
where Æ(H) is the complete elliptic integral of the first kind,
defined as
,/%
Æ(H) = @
0
Cù
% − H% WYU% ù
and 2' = ./ . + 2' ; the passband ripple is /* log +/ . + y' .
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Elliptic Filters …
q Computer programs are available for designing elliptic filters
from the frequency specifications indicated above.
q Butterworth & Chebyshev filters might be preferable in some
applications because they possess better phase response
characteristics.
q The phase response of elliptic filters is more nonlinear in the
passband than a comparable Butterworth filter or a Chebyshev
filter, especially near the band edge.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Bessel Filters
q Bessel filters are a class of all-pole filters that are characterized
by the system function
! à =
where âR à
%
âR à
is the íth-order Bessel polynomial.
q These polynomials can be expressed in the form
R
âR à = N 6- à-/0
where the coefficients {6- } are given as 6- =
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:í − O !
í−O !
:R'- O!
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Bessel Filters …
q Alternatively, the Bessel polynomials may
be generated recursively from the relation
âR à = :í − % âR'$ à + à% âR'% à
with Ø" ã = . and Ø+ ã = ã + . as initial
conditions.
q An important characteristic of Bessel
filters is the linear-phase response over
the passband of the filter.
q An example showing a comparison of the
magnitude and phase responses of a Bessel
filter and Butterworth filter of order
Q = ) is as shown.
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Prof. Hesham Tolba
Digital Signal Processing
Magnitude & phase responses of
Bessel & Butterworth filters of
order R = =.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Bessel Filters …
q The Bessel filter has a larger
transition bandwidth.
q Its phase is linear within the
passband.
q It should be emphasized that the
linear-phase characteristics of the
analog filter are destroyed in the
process of converting the filter into
the digital domain by means of the
transformations described previously.
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Magnitude & phase responses of
Bessel & Butterworth filters of
order R = =.
Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Some Examples of Digital Filter Designs Based on the
Bilinear Transformation
q
A LPF is designed to meet specifications
of
q a maximum ripple of .// dB in the
passband,
q e*-dB attenuation in the stopband,
q a passband edge frequency of %3 = *. /p'
q a stopband edge frequency of %4 = *. \*'
q
Example 1: A Butterworth filter of order
Q = \q is required to satisfy the
specifications.
q
Its frequency response characteristics
are as shown.
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Prof. Hesham Tolba
Digital Signal Processing
Frequency response characteristics
of a 37-order Butterworth filter.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Some Examples of Digital Filter Designs Based on the
Bilinear Transformation …
q Example 2: If a Chebyshev filter I
used, a filter of order Q = .\
satisfies the specifications.
q The frequency response
characteristics for a type I
Chebyshev filter are as shown.
q The filter has a passband ripple of
*. \. dB.
Frequency response characteristics of
a 13-order Type I Chebyshev filter.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Some Examples of Digital Filter Designs Based on the
Bilinear Transformation …
q Example 3: An elliptic filter of order
Q = q is designed which also satisfies
the specifications.
q The numerical values for the filter
parameters are as listed
q The resulting frequency specifications
are as shown.
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Digital Signal Processing
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Prof. Hesham Tolba
Digital Signal Processing
Frequency response characteristics
of a 7-order Elliptic filter.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Some Examples of Digital Filter Designs Based on the
Bilinear Transformation …
q The following notation is used for the parameters in the
function ,(=):
X
K ü, ( + K ü, % 8'$ + K ü, : 8'%
!(8) = Ç
% + 6 ü, % 8'$ + 6 ü, : 8'%
?/$
q It is a simple matter to convert a LP analog filter into a BP,
BS, or HP analog filter by a frequency transformation, as will
be seen later.
q The bilinear transformation is then applied to convert the
analog filter into an equivalent digital filter.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations
q It is a simple matter to take a LP prototype filter (Butterworth,
Chebyshev, elliptic, Bessel) and perform a frequency transformation
to design a HP or a BP or a BS filter.
q One possibility is to perform the frequency transformation in the
analog domain and then to convert the analog filter into a
corresponding digital filter by mapping à-plane ➞ ç-plane.
q An alternative approach is first to convert the analog LP filter into
a LP digital filter and then to transform the LP digital filter into
the desired digital filter by a digital transformation.
q These two approaches yield different results, except for the bilinear
transformation, in which case the resulting filter designs are
identical.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Analog Domain …
q Suppose that we have a LPF with passband edge frequency
ï3 and
we wish to convert it to another LPF with passband edge
frequency ïd3.
q The transformation that accomplishes this is
ã⟶
ï3
ã,
ïd3
(©VbÉÄSS hV ©VbÉÄSS)
q Thus we obtain a lowpass filter with system function
,U (ã) = ,3 ï3/ïd3 ã , where ,3(ã) is the system function of the
prototype filter with passband edge frequency ï3.
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Analog Domain …
q If we wish to convert an LPF into an HPF with passband edge
frequency ïd3, the desired transformation is
ï3ïd3
ã⟶
,
ã
(©VbÉÄSS hV cT™cÉÄSS)
q The system function of this HPF is
,R (ã) = ,3 ï3ïd3/ã
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Analog Domain …
å! into
a BPF, with upper and lower band edge frequencies åY & åN , can be
accomplished by first converting the LPF into another LPF having a
band edge frequency åAC = % and then performing the transformation
q Converting an LPF analog filter with passband edge frequency
à⟶
à% + åN åY
,
à åY − åN
(öQa°¢WW ^Q £¢UR°¢WW)
q The same result can be accomplished in a single step by means of the
transformation
à ⟶ åC
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Digital Signal Processing
Prof. Hesham Tolba
à% + åN åY
,
à åY − åN
Prof. Hesham Tolba
(öQa°¢WW ^Q £¢UR°¢WW)
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Analog Domain …
q Thus we obtain
,l(ã) = ,3
ã' + ïU ïV
ï3
ã ïV − ïU
ï3 into
a BS filter, the transformation is simply the inverse of
ã ⟶ ã' + ïU ïV ⁄ã ïV − ïU with the additional factor ï3 serving to
normalize for the band-edge frequency of the LPF.
q Converting an LP analog filter with band-edge frequency
q Thus the transformation is
à ⟶ åC
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à åY − åN
,
à% + åN åY
(öQa°¢WW ^Q £¢URW^Q°)
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Analog Domain …
q This leads to
,l4(ã) = ,3 ï3
ã ïV − ïU
ã' + ïU ïV
q All the above mappings are
summarized in the shown table.
Frequency Transformations for Analog Filters
(Prototype LPF has Band Edge Frequency `&).
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Transform the single-pole lowpass Butterworth filter with
system function
,(ã) =
ï3
ã + ï3
into a bandpass filter with upper and lower band edge
frequencies ïV and ïU , respectively.
q Solution: The desired transformation is given by
ã ⟶ ï3
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ã' + ïU ïV
ã ïV − ïU
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Thus we have
.
ã' + ïU ïV
ã ïV − ïU + .
ïV − ïU ã
= '
ã + ïV − ïU ã + ïU ïV
, ã =
q The resulting filter has a zero at
^=
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Digital Signal Processing
Prof. Hesham Tolba
ã = * and poles at
− W6 − W5 ± W)6 + W)5 − 0W6 W5
5
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Digital Domain
q Frequency transformations can be performed on a digital LPF to
convert it to either a BP, BS, or HP filter.
=*+ by a rational
function ± = , which must satisfy the following properties:
1. The mapping =*+ ⟶ ± =*+ must map points inside the unit
circle in the ñ-plane into itself.
2. The unit circle must also be mapped into itself.
q This implies that for x = .,
q The transformation involves replacing the
*+
J*12 = ± J*12 ≡ ± %
= ± % J16Km n 2
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Digital Domain …
q We must have
± %
= . for all %, i.e., the mapping must be all
pass.
q Hence it is of the form
&
±
=*+
= ±Ç
-/$
8'$ − 6% − 6- 8'$
where 7- < . to ensure that a stable filter is transformed
into another stable filter (i.e., to satisfy condition 1).
q From the above form, we obtain the desired set of digital
transformations for converting a prototype digital LPF into
either a BP, a BS, an HP, or another LP digital filter.
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Prof. Hesham Tolba
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Frequency Transformations in the Digital Domain …
q These transformations are
tabulated as shown.
Frequency Transformations for digital Filters
(Prototype LPF has Band Edge Frequency 2&).
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example
q Convert the single-pole lowpass Butterworth filter with system
function
,(=) =
*. /)p . + =*+
. − *. p*s=*+
into a BPF with upper and lower cutoff frequencies %V and %U ,
respectively. The LPF has 3-dB bandwidth, %3 = *. /'.
q Solution:
q The desired transformation is
=*+ ⟶
=*' − 7+ =*+ + 7'
7' =*' − 7+ =*+ + .
where 7+ and 7' are defined in the previous Table.
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Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q Substitution into
,(=) yields
=*' − 7+ =*+ + 7'
7' =*' − 7+ =*+ + .
,(=) =
=*' − 7+ =*+ + 7'
. + *. p*s . −
7' =*' − 7+ =*+ + .
*. /)p . − 7' . − =*'
=
. + *. p*s7' − .. p*s7+ =*+ + 7' + *. p*s =*'
*. /)p . −
= = ±. and a pair
of poles that depend on the choice of %V and %U .
q Note that the resulting filter has zeros at
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Example …
q For example, suppose that
q Since
%V = \'/p and %U = /'/p.
%3 = *. /', we find that Æ = ., 7' = * and 7+ = *.
q Then
*. /)p . − =*'
,(=) =
. + *. p*s=*'
q This filter has poles at
% = '//.
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Prof. Hesham Tolba
= = ±5*. q.\ and hence resonates at
Prof. Hesham Tolba
Digital Signal Processing
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General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations
q Frequency transformation can be performed either in the analog
domain or in the digital domain.
q Some caution must be exercised depending on the types of
filters being designed.
q The impulse invariance method and the mapping of derivatives
are inappropriate to use in designing HP and many BP filters,
due to the aliasing problem.
q Analog frequency transformation followed by conversion into
the digital domain by use of these two mappings would not be
employed.
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Digital Signal Processing
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Table of contents
General Considerations
Design of FIR Filters
Design of IIR Filters
Frequency Transformations
Observations …
q It is much better to perform the mapping from an analog LPF
into a digital LPF by either of these mappings, and then to
perform the frequency transformation in the digital domain.
q Thus the problem of aliasing is avoided.
q In the case of the bilinear transformation, where aliasing is
not a problem, it does not matter whether the frequency
transformation is performed in the analog domain or in the
digital domain.
q In this case only, the two approaches result in identical
digital filters.
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Digital Signal Processing
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