Filtering
Shean‐Jen Chen @ 12‐8‐2014 1. Introduction: System, Convolution, FT Properties
2. Frequency Response of Systems by Linear Constant‐Coefficient Difference Equations (LCCDE)
3. Magnitude‐Phase Representation
4. Ideal & Nonideal Frequency‐Selective Filters
5. 1st‐Order & 2nd‐Order CT Systems 6. Passive & Active Filters
7. Infinite Impulse Response (IIR) Filters
8. Frequency Transformations
9. z‐Transform
10. Mapping CT Filters to DT Filters
11. Summary
1
1. Introduction: System, Convolution, FT (1)
• Filtering: Change the relative amplitudes of the frequency
components in a signal or perhaps eliminate some frequency
components.
x(t)
Input
h(t)
System
y(t)
Output
F.T.
x(t )  h(t )  y (t )  X ( j)  H ( j)  Y ( j)
Comments:
(i) The spectrum of the output is that of the input multiplied by the frequency
response of the system H ( j) . LTI systems that change the shape of the
spectrum are often referred to as frequency‐shaping filters.
(ii) Filtering can be conveniently accomplished through the use of such
systems with an appropriately chosen frequency response, as frequency‐
selective filters.
(iii) Convolution: is to find the response y(t) of a system to an excitation x(t).
(iv) Fourier Transform: is especially useful for problems in the steady state. 2
232
1. Introduction: System, Convolution, FT (2)
• Frequency‐Shaping Filters: In audio system, a filter is included to
permit the listener to modify the relative amounts of low‐
frequency energy (Bass) and high‐frequency energy (Treble).
Comment: In high‐fidelity (Hi‐Fi) audio systems, a filter is included in the
preamplifier to compensate for the freq.‐response characteristics of the
speakers. In the following figures, the equalizing circuits are designed to
compensate for the freq.‐response of the speakers and the listening room.
(a)
(c)
(b)
234
1. Introduction: System, Convolution, FT (3)
Another class of frequency‐shaping filters often encountered is
that for which the filter output is the derivative of the filter input,
i.e., y(t) = dx(t)/dt . With x(t) of the form x(t )  e jt , y(t) will be
y (t )  je jt , from which it follows that the frequency response is
H ( j )  j 
－ CT analog differentiator: amplifies high frequencies (enhances sharp edges)
－ DT with cutoff frequency
1. Introduction: System, Convolution, FT (4)
• System Properties:




y (t )   x ( )h (t   )d    h ( ) x (t   )d 
 y (t )  x (t )  h(t )  h(t )  x (t )
－
－
－
－
－
－
Memory
Invertibility
Causality
Stability
Time Invariance
Linearity

y (t )   x( )h(t   )d  if x(t )  0 for t  0, i.e., a causal input
0
t
  x( )h(t   )d  if h(t )  0 for t  0, i.e., a causal system
0
1. Introduction: System, Convolution, FT (5)
1
Synthesis Eq.: x(t ) 
2



X ( j)e jt d

Analysis Eq.: X ( j)   x(t )e jt dt

• Parseval’s Relation
jΩ
jΩ
Total energy
in the time-domain
Total energy
in the frequency-domain
- Spectral density
• Convolution
Y ( j)  F  h(t )  x(t )   H ( j) X ( j)
• Multiplication
1
Y ( j)  F  f1 (t ) f 2 (t )  
F1 ( j)  F2 ( j)
2
6
1. Introduction: System, Convolution, FT (6)
Example: To find the signal x(t) whose Fourier transform is
X(jΩ)
X ( j ) 

1,
0,
1 W jt
e d


W
2
sin Wt W
 Wt 

 sin c  
t

 
x(t ) 
 W
 W
I.F.T.
Ω
X1(jΩ)
X3(jΩ)
X2(jΩ)
Ω
Ω
7
Ω
1. Introduction: System, Convolution, FT (7)
‐ Duality:
X1(jΩ)
Ω
X2(jΩ)
Ω
8
1. Introduction: System, Convolution, FT (8)
• Example: Cascading filtering operations
H(jΩ)
H ( j )  H 2 ( j )  H1 ( j )
H ( j)  H1 ( j)  H1 ( j) if H 2 ( j)  H1 ( j)
2
 H ( j)  H1 ( j) Sharper frequency selectivity
• Example:
X(jΩ)
H(jΩ)
Y(jΩ)
 y (t )  x(t )
Y ( j )  X ( j )
1. Introduction: System, Convolution, FT (9)
• Example:
h(t )  e  t u (t ) & x(t )  e 2t u (t )
 y (t )  h(t )  x(t )
1
1
Y ( j )  H ( j )  X ( j ) 

(1  j) (2  j)
A rational function of jΩ, ratio of polynomials of jΩ
by partial fraction expansion
1
1
Y ( j) 

(1  j) (2  j)
inverse FT
y (t )  e  t  e 2t  u (t )
330
2. Frequency Response by LCCDE (1)
A particularly important and useful class of continuous‐time LTI
systems is those for which the input and output satisfy a linear
constant‐coefficient differential equation (LCCDE) of the form
d k y (t ) M
d k x(t )
  bk
ak

k
k
dt
dt
k 0
k 0
N
Nth‐order LCCDE
To determine the frequency response H(jΩ) for an LTI system
described by the differential equation
N
M
d k y (t ) 
d k x(t ) 
 F   ak
  F  bk

k
k
dt
dt
 k 0

 k 0

F.T.

N
M
k
a
(
j

)
Y
(
j

)

b
(
j

)
X ( j )
 k
k
k 0
k
k 0
331
2. Frequency Response by LCCDE (2)
M
N



 Y ( j )   ak ( j) k   X ( j)   bk ( j) k 
 k 0

 k 0

 H ( j ) 
Y ( j )

X ( j )
k
b
(
j

)
 k 0 k
M
k

a
(
j
)
 k 0 k
N
Frequency response for an LTI system by Nth‐order LCCDE
Example: Consider a table LTI system characterized by the
differential equation with a ＞ 0. dy (t )
dt
 ay (t )  x(t )
From the equation, the frequency response is H ( j) 
1
j  a
Comparing this with the result, we see that the above equation is
 at
e
u (t ) . The impulse response of the
the Fourier transform of
system is then recognized as h(t )  e  at u (t )
427
3. Magnitude‐Phase Representation (1)
The transform Y(jΩ) of the output of an LTI system is related to the transform X(jΩ) of the input to the system by the equation
Y ( j  )  H ( j ) X ( j )
 Magitude: Y ( j)  H ( j) X ( j)
& Phase: Y ( j)  H ( j)  X ( j)
H ( j) is as the gain & H ( j) is as the phase shift of the system.
• The system has unit gain and linear phase—i.e.,
H ( j)  1 & H ( j)   with   0 & constant
• The system with this frequency response characteristic produces an output that is simply a time shift of the input—i.e.,
y (t )  x(t  t0 )
F .T .

Y ( j)  e  jt0 X ( j)
 H ( j )  1 & H ( j)  t0
429
3. Magnitude‐Phase Representation (2)
• Outputs with linear & nonlinear phase, unit gain System:
 Output with linear phase system
 Input
H ( j)  10
 Output with nonlinear phase
H ( j)  
 Output with nonlinear & linear
430
3. Magnitude‐Phase Representation (3)
• Result: Linear phase ⇔ simply a rigid shift in time, no distortion
– Nonlinear phase ⇔ distortion as well as shift
• By taking the band to be very small, we can accurately
approximate the phase of this system in the band with the linear
approximation
H ( j)    
 Y ( j )  X ( j) H ( j) e  j e  j
• Group Delay at each frequency equals the negative of the slope
of the phase at that frequency; i.e., the group delay is defined as
d
 ( )  
H ( j)
d
  ()= for linear phase system
432
3. Magnitude‐Phase Representation (4)
• Example: Consider the impulse response of an all‐pass system
with a group delay that varies with frequency.
3
H ( j )   H i ( j )
i 1
1   j / i   2 j i   / i 
2
where H i ( j) 
1   j / i   2 j i   / i 
2
with F1  50 Hz &  1  0.066, F2  100 Hz &  2  0.033,
F3  300 Hz &  3  0.058
Since the numerator of each of the factors H i ( j) is the complex
conjugate of the corresponding denominator, it follows
that H i ( j)  1 . Consequently, we may also conclude that
H ( j)  1 all‐pass
433
3. Magnitude‐Phase Representation (5)
3
 2 i   / i  
H i ( j)  2 arctan 
 & H ( j)   H i ( j)
2
i 1
1    / i  
If the values of H ( j) are restricted to lie between –π and π, we
obtain the principal‐phase function, i.e., the phase modulo 2π. This
function contains discontinuities of size 2π at various frequencies,
making the phase function non‐differentiable at those points.
However, the addition or subtraction of any integer multiple of 2π
to the value of the phase at any frequency leaves the original
frequency response unchanged. Thus, by appropriately adding or
subtracting such integer multiples of 2π from various portions of
the principal phase, we obtain the unwrapped phase. The group
delay as a function of frequency may now be computed as
d
 ( )  
H ( j)
d
433
3. Magnitude‐Phase Representation (6)
Principal phase
phase modulo 2π
Unwrapped phase
by appropriately adding or subtracting such integer multiples of 2π
433
3. Magnitude‐Phase Representation (7)
Observe that frequencies in the close vicinity of 50 Hz experience
greater delay then frequencies in the vicinity of 150 Hz or 300 Hz.
Group delay
Impulse response
The effect of such nonconstant
group
delay
can
be
observed in the impulse response
of the LTI system.
Since the all‐pass system has
nonconstant group delay, different
frequencies in the input are
delayed by different amounts.
This phenomenon is referred to as
dispersion. The group delay is
highest at 50 Hz.
Consequently, the latter parts of
the impulse response to oscillate
at lower frequencies near 50 Hz.
436
3. Magnitude‐Phase Representation (8)
• Log‐Magnitude and Phase Plots: it is often convenient to use a
logarithmic scale for the magnitude of the Fourier transform.
log Y ( j)  log H ( j)  log X ( j)
For example, on a linear scale, the detailed magnitude
characteristics in the stop band of a frequency‐selective filter
with high attenuation are typically not evident, whereas they
are on a logarithmic scale.
If h(t) is real, then H ( j) is an even function of Ω and H ( j) is
an odd function of Ω. Because of this , the plots for negative Ω
are superfluous and can be obtained immediately from the plots
for positive Ω.
438
3. Magnitude‐Phase Representation (9)
It often allows a much wider range of frequencies to be displayed
than does a linear frequency scale.
 Bode Plot
On a logarithmic frequency scale,
the shape of a particular response
curve doesn’t change if the
frequency is scaled.
Furthermore for continuous‐time
LTI
systems
described
by
differential
equations,
an
approximate sketch of the log
magnitude vs. log frequency can
often be easily obtained through
the use of asymptotes.
236
4. Ideal & Nonideal Freq.‐Selective Filters (1)
․ Frequency‐selective filters are a class of filters specifically
intended to accurately or approximately select some bands of
frequencies and reject others.
․ a lowpass filter (LPF) is a filter that passes low frequencies—i.e.,
frequencies around Ω = 0 —and attenuates or rejects higher
frequencies. A highpass filter (HPF) is a filter that passes high
frequencies and attenuates or rejects low ones, and a bandpass
filter (BPF) is a filter that passes a band of frequencies and
attenuates frequencies both higher and lower than those in the
band that is passed.
․ Idea filter are quite useful in describing idealized system
configurations for a variety of applications. 理想濾波器實際上
是無法實現的，必須用近似的可實現濾波器來代替。
4. Ideal & Nonideal Freq.‐Selective Filters (2)
• A filter is a circuit that is designed to pass signals with desired frequencies and reject or attenuate others.
• Passive filter consists of only passive element R, L and C.
• Active filter if it consists of active elements (such as transistors and op amps) in addition to passive elements R, L, and C.
• There are four types of filters.
Lowpass filter (LPF) Highpass filter (HPL) Bandpass filter (BPF) Bandstop filter (BSF)
4. Ideal & Nonideal Freq.‐Selective Filters (3)
• Frequency Response H(jΩ) of a system: 1. The transfer function of a system is the frequency‐dependent ratio of the Fourier transform of an output Y(jΩ) to the Fourier transform of an input X(jΩ).
Y ( j )
H ( j) 
X ( j )
2. It is the variation in a system’s behavior H ( j)  H ( j) e jH ( j )
with change in signal frequency and may also be considered as the variation of the magnitude and phase with frequency.
3. A zero, as a root of the numerator polynomial, is N ( j )
a value that results in a zero value of the function. H ( j) 
D ( j)
A pole, as a root of the denominator polynomial, is a value for which the function is infinite.
24
4. Ideal & Nonideal Freq.‐Selective Filters (4)
• Ideal LPF with zero phase
H ( j)
1,
H ( j )  
0,
  c
I.F.T.
c

c
sin c t
,
t
 > c

c
h(t ) 

• Ideal LPF with linear phase


c
c
The width of the filter passband is proportional to
c , while the width of the main lobe of the 1 / c
impulse is proportional to .
H ( j)
c
I.F.T.
 c
c


H ( j)  
c
 c

An ideal filter with linear phase over the
passband introduces only a simple time
shift relative to the response of the ideal
25
LPF with zero phase characteristic.
440
4. Ideal & Nonideal Freq.‐Selective Filters (5)
• Step responses s(t) of the ideal lowpass filter
理想低通濾波器具有一些不好的步級響
應特性：超過終值的超越量呈現出振盪
現象，常稱為「振鈴」。


c

步級響應的「上升時間」，為濾波器響
應時間的一種量測，與濾波器的頻寬有
反比關係。
The characteristics of ideal filter are not
always desirable in practice.
對於頻譜重疊的訊號的濾波，一般常用
的是從通帶到止帶的逐漸過渡的濾波器
c
440
4. Ideal & Nonideal Freq.‐Selective Filters (6)
• Nonideal Filters
 The ideal frequency‐selective characteristics are desirable,
they may not be attainable. 對於理想低通濾波器在輸入的
不連續點處，將產生超越量而呈現振鈴現象，此種時域表
現可能不符需求。理想低通濾波器是非因果的，而因果性
是實現的必要條件，故應尋求理想濾波器的因果近似系統
。
 愈精確近似理想的濾波，往往結構較複雜，成本較高，故
常使用較單純的濾波器。
 For all of these reasons, nonideal filters are of considerable
practical importance, and the characteristics of such filters are
frequently specified or quantified in terms of several
parameters in both the frequency and time domain.
445
4. Ideal & Nonideal Freq.‐Selective Filters (7)
H ( j)
p
s

 In addition to the specification of magnitude characteristics in
the frequency domain, in some cases the specification of phase
characteristics is also important.
 在相位特性上，濾波器常常期望具有線性或接近線性的相位
移特性。
448
5. 1st‐Order & 2nd‐Order CT Systems (1)
• The differential equation for a first‐order system is often
expressed in the form dy (t )

 y (t )  x(t )
dt
where τ is the time constant.
• The corresponding frequency response for the first‐order
1
system is
H ( j ) 
j  1
• and the impulse response is
h(t ) 
1
e  t / u (t )

• The step response of the system is
s (t )  h(t ) * u (t )  1  e  t /  u (t )
449
5. 1st‐Order & 2nd‐Order CT Systems (2)
20 log10 H ( j )  10 log10 ( ) 2  1
 20 log10 ( )
 20 log10 ()  20 log10 ( ) for   1 / 
Ω
For the first‐order system, the low‐
and high‐frequency asymptotes of the log magnitude are straight lines.
一階系統的低頻和高頻漸近線為
直線。低頻漸近線為 0 dB 水平線
。高頻漸近線為Ω每十倍頻減少20
dB的斜直線。
H ( j )   tan 1 ( )
Ω
0,

 ( / 4)  log10 ( )  1 ,
 / 2,

  0.1 / 
0.1/     10 / 
  10 / 
448
5. 1st‐Order & 2nd‐Order CT Systems (3)
• The differential equation for a second‐order system is often
expressed in the form
d 2 y (t )  b  dy (t )  k 
1
 
   y (t )  x(t )
2
m
dt
 m  dt
m
set  n 
k
b
& 
m
2 km
• The corresponding frequency response for the first‐order
system is
 2n
 2n
H ( j ) 

2
2
( j)  2 n ( j)   n ( j  c1 )( j  c2 )
where c1 & c2 are poles with c1,2   n   n  2  1
n
M
M
where M 
 H ( j ) 

j  c1 j  c2
2  2 1
448
5. 1st‐Order & 2nd‐Order CT Systems (4)
• and the impulse response is
h(t )  M ec t  ec t  u (t )
1
h(t ) 
 n e  n t
2 j 1 
2
e
j ( n 1 2 ) t
2
e
 j ( n 1 2 ) t
 u(t )
 n e   n t 

sin( n 1   2 )t  u (t )

1  2 
• Thus, for 0＜ζ＜1 , the second‐order system has in impulse
response that has damped oscillatory behavior, and this case
the system is referred to as being under‐damped.
• If ζ ＞1, both c1 and c2 are real and negative, and the impulse
response is the difference between two decaying exponentials.
In this case, the system is overdamped.
• The case of ζ = 1, when c1 = c2, is called the critically damped.
453
5. 1st‐Order & 2nd‐Order CT Systems (5)
• The step response of the system is

 ec1t ec2t  

s (t )  h(t ) * u (t )  1  M 
  u (t ) for   1
c2  

 c1
Impulse response
of 2nd‐order
decreases by 40 dB for every increase in Ω of a factor of 10.
20 log10
2 2
2

   

2   
H ( j )  10 log10  1  
   4 
 
 n  
    n  

Step response
of 2nd‐order
 2   /  n 
H ( j)   tan 1 
 1    /  2
n





6. Passive & Active Filters (1)
• In many applications, frequency‐selective filtering is
accomplished through the use of LTI systems described by linear
constant‐coefficient differential or difference equations.
• 頻率選擇濾波器常用由線性常係數微分或差分方程述的LTI系
統來完成。主要原因有三：
許多實際系統可用微分或差分方程表示
利用微分或差分方程表示的系統可以很容易實現
以微分或差分方程表示的系統，具有寛廣彈性設計範圍
• 1st‐order Passive LPF with RC by LCCDE:
dvo (t )
RC
 vo (t )  vs (t )
dt
I . F .T .
Vo ( j)
1 jC
1
1  t / RC
e
u (t )
 H ( j ) 


 h(t ) 
Vi ( j) R  1 jC 1  jRC
RC
F .T .
6. Passive & Active Filters (2)
For the RC circuit, the frequency response is Vo ( j) / Vs ( j)
H ( j) 
1
jC
Vo ( j)
1 jC
1


Vi ( j) R  1 jC 1  jRC
 H ( j ) 
1
1    0 
2
, H ( j)   tan 1
35

, 0  1 RC
0
6. Passive & Active Filters (3)
• 1st‐order Passive HPF with RC by LCCDE:
RC
F .T .
dvo (t )
dv (t )
 vo (t )  RC s
dt
dt
 H ( j ) 
Vo ( j)
jRC

Vi ( j) 1  jRC
A HPF is also made of a RC circuit, with the output taken off the resistor. The
cutoff frequency will be the same as the LPF. The difference being that the
frequencies passed go from Ωc to infinity.
• 2nd‐order Passive BPF & BSF with LCR:
 The RLC series resonant circuit provides a
BPF when the output is taken off R.
 A BSF can be created from a RLC circuit by
taking the output from the LC series
combination.
6. Passive & Active Filters (4)
• Resonance is a condition in an RLC circuit in which the
capacitive and inductive reactance are equal in magnitude,
thereby resulting in purely resistive impedance.
1
Z ( j  )  R  j L 
j C
1
 R  j ( L 
)
C
Resonance  Im( Z )  L 
1
0
C
Resonance frequency:
o 
1
1
rad/s or Fo 
Hz
2 LC
LC
37
6. Passive & Active Filters (5)
• The features of series resonance:
1. The impedance is purely resistive, Z = R;
2. The supply voltage Vs and the current I are in phase, so cosθ = 1;
3. The magnitude of the frequency response H(jΩ) = Z(j Ω) is minimum;
4. The inductor voltage and capacitor voltage can be much more than the source voltage.
• The frequency response of the resonance circuit current is
I | I |
Vm
R 2  ( L  1/  C ) 2
• Bandwidth B = ω2-ω1
38
6. Passive & Active Filters (6)
• Quality Factor: Q  2
P eak energy stored in the circuit  o L
1


 oC R
E nergy dissipated by the circuit
R
in one period at resonance
• The relationship between the B, Q, & Ω o:
R o
 o2CR
B 
L Q
• The quality factor Q is the ratio of its resonant frequency to its bandwidth.
• If the bandwidth is narrow, the quality factor of the resonant circuit must be high. • If the band of frequencies is wide, the quality factor must be low. 39
6. Passive & Active Filters (7)
• Passive filters have a few drawbacks:
– cannot create gain > 1
– do not work well for frequencies below the audio range
– require inductors, which tend to be bulky and more
expensive than other components
• It is possible, using op‐amps, to create all the common active
filters. Their ability to isolate input and output also makes them
very desirable.
•
1st‐order
Active Filters with op‐amps:
 LPF
H (j)  
Rf
 Rf 
1
vo  vi 1+
 & Fc 
2 RC
 Ri 
1
1
& Fc 
2 R f C f
Ri 1  jC f R f
 HPF
H (j)  
One pole
jCi R f
1  jCi Ri
& Fc 
1
2 Ri Ci
6. Passive & Active Filters (8)
• 1st‐order Active BPF with op‐amps:
To avoid the use of an inductor, it is possible to use a cascaded series of lowpass active filter into a highpass active filter.
To prevent unwanted signals passing, their gains are set to unity, with a final stage for amplification.
6. Passive & Active Filters (9)
• 1st‐order Active BSF with op‐amps:
Creating a bandstop filter requires using a lowpass and highpass filter in parallel. Both output are fed into a summing amplifier.
It will function by amplifying the desired signals compared to the signal to be rejected.
1 
1
RC1
2 
1
RC2
6. Passive & Active Filters (10)
• 1st‐order Active Filters with op‐amps:
 HPF
 Rf 
1
&

vo  vi 1+
F

c
R
2 RC
i 

• 2nd‐order Active Filters :
 Sallen‐Key LPF
vi
 Rf 
1
vo  vi 1+
 & Fc 
R
2 R1 R2C1C2
i 

448
7. Infinite Impulse Response (IIR) Filters (1)
• IIR filter design will be treated as magnitude‐only design
Vout  j 
2
2
Pout

 H  j 
2
Pin
Vin  j 
on principles of maximum transfer of power
H  j 
2
2
1
 H  j   1,
  p
2
1 
2
1
0  H  j   2 ,  s  
A
1
1
1  2
Analog LPF specifications on the magnitudes‐square response
1 A2
p
s

448
7. Infinite Impulse Response (IIR) Filters (2)
Find H  s  ?
H  j   H  j  H   j   H  j  H   j 
2
 H  s  H   s  s  j
s-plane
o o
x x
σ
X x
o o
H  j 
The poles and zeros of are distributed in a mirror‐image symmetry with respect to the jΩ axis.
Poles and zeros occur in complex conjugate pairs (symmetry w.r.t. the real axis)
2
jΩ
H  s  H   s   H  j 
2
 s j
A causal and stable filter: all poles of H(s) must lie within the left half‐plane (LHP) → assign all LHP poles of H(s)H(-s) to H(s)
448
7. Infinite Impulse Response (IIR) Filters (3)
• Type I “Butterworth Filters” (Maximally flat filter)
Nth‐order lowpass filter: H  j  
1
2
H ( j )
N→ ∞
smaller N
large N
Monotonically decreasing
now, H  j  
• N: filter order
• Ωc: cut‐off freq. = 3 dB freq.
2
• H  j  approaches an ideal LPF as N → ∞
H  j c  
Maximum flat at Ω=0
2

c
1
 s 
1 

j

 c

1 


 c
1
H  j 
 20 log10 c  3 dB
2
 s 
for ploes  1  

j

 c
2N
s  j
2N
2N
0
448
7. Infinite Impulse Response (IIR) Filters (4)
(2 n 1)
1
1

j
s
N
2

  1 2 N  s  jc  1 2 N  jc e
j c
(2 n 1)
1
j
 

j
 j (2 n 1)
2N
2
N
, n  1, 2,..., N   1  e
 1  e  e



 s p  c e
jΩ
If N  2, s p  c e
x x
Ωc
x x
 (2 n 1)
j j

2
2N
σ
1
j 
4
 c e
, c e
  2 n 1 
j 

 2 2N 
, n  1, 2,..., N
3
j 
4
In general, poles are equally spaced at 2N
points around circle of radius Ωc with symmetry about π/2.
→ sp is a pole, then –sp is also pole
448
7. Infinite Impulse Response (IIR) Filters (5)
For stable filter, choose H(s) to have all LHP poles
k
H (s) 

( s  s p1 )( s  s p2 )( s  s p3 )
cN

LHP poles
s  s 
pk
k is chosen such that H(j0)=1
Now, N  2 & c  1  s p  e  j 4 , e  j 3
 LHP poles = e  j 3 4 
4
1  j
2
k
 H (s) 
 1  j  1  j 
s
 s 

2 
2 

k
1
 2
= 2
( H ( j 0)  k  1)
s  2s  1 s  2s  1
448
7. Infinite Impulse Response (IIR) Filters (6)
B( s) 
c
c
s  c
dy (t )
 c y (t )  c x(t )
dt
B( s) 
c2
s 2  2c s  c2
c
d 2 y (t )
dy (t )
2
2

2



y
(
t
)


x(t )
c
c
c
2
dt
dt
3c
B( s)  3
s  2c s 2  2c2 s  3c
c
d 3 y (t )
d 2 y (t )
2 dy (t )
3
3

2


2



y
(
t
)


c
c
c
c x (t )
3
2
dt
dt
dt
c
448
7. Infinite Impulse Response (IIR) Filters (7)
• Example: Find N & Ωc (Typical LPF spec.)
H  j 
 1 dB
H  j 
 20 dB
if 0    1 : 20log10
0.89
   2 : 20log10
H  j 
10
 10log
2
H  j 
10
 1 & 10log
2
Ω
Ω2

1

1 


 c
 20
 H  j   100.1 & H  j   102
0.1
Ω1
2
2N
 10
0.1
&
2
1

1 


 c
2N
 102
Satisfies freq. spec., we only need to worry about Ω1 & Ω2
 1 
1  


 c
2N
 10
0.1
 2 
& 1 


 c
2N
 102
2 Eqns. & 2 unknown → to solve for N & Ωc
448
7. Infinite Impulse Response (IIR) Filters (8)
448
8. Frequency Transformations (1)
741
9. z‐Transform (1)
The z‐transform of a general discrete‐time signal x[n] is defined as

X ( z) 


x[n]z  n
n 
where z is a complex variable. For convenience, the z‐transform
of x[n] will sometimes be denoted as Z {x[n]} and the relationship
between x[n] and its z‐transform indicated as
Z
x[n] 
 X ( z)
The complex variable z in polar form as
z  re j
with r as the magnitude of z and ω as the angel of z.
j
 X (re ) 


n 
x[n](re j )  n
in terms of r and ω
741
10. z‐Transform (2)
j
 X (re ) 

n
 j n
x
[
n
]
r
e


n 
X (re j ) is the Fourier transform of the sequence x[n] multiplied
by a real exponential r  n ; that is,
X (re j )  F  x[n]r  n 


n
For r = 1, or equivalently, z  1 , X ( z )   x[n]z
n 
the Fourier transform; that is,
X ( z)
z  e j
 X (e j )  F  x[n]
reduces to
448
10. Mapping CT Filters to DT Filters (1)
• Bilinear Transformation: preserves the system representation
from analog to digital domain (impulse invariance
transformation). s  2 1  z 1  z  2 T  s  T sz  T s  z  1  0
T 1  z 1
2 T s
2
2
H ( z )  H a ( s ) s  2 1 z 1
T 1 z 1
Properties of bilinear transformation:
(i) Rational functions of s map into rational function of z
2
 2






z
Re(
s
)
j
Im(
s
)
Re(
s
)
j
Im(
s
)
(ii) If Re(s) < 0, then  T
 T


z 1
 
If Re(s) > 0, then
∴ LHP maps inside the unit circle
→ stability is preserved
2 T  j
z 
2 T  j
(iii) If Re(s) = 0 → s = jΩ, then
→ jΩ axis maps into unit circle  a a  1

 z 1
448
10. Mapping CT Filters to DT Filters (2)
• Exact relationship:
2 1  z 1
2 1  e  j
s
 j 
1
T 1 z
T 1  e 1 j
 j 2
e j  2  e  j  2  2 j sin  2

2e


 j 2
j 2
 j 2
Te
 e  e  T cos  2


bilinear transformation
2

tan
T
2
T
  2 tan
2
1
narrower
249
11. Summary (1)



徵詳細地去檢視LTI系統的特性及它們對訊號的影響。尤其是，我們小
心地檢視訊號與系統的大小及相位的特性，並且也介紹了大小的數值
及LTI系統的波德圖。我們也討論到訊號與系統在相位與相位失真方面
的衝擊 。這個檢視引領我們了解到線性相位特性所扮演的特殊角色，
它在全頻域上將形成一個固定的延遲，且引導出非線性相位特性系統
的非固定群延遲與分散的概念。
頻率選擇濾波器及其時域與頻域的折衷考慮：我們檢視了理想與非理
想頻率選擇器兩者，並見到了時域與頻域的考量、因果性的限制及實
現上的爭論點，這些常可讓我們在過渡頻帶及通帶與止帶的容忍度上
，對非理想濾波器做出更好的抉擇。
我們也針對一階及二階的連續時間系統在時域—頻域特性上做更詳細
的檢視。特別注意到在系統的響應時間與頻域頻寬之間的折衷考慮。
由於一階及二階系統是建立更複雜、更高階LTI系統的基本方塊，在實
務上對於那些基本系統所發展的洞察力是相當有用的。以傅立葉分析
工具居於先導地位的例子，以及這些工具提供的洞察力，說明了在LTI
系統的分析與設計上，傅立葉分析的方法是具有可觀的價值的。