International Journal of Biomedical Signal Processing, 2(1), 2011, pp. 49-53
A SIMPLE APPROACH TO DESIGN LINEAR PHASE IIR FILTERS
Shivani Duggal1 and D. K. Upadhyay2
1
Guru Tegh Bahadur Institute of Technology Delhi, E-mail: duggal_shivani@yahoo.com
2
ECE Division N.S.I.T. Dwarka, E-mail: upadhyay_d@rediffmail.com
Abstract: This paper proposed a simple technique to design Linear Phase Infinite Impulse Response (IIR)
filter. The design problem is formulated as linear equation problem, which uses a frequency domain sampling
of the desired Magnitude and Phase response of filters along with optimal solution of linear equations to
obtain the filter design. The proposed method can give the best result if we increase the number of samples.
We have presented the filter design solving methodology along with the simulation results.
Keywords: Finite impulse response, Infinite impulse response, filter design, linear phase filter, Frequency
domain sampling.
I.
INTRODUCTION
Finite Impulse Response (FIR) filters are widely
used in many digital signal processing applications,
especially in the field of communications, because
they can exhibit linear phase characteristics. A filter
with a nonlinear phase characteristic causes
distortion to its input signal, since the various
frequency components in the signal will be delayed,
in general, by amounts which are not proportional
to their frequency, thus altering their mutual
harmonic relationships. A distortion of this form is
undesirable in many practical digital signal
processing applications such as music, video, data
transmission and biomedicine and can be avoided
by using filters with linear phase over frequency
range of interest [10]. Unfortunately, in many cases
the order of FIR filters is high for practical
implementation. In general, the number of delay
elements and multipliers needed for an FIR design
is much higher compared to similar Infinite Impulse
Response (IIR) implementations. Usually, IIR filters
offer computational efficiency and improved
selectivity in comparison with FIR filters. But the
design of the IIR filter has restriction Phase response
is not physically realizable. Some methods in the
History try to achieve a good approximation of the
linear phase, but results in extensive and complex
hardware. We introduce a concept of using the
frequency domain samples of the desired
magnitude and phase response of the filter and
using the same to obtain the filter coefficients to
implement the filter. The given magnitude and
phase responses of the filter can be sampled at a
convenient sampling frequency and the number of
samples can be conveniently chosen. Let us consider
the standard expression [2] for the IIR filter transfer
function.
H ( z) =
Y ( z) b0 + b1 z −1 ...bN z − N
=
U( z) 1 + a1 z −1 ...aD z − D
(1)
Where U (z) is the z-transform of the input signal,
Y (z) is the z-transform of the output signal, and ‘a’
and ‘b’ are real valued coefficients. From the above
transfer function, we can write the time-domain
difference equation as,
y(k)=-a1 y(k-1)- a2 y(k-2)-.... aD y(k-D)
+b0 u(k) + b1 u(k-1) +... bN u(k-N).
(2)
where the a and b coefficients are exactly the same
as in the transfer function above, k is the time index,
u (k) and y (k) are the current values of the input
and output (respectively), u(k – N) was the input
value N samples in the past, and y(k – D) was the
output value D samples in the past. Generally, if,
frequency response of a system at a frequency ω1 is
given in magnitude/phase form as A 1 ∠ϕ 1 the
output amplitude will be A1 times the input
amplitude and the output phase will be shifted an
angle ϕ1 relative to the input phase when a steadystate sine or cosine wave of frequency ω1 is applied
to the system. For instance, if the input to the system
described above at time instance ‘k’ is u 1(k) =
Cos(kω1ts), where ts is the sampling period, then the
output will be y1(k) = A1 Cos(kω1ts + ϕ1). The input
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International Journal of Biomedical Signal Processing
and output values at any sample time can be
determined in a similar manner. The final sampled
values in the matrix form with time index (k=0) is,
A=
7 2 


2 3
 4 5 
B=
1 
 
2 
 3 
Possible Selections:
(3)
7 2 
A1 = 
 B1 =
2 3
 1
 2 3
 2  , A 2 =  4 5  B2 =
 


 4 5
A3 = 
 B3 =
7 2 
The above equation is similar to AX ≈ B. the
matrix ‘A’ contains the sampled coefficients at
different frequencies and the column vector ‘X’
contains the filter coefficients. The first ‘D’ values
in ‘X’ represent the coefficients a1, a2, etc in
denominator and the next ‘N’ values give the b0,
b1, b2, etc coefficients in numerator. The column
vector ‘Y’ is the desired output value (both
amplitude and phase) at specific frequencies. Our
aim now is to solve the above problem, and get the
filter coefficients given by the vector ‘X’. Standard
methods use the inverse of the matrix ‘A’ .If the
inverse of the matrix does not exist, then Pseudo
inverses [9] can be used. To make the procedure
simple, we introduce the concept of optimal
solution of linear equation into this context.
II. DESIGN METHODOLOGY
Now we can move on to formulate the problem to
find the filter coefficients. The knowledge of desired
frequency response (both magnitude as well as
phase) is the only requirement. Here we try to
reduce the error to get the best solution by
minimizing this error parameter. The algorithm
which we follow to get the best solution is as given.
A. Algorithm for Low Pass Filter
1. Generate sample table (Phase, Amplitude,
Frequency) for Low Pass Filter from desired
magnitude and phase response.
2. Select order N for numerator and D for
denominator.
3. Compute the matrices ‘A’ & ‘B’ Which is in the
same form as (3) using values of sample table
and N, D for particular selected value of sample
time ts.
4. Select all possible combinations of equations
from matrix ‘A’ such that we will be able to form
square matrix & accordingly ‘B’ is selected.
For example:
2 
3
 
 3
1 
 
5. For each Square matrix computed we will be
able to find trial solution of ‘X’ by
Xn= Inv (An)*Bn.
(4)
6. Using all trial solutions of ‘X’ we find error by
€n = (An*Xn - Bn) for all values of n
(5)
7. We select trial ‘X’ solution as desired filter
coefficients for which error €n is minimum.
B. Algorithm for Band Pass and High Pass Filters
1. Same as step 1 of low pass filter the only
difference here we generate sample table (Phase,
Amplitude, Frequency) for Band Pass and High
Pass Filters respectively using desired
magnitude and phase response of Band Pass
and High Pass filters.
2. Select order N for numerator and D for
denominator.
3. Compute the matrices ‘A’ & ‘B’ as in Low Pass
Filter using values of sample table and N, D for
particular selected value of sample time ts.
Steps 4, 5, 6 and 7 are repeated for BP/HP as in
Low pass filter.
III. SIMULATION RESULTS
We have taken few examples [11], [12], [13] of low
pass; band pass and high pass filter we have chosen
a second order filter, with N = D = 2 , sampling rate
ts is 10-3s and solved the above formulated problem
using Matlab.
A. Simulation Results Low Pass Filter
The desired response of Low pass IIR Filter has the
following: Specifications fs1at 100 Hz, fs2 at 350 Hz
and the total frequency is 500 Hz. The signal is
sampled corresponding to phase, amplitude and
frequency. We have experimented with 16 samples.
Using, the same procedure as discussed in
Algorithm for Low Pass Filter, we obtained the
Filter coefficients as given in Table 1.
51
A Simple Approach to Design Linear Phase IIR Filters
Table 1
Obtained Coefficients - LPF
a0
1
a1
a2
b0
b1
b2
0.1995
-0.3855
0.2657
0.4077
0.1408
The obtained transfer function (6) are used to
plot the magnitude and phase response and the
results are compared with the desired one as shown
in Figure 1.
OBTAINED TRANSFER FUNCTION
Y ( z)
HLP(Z) U( z) =
0.2657 + 0.4077 z −1 + 0.1408 z −2
1 + 0.1995 z −1 − 0.3855 z −2
(6)
Results Discussion (Low Pass Filter)
The obtained response better approximates the
desired one with just 16 samples and a second order
filter. It can be inferred that the phase response
exhibits almost linear characteristics in our pass
band which is 0 - 100Hz. We are not concerned of
the phase relation in the stop band, but the phase is
a linear curve in stop band also. The small deviation
in the response in the transition from the pass band
to stop band can be rectified by increasing the
number of sample points in this region.
Figure 1b: Comparison of Desired and Obtained Phase
Responses of the Low Pass Filter
With this method, a linear phase response can
be achieved even with an IIR filter.
B. Simulations Results Band Pass
We now discuss the second example of designing a
band pass filter. A Band pass IIR Filter was designed
for the given Specifications: fs1at 100 Hz, fs2 at 350
Hz and the total frequency is 500 Hz. The signal is
sampled corresponding to amplitude, phase and
frequency. We have experimented with 16 samples.
Using, the same procedure as discussed in
Algorithm for Band Pass Filter, we obtained the
Filter coefficients as given in Table 2.
Table 2
Obtained Coefficients - BPF
a0
a1
a2
b0
b1
b2
1
-0.4705
0.0729
0.5132
0.0329
-0.4287
The obtained transfer function (7) are used to
plot the magnitude and phase response and the
results are compared with the desired one as shown
in Figure 2.
OBTAINED TRANSFER FUNCTION
Figure 1a: Comparison of Desired and Obtained Magnitude
Responses of the Low Pass Filter
H BP (Z ) =
Y ( z) 0.5132 + 0.0329 z −1 − 0.4287 z −2
=
(7)
U( z)
1 − 0.4705 z −1 + 0.0729 z −2
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International Journal of Biomedical Signal Processing
samples. By Increasing the Number of samples, the
obtained response may be further improved.
C. Simulations Results High Pass
The Third example we have considered is that of a
High pass filter. The procedure similar to the
previous examples is followed here also. A High
pass IIR Filter was designed for the given
Specifications: stop band1at 100 Hz, stop band2 at
350 Hz and the total frequency is 500 Hz. The signal
is sampled corresponding to amplitude, phase and
frequency. We have experimented with 16 samples.
Using the same procedure as discussed in
Algorithm for High Pass we obtained the Filter
coefficients as given in Table 3.
Table 3
Obtained Coefficients - HPF
Figure 2a: Comparison of Desired and Obtained Magnitude
Responses of the Band Pass Filter
a0
a1
a2
b0
b1
b2
1
-0.3407
0.0671
0.4222
-0.7052
0.2805
OBTAINED TRANSFER FUNCTION
H BP (Z ) =
Y ( z) 0.4222 − 0.7052 z −1 + 0.2805 z −2
=
(8)
U ( z)
1 − 0.3407 z −1 + 0.0671 z −2
The obtained transfer function (8) are used to
plot the magnitude and phase response and the
results are compared with the desired one as shown
in Figure 3.
Figure 2b: Comparison of Desired and Obtained Phase
Responses of the Band Pass Filter
Results Discussion (Band Pass Filter)
The phase response in the pass band, that is 100 Hz
– 350 Hz, is quite linear with respect to frequency.
The small deviation in the response in the
transition from the pass band to stop band can be
rectified by increasing the number of sample points
in this region. We have experimented with 16
Figure 3a: Comparison of Desired and Obtained Magnitude
Responses of the High Pass Filter
53
A Simple Approach to Design Linear Phase IIR Filters
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T. W. Parks and C. S. Burrus. Digital Filter Design.
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Figure 3b: Comparison of Desired and Obtained Phase
Responses of the High Pass Filter
Results Discussion (High Pass Filter)
The obtained response better approximates the
desired one with just 16 samples and a second order
filter. It can be inferred that the phase response
exhibits almost linear characteristics in our pass
band which is 100-350Hz. The small deviation in
the response in the transition from the pass band to
stop band can be rectified by increasing the number
of sample points in this region.
IV. CONCLUSION
This methodology for design of linear phase IIR
filters is very simple. This requires only very low
computational resources in spite of a very good
approximation of the linear phase response. The
design problem is formulated as linear equation
problem, which uses a frequency domain sampling
of the desired Magnitude and Phase response of
filters along with optimal solution of linear
equations to obtain the filter design. The proposed
method can give the best result if we increase the
number of samples. The results can be further
improved by proper choice and position of sample
points. It is very simple technique that may be
effectively adopted by the design engineers.
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