# FIR Filter Design by nikeborome

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```									 Chapter 5

FIR Filter Design
Objectives
•   Describe the general approach to filter design and the design equation for
FIR filters.
•   Define phase distortion and show how it affects signals.
•   Demonstrate how linear phase response eliminates phase distortion.
•   Demonstrate the symmetry condition on FIR impulse response that results
in linear phase.
•   Derive the impulse responses of ideal low-pass and high-pass linear phase
filters.
•   Describe and demonstrate the window design method for linear phase FIR
filters.
•   Demonstrate the design of low-pass, high-pass, band-pass, and band-reject
FIR filters.
•   Describe and demonstrate the sampling method of linear phase FIR filter
design.
•   Demonstrate the tools in MATLAB for optimized FIR filter design using the
Parks-McClellan algorithm.
Design of Frequency Selective FIR
Filters
• For frequency selective filters the design problem is to
find the impulse response, h[n], for a desired frequency
response, H(Ω)
• In principle, this can be done using the inverse
discrete-time Fourier transform (DTFT):

H ()    
n 
h[n]e  jn

1         
  H ()e
jn
h[n]                                 d   for -  n  
2         
Phase Distortion
• Phase distortion results from a variable
time delay (phase delay) for different
frequency components of a signal.
• If the phase response of a filter is a linear
function of frequency, then the phase
delay is constant for all frequencies and no
phase distortion occurs.
Linear Phase

An input signal (blue sinusoid) processed in the pass-
band of a filter (output red sinusoid) will, in general,
experience a phase change (θ1 to θ2) and a time delay (t1
to t2).
Linear Phase (Analog Filter)
ωt1 + θ1 = ωt2 + θ2            The output amplitude equals the input
amplitude in the pass-band

ω(t2 - t1 ) + (θ2 - θ1 ) = 0
(t2 - t1 ) = D t = Phase delay
(θ2 - θ1 ) = D θ = Phase change

ωD t + D θ = 0

Dθ    Dθ
Dt = -      =-
ω     2πf
This is the phase change condition for constant
D θ = - kf              phase delay independent of frequency: the phase
change is a linear function of frequency
Linear Phase (Digital Filter)
The output amplitude equals the input
W n + Dθ = 0
D                     amplitude in the pass-band

t  nTs
t
n      tf s  sample delay           Δn is a constant if Δt is constant
Ts

n  

n  k
The condition for Δn being constant is that
   k               Δθ be a linear function of Ω

dD θ
Group delay º -          A constant group delay is a signature of linear
dW    phase
Constant Phase vs. Linear Phase
>> t=0:.04/1000:.04; % The small increment of t approximates an analog signal
>> f0=50;
>> term1=(2/pi)*sin(2*pi*(f0)*t);
>> term2=(2/(3*pi))*sin(2*pi*(3*f0)*t);
>> term3=(2/(5*pi))*sin(2*pi*(5*f0)*t);
>> term4=(2/(7*pi))*sin(2*pi*(7*f0)*t);
>> s=term1+term2+term3+term4;
>> plot(t*1000,s);title('4 Term Square Wave');xlabel('milliseconds');

4 Term Square Wave
0.6

0.4
Zero phase
0.2                                                                    change for each
sinusoid
0

-0.2

-0.4

-0.6

-0.8
0   5     10     15         20     25   30   35     40
milliseconds
Constant Phase vs. Linear Phase
>> term1=(2/pi)*sin(2*pi*(f0)*t+pi/6);
>> term2=(2/(3*pi))*sin(2*pi*(3*f0)*t+pi/6);
>> term3=(2/(5*pi))*sin(2*pi*(5*f0)*t+pi/6);
>> term4=(2/(7*pi))*sin(2*pi*(7*f0)*t+pi/6);
>> s=term1+term2+term3+term4;
>> plot(t*1000,s);
>> title('4 Term Square Wave with Constant 30 deg Phase');xlabel('milliseconds');

4 Term Square Wave with Constant 30 deg Phase
1

0.8

0.6

0.4                                                                      A constant phase
0.2                                                                      change of +30
0                                                                      degrees for each
-0.2                                                                     sinusoid results in
-0.4                                                                     phase distortion.
-0.6

-0.8

-1
0   5       10      15         20     25      30        35   40
milliseconds
Constant Phase vs. Linear Phase
>> term1=(2/pi)*sin(2*pi*(f0)*t-.01*pi*(f0));
>> term2=(2/(3*pi))*sin(2*pi*(3*f0)*t-.01*pi*(3*f0));
>> term3=(2/(5*pi))*sin(2*pi*(5*f0)*t-.01*pi*(5*f0));
>> term4=(2/(7*pi))*sin(2*pi*(7*f0)*t-.01*pi*(7*f0));
>> s=term1+term2+term3+term4;
>> plot(t*1000,s);title('4 Term Square Wave with 5ms Phase Delay');xlabel('milliseconds')
4 Term Square Wave with 5ms Phase Delay
0.6

0.4                                                                  Computed with a constant
0.2
phase (time) delay of 5 ms
and linear phase. Linear
0
phase results in no phase
-0.2
distortion.
Dθ     - kf    k
-0.4                                                               D t = - .005 = -       =-      =
2πf    2πf    2π
-0.6                                                               k = - .01π

-0.8
0   5    10       15         20     25     30     35   40   D θ = - .01πf
milliseconds
Sufficient Condition for Linear
Phase
• If a FIR filter consists of an odd number of
coefficients, M + 1, where M is even, and is
symmetrical about the M/2 term, the filter has a
linear phase response of Δθ = −(M/2)Ω. The
filter will have a group delay of M/2
• This is termed a “type I” filter (odd length and
positive symmetry). Other types have different
permutations of length and symmetry. This type
is the most common and easiest to design.
Sufficient Condition for Linear
Phase
Shifted Symmetrical Impluse Response
3

2                                                          1. Odd number of impulse
response values
1
2. Symmetry point: n=2
3. Equal values of the
h[n]

0
-1
the symmetry point
4. Group delay = 2
-2

-3
-1    0      1        2         3         4      5   6
n
Sufficient Condition for Linear
Phase
>> h=[-1,1,2,1,-1];
>> fvtool(h,1)

Magnitude and Phase Responses                                                                                      Group Delay
4.5                                                                       0                                3

2.8

3.6                                                                       -60                             2.6

Phase (in samples)
2.4

Group delay(degrees)
2.7                                                                       -120                            2.2
Magnitude

2

1.8                                                                       -180                            1.8

1.6
0.9                                                                       -240
1.4

1.2
0                                                                        -300
0    0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9                                    1
0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)
Normalized Frequency ( rad/sample)
Sufficient Condition for Linear Phase
The Running Average Filter
>> n=0:6;
>> raf7=(1/7)*ones(1,7);
>> stem(n,raf7)
>> xlabel('n');ylabel('h[n]');
>> title('Impulse Response of a 7-Point Running Average Filter')
>> axis([-1 7 0 .16])
>> fvtool(raf7,1)
Magnitude and Phase Responses
Impulse Response of a 7-Point Running Average Filter
0.16                                                                                    1                                                                       100

0.14
0.8                                                                      40
0.12

Phase (degrees)
0.1                                                                                   0.6                                                                      -20
Magnitude
h[n]

0.08

0.4                                                                      -80
0.06

0.04
0.2                                                                      -140
0.02

0                                                                                     0                                                                       -200
-1   0          1        2        3        4        5           6   7                     0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
n                                                                       Normalized Frequency ( rad/sample)
The Ideal Low-Pass Filter

H(Ω)

1

-π   -Ω0          Ω0   π
Impulse Response of the Ideal LP
Filter
1 
2 
0                           0
h[n]          H ()e jn d    h[0] 
1
        e j 0 d  
1
        d 
2   0                     2   0


1 - 0     0       1                 1
0  (0 )  (20 )  0
H() =                         2                2       
0 otherwise
1 0 jn                        0
n0
h[n] 
2  0 e d                   

h[n]  
 1 sin( n)                   n0
1     jn 0                        n


0
e
2 jn       0
sin( x)
1 1                          sinc( x) 
        e jn0  e  jn0                         x
n 2 j                   
1              1 
h[n]       sin(0 n)          n  sin c(0 n)
or                                       n              n  0

1                       or
h[n]     sin( 0 n)                     0
n                       h[n]         sinc(0 n) -  n  

Impulse Response of the Ideal LP
Filter
• The impulse response of the ideal low pass filter is:

0
h[n]         sinc(0 n) -  n  


• It is easily seen that this impulse response has linear phase
because it has the symmetry property of h[-n] = h[n]
• As given, however, this impulse response is not computable: it is
infinite and anti-causal
• A finite set of M+1 h[n] values can be delayed by M/2 samples to
create a finite and causal h[n]
• Creating the finite and causal impulse response will affect some filter
properties, but not its linear phase response
Example:
21 Coefficient LP Filter with Ω0 = π/4
>> n=0:20;
>> omega=pi/4; % This is the cut-off frequency
>> h=(omega/pi)*sinc(omega*(n-10)/pi); % Note the 10 step shift
>> stem(n,h)
>> title('Sample-Shifted LP Impulse Response')
>> xlabel('n')                                  Note the linear phase properties
>> ylabel('h[n]')                               of the impulse response
>> fvtool(h,1)
Sample-Shifted LP Impulse Response                                                        Magnitude Response
0.25                                                                            1.4

0.2                                                                            1.2

0.15                                                                             1
Magnitude

0.1                                                                            0.8
h[n]

0.05                                                                            0.6

0                                                                            0.4

-0.05                                                                           0.2

-0.1                                                                            0
0   2   4      6     8      10    12     14      16   18   20                 0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
n                                                                 Normalized Frequency ( rad/sample)
Example:
201 Coefficient LP Filter with Ω0 = π/4
>> n=0:200;     % This sets the order of the filter where length(n)=201
>> omega=pi/4;
>> h=(omega/pi)*sinc((n-100)*omega/pi);      %Note the sample shift of 100 samples
>> fvtool(h,1)

Magnitude Response
1.4
Higher order = sharper
1.2
transition
1
Magnitude

0.8

0.6
Side-lobe “ripple”
(Gibbs phenomenon)
0.4
due to abrupt
0.2
truncation of the
impulse response
0
0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)
Tapering Windows
• Used to taper the abrupt truncation of the
impulse response towards zero
• Example: Hamming window

æ 2πn ÷ö
w[n] = .54 + .46 cos ç
ç N - 1÷
÷
è      ø

hW [n] = w[n]h[n]
Effect of Tapering Windows
>> n=0:20;                                                                  >> stem(n,w)
>> omega=pi/4;                                                              >> title('Hamming Window')
>> h=(omega/pi)*sinc((n-10)*omega/pi);                                      >> figure,stem(n,h,'ko')
>> w=0.54+0.46*cos(2*pi*(n-10)/20);                                         >> hold
% This is the Hamming window. Note that                                     >> stem(n,hw,'bd')
% N-1 = 20 in this case                                                    >> title('Comparison of Rectangular and Hamming
>> hw=h.*w;                                                                       Window Impulse Response')
% Note the use of “ .* “ to multiply h and w                                >> legend('Rectangular','Hamming')
% sample-by-sample                                                          >> fvtool(hw,1)
Comparison of Rectangular and Hamming Window Impulse Response
0.25
Rectangular
Hamming
0.2

0.15

0.1

0.05

0

-0.05

-0.1
0      2      4      6     8     10     12    14     16     18      20
Effect of Tapering Windows
Comparison of Rectangular and Hamming Window Impulse Response
0.25
Magnitude Response
1.4
Rectangular
Hamming
0.2
1.2

0.15
1

0.1

Magnitude
0.8

0.05                                                                                         0.6

0                                                                                         0.4

-0.05                                                                                        0.2

-0.1                                                                                         0
0      2      4      6     8     10     12    14     16     18      20                     0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)
Effect of Tapering Windows
Magnitude Response (dB)
20

0

-20

-40
Magnitude (dB)

-60

-80

-100

-120

-140
0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)
The Ideal High Pass Filter

1

-π   -Ω0          Ω0   π
H(Ω)
The Ideal High Pass Filter
1 π
ò- π H(W)e dW
jWn
h[n] =
2π
1 - W0 jWn           1 p jWn
2π ò- π              2π òW0
=              e dW+             e dW
1 1 jWn - W0      1 1 jWn π
=           e -π +            e W
2π jn             2π jn      0

1 1 é - jW0n - jπn ù 1 1 é jπn
=            e    - e       +       e - e jW0n ù
πn 2 j ë             û πn 2 j ë           û
1 1 é jW0n                1 1 é jπn - jπn ù
=           - e + e- jW0n ù+
ë                        e
û πn 2 j ë - e       û
πn 2 j
1              1
=       sin(πn) -       sin(W n)
0
πn            πn

W
h[n] = sinc(πn) - 0 sinc(W n)
0                       - ¥ < n< ¥
π
Design Example:
201 Coefficient HP with Ω0 = 3π/4
>> n=0:200;
>> omega=3*pi/4;
>> h=sinc(n-100)-(omega/pi)*sinc(omega*(n-100)/pi); % Both terms shifted by 100
% samples
>> hw=h.*blackman(201)';
>> fvtool(hw,1)
Magnitude Response (dB)
50                                                                       The Blackman tapering
window provides greater
0                                                                       suppression of stop-
band side-lobes
Magnitude (dB)

-50

-100

-150

-200
0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)
Band-Stop and Band-Pass Filters
HBR = HLP + HHP
1

hbr = hLP + hHP
0       ΩL                                    ΩH   π

HLP                            HHP

1

HLPH - HLPL

hbp = hLPH - hLPL
0        ΩL                               ΩH   π

HLPL                     HLPH
Band-Stop and Band-Pass
Examples
>> n=0:200;
>> fs=2000;
>> omegaL=2*pi*400/fs;
>> omegaH=2*pi*600/fs;
>> hLPL=(omegaL/pi)*sinc(omegaL*(n-100)/pi).*hamming(201)';
>> hLPH=(omegaH/pi)*sinc(omegaH*(n-100)/pi).*hamming(201)';
>> hHP=(sinc(n-100)-(omegaH/pi)*sinc(omegaH*(n-100)/pi)).*hamming(201)';
>> h_bandpass=hLPH-hLPL;
>> h_bandreject=hLPL+hHP;
>> fvtool(h_bandpass,1)
>> fvtool(h_bandreject,1)

Magnitude Response (dB)
Magnitude Response (dB)                                         20
20

0                                                                                          0

-20                                                                                        -20
Magnitude (dB)

-40
Magnitude (dB)

-40

-60
-60
-80

-80
-100

-120                                                                                       -100

-140                                                                                       -120
0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9                           0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)                                                       Normalized Frequency ( rad/sample)
Effect of Group Delay
(Order 200 Filter)
>> m=1:250; % The sinusoid will have a length of 250 samples
>> f=2*pi*100/fs;
>> x=sin(f*m);
>> y=filter(hLPL,1,x); % The filter command executes the filter on the signal x
>> subplot(2,1,1),plot(x),title('Input 100 Hz Sinusoid'),axis([0,250,-2,2]);
>> subplot(2,1,2),plot(y),title('Low-Pass Filter Output'),axis([0,250,-2,2]);

Input 100 Hz Sinusoid
2

1

0

-1

-2
0      50          100           150      200            250

Low-Pass Filter Output
2

1

0

-1

-2
0      50          100           150      200            250
Sampling Method of FIR Design
Basic Theory
• The sampling method is based on the principle that the DFT is a
sample of the DTFT
• To see this, take the DFT of the frequency response of a filter:

>> n=0:10; % Design a low pass filter by the window method
>> omega=pi/4;
>> h1=(omega/pi)*sinc(omega*(n-5)/pi);
>> dtft_demo(h1,0,2*pi,512); % Display the DTFT of the filter
>> hold
>> [H1,f]=dft_demo(h1); % Take the DFT of the filter
>> stem(f/pi,abs(H1),'k');
>> legend('DTFT of h1','DFT of h1')
>> title('Fourier Transforms of the Impulse Response h1')
>> hold off
DFT and DTFT of a Filter
Fourier Transforms of the Impulse Response h1
1.4
DTFT of h1
DFT of h1
1.2

1

0.8

0.6

0.4

0.2

0
0   0.2   0.4   0.6     0.8        1      1.2   1.4   1.6     1.8      2
Units of Pi
Sampling Method of FIR Design
Basic Theory
• Evenly spaced samples of the frequency response in the frequency
range Ω = 0 to 2π represents the discrete Fourier transform (DFT) of
a finite impulse response of the same length

N 1        j 2
k
H [k ]   h[n]e
n
N

n 0

k  0,1, 2,...N  1

• Therefore, given the frequency response, the impulse response can
be computed from the inverse DFT of the frequency response
1 N 1       k
j 2 n
h[n]   H [k ]e N
N k 0
n  0,1,...N  1
Steps in a Sampling Design
Using the Custom M-files
• Step 1: Determine the critical frequency Ω0
• Step 2: Determine the order M, where M is even
• Step 3: Construct a vector of M+1 real-valued
frequency response values evenly spaced from
Ω = 0 to 2π. (Custom M-files help do this)

[H,omega]=selectH_lp(Ω0,M+1) or
[H,omega]=selectH_hp(Ω0,M+1)
Steps in a Sampling Design
• Step 4: Create a causal frequency response by
delaying the response by M/2 steps
H_delay=exp(-j*omega*M/2).*H
• Step 5: Compute the impulse response of the
delayed frequency response using the inverse
discrete Fourier transform
h=inv_dft_demo(H_delay)
• Step 6: Window the impulse response with a
tapering window function; for example:
hw=h.*hamming(length(h))'
Sampling Design Example 1

Design an order 24 HP filter with a cut-off
frequency of π/2
• Generate and plot the sampled frequency
response:
>> [x,f]=selectH_hp(pi/2,25);
>> stem(f/pi,x);
>> title('Samples of the Frequency Response of a N=25
HP Filter with pi/2 Cutoff')
>> xlabel('Digital Frequency in Units of Pi')
>> axis([0 2 0 1.5]);
Design Example 1
Sampled Frequency Response
Samples of the Frequency Response of a N=25 HP Filter with pi/2 Cutoff
1.5

1

0.5

0
0       0.2    0.4     0.6      0.8    1      1.2     1.4   1.6   1.8        2
Digital Frequency in Units of Pi
Design Example 1
Impulse Response
• Compute the causal frequency response:

>> M=length(x)-1;
>> H=exp(-j*f*M/2).*x;

• Compute the impulse response with the inverse
DFT and plot the resulting frequency response:

>> hhp1=inv_dft_demo(H);
>> fvtool(hhp1,1)
Design Example 1
Frequency Response
Magnitude Response
1.4

1.2

1
Magnitude

0.8

0.6

0.4

0.2

0
0    0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)
Design Example 1
Compare the sampling design with an ideal window design

>> n=0:24;
>> omega=pi/2;
>> hhp2=sinc(n-12)-(omega/pi)*sinc(omega*(n-12)/pi);
>> [Hhp1,freq]=dtft_demo(hhp1,0,pi,512);
>> [Hhp2,freq]=dtft_demo(hhp2,0,pi,512);
>> plot(freq/pi,abs(Hhp1)); %This plots the magnitude of Hhp1 versus
frequency
>> hold
>> plot(freq/pi,abs(Hhp2),'--k') %This plots the magnitude of Hhp2
versus frequency
>> xlabel('Units of Pi')
>> title('Comparison of Ideal Window and Sampling Method for HP
Design')
>> legend('Sampling Method – hhp1','Windowed Method –hhp2')
Design Example 1
Comparison of Ideal Window and Sampling Method for HP Design
1.4
Sampling Method – hhp1
Windowed Method –hhp2
1.2

1

0.8

0.6

0.4

0.2

0
0    0.1   0.2    0.3    0.4       0.5     0.6      0.7   0.8    0.9      1
Units of Pi
Sampling Design 2

• Specifications:
– low-pass filter of order 100,
– a cut-off frequency 500 Hz
– sampling frequency of 3000 Hz
– Hamming window.
Sampling Design 2
MATLAB Code
>> n=0:100;
>> fs=3000;
>> fc=500;
>> omega_cutoff=2*pi*fc/fs;
>> [x,f]=selectH_lp(omega_cutoff,length(n)); % Create the response samples
>> M=length(x)-1;
>> H=exp(-j*f*M/2).*x; % Compute the causal filter frequency response
>> h=inv_dft_demo(H); % Compute the impulse response
>> h_hamming=h.*hamming(length(n))'; % Window with a Hamming window
>> subplot(2,1,1),dtft_demof(h,0,1500,512,3000); % Plot the magnitude
response
>> title('Sampling Design - Rectangular Window')
>> subplot(2,1,2),dtft_demof(h_hamming,0,1500,512,3000);
>> title('Sampling Design - Hamming Window')
Sampling Design 2
Results
Sampling Design - Rectangular Window
1.5

1

0.5

0
0           500                   1000        1500
Hz
Sampling Design - Hamming Window
1.5

1

0.5

0
0           500                   1000        1500
Hz
Using FIR2 for Sampling Designs
•   >> B=fir2(N,F,A)
•   N = order
•   F = Frequency break points, Ω in units of π
•   A = Amplitudes corresponding to the break points in F

F = [0, 0.3, 0.3, 1]
A = [1, 1, 0, 0]

0           0.3π                           π
Sampling Design with FIR2
>> N=100; % Set filter order
>> fs = 3000; % Set sampling frequency
>> fc=500; % Set cutoff frequency
>> F=[0,2*fc/fs,2*fc/fs,1]; % Vector of frequency break
points (omega values in units of pi)
>> A=[1,1,0,0]; % Amplitudes corresponding to F
>> B=fir2(N,F,A); % Compute impulse response
>> dtft_demof(B,0,1500,512,3000);
>> title('Sampling Design Example with MATLAB FIR2')

Note: By default FIR2 uses a Hamming window
Sampling Design with FIR2
Results
Sampling Design Example with MATLAB FIR2
1.4

1.2

1

0.8

0.6

0.4

0.2

0
0          500                  1000           1500
Hz

The values for order, transition width, and ripple cannot be independently
specified. Specifying two of the parameters forces a particular value for
the third. For a specified ripple and transition width the order is “optimal”
in a Parks-McClellan design.
A Parks-McClellan Design for a
Low-Pass Filter
• Filter specifications:
– Order 20 low-pass
– Critical frequency π/4
– Transition width 0.2π
• By specifying the order and the transition width, the forced
parameter is the pass-band and stop-band ripple

>> f=[0 .15 .35 1];   % This sets the band edges with normalized frequencies
% The transition is between the .15 and .35 values = 0.20
>> a=[1 1 0 0];       % This sets the desired amplitude response in the pass-band and
% stop-band by specifying amplitudes at the band edges.
>> N=20;               % This sets the filter order. The filter length will be N + 1.
>> h=firpm(N,f,a);
>> fvtool(h,1)
Parks-McClellan Low Pass
Results
Magnitude Response (dB)
20

0

-20
Magnitude (dB)

-40

-60

-80

-100

-120
0   0.1   0.2   0.3      0.4     0.5     0.6     0.7   0.8   0.9
Normalized Frequency ( rad/sample)

To improve the ripple performance, either the order must
be increased or the transition width must be increased
Comparison of the 3 Methods
>> % Ideal Windowed Design
>> n=0:20;
>> omega=pi/4;
>> hwin=(omega/pi)*sinc(omega*(n-10)/pi).*blackman(21)';
>> % Sample Design
>> [H,f]=selectH_lp(pi/4,21); %This is a custom M-file for generating response samples
% and frequencies for a low-pass filter
>> M=20;
>> Hk=exp(-j*f*M/2).*H;
>> hs=inv_dft_demo(Hk);
>> hsamp=hs.*blackman(21)'; % Blackman window the sampled response
>> % Optimal Design
>> f=[0,.15,.35,1];
>> a=[1,1,0,0];
>> w=[1,1];
>> N=20;
>> hopt=firpm(N,f,a,w);
Comparison of the 3 Methods (order 20)

Best transition performance for a given order
Optimal Design Example
A Multiple Band-Pass Filter
• Filter specifications:
– Sampling frequency 8 kHz
– Band-pass at:
• 300 to 400 Hz
• 600 to 700 Hz
• 2000 to 3000 Hz
– Transition width 25 Hz
– -40 dB ripple in both the pass-bands and stop-bands
• With this design the forced parameter is the filter
order; in this case it was found by
experimentation to be 700.
Multiple Band-Pass MATLAB Code
Optimal Parks-McClellan Filter
>> fs=8000;          % This is the sampling rate
>> f1=2*300/fs; % The following lines define the frequency edges
>> f2=2*400/fs;
>> f3=2*600/fs;
>> f4=2*700/fs;
>> f5=2*2000/fs;
>> f6=2*3000/fs;
>> trans=2*25/fs; % This defines the transition width
% The following vector defines the stop-band and pass-band edges
>> f=[0,f1-trans,f1,f2,f2+trans,f3-trans,f3,f4,f4+trans,f5-trans,f5,f6,f6+trans,1];
% The following vector gives the stop-band and pass-band amplitudes
>> a=[0,0,1,1,0,0,1,1,0,0,1,1,0,0];
>> N=700;             % This is the filter order which was found by trial-and-error
>> hmb=firpm(N,f,a); % The impulse response found by the Parks-McClellan
algorithm
>> fvtool(hmb,1)
Multiple Band-Pass Results
Optimal Parks-McClellan Filter
Magnitude Response (dB)
20

0

-20
Magnitude (dB)

-40

-60

-80

-100

-120
0   0.5   1    1.5          2        2.5   3   3.5
Frequency (kHz)

Stop-Band Response can be improved by either
higher filter order or a relaxed transition specification
Summary
• FIR filters allow the design of linear phase
filters, which eliminate the possibility of
signal phase distortion.
• Three methods of linear phase FIR design
were discussed:
– The ideal window method
– The sampling method
– The optimal Parks-McClellan method

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