ECE 569 Digital Signal Processing Project

ECE 569 Digital Signal Processing
Project 1 Report
Name: Weixiong Wang No. A20332258
PART A
Design Object:
1.Design a linear phase ,bandpass FIR filter to meet the following specifications:
 Passband of ];5.0;25.0[  p ;
 Stopband of ];,55.0[]2.0,0[  s
 Passband tolerance ;1.0p
 Stopband tolerance .05.0s
Design the FIR filter by using
1.Least-squares error minimization.
2.Equiripple design by Remez algorithm.
For same specifications,design IIR filter by using
3.Butterworth filter.
4.Chebyshev type Ι (or type ΙΙ) filter.
2.Use provided function “box_plot” plot filter specification graphs.
Design Procedure:
 Least-squares FIR filter.
Functional Matlab Function: firls
In the design process,setting the band edges with different order N to meet the specifications.As
from N=80,gradually decrease the order until find a minimal order for the filter to meet band
edges adjusted to yield the best response.Figure A_1.1 shows the result.for N=35.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.2
0.4
0.6
0.8
1
Normalized Frequency: 0.2556152
Magnitude: 1.017338
Magnitude: 1.008029
Normalized Frequency ( rad/sample)
Magnitude
Magnitude Response and Phase Response
-31.3438
-25.1772
-19.0107
-12.8442
-6.6776
-0.5111
Phase(radians)
Least Square: Magnitude
Least Square: Phase
Figure A_1.1
 Equiripple FIR filter.
Functional Matlab Function: firpm
In the design process,setting the band edges with different order N to meet the
specifications.Then filter order was gradually reduced and obviously we will find the ripples are
equal satisfy the requirement of equiripple FIR filter.
The smallest N for us could adjust bandedges to let the equiripple in passband&stopband was
N=36 as shown in Figure A_2.1.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
5
10
15
Magnitude: 13.80579
Magnitude: 13.74899
Magnitude
-19.9323
-12.3817
-4.8311
2.7195
Phase(radians)
Equiripple: Magnitude
Equiripple: Phase
Figure A_2.1
FIR Filter Conclusion:
For an efficient and optimized digital FIR filter design, least-squares design and the equi-ripple

design are two extreme points on a trade-off curve between maximum error and error energy.
Equiripple filter has equal ripples in passband & stopband, which means the signal distortion
that happens at the edge of the passband due to presence of a large ripple is avoided in equiripple
design,but equiripple design has a large transition band, thus limiting the total passband
width.What’s more,equiripple filters seek to minimize the maximum error between the desired
filter response and the designed approximation.
Least Squares design minimizes the error energy, its maximum error is relatively large, and the
opposite holds for the equi-ripple design.And the passband ripple are not equi-ripple & exhibit a
spike at the passband edge,which causes signal distortion at the edge.On the other
hand,Least-squares filters seek to minimize the total squared error betwen the desired filter
response and the designed approximation.
 Butterworth IIR filter
Functional Matlab Function: butter buttord
buttord was used to get the minimum order of a digital or analog Butterworth filter required to
meet a set of filter design specifications.
By using the buttord function,we can get N=11 as the minimal order.As shown in Figure A_3.1.
Here we can find the band edge are about to meet the required specification of the filter.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-250
-200
-150
-100
-50
0
Magnitude: -0.8773566
Magnitude(dB)
Magnitude Response (dB)
Figure A_3.1
(In this figure we can find that there is no ripples in the passband edge is about from
5.025.0 21  pp fandf )
Then we see the phase response of this filter,we can find it’s not a straight line.That’s because it’s
a IIR filter,the phase response in no more a straight line.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.2
0.4
0.6
0.8
1
Magnitude
-35.9581
-29.115
-22.2719
-15.4288
-8.5857
-1.7426
Phase(radians)
Butterworth: Magnitude
Butterworth: Phase
Figure A_3.2
 Chebyshev type I IIR filter
Functional Matlab Function: cheby1 cheb1ord
cheb1ord calculates the minimum order of a digital or analog Chebyshev Type I filter required to
meet a set of filter design specifications.
By using the cheb1ord function,we can get N=5 as the minimal order.As shown in Figure A_4.1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-160
-140
-120
-100
-80
-60
-40
-20
0
Magnitude: -0.001287614
Magnitude(dB)
Magnitude Response (dB)
Figure A_4.1
(In this figure we can find that there is no ripples in the passband edge is about from
5.025.0 21  pp fandf )
Figure A_4.2 is the magnitude and phase response,we can find that phase decrease greatly in the
passband magnitude,and because it’s a IIR filter, the phase response in no more a straight line.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.2
0.4
0.6
0.8
1
Magnitude
-14.4231
-11.1671
-7.9111
-4.6552
-1.3992
1.8568
Phase(radians)
Chebyshev1: Magnitude
Chebyshev1: Phase
Figure A_4.2
IIR Filter Conclusion:
The Butterworth filter is maximally flat It optimal is in the sense that of all possible filters with
a monotonic passband and a monotonic stop band, it has the minimum attenuation in the pass
band.
The Chebyshev Type I filter minimizes the absolute difference between the ideal and actual
frequency response over the entire passband by incorporating an equal ripple of Rp dB in the
passband. Stopband response is maximally flat. The transition from passband to stopband is more
rapid than for the Butterworth filter.
The Butterworth filter is completely defined mathematically by 2 parameters: Cutoff frequency
and number of poles. The Chebyshev filter has a third parameter: Passband Ripple.

box_plot
Below are the filter specifications as magnitude response graph of the each filters.
Figure A_5.1 is the filter specification graph of least square filter.Figure A_5.2 is the phase
response.
Least Square Filter
0 1 2 3 4 5 6
0.2
0.4
0.6
0.8
1
1.2
 [rad/sample]
|H(w)|
Filter magnitude response
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
 [rad/sample]
|H(w)|
Filter magnitude response-logscale
Figure A_5.1
Filter magnitude response with design specifications.
Left: Magnitude response Right: Magnitude response on a logarithmic scale
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-15
-10
-5
0
Phase(radians)
Phase Response
Figure A_5.2
Equiripple Filter
Figure A_5.3 is the filter specification graph of equiripple filter.Figure A_5.4 is the phase
response.

0 1 2 3 4 5 6
0.2
0.4
0.6
0.8
1
1.2
 [rad/sample]
|H(w)|
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
 [rad/sample]
|H(w)|
Figure A_5.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-20
-15
-10
-5
0
Phase(radians)
Phase Response
Figure A_5.4
Butterworth Filter
response.
0 1 2 3 4 5 6
0.2
0.4
0.6
0.8
1
1.2
 [rad/sample]
|(H(w)|
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
 [rad/sample]
|H(w)|
Figure A_5.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-35
-30
-25
-20
-15
-10
-5
Phase(radians)
Phase Response
Figure A_5.6
Cheby1 Filter
response.
0 1 2 3 4 5 6
0.2
0.4
0.6
0.8
1
1.2
 [rad/sample]
|H(w)|
0 1 2 3 4 5 6
10
-4
10
-3
10
-2
10
-1
10
0
 [rad/sample]
|H(w)|
Figure A_5.7

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
-14
-12
-10
-8
-6
-4
-2
0
2
Phase(radians)
Phase Response
Figure A_5.8
PART B
Design Procedure:
The phase delay for the IIR filter is the average number of group delay I got from fvtool.
Below are the BER performances of 4 filters under different signal type,as shown in
figure B_1
Filter
Type
Signal
Type
clean Interfere noise both
Least Square
0% 50.19% 5.5011% 50.19%
0% 0.41016% 3.8015% 4.2817%
Equiripple
0% 50.19% 5.5011% 50.19%
0% 0.67027 4.3417% 4.952
Butterworth
0% 50.19% 5.5011% 50.19%
0% 0.30012% 5.3621% 4.932%
Chebyshev1
0% 50.19% 5.5011% 50.19%
0% 0.82016% 6.4013% 5.6511%
Figure B_1

Difference Analysis:
From the figure,we can see that for different types signal,
For FIR filter,Least Square filter always has the minimal BER,that’s because this
filter can reduce the error energy mostly in 4 kinds filters.Both two FIR filters are
linear phase filter,so this won’t cause "phase distortion" or "delay distortion",prevent more error
inr the bit transformation.
For IIR filter,the BER for Butterworth filter is less than Cheby1,that’s because
butter has both flat passband and stopband while Cheby1 has more ripples in the
passband,which will somehow influence the BER performance.And under same
passband tolerance Cheby1 arrive a sharper slope than Butter,this result in a narrower
bandwith for Cheby1 and increase the error of signal.
Question:
From the table we can find that the BER under “noise” for Cheby1 is larger than
6%,and I tried other value of the delay,found that 11 is the minimal value I can get. I
don’t know which part I am wrong.

ECE 569 Digital Signal Processing Project

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