An N point DFT can be considered to be a set of N filters; each filter tuned to detect one...

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An N point DFT can be considered to be a set of N filters; each filter tuned to detect one particular frequency. Consider the filter corresponding to the DC component with N=11. a) Plot the frequency response of this filter. b) How many times does the magnitude of this filter go to zero around the unit circle? c) Plot the zeros of the filter (use roots and zplane in Matlab) (3 marks) 2. An FIR filter with symmetric or antisymmetric coefficients will have linear phase. Show that for such a filter the zeros will either be 1) at z = 1, 2) at z = -1, 3) in reciprocal pairs on the real axis, 4) in conjugate pairs on the unit circle or 5) in reciprocal conjugate quads. Hint – For symmetric coefficients show that Hz z Hz M ( ) ( ( )) ? ? ?1 and examine the zero locations. (3 marks) 3. Design a LP FIR filter to meet the following specifications using
	
Document Preview: ELEC4620/7462 Digital Signal Processing 
Assignment 2 (rev. 1) 
(Due: 5pm, Friday 01/09/2017)

1. An N point DFT can be considered to be a set of N filters; each filter tuned to detect one 
particular frequency.  Consider the filter corresponding to the DC component with N=11. 
a) Plot the frequency response of this filter. 
b) How many times does the magnitude of this filter go to zero around the unit circle? 
c) Plot the zeros of the filter (use roots and zplane in Matlab) 
(3 marks) 
 
2. An FIR filter with symmetric or antisymmetric coefficients will have linear phase.  Show 
that for such a filter the zeros will either be 1) at z = 1, 2) at z = -1, 3) in reciprocal pairs on 
the real axis, 4) in conjugate pairs on the unit circle or 5) in reciprocal conjugate quads.  
Hint – For symmetric coefficients show that 
 
?M?1
 
Hz ()?z (H(z ))
 
and examine the zero locations. 
(3 marks) 
 
3. Design a LP FIR filter to meet the following specifications using the window method.  Use a 
Blackman window. 
 Fs  = 20 kHz 
Fc = 5.0 kHz (3 dB down) 
Attenuation = 60 dB at 7 kHz  
Give all the relevant plots (impulse, frequency responses) and the performance of the final 
filter. Compare this filter to one designed using the optimal method 
(4 marks) 
 
4. Design a HP FIR filter to meet the following specifications using the window method.  Use 
a Kaiser window. 
Fs = 20 kHz 
Fc = 3.7 kHz (1 dB down) 
Attenuation = 80 dB at 3.5 kHz 
Give all the relevant plots (impulse, frequency responses) and the performance of the final 
filter. Compare this filter to one designed using the optimal method. 
(4 marks) 
 
5. Consider the impulse response of the following low pass filter designed using the Parks-
McClelland method 
 
h(n) = XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX XXXXXXXXXX 
 
Take the Fourier Transform of this sequence and express this result as a polynomial in powers of 
cos ????Substitute x = cos ??to express the result as a polynomial in x. ...

David · Accepted Answer

1. An N point DFT can be written as 




10
0
2
)()(
n
n
N
k
jn
enxkX

. If the frequency response of a filter 
has an impulse at a particular frequency, the output will have only 1 frequency and the rest 
frequencies will be filtered out.
To find the DC component, k will be equal to zero and hence X[0] = x[n]
Frequency response: 
A. 
 
 
B. Since it is a 11 point DFT, there will be 11 samples in the frequency domain. 1 sample will 
correspond to the desired frequency response. The rest samples will correspond to ‘0’ response. 
Hence, 10 is the answer.
C. There will be no zeros for this filter since only DC component is available.
2. For any linear phase FIR filter, )()( 1 zHzzH N  
‘+’ applies when the coefficients are symmetric and ‘–‘applies when the coefficients are anti-
symmetric. 
So, if zoz   is a zero of the filter, then 
zo
z
1
 is also a zero of the filter. 
Case-1: 
Zero at z=1 
If z=1 is a zero, then by using the above argument, z=(1/1)=1 is also a zero of the filter.
Case-2: 
Zero at z=-1 
If z=-1 is a zero, then by using the above argument, z=(1/-1)=-1 is also a zero of the filter.
Case-3: 
Let z=a be a zero on the real axis. Then, 
a
z
1
 will also be a zero. Also we can note here that 
since ‘a’ is real, (1/a) is also real and their magnitudes are reciprocal in nature. Hence real zeros 
occur in reciprocal pairs.

An N point DFT can be considered to be a set of N filters; each filter tuned to detect one particular frequency. Consider the filter corresponding to the DC component with N=11. a) Plot the frequency...

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