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Summary: amplitude formulas

Type A
I M h M 2 n 0 M 1 h n M n
II M 2 n 0 N 2 1 h n M n
III M 2 2 n 0 M 1 h n M n
IV M 2 2 n 0 N 2 1 h n M n

where M N 1 2

Amplitude response characteristics

To analyze or design linear-phase FIR filters, we need to know the characteristics of the amplitude response A .

Type Properties
I A is even about 0 A A
A is even about A A
A is periodic with 2 A 2 A
II A is even about 0 A A
A is odd about A A
A is periodic with 4 A 4 A
III A is odd about 0 A A
A is odd about A A
A is periodic with 2 A 2 A
IV A is odd about 0 A A
A is even about A A
A is periodic with 4 A 4 A

Evaluating the amplitude response

The frequency response H f of an FIR filter can be evaluated at L equally spaced frequencies between 0 and using the DFT. Consider a causal FIR filter with an impulse response h n of length- N , with N L . Samples of the frequency response of the filter can be written as H 2 L k n 0 N 1 h n 2 L n k Define the L -point signal g n 0 n L 1 as g n h n 0 n N 1 0 N n L 1 Then H 2 L k G k DFT L g n where G k is the L -point DFT of g n .

Types i and ii

Suppose the FIR filter h n is either a Type I or a Type II FIR filter. Then we have from above H f A M or A H f M Samples of the real-valued amplitude A can be obtained from samples of the function H f as: A 2 L k H 2 L k M 2 L k G k W L M k Therefore, the samples of the real-valued amplitude function can be obtained by zero-padding h n , taking the DFT, and multiplying by the complex exponential. This can be written as:

A 2 L k DFT L
    h n 0 L - N
W L M k

Types iii and iv

For Type III and Type IV FIR filters, we have H f M A or A H f M Therefore, samples of the real-valued amplitude A can be obtained from samples of the function H f as: A 2 L k H 2 L k M 2 L k G k W L M k Therefore, the samples of the real-valued amplitude function can be obtained by zero-padding h n , taking the DFT, and multiplying by the complex exponential.

A 2 L k DFT L
    h n 0 L - N
W L M k

Evaluating the amp resp (type i)

In this example, the filter is a Type I FIR filter of length 7. An accurate plot of A can be obtained with zero padding.

The following Matlab code fragment for the plot of A for a Type I FIR filter.

h = [3 4 5 6 5 4 3]/30;N = 7; M = (N-1)/2;L = 512; H = fft([h zeros(1,L-N)]); k = 0:L-1;W = exp(j*2*pi/L); A = H .* W.^(M*k);A = real(A); figure(1)w = [0:L-1]*2*pi/(L-1);subplot(2,1,1) plot(w/pi,abs(H))ylabel('|H(\omega)| = |A(\omega)|') xlabel('\omega/\pi')subplot(2,1,2) plot(w/pi,A)ylabel('A(\omega)') xlabel('\omega/\pi')print -deps type1

The command A = real(A) removes the imaginary part which is equal to zero to within computerprecision. Without this command, Matlab takes A to be a complex vector and the following plot command will not be right.

Observe the symmetry of A due to h n being real-valued. Because of this symmetry, A is usually plotted for 0 only.

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Evaluating the amp resp (type ii)

The following Matlab code fragment produces a plot of A for a Type II FIR filter.

h = [3 5 6 7 7 6 5 3]/42;N = 8; M = (N-1)/2;L = 512; H = fft([h zeros(1,L-N)]); k = 0:L-1;W = exp(j*2*pi/L); A = H .* W.^(M*k);A = real(A); figure(1)w = [0:L-1]*2*pi/(L-1);subplot(2,1,1) plot(w/pi,abs(H))ylabel('|H(\omega)| = |A(\omega)|') xlabel('\omega/\pi')subplot(2,1,2) plot(w/pi,A)ylabel('A(\omega)') xlabel('\omega/\pi')print -deps type2

The imaginary part of the amplitude is zero. Notice that A 0 . In fact this will always be the case for a Type II FIR filter.

An exercise for the student: Describe how to obtain samples of A for Type III and Type IV FIR filters. Modify the Matlab code above for these types. Do you notice that A 0 always for special values of ?

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Modules for further study

  • Zero Locations of Linear-Phase Filters
  • Design of Linear-Phase FIR Filters by Interpolation
  • Linear-Phase FIR Filter Design by Least Squares

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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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