5.4 Linear-phase fir filters: amplitude formulas

Intro to digital signal processing Page 1 / 1

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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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