Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 345

subplot(2,1,1);loglog(f,Axk(1:N/2 þ 1));
title( ' Audio spectrum ' );
axis([10 100000 0.00001 100]);grid;
Ayk ¼ 2*abs(fft(y))/N; Ayk(1) ¼ Ayk(1)/2; % One-sided amplitude
% spectrum of the output

subplot(2,1,2);loglog(f,Ayk(1:N/2 þ 1));
xlabel( ' Frequency (Hz) ' );
title(

);
axis([10 100000 0.00001 100]);grid;

Equalized audio spectrum

' '

8.6 IMPULSE-INVARIANT DESIGN METHOD
We illustrate the concept of the impulse-invariant design in Figure 8.27 . Given the transfer function of
a designed analog filter, an analog impulse response can be easily found by the inverse Laplace
transform of the transfer function. To replace the analog filter by the equivalent digital filter, we apply
an approximation in the time domain in which the digital filter impulse response must be equivalent to
the analog impulse response. Therefore, we can sample the analog impulse response to get the digital
impulse response, and take the z-transform of the sampled analog impulse response to obtain the
transfer function of the digital filter.
The analog impulse response can be achieved by taking the inverse Laplace transform of the analog
filter HðsÞ , that is,

h t ¼ L 1

ðHðsÞÞ (8.37)

Now, if we sample the analog impulse response with a sampling interval of T and use T as a scale
factor, it follows that

T$h n ¼ T$hðtÞj t¼nT ; n 0

(8.38)

h ()

() t

t t

X () 1

H () ht L Hs ()

1

()

h ( )

() n

n n

H ()

ADC

X () 1

Thn Tht () () tnT

Hz ZThn

FIGURE 8.27
Impulse-invariant design method.