Filter Design by Impulse Invariance & Bilinear Transformation | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

Filter design by impulse invariance

In the impulse variance design procedure the impulse response of the impulse response of the discrete time system is proportional to equally spaced samples of the continues time filter, i.e.,

where T_d represents a sampling interval, since the specifications of the filter are given in discrete time domain, it turns out that T_d has no role to play in design of the filter. From the sampling theorem we know that the frequency response of the discrete time filter is given by

Since any practical continuous time filter is not strictly bandlimited there issome aliasing. However, if the continuous time filter approaches zero at high frequencies, the aliasing may be negligible. Then the frequency response of the discrete time filter is

We first convert digital filter specifications to continuous time filter specifications. Neglecting aliasing, we get  specification by applying the relation

        (9.2)

where  is transferred to the designed filter H(z), we again use equation (9.2) and the parameter T_dcancels out.

Let us assume that the poles of the continuous time filter are simple, then

The corresponding impulse response is

Then

The system function for this is

We see that a pole at  s= s_k in the s-plane is transformed to a pole at z = e ^sk T_d in the z-plane. If the continuous time filter is stable, that is Re {s_k} < 0 then the magnitude of e^sk^Td will be less than 1, so the pole will be inside unit circle. Thus the causal discrete time filter is stable. The mapping of zeros is not so straight forward.

Example:

Design a lowpass IIR digital filter H(z) with maximally flat magnitude characteristics. The passband edge frequency   with a passband ripple not exceeding 0.5dB. The minimum stopband attenuation at the stopband edge frequency  is 15 dB. 

We assume that no aliasing occurs. Taking T_d = 1 , the analog filter has   , the passband ripple is 0.5dB, and minimum stopped attenuation is 15dB. For maximally flat frequency response we choose Butterworth filter characteristics. From passband ripple of 0.5 dB we get

at passband edge.

From this we get  

From minimum stopband attenuation of 15 dB we get

at stopped edge A² = 31.62 

The inverse discrimination ratio is given by

and inverse transition ratio 1/k is given by

Since N must be integer we get N=4. By  we get 

The normalized Butterworth transfer function of order 4 is given by

This is for normalized frequency of 1 rad/s. Replace s by  from this we get

Bilinear Transformation

This technique avoids the problem of aliasing by mapping  axis in the s-plane to one revaluation of the unit circle in the z-plane.

If  H_a(s) is the continues time transfer function the discrete time transfer function is detained by replacing s with

  (9.3)

Rearranging terms in equation (9.3) we obtain.

Substituting   , we get

If , it is then magnitude of the real part in denominator is more than that of the numerator and so. Similarly if , than  for all. Thus poles in the left half of the s-plane will get mapped to the poles inside the unit circle in z-plane. If  then

So,  we get

rearranging we get

or

   (9.5)  
or

       (9.6)

The compression of frequency axis represented by (9.5) is nonlinear. This is illustrated in figure 9.4.

Because of the nonlinear compression of the frequency axis, there is considerable phase distortion in the bilinear transformation.

Example

We use the specifications given in the previous example. Using equation (9.5) with T_d = 2 we get