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

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

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

The corresponding impulse response is

The system function for this is

We see that a pole at in the s-plane is transformed to a pole at   T_d in the z-plane. If the continuous time filter is stable, that is , then the magnitude of 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.

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

We assume that no aliasing occurs. Taking , 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

The inverse discrimination ratio is given by

and inverse transition ratio 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 to get , from this we get