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Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Digital Signal Processing - Electrical Engineering (EE)

Electrical Engineering (EE): Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Digital Signal Processing - Electrical Engineering (EE)

The document Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Digital Signal Processing - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
All you need of Electrical Engineering (EE) at this link: Electrical Engineering (EE)

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 Td represents a sampling interval, since the specifications of the filter are given in discrete time domain, it turns out that Td 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 Tdcancels 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= sk in the s-plane is transformed to a pole at z = e sk Td in the z-plane. If the continuous time filter is stable, that is Re {sk} < 0 then the magnitude of eskTd 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 Td = 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 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 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  Ha(s) is the continues time transfer function the discrete time transfer function is detained by replacing 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 Td = 2 we get The document Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Digital Signal Processing - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
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