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Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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Filter design by impulse invariance

In the impulse variance design procedure the impulse response of the impulse responseFilter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) of the discrete time system is proportional to equally spaced samples of the continues time filter, i.e.,

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

We first convert digital filter specifications to continuous time filter specifications. Neglecting aliasing, we get Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) specification by applying the relation

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)        (9.2)

where Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) 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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The corresponding impulse response is

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Then

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The system function for this is

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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 Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  with a passband ripple not exceeding 0.5dB. The minimum stopband attenuation at the stopband edge frequency Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) is 15 dB. 

We assume that no aliasing occurs. Taking Td = 1 , the analog filter has  Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) , 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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

at passband edge.

From this we get  Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

From minimum stopband attenuation of 15 dB we get

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

at stopped edge A2 = 31.62 

The inverse discrimination ratio is given by

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

and inverse transition ratio 1/k is given by

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Since must be integer we get N=4. By Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) we get  Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

The normalized Butterworth transfer function of order 4 is given by

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

This is for normalized frequency of 1 rad/s. Replace by Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) from this we get

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Bilinear Transformation

This technique avoids the problem of aliasing by mapping Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) 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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)  (9.3)

Rearranging terms in equation (9.3) we obtain.

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

Substituting  Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) , we get

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

If Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE), it is then magnitude of the real part in denominator is more than that of the numerator and so. Similarly if Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE), than Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) 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 Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) then

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

So, Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE) we get

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

rearranging we get

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
or

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)   (9.5)  
or

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)       (9.6)

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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)
Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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

Filter Design by Impulse Invariance & Bilinear Transformation Notes | Study Signals and Systems - Electronics and Communication Engineering (ECE)

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