Time Domain invariance

Given an analog filter's response to a specific input we require that the response of the digital filter (to be designed) to the digital version of the analog input should be the same as the analog response at sampling instants. If the input is an impulse function the corresponding design is called an impulse invariant design, if the input is a step function the corresponding design is called a step invariant design.

Impulse-invariant design If h_a(t) represents the response of an analog filter H_a(s) to a unit impulse δ(t), then the unit sample response of a discrete-time filter used in an A/D - H(z) - D/A structure is selected to be the sampled version of h_a(t). That is, we are preserving the response to an impulse. Therefore the discrete-time filter is characterized by the system function, H(z), given by

 

If we are given an analog filter with system function H_a(s) the corresponding impulse-invariant digital filter, H(z), is seen from above to be

 
Note that at this point we have not specified how H_a(s) was obtained, but rather we have shown how to obtain the digital filter H(z) from any given H_a(s) using impulse invariance.

Example: [Low pass filter] For the analog filter   find the H(z) corresponding to the impulse invariant design using a sample rate of 1/T samples/sec.

Solution The analog system's impulse response is   

The corresponding h(n) is then given by

 
where, as previously, we have set  . The discrete-time filter, then, is given by the z- transform of h(n)

The discrete-time filter, then, is given by the z-transform of h(n)
 
which has a pole at   

In effect, the pole at s = -α in the s-plane is mapped to a pole at  in the z-plane. (HW What is the difference equation?)

Email based Time domain invariance assignment help - Time domain invariance homework help at Expertsmind

Are you finding answers for Time domain invariance based questions? Ask Time domain invariance questions and get answers from qualified and experienced  Digital signal processing tutors anytime from anywhere 24x7. We at www.expertsmind.com offer Time domain invariance assignment help -Time domain invariance homework help and  Digital signal processing  problem's solution with step by step procedure.

Why Expertsmind for Digital signal processing assignment help service

1.     higher degree holder and experienced tutors

2.     Punctuality and responsibility of work

3.     Quality solution with 100% plagiarism free answers

4.     On Time Delivery

5.     Privacy of information and details

6.     Excellence in solving Digital signal processing queries in excels and word format.

7.     Best tutoring assistance 24x7 hours