Z Transform, System Response.

Cuthbert Nyack
The above diagram shows the relation between the input and output of a sampled system. Sampled system is characterised by a transfer function H(z) and Impulse Response h(n) related by the following equation.
The input can be expressed as a sequence of samples x(n) or its Z transform X(z), both of which are related by the following equation.
The Z transform of the output Y(z) is obtained from the following equation:-
and its inverse y(n) by:-
The output sequence y(n) can be obtained in the time domain by taking the convolution of the input sequence with the impulse response of the system.
The Frequency spectrum of the input and output are given by the following 2 equations.
It is generally the case that a system response can be obtained in the frequency domain by finding the inverse of the output spectrum. However because it is continuous, the DTFT is not suitable for this. Instead system response in the frequency domain is obtained via the DFT or the computationally efficient FFT.
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© 2000 Cuthbert A. Nyack.