# 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.