Relationships among system representations

A discrete time linear shift invariant system is characterized by the unit sample response of it, a difference equation, a system function, or the frequency response. Suppose that the system is described by linear constant coefficient difference equation

System function Take z-transform of both sides of above equation

The system function is H(z) = 

Unit sample response If x(n) = δ(n) then X(z) = ?[x(n)] = ?[δ(n)] = 1. The corresponding y(n) is unit sample response h(n). We have

Y ( z) / X ( z) = H(z),  or          Y(z) = H(z).X(z) = H(z).1 = H(z)

Therefore, given H(z), system function, unit sample response is h(n) = ?^-1[H(z)].

The difference equation from the H(z) The system function H(z) is the 1st written in terms of negative powers of z and set equal to Y ( z)/X ( z) . Then cross-multiply and take inverse z-transform to obtain the difference equation.

Frequency response of system is Fourier transform (DTFT) of the unit sample response h(n):

Compare this with system function H(z) defined as z-transform of the unit sample response h(n)

Therefore the frequency response, if it exists, can be obtained by replacing z in H(z) by e jw as shown below:

The system is BIBO-stable.

The above relationships for the stable, causal system represented by a linear constant coefficient difference equation are shown in diagram below.

Example: Find out the impulse response of y(n) = a y(n-1) + x(n).

Solution Note that we have solved this in the time domain previously. Taking z-transform of both the sides (having zero initial conditions),

Assume causality. From the table of transforms, h(n) = ?^-1[H(z)] = aⁿ u(n).

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