The discrete-time (DT) signal, which is a series of real or complex numbers, is transformed into a complex frequency-domain (z-domain or z-plane) representation using the Z-transform in signal processing.
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Z Transform of a delta or unit impulse function
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Example of Z Transform
For a real world example, when we send a unit impulse signal for testing input we receive multiple impulse responses at receiver due to different multipath.
Let's assume, data signal x[n] = [2 -5 1 3]
and channel impulse responses h[n] = [-1 4 2]
Now simply multiply the data signal and channel co-efficients learned in elementary school
3 1 -5 2
2 4 -1
-----------------------
6 14 -9 -17 13 -2
It can be represented as
x[n]*h[n] = 6z^(-5) + 14z^(-4) - 9z^(-3) - 17z^(-2) + 13z^(-1) - 2z
After computing discrete time z transform it is defined as simple multiplication of X(z) and H(z).