# Computing the Z-transform

1. For a discrete-time signal x[n] with the z-Transform X(z) = z/(8z^2-2z-1), find the z-Transform, V(z) for the signal v[n] = x[n] * x[n].

2. Compute the unit-pulse response h[n] for the discrete-time system y[n+2]2y[n+1]+y[n]=x[n] (for n = 0, 1, 2, 3).

3. Compute the z-transform of the discrete time signal defined by x[n] = δ[n] +5δ[n- 1].

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#### Solution Preview

1.

Because,

v[n] = = x[n] * x[n]

Hence,

V(z) = X(z).X(z) = ( z/(8z^2-2z-1)).( z/(8z^2-2z-1)) = z^2/(8z^2 - 2z - 1)^2 --Answer

2.

y[n+2] - 2y[n+1]+y[n]=x[n]

=> y[n] = - y[n+2] + 2y[n+1] + x[n]

z- transform:

=> Y(z) = - z^2 ...

#### Solution Summary

By using properties of z-transform, z-transform of two X(z) discrete-time signals are derived. Also, unit-pulse response of given function is derived.

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