##
*Z* Transform of Convolution

From Eq.(5.5), we have that the output from a linear
time-invariant filter with input and impulse response is given
by the *convolution*of and ,

*i.e.*,

where ``'' means convolution as before. Taking the

*z*transform of both sides of Eq.(6.3) and applying the convolution theorem from the preceding section gives

where H(z) is the

*z*transform of the filter impulse response. We may divide Eq.(6.4) by to obtain

*z*transform of an impulse response is equal to the transfer function of the filter, provided the filter is linear and time invariant.

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Z Transform of Difference Equations

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Shift and Convolution Theorems