## 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.,

 (7.3)

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

 (7.4)

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

This shows that, as a direct result of the convolution theorem, the 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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