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 filterimpulse 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.