TABLE OF CONTENTS One of the main reasons to study the z-transform is to use it to help analyze systems that "run on" samples of data rather than continuously arriving data. In such systems the form of the sequence with which one works is often e( k) = f( kT). That is, the elements of the sequence are actually samples of a function from which a sample is acquired every T seconds.
Suppose that one samples the unit step function. That is, suppose that e( k) = u( kT). As this sequence is all ones (for non-negative values of k) its z-transform is:
Suppose that the function that is being sampled is 5 sin( ? t). Then the sequence whose z-transform we must find is e( k) = 5 sin( ? Tk). Using the transforms we have already calculated, we find that:
Suppose that the function being sampled is the ramp. That is, let e( t) = tu( t). The samples are:
Thus, the z-transform of the the samples must be ? zT times the derivative of the z-transform of the unit step. That is, the z-transform is:
