Introduction to the z-transform

Given a discrete time LTI system with impulse response , recall that the response (if it's well defined) to the input signal is , where is the transfer function of the system. Now let's calculate the response to the input signal in terms of the impulse response:

From this, we see that the summation above in the square brackets must be the transfer function:

The right hand side of the above equation is known as the z-transform of the signal . Thus, the transfer function of an LTI system is the z-transform of the system's impulse response. The region of convergence of the transfer function is thus the set of complex numbers for which the z-transform of the impulse response converges.