The Z-Transform Transfer Function

Although the time-domain convolution is the most basic relationship of the input to the output for linear systems, the z-transform is a close second in importance. It gives different insight and a different set of tools for analysis and design of linear time-invariant discrete-time systems.

If our system in linear and time-invariant, we have seen that its output is given by convolution. y ( n ) = ∑ m = − ∞ ∞ h ( n − m ) x ( m ) y ( n ) = ∑ m = − ∞ ∞ h ( n − m ) x ( m ) Assuming that h ( n ) h ( n ) is such that the summation converges properly, we can calculate the output to an input that we already know has a special relation with discrete-time transforms. Let x ( n ) = z n x ( n ) = z n which gives y ( n ) = ∑ m = − ∞ ∞ h ( n − m ) z m y ( n ) = ∑ m = − ∞ ∞ h ( n − m ) z m With the change of variables of k = n − m k = n − m , we have y ( n ) = ∑ k = − ∞ ∞ h ( k ) z n − k = [ ∑ k = − ∞ ∞ h ( k ) z − k ] z n y ( n ) = ∑ k = − ∞ ∞ h ( k ) z n − k = [ ∑ k = − ∞ ∞ h ( k ) z − k ] z n or y ( n ) = H ( z ) z n y ( n ) = H ( z ) z n We have the remarkable result that for an input of x ( n ) = z n x ( n ) = z n , we get an output of exactly the same form but multiplied by a constant that depends on z z and this constant is the z-transform of the impulse response of the system. In other words, if the system is thought of as a matrix or operator, z n z n is analogous to an eigenvector of the system and H ( z ) H ( z ) is analogous to the corresponding eigenvalue.

We also know from the properties of the z-transform that convolution in the n n domain corresponds to multiplication in the z z domain. This means that the z-transforms of x ( n ) x ( n ) and y ( n ) y ( n ) are related by the simple equation Y ( z ) = H ( z ) X ( z ) Y ( z ) = H ( z ) X ( z ) The z-transform decomposes x ( n ) x ( n ) into its various components along z n z n which passing through the system simply multiplies that value time H ( z ) H ( z ) and the inverse z-transform recombines the components to give the output. This explains why the z-transform is such a powerful operation in linear discrete-time system theory. Its kernel is the eigenvector of these systems.

The z-transform of the impulse response of a system is called its transfer function (it transfers the input to the output) and multiplying it times the z-transform of the input gives the z-transform of the output for any system and signal where there is a common region of convergence for the transforms.