This chapter will address questions about the z-transform, an analytical tool for systems. What is z? How does this transform work? How can it be used to combine filters? Why do delay units sometimes have a z ^?1 symbol? How does this transform relate to other transforms we have seen? We will answer these questions and more in the following sections.

The z-transform is a generalized version of the Fourier transform. Like the Fourier transform, it allows us to represent a time-domain signal in terms of its frequency components. Instead of accessing signal values in term of n, a discrete index related to time, we can know the response of the signal for a given frequency (as we did with the Fourier transform). The difference is that we can also specify a magnitude with the z-transform.

The z-transform serves two purposes. First, it provides a convenient way to notate the effects of filters. So far, we have used the coefficients, h[ n] = { a, b, c, d}, to describe how the output y[ n] relates to the input x[ n]. In z-transform notation, we say Y( z) = H( z) X( z), where H( z) is the z-transform of h[ n]. We can think of H( z) as something that operates on X( z) to produce the...