The sampling theorem provides us with a tool to map a continuous-time signal to a sequence of discrete-time samples taken at a given sampling rate. By choosing a different sampling rate, the same continuous-time signal can be mapped to an arbitrary number of different discrete-time signals. What is the relationship between these sequences as seen purely from the discrete-time world? Otherwise stated: can we transform one into another without the need to interpolate back to continuous time? These are the questions that multirate theory sets out to answer.

While the primary application of multirate theory is implicit sampling rate conversion, we will see that multirate plays an important role in a variety of applied signal processing algorithms, from the design of efficient filters, to spectral shaping for communication systems, to data compression standards. From a more theoretical point of view, multirate theory is at the cornerstone of advanced analysis techniques which go under the name of time-frequency decompositions.

The fundamental operations in multirate signal processing are upsampling and downsampling; upsampling increases the number of samples in a sequence, while downsampling reduces it.

We will examine each operation in turn, both in the time and in the frequency domains. We will see that, while upsampling is a "safe" operation that can always be undone, downsampling involves a potential loss of data and therefore it must be performed with caution.

The sampling rate changes introduced by up- and downsampling represent a sort of "discrete-time sampling theorem" and the spectral characteristics of each operation will require the use of an appropriate filter to either remove unwanted spectral components or to prevent aliasing.

Finally, we will show how upsampling and downsampling can be combined to perform a fractional sampling rate change in the discrete-time domain, as we would need to do for instance if we needed to convert a CD-audio standard to DVD-audio.