Multirate filter banks produce multiple output signals by filtering and subsampling a single input signal, or conversely, generate a single output by upsampling and interpolating multiple inputs. Two of their main applications are subband coders for speech processing and transmultiplexers for telecommunications. Below, we derive a theoretical framework for the analysis, synthesis, and computational complexity of multirate filter banks. The use of matrix notation leads to basic results derived from properties of linear algebra. Using rank and determinant of filter matrices, it is shown how to obtain aliasing/ crosstalk-free reconstruction, and when perfect reconstruction is possible. The synthesis of filters for filter banks is also explored, three design methods are presented, and finally, the computational complexity is considered.