In this thesis, we study periodic and multirate systems. In particular we are interested in rate changers, nonuniform filter banks, and nonuniform transmultiplexers.

We consider the general characteristics of multirate building blocks and the approximation of these systems by systems that have a different but compatible period. In the ℋ₂ model-matching case, we show how to obtain the optimal approximant. Then, we use these multirate building blocks in the design of nonuniform filter banks and transmultiplexers.

Contrary to uniform filter banks, for nonuniform filter banks, if we use LTI filters in the analysis and synthesis filter banks, it may not be possible to achieve perfect-reconstruction. But, by using general multirate building blocks, we can eliminate some of the design constraints.

We assume that the analysis filter bank consists of finite impulse response (FIR) filters. The initial FIR analysis filters are designed according to the characteristics of the input. By the design procedure, the FIR synthesis filters are found so that the norm of the error system is minimized over all synthesis filters that have a prespecified order. Then, the synthesis filters obtained in the previous step are fixed and the analysis filters are found similarly. By iteration, the norm of the error system decreases until it converges to its final value.

In the ℋ₂ design of nonuniform filter banks, the problem is formulated in the frequency domain. In this approach, the frequency selectivity of filters can also be considered by adding penalty terms. At each iteration, the least squares solution of a system of linear equations yields the local optimal filters.

In the ℋ_∞ design of nonuniform filter banks, the problem is formulated in the state-space framework. The problem is then posed as an optimization problem based on linear matrix inequalities, which can be solved effectively by interior point methods.

Nonuniform transmultiplexers can be used for the conversion of signals that have different sampling rates, between the time-division multiplexing (TDM) format and the frequency-division multiplexing (FDM) format. Here, we have posed the design of nonuniform transmultiplexers as an iterated semidefinite programming problem.