Signals, Systems, and Signal Processing

This chapter introduces state-space descriptions for computational graphs (structures) representing discrete-time LTI systems. They are not only useful in theoretical analysis, but can also be used to derive alternative structures for a transfer function starting from a known structure. The chapter considers systems with possibly multiple inputs and outputs (MIMO systems); systems with a single input and a single output (SISO systems) are special cases. General expressions for the transfer matrix and impulse response matrix are derived in terms of state-space descriptions. The concept of structure minimality is discussed, and related to properties called reachability and observability. It is seen that state-space descriptions give a different perspective on system poles, in terms of the eigenvalues of the state transition matrix. The chapter also revisits IIR digital allpass filters and derives several equivalent structures for them using so-called similarity transformations on state-space descriptions. Specifically, a number of lattice structures are presented for allpass filters. As a practical example of impact, if such a structure is used to implement the second-order allpass filter in a notch filter, then the notch frequency and notch quality can be independently controlled by two separate multipliers.