Signals, Systems, and Signal Processing

This chapter introduces recursive difference equations. These equations represent discrete-time LTI systems when the so-called initial conditions are zero. The transfer functions of such LTI systems have a rational form (ratios of polynomials in z). Recursive difference equations offer a computationally efficient way to implement systems whose outputs may depend on an infinite number of past inputs. The recursive property allows the infinite past to be remembered by remembering only a finite number of past outputs. Poles and zeros of rational transfer functions are introduced, and conditions for stability expressed in terms of pole locations. Computational graphs for digital filters, such as the direct-form structure, cascade-form structure, and parallel-form structure, are introduced. The partial fraction expansion (PFE) method for analysis of rational transfer functions is introduced. It is also shown how the coefficients of a rational transfer function can be identified by measuring a finite number of samples of the impulse response. The chapter also shows how the operation of polynomial division can be efficiently implemented in the form of a recursive difference equation.