Signals, Systems, and Signal Processing

This chapter introduces recursive difference equations where the initial conditions are nonzero. The output of such a system is studied in detail. One application is in the design of digital waveform generators such as oscillators, and this is explained in considerable detail. The coupled-form oscillator, which simultaneously generates synchronized sine and cosine waveforms at a given frequency, is presented. The chapter also introduces another application of recursive difference equations, namely the computation of mortgages. It is shown that the monthly payment on a loan can be computed using a first-order recursive difference equation. The equation also allows one to calculate the interest and principal parts of the payment every month, as shown. Poles play a crucial role in the behavior of recursive difference equations with zero or nonzero initial conditions. Many different manifestations of the effect of a pole are also summarized, including some time-domain dynamical meanings of poles.