Summary

This chapter has introduced many of the concepts associated with digital filters, such as signal representations, filter representations, difference equations, signal flow graphs, software implementations, sine-wave analysis (real and complex), frequency response, amplitude response, phase response, and other related topics. We used a simple filter example to motivate the need for more advanced methods to analyze digital filters of arbitrary complexity. We found even in the simple example of Eq. (1.1) that complex variables are much more compact and convenient for representing signals and analyzing filters than are trigonometric techniques. We employ a complex sinusoid $A e^{j(\omega nT+\phi)}$ having three parameters: amplitude, phase, and frequency, and when we put a complex sinusoid into any linear time-invariant digital filter, the filter behaves as a simple complex gain $H(e^{j\omega T})= G(\omega)e^{j\Theta(\omega)}$, where the magnitude $G(\omega)$ and phase $\Theta(\omega)$ are the amplitude response and phase response, respectively, of the filter.