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.

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