Amplitude Response

Since the frequency response is a complex-valued function, it has a magnitude and phase angle for each frequency. The magnitude of the frequency response is called the amplitude response (or magnitude frequency response), and it gives the filter gain at each frequency $\omega$ .

In this example, the amplitude response is

$$\left\vert H(e^{j\omega T})\right\vert = \left\vert\frac{1 + g_1 e^{-jM_1\omega T}}{1 + g_2 e^{-jM_2\omega T}}\right\vert \protect$$ (4.5)

which, for $g_1=g_2=1$ , reduces to

$$\left\vert H(e^{j\omega T})\right\vert = \frac{\left\vert\cos\left(M_1\omega T/2\right)\right\vert}{\left\vert\cos\left(M_2\omega T/2\right)\right\vert}.$$

Figure 3.10a shows a graph of the amplitude response of one case of this filter, obtained by plotting Eq.(3.5) for $\omega T \in[-\pi,\pi]$ , and using the example settings $g_1 = 0.5^3$ , $g_2 = 0.9^5$ , $M_1 = 3$ , and $M_2=5$ .

Figure 3.10: Frequency response of the example filter $y(n) = x(n) + 0.5^3 x(n-3) - 0.9^5 y(n-5)$ . (a) Amplitude response. (b) Phase response.