Polar Form of the Frequency Response

When the complex-valued frequency response is expressed in polar form, the amplitude response and phase response explicitly appear:

$$\zbox {H(e^{j\omega T}) = G(\omega)e^{j\Theta(\omega)}} \protect$$ (8.2)

Writing the basic frequency response description

$$Y(e^{j\omega T}) = H(e^{j\omega T})X(e^{j\omega T}) \protect$$

(from Eq. (7.1)) in polar form gives

$$\begin{eqnarray*} Y(e^{j\omega T}) &=& \left\vert Y(e^{j\omega T})\right\vert e^... ...ight\vert\right] e^{j[\angle X(e^{j\omega T})+ \Theta(\omega)]} \end{eqnarray*}$$

which implies

$$\begin{eqnarray*} \left\vert Y(e^{j\omega T})\right\vert &=& G(\omega) \left\ver... ...{Y(e^{j\omega T})} &=& \Theta(\omega) + \angle X(e^{j\omega T}). \end{eqnarray*}$$

Equation (7.2) gives the frequency response in polar form. For completeness, recall the transformations between polar and rectangular forms (i.e., for converting real and imaginary parts to magnitude and angle and vice versa). The real part of a complex number $z$ is written as re$\left\{z\right\}$, and the imaginary part of $z$ is denoted im$\left\{z\right\}$.

$$\begin{eqnarray*} G(\omega) &\isdef & \left\vert H(e^{j\omega T})\right\vert = \... ...ga T})\right\}}{\mbox{re}\left\{H(e^{j\omega T})\right\}}\right] \end{eqnarray*}$$

Going the other way from polar to rectangular (using Euler's formula),

$$\begin{eqnarray*} \mbox{re}\left\{H(e^{j\omega T})\right\} &=& G(\omega) \cos[\T... ...ft\{H(e^{j\omega T})\right\} &=& G(\omega) \sin[\Theta(\omega)]. \end{eqnarray*}$$

Application of these formulas to some basic example filters will be carried out in §10.1. A matlab listing for computing the frequency response of any IIR filter is given in §7.5.1 below.