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:

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

(8.3)

Writing the basic frequency response description

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

(from Eq. $\,$ (7.2)) 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

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

This states explicitly that the output magnitude spectrum equals the input magnitude spectrum times the filter amplitude response, and the output phase equals the input phase plus the filter phase at each frequency $\omega$ .

Equation (7.3) 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):

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

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 are carried out in Appendix B. Some useful trig identities are summarized in Appendix A. A matlab listing for computing the frequency response of any IIR filter is given in §7.5.1 below.

Subsections

- Separating the Transfer Function Numerator and Denominator

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Frequency Response Analysis

             Separating the Transfer Function Numerator and Denominator