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From Sinewaves to Arbitrary Signals

By superposition, we may readily generalize complex sinewave analysis to the case in which $ x(n)$ is an arbitrary superposition of input sinusoids:

$\displaystyle x(n) \isdef \int_{-\pi/T}^{\pi/T}A(\omega) e^{j[\omega n T + \phi(\omega)]} d\omega

(This is formally $ 2\pi$ times the inverse DTFT of $ X(e^{j\omega T})\isdef A(\omega) e^{j\omega n T + \phi(\omega)}$.) Passing this superposition of sinusoids through any LTI filter $ {\cal L}_n$ having transfer function $ H(z)$ yields, by linearity,

y(n) &=& {\cal L}_n\left\{\int_{-\pi/T}^{\pi/T} X(e^{j\omega ...
... \int_{-\pi/T}^{\pi/T} H(e^{j\omega T}) X(e^{j\omega T}) d\omega

where $ G(\omega)$ is the amplitude response, and $ \Theta(\omega)$ the phase response of the LTI filter. We have thus shown by superposition that, given any input signal $ x$, the output spectrum $ Y(e^{j\omega T})$ is equal to the input spectrum $ X(e^{j\omega T})$ multiplied by the frequency response $ H(e^{j\omega T})$, where the frequency response can be measured one frequency at a time using a sinusoidal input signal.

In contrast to the polar representation of frequency response $ H(e^{j\omega T})=
G(\omega)e^{j\Theta(\omega)}$, the real and imaginary parts do not have such intuitively appealing individual interpretations. Consequently, the polar form is usually preferred for expressing filter responses as a function of frequency.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (August 2006 Edition).
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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