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Complex Sinewave Analysis Revisited

Consider again the sine-wave analysis method used in Chapter 11.3.1) for the simplest lowpass filter. We let the input signal be a complex sinusoid

$\displaystyle x(n) = A e^{j\omega_0 n T+ \phi}

having amplitude $ A$ and phase $ \phi$. Recall that for complex sinusoidal inputs, the amplitude response is measurable, after any transient response has settled down, as the instantaneous output amplitude divided by the instantaneous input amplitude, or $ G(\omega_0) = \left\vert y(n)\right\vert/\left\vert x(n)\right\vert$. (This simple formula for $ G(\omega_0)$ holds only for complex sinusoidal inputs, since a constant-amplitude signal is required.) Secondly, the phase response is given by the phase of the output sinusoid minus the phase of the input sinusoid, or $ \Theta(\omega) = \angle y(n) - \angle x(n)$. Thus, the output $ y(n)$ must be

y(n) &=& [G(\omega_0)A ] e^{j[\omega_0 n T + \phi + \Theta(\o...
...T}) A e^{j[\omega_0 n T + \phi]}\\
&=& H(e^{j\omega_0T}) x(n).

This shows that the output of an LTI filter in response to a complex sinusoid at frequency $ \omega_0$ is obtained by (1) scaling by $ G(\omega_0)$ and phase-shifting by $ \Theta(\omega_0)$, or, equivalently, (2) multiplying the input complex sinusoid by the (complex) frequency response at frequency $ \omega_0$.

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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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