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Sinusoids at the Same Frequency

An important property of sinusoids at a particular frequency is that they are closed with respect to addition. In other words, if you take a sinusoid, make many copies of it, scale them all by different gains, delay them all by different time intervals, and add them up, you always get a sinusoid at the same original frequency. This is a nontrivial property. It obviously holds for any constant signal $ x(t)=c$ (which we may regard as a sinusoid at frequency $ f=0$ ), but it is not obvious for $ f\neq 0$ (see Fig.4.2 and think about the sum of the two waveforms shown being precisely a sinusoid).

Since every linear, time-invariant (LTI4.2) system (filter) operates by copying, scaling, delaying, and summing its input signal(s) to create its output signal(s), it follows that when a sinusoid at a particular frequency is input to an LTI system, a sinusoid at that same frequency always appears at the output. Only the amplitude and phase can be changed by the system. We say that sinusoids are eigenfunctions of LTI systems. Conversely, if the system is nonlinear or time-varying, new frequencies are created at the system output.

To prove this important invariance property of sinusoids, we may simply express all scaled and delayed sinusoids in the ``mix'' in terms of their in-phase and quadrature components and then add them up. Here are the details in the case of adding two sinusoids having the same frequency. Let $ x(t)$ be a general sinusoid at frequency $ \omega$ :

$\displaystyle x(t) \isdef A\sin(\omega t+\phi)

Now form $ y(t)$ as the sum of two copies of $ x(t)$ with arbitrary amplitudes and phase offsets:

y(t) &\isdef & g_1 x(t-t_1) + g_2 x(t-t_2) \\
&=& g_1 A \sin[\omega (t-t_1) + \phi]
+ g_2 A \sin[\omega (t-t_2) + \phi]

Focusing on the first term, we have

g_1 A \sin[\omega (t-t_1) + \phi]
g_1 A \sin[\omega t + (\phi - \omega t_1)] \\
\left[g_1 A \sin(\phi-\omega t_1)\right] \cos(\omega t) + \\
& &\left[g_1 A \cos(\phi-\omega t_1)\right] \sin(\omega t) \\
&\isdef & A_1 \cos(\omega t) + B_1 \sin(\omega t).

We similarly compute

$\displaystyle g_2 A \sin[\omega (t-t_2) + \phi]
A_2 \cos(\omega t) + B_2 \sin(\omega t)

and add to obtain

$\displaystyle y(t) = (A_1+A_2) \cos(\omega t) + (B_1+B_2) \sin(\omega t).

This result, consisting of one in-phase and one quadrature signal component, can now be converted to a single sinusoid at some amplitude and phase (and frequency $ \omega$ ), as discussed above.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University