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 (which we may regard as a sinusoid at frequency ), 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 (LTI ^4.2) 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 be a general sinusoid at frequency $\omega$ :

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

Now form

as the sum of two copies of

with arbitrary amplitudes and phase offsets:

$\begin{eqnarray*} y(t) &\isdef & g_1 x(t-t_1) + g_2 x(t-t_2) \\ &=& g_1 A \sin[\omega (t-t_1) + \phi] + g_2 A \sin[\omega (t-t_2) + \phi] \end{eqnarray*}$

Focusing on the first term, we have

$\begin{eqnarray*} g_1 A \sin[\omega (t-t_1) + \phi] &=& g_1 A \sin[\omega t + (... ...omega t) \\ &\isdef & A_1 \cos(\omega t) + B_1 \sin(\omega t). \end{eqnarray*}$

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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Previous: In-Phase & Quadrature Sinusoidal Components
Next: Constructive and Destructive Interference

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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Mathematics of the DFT
    Sinusoids and Exponentials
       Sinusoids
          Sinusoids at the Same Frequency