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Sums of sinusoids

This page largely based on http://math.stackexchange.com/a/1239123 with thanks.

Sinusoids

Paraphrasing Wolfwram mathworld - a sinusoid is a function of some variable, say \(t\), that is similar to the sine function but may be shifted in phase, frequency, amplitude, or any combination of the three.

The general formula for a sinusoid function is:

(1)\[f(t) = A \sin(2 \pi f t + \theta) = A \sin(\omega t + \theta)\]

where:

  • \(A\) is the amplitude — the maximum value of the function;

  • \(f\) is the ordinary frequency — the number of cycles per unit of \(t\);

  • \(\omega = 2 \pi f\) is the angular frequency — the number of radians per unit of \(t\);

  • \(\theta\) is the phase offset (in radians).

The standard sine function \(f(t) = \sin(t)\) is a special case of a sinusoid, with \(A = 1\), \(f = 1 / 2 \pi\), \(\theta = 0\).

The standard cosine function \(f(t) = \cos(t)\) is a special case of a sinusoid, with \(A = 1\), \(f = 1 / 2 \pi\), \(\theta = -pi / 2\).

The sum of sinusoids with the same frequency is also a sinusoid

Remembering The angle sum rule, we can write any sinusoid as a weighted sum of a sine and a cosine:

(2)\[\begin{split}A \sin(\omega t + \theta) = \\ A \sin(\omega t) \cos(\theta) + A \cos(\omega t) \sin(\theta) = \\ A' \sin(\omega t) + A'' \cos(\omega t)\end{split}\]

where:

(3)\[\begin{split}A' = A \cos(\theta) \\ A'' = A \sin(\theta)\end{split}\]

Equation (2) also points us to the fact that any weighted sum of a sine and cosine can be written as a single sinusoid. For any \(A', A''\), we can solve equations (3) for \(A, \theta\):

\[\begin{split}A = \sqrt{A'^2 + A''^2} \\ \tan(\theta) = \frac{A''}{A'}\end{split}\]

Thus, any sum of sinusoids, of the same frequency and therefore the same input \(\omega t\), is also a sinusoid:

\[A \sin(\omega t + \theta) + B \sin(\omega t + \phi) = (A' + B') \sin(\omega t) + (A'' + B'') \cos(\omega t)\]