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
The general formula for a sinusoid function is:
where:
is the amplitude — the maximum value of the function; is the ordinary frequency — the number of cycles per unit of ; is the angular frequency — the number of radians per unit of ; is the phase offset (in radians).
The standard sine function
The standard cosine function
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:
where:
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
Thus, any sum of sinusoids, of the same frequency and therefore the same
input