Sinusoids

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Sinusoids

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Sinusoids

A sinusoid is any function of time having the following form:

\

where all variables are real numbers, and

\



The term “peak amplitude” is often shortened to “amplitude,” e.g., “the amplitude of the sound was measured to be 5 Pascals.” Strictly speaking, however, the “amplitude” of a signal $x$ is its instantaneous value $x(t)$ at any time $t$. The peak amplitude $A$ satisfies $x(t)\. The “instantaneous magnitude” or simply “magnitude” of a signal$x(t)$ is given by $\, and the peak magnitude is the same thing as the peak amplitude.

Note that Hz is an abbreviation for Hertz which physically means “cycles per second.” You might also encounter the older (and clearer) notation “c.p.s.” for cycles per second.

Since $\ is periodic with period $2\, the phase $\ is indistinguishable from the phase $\. As a result, we may restrict the range of $\ to any length $2\ interval. When needed, we will choose

\

i.e., $\. You may also encounter the convention $\.



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