Example Sinusoids

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

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Example Sinusoids

Figure 5.1 plots the sinusoid $A \, for $A=10$, $f=2.5$, $\, and$t\. Study the plot to make sure you understand the effect of changing each parameter (amplitude, frequency, phase), and also note the definitions of “peak-to-peak amplitude” and “zero crossings.”

Figure 5.1:An example sinusoid.
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The Mathematica code for generating this figure is listed in §5.4.

A “tuning fork” vibrates approximately sinusoidally. An “A-440” tuning fork oscillates at $440$ cycles per second. As a result, a tone recorded from an ideal A-440 tuning fork is a sinusoid at $f=440$ Hz. The amplitude$A$ determines how loud it is and depends on how hard we strike the tuning fork. The phase $\ is set by exactly when we strike the tuning fork (and on our choice of when time 0 is). If we record an A-440 tuning fork on an analog tape recorder, the electrical signal recorded on tape is of the form

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As another example, the sinusoid at amplitude $1$ and phase $\ (90 degrees) is simply

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Thus, $\ is a sinusoid at phase 90-degrees, while $\ is a sinusoid at zero phase. Note, however, that we could just as well have defined $\ to be the zero-phase sinusoid rather than $\. It really doesn’t matter, except to be consistent in any given usage. The concept of a “sinusoidal signal” is simply that it is equal to a sine or cosine function at some amplitude, frequency, and phase. It does not matter whether we choose $\or $\ in the “official” definition of a sinusoid. You may encounter both definitions. Using $\ is nice since “sinusoid” in a sense generalizes $\. However, using $\ is nicer when defining a sinusoid to be the real part of a complex sinusoid (which we’ll talk about later).

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