The Truncated Cosine

This page will seek the Fourier Transform of the truncated cosine, which is given in Equation [1] and plotted in Figure 1.
This function is a cosine function that is windowed - that is, it is multiplied by the
box or rect function.
Figure 1. The Truncated Cosine given by Equation [1], for
Let's find the Fourier Transform of this function. To start, we can rewrite the function
To start, we can find the Fourier Transform of
The Fourier Transform of the Box Function can be recalled, to determine
Now, the Fourier Transform of the multiplication of two function can be found by
The convolution of
Equation [6] is valid for all functions
Now it's just algebra time. The last line can be rewritten as:
To further simplify the above, recall the trigonometric identities:
Hence, Term 1 = 0, and Term 2 can be simplified. The result is:
And there you have the result. The Fourier Transform
Figure 2. The Fourier Transform of the Truncated Cosine for
Fin.
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[Equation 1]
*W*=2.*g(t)*
as the product of two other functions:
[Equation 2]
*h(t)* by recalling the
Fourier Transform of the Cosine Function, we can
determine *H(f)*:
[Equation 3]
*K(f)*:
[Equation 4]
*convolving* their individual Fourier Transforms.
This is simply the modulation property of the Fourier Transform:
[Equation 5]
*H(f)* and *K(f)* might seem difficult, but recall the property of
the dirac-delta impulse function:
[Equation 6]
*f(t)*, which will make Equation [5] simple to evaluate:
[Equation 7]
[Equation 8]
*G(f)* is plotted in Figure 2:*W*=2

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