The Truncated Cosine

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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.

truncated cosine function
[Equation 1]

cosine times rectangle function

Figure 1. The Truncated Cosine given by Equation [1], for W=2.

Let's find the Fourier Transform of this function. To start, we can rewrite the function g(t) as the product of two other functions:

rewritten as the product of functions
[Equation 2]

To start, we can find the Fourier Transform of h(t) by recalling the Fourier Transform of the Cosine Function, we can determine H(f):

cosine function
[Equation 3]

The Fourier Transform of the Box Function can be recalled, to determine K(f):

fourier transform of rect function
[Equation 4]

Now, the Fourier Transform of the multiplication of two function can be found by convolving their individual Fourier Transforms. This is simply the modulation property of the Fourier Transform:

modulation for Fourier Transforms
[Equation 5]

The convolution of H(f) and K(f) might seem difficult, but recall the property of the dirac-delta impulse function:

convolution with impulse
[Equation 6]

Equation [6] is valid for all functions f(t), which will make Equation [5] simple to evaluate:

derivation of fourier transform

Now it's just algebra time. The last line can be rewritten as:


To further simplify the above, recall the trigonometric identities:

trigonometric identities
[Equation 7]

Hence, Term 1 = 0, and Term 2 can be simplified. The result is:

the Fourier Transform
[Equation 8]

And there you have the result. The Fourier Transform G(f) is plotted in Figure 2:

end fourier transform result

Figure 2. The Fourier Transform of the Truncated Cosine for W=2


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