The Fourier Transform of the 

Right-Sided Sine and Cosine Functions

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The Right-Sided Cosine Function

The right-sided cosine function is just the regular cosine function, multiplied by the unit step function. That is, it is zero for t<0 and a cosine for positive t. The equation for the right-sided cosine is given in [1]:

calculation of right sided cosine fourier transform
[Equation 1]

This function is plotted in Figure 1 for A=2:

plot of cosine function times unit step

Figure 1. Plot of Right-Sided Cosine Function for A=2.

The Fourier Transform can be found by noting the Fourier Transforms of the unit step and the cosine:

Fourier transform of unit step
[Equation 2]

fourier transform of cosine
[Equation 3]

Using Equations [2] and [3] along with the modulation property of Fourier Transforms, we obtain the result:

fourier transform of right-sided cosine

The plot of the magnitude of the Fourier Transform of Equation [1] is given in Figure 2. Note that the vertical arrows represent dirac-delta functions.

Figure 2. Plot of Absolute Value of Fourier Transform of Right-Sided Cosine Function.

The Right-Sided Sine Function

The right-sided Sine function can be obtained in the same way. This function is mathematically written in Equation [5]:

right sided sine for Fourier analysis
[Equation 5]

The plot of the right-sided Sine Function is given in Figure 3 for A=2:

plot of step function times sine function

Figure 3. Plot of Step Function times the Sine Function for A=2.

Recall the Fourier Transform of the Sine Function:
fourier transform of sine function
[Equation 6]

And we can procede exactly as before to obtain the final result:

fourier transform of right sided sine function

Equation 7 gives the Fourier Transform of the right-sided sine function. The absolute value of this function is plotted in Figure 4:

plot of FT of step function times sine

Figure 4. Plot of the Absolute Value of The Right-Sided Sine Function.

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