The Fourier Transform of the Quadratic Sinusoids
The Quadratic Cosine FunctionAre you interested in the sinusoidal functions with a quadratically varying phase? This is the right page. The function is a cosine or sine function with an argument of tsquared instead of t. The cosine version is given in Equation [1], and plotted in Figure 1 below:
Figure 1. Plot of QuadraticCosine for k=1. The Fourier Transform for the cos(t^2) function can be found fairly easily using the Fourier Transform for the Complex Gaussian. We begin by rewriting the cosine function using Euler's formula:
Assume k is positive. If k is negative, the result will still be the same, which you can work through [note that the cos(x)=cos(x)]. If k is zero, then we have a constant and not a quadraticallyvarying sinusoid. The Fourier Transform can be found since we know the Fourier Transform of the complex Gaussian. The terms in Equation [2] can have the Fourier Transform taken:
The other term in Equation [2] can be found by using the Fourier Transform of the Complex Gaussian for the case of a negative real constant, simply by manipulating the signs:
The end result can then be obtained:
If k is negative, the Fourier Transform should not change, because cos(kt^2) is an even function. Hence, the general result is:
This function is plotted in Figure 2:
Figure 2. Plot of Fourier Transform of QuadraticCosine [Eq 6] for k=1.
The Quadratic Sine FunctionThe quadratic sine function is given in Equation [7], and plotted in Figure 3:
Figure 3. Plot of QuadraticSine for k=1. The Fourier Transform of this function can be found using the same technique. That is rewriting the sine function as:
For positive k, we can work through the above math as before and get the result:
To get the general result (including negative k values), recall that sin(t)=sin(t). The end result:
Equation [10] is plotted in Figure 4 for k=1:
Figure 4. Plot of Fourier Transform of Quadratic Sine (Eq [10]) for k=1.
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