The Fourier Transform of the Complex Gaussian
The Complex Gaussian
The complex Gaussian is a Gaussian Function with a imaginary argument. It is also can be viewed as a sinusoid with a phase that increases quadratically with time. This function arises in the study of optics. The function is given in Equation , where k is a positive real constant:
To start, let's rewrite the complex Gaussian h(t) in terms of the ordinary Gaussian function g(t):
Now, we'd like to use the scaling property of the Fourier Transform directly, but note that the following equation only holds for c being real-valued (not a complex constant, as we are using in Equation ):
The magnitude sign in Equation  arises because if c is real and negative, the integration limits will flip and the 1/c becomes -1/c (see Scaling Proof for Fourier Transforms. Hence, the generalization of Equation  to complex numbers is not valid.
However, we are a little bit lucky in this case. The Gaussian Function of Equation  is well-defined when the argument becomes complex - that is, the complex exponential is well understood. This isn't the case in general - for instance, what is u(it) - the step function evaluated at a complex argument? There is no simple or consistent way to define it.
In this case, we have the constant c being given by:
I apologize for the lack of rigor on this next statement, but basically we need the real and imaginary parts of the scaling constant c to be positive in order to avoid any additional minus signs that will show up in Equation . This is because if the real and imaginary parts of c are positive, we won't need to do a sign change in the integration limits. Assuming k is positive, the resulting Fourier Transform follows since we know G(f), the Fourier Transform of the Gaussian already:
If on the other hand, k is negative, then the constant c in Equation  will will be given by:
Hence, k has a negative real part and a positive imaginary part. By multiplying the constant c in Equation  by -i, we can force c to have a positive real and imaginary part. Hence, we can find the Fourier Transform of the complex Gaussian for the negative k case:
Hence, the general solution for the Fourier Transform is:
Note that if k=0, then the complex Gaussian is simply a constant, so the Fourier Transform will be the dirac-delta functional.
Next: Quadratic Sinusoids
Previous: Right-Sided Sinusoids
The Fourier Transform (Home)