## Fourier Transform of Complex Gaussian

Theorem:

 (D.16)

Proof: [202, p. 211] The Fourier transform of is defined as

 (D.17)

Completing the square of the exponent gives

Thus, the Fourier transform can be written as

 (D.18)

using our previous result.

### Alternate Proof

The Fourier transform of a complex Gaussian can also be derived using the differentiation theorem and its dual (§B.2).D.1

Proof: Let

 (D.19)

Then by the differentiation theorem (§B.2),

 (D.20)

By the differentiation theorem dual (§B.3),

 (D.21)

Differentiating gives

 (D.22)

Therefore,

 (D.23)

or

 (D.24)

Integrating both sides with respect to yields

 (D.25)

In §D.7, we found that , so that, finally, exponentiating gives

 (D.26)

as expected.

The Fourier transform of complex Gaussians (chirplets'') is used in §10.6 to analyze Gaussian-windowed chirps'' in the frequency domain.

Next Section:
Why Gaussian?
Previous Section:
Gaussian Integral with Complex Offset