Signals & Systems – Conjugation and Autocorrelation Property of Fourier Transform

Statement − The conjugation property of Fourier transform states that the conjugate of function x(t) in time domain results in conjugation of its Fourier transform in the frequency domain and ω is replaced by (−ω), i.e., if

Then, according to conjugation property of Fourier transform,

Proof

From the definition of Fourier transform, we have

Taking conjugate on both sides, we get

Now, by replacing (ω) by (−ω), we obtain,

Or, it can also be represented as,

Autocorrelation Property of Fourier Transform

The autocorrelation of a continuous-time function 𝑥(𝑡) is defined as,

Statement − The autocorrelation property of Fourier transform states that the Fourier transform of the autocorrelation of a single in time domain is equal to the square of the modulus of its frequency spectrum. Therefore, if

Then, by the autocorrelation property of Fourier transform,

Proof

By the definition of autocorrelation, we have,

Then, from the definition of Fourier transform, we get,

By interchanging the order of integration, we get,

Substituting [(𝑡 − 𝜏) = 𝑢] in the second integration,

Or, it can also be represented as,