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Signals & Systems – Conjugation and Autocorrelation Property of Fourier Transform
Fourier Transform
For a continuous-time function x(t), the Fourier transform of x(t) can be defined as,
$$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$
Conjugation 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
$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$
Then, according to conjugation property of Fourier transform,
$$\mathrm{x^*(t)\overset{FT}{\leftrightarrow}X^*(-\omega)}$$
Proof
From the definition of Fourier transform, we have
$$\mathrm{X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt}$$
Taking conjugate on both sides, we get
$$\mathrm{X^*(\omega)=[\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt]^*}$$
$$\mathrm{\Rightarrow X^*(\omega)=\int_{-\infty}^{\infty}x^*(t)e^{j\omega t}dt}$$
Now, by replacing (ω) by (−ω), we obtain,
$$\mathrm{X^*(-\omega)=\int_{-\infty}^{\infty}x^*(t)e^{-j\omega t}dt=F[x^*(t)]}$$
$$\mathrm{\therefore F[x^*(t)]=X^*(-\omega)}$$
Or, it can also be represented as,
$$\mathrm{x^*(t)\overset{FT}{\leftrightarrow}X^*(-\omega)}$$
Autocorrelation Property of Fourier Transform
The autocorrelation of a continuous-time function 𝑥(𝑡) is defined as,
$$\mathrm{R(\tau)=\int_{-\infty}^{\infty}x(t)x^*(t-\tau)dt}$$
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
$$\mathrm{x(t)\overset{FT}{\leftrightarrow}X(\omega)}$$
Then, by the autocorrelation property of Fourier transform,
$$\mathrm{R(\tau)\overset{FT}{\leftrightarrow}|X(\omega)|^2}$$
Proof
By the definition of autocorrelation, we have,
$$\mathrm{R(\tau)=\int_{-\infty}^{\infty}x(t)x^*(t-\tau)dt}$$
Then, from the definition of Fourier transform, we get,
$$\mathrm{X(\omega)=F[R(\tau)]=\int_{-\infty}^{\infty}[\int_{-\infty}^{\infty}x(t)x^*(t-\tau)dt]e^{-j\omega t}dt}$$
By interchanging the order of integration, we get,
$$\mathrm{F[R(\tau)]=\int_{-\infty}^{\infty}x(t)[\int_{-\infty}^{\infty}x^*(t-\tau)e^{-j\omega t}d\tau]dt}$$
Substituting [(𝑡 − 𝜏) = 𝑢] in the second integration,
$$\mathrm{\tau=(t-u)\:and\:d\tau=du}$$
$$\mathrm{\therefore F[R(\tau)]=\int_{-\infty}^{\infty}x(t)[\int_{-\infty}^{\infty}x^*(u)e^{-j\omega(t-u)}du]dt}$$
$$\mathrm{ \Rightarrow F[R(\tau)]=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt\int_{-\infty}^{\infty}x^*(u)e^{j\omega u}du}$$
$$\mathrm{ \Rightarrow F[R(\tau)]=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt[\int_{-\infty}^{\infty}x(u)e^{-j\omega u}du]^*}$$
Therefore
$$\mathrm{F[R(\tau)]=X(\omega)X^*(\omega)=|X(\omega)|^2}$$
Or, it can also be represented as,
$$\mathrm{R(\tau)\overset{FT}{\leftrightarrow}|X(\omega)|^2}$$
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