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Fourier Transform Pairs of Conjugation and Time Reversal

November 17, 2021

Introduction

Too often I find myself asking the questions “what is the Fourier transform of $x(-t)$ ”? “What is the inverse Fourier transform of $X^*(-f)$ ?” Finding them in a book or on Wikipedia often takes too long so to save myself the time and hopefully be useful to others, I’ve provided the derivations for the Fourier transforms of x(t), $x^*(t)$ , x(-t) and $x^*(-t)$ below.

Fourier Transform

The Fourier transform of x(t) is given by

(1) $\begin{equation*}\mathcal{F}\{ x(t) \} = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} ~ dt = X(f).\end{equation*}$

Fourier Transform of Conjugate

The Fourier transform of $x^*(t)$ , the conjugate of x(t), is given by

(2) $\begin{equation*}\mathcal{F}\{ x^*(t) \} = \int_{-\infty}^{\infty} x^*(t) e^{-j2\pi f t} ~ dt.\end{equation*}$

A double-conjugate is applied to (2)

(3) $\begin{equation*}\left( \left( \int_{-\infty}^{\infty} x^*(t) e^{-j2\pi f t} ~ dt \right)^* \right)^*\end{equation*}$

and the inner conjugate is simplified as

(4) $\begin{equation*}\left( \int_{-\infty}^{\infty} x(t) e^{j2\pi f t} ~ dt \right)^*.\end{equation*}$

A variable substitution is made for the frequency $\psi$ ,

(5) $\begin{equation*}f = -\psi\end{equation*}$

such that (4) is written as

(6) $\begin{equation*}\left( \int_{-\infty}^{\infty} x(t) e^{-j2\pi \psi t} ~ dt \right)^*.\end{equation*}$

Using (1) the integral in (6) is simplified to

(7) $\begin{equation*}\left( X(\psi) \right)^* = \left( X(-f) \right)^*\end{equation*}$

such that

(8) $\begin{equation*}\mathcal{F} \{x^*(t) \} = X^*(-f).\end{equation*}$

Fourier Transform of Time Reversal

The Fourier transform of x(-t), which is time reversed x(t), is given by

(9) $\begin{equation*}\mathcal{F}\{ x(-t) \} = \int_{-\infty}^{\infty} x(-t) e^{-j2\pi f t} ~ dt.\end{equation*}$

Substitute $\tau = -t$ such that (9) is written as

(10) $\begin{equation*}-\int_{\infty}^{-\infty} x(t) e^{j2\pi f \tau} ~ d\tau\end{equation*}$

where

(11) $\begin{equation*}dt = -d\tau.\end{equation*}$

Swapping the integration limits in (10) cancels the negation so the integral is written as

(12) $\begin{equation*}\int_{-\infty}^{\infty} x(\tau) e^{j2\pi f \tau} ~ d\tau.\end{equation*}$

Substitute for the frequency f,

(13) $\begin{equation*}f = -\psi\end{equation*}$

such that (12) is written as

(14) $\begin{equation*}\int_{-\infty}^{\infty} x(\tau) e^{-j2\pi \psi \tau} ~ d\tau.\end{equation*}$

Using (1) the Fourier transform of x(-t) is written as

(15) $\begin{equation*}\mathcal{F} \{ x(-t) \} = X(\psi)\end{equation*}$

which is simplified as

(16) $\begin{equation*}\mathcal{F} \{ x(-t) \} = X(-f).\end{equation*}$

The blog on negative frequency showed that positive frequencies rotate in a counter-clockwise direction around the unit circle. My interpretation of (16) is that reversing the time, x(-t), also reverses the direction rotation around the unit circle for positive frequencies, which is why the frequencies in X(-f) are reversed as well.