[Jupyter notebook]

Some definitions $\begin{equation} \def\flecheTF{\mathop{\rightharpoonup}\limits_{\mbox{$\leftharpoondown $}}} \def\TFI#1#2#3{{\displaystyle{\int_{-\infty}^{+\infty} #1 ~e^{j2\pi #2 #3} ~\dr{#2}}}} \def\TF#1#2#3{{\displaystyle{\int_{-\infty}^{+\infty} #1 ~e^{-j2\pi #3 #2} ~\dr{#2}}}} \def\tf#1{{\mathrm{FT}\left\{ #1 \right\}}} \def\sinc#1{{\mathrm{sinc}\left( #1 \right)}} \def\rect{\mathrm{rect}} \def\egalpardef{\mathop{=}\limits^\triangle} \def\dr#1{\mathrm{d}#1} \end{equation}$

1 Symmetries of the Fourier transform.¶

Consider the Fourier pair $\begin{equation} x(n) \flecheTF X(f). \end{equation}$ When $x(n)$ is complex valued, we have $\begin{equation} \fbox{$x^*(n) \flecheTF X^*(-f)$}. \end{equation}$ This can be easily checked beginning with the definition of the Fourier transform: $\begin{eqnarray*} \tf{x^*(n)} & = & \sum_n {x^*(n)}e^{-j2\pi fn}, \\ & = & \left(\int_{[1]}{x(n)}e^{j2\pi fn} \dr{f} \right)^*, \\ & = & X^*(-f). \end{eqnarray*}$ In addition, for any signal $x(n)$ , we have $\begin{equation} \fbox{$x(-n) \flecheTF X(-f)$}. \end{equation}$ This last relation can be derived directly from the Fourier transform of $x(-n)$ $\begin{equation} \tf{x(-n)} = \TF{x(-n)}{t}{f}, \end{equation}$ using the change of variable $-t \rightarrow t$ , we get $\begin{eqnarray*} \tf{x(-n)} & = & \TFI{x(n)}{t}{f}, \\ & = & X(-f). \end{eqnarray*}$

using the two last emphasized relationships, we obtain $\begin{equation} \fbox{$x^*(-n) \flecheTF X^*(f)$}. \end{equation}$ To sum it all up, we have $\begin{equation} \fbox{ $ \begin{array}{lll} x(n) & \flecheTF & X(f) \\ x(-n) & \flecheTF & X(-f) \\ x^*(n) & \flecheTF & X^*(-f) \\ x^*(-n) & \flecheTF & X^*(f) \end{array} $} \end{equation}$

These relations enable to analyse all the symetries of the Fourier transform. We begin with the Hermitian symmetry for real signals: $\begin{equation} \fbox{$X(f) = X^*(-f)$} \end{equation}$ from that, we observe that if $x(n)$ is real, then [prop:prop-sym]

the real part of $X(f)$ is even,
the imaginary part of $X(f)$ is odd,
the modulus of $X(f)$ , $|X(f)|$ is even,
the phase of $X(f)$ , $\theta(f)$ is odd.

Moreover, if $x(n)$ is odd or even ( $x(n)$ is not necessarily real), we have [prop:tf-sym] $\begin{equation} \fbox{ $ \begin{array}{lllll} \text{[even] } & x(n)=x(-n) & \flecheTF & X(f)=X(-f) & \text{ [even]}\\ \text{[odd] } & x(n)=-x(-n) & \flecheTF & X(f)=-X(-f) & \text{ [odd]} \end{array} $} \end{equation}$

The following table summarizes the main symmetry properties of the Fourier transform:

$\begin{equation} \begin{array}{||l|l|l|l|l||} \hline\hline \mathbf{x(n)} & \mathbf{\text{Symmetry}} & \mathbf{\text{Time}} & \mathbf{\text{Frequency}} & \mathbf{\text{consequence on $X(f)$}}\\ \hline\hline \text{real} & \text{any} & x(n)=x^*(n) & X(f) = X^*(-f) & \text{Re. even, Im. odd} \\ \hline \text{real} & \text{even} & x(n)=x^*(n)=x(-n) & X(f) = X^*(-f)=X(-f) & \text{Real and even} \\ \hline \text{real} & \text{odd} & x(n)=x^*(n)=-x(-n) & X(f) = X^*(-f)=-X(-f) & \text{Imaginary and odd} \\ \hline \text{imaginary} & \text{any} & x(n)=-x^*(n) & X(f) = -X^*(-f) & \text{Re. odd, Im. even} \\ \hline \text{imaginary} & \text{even} & x(n)=-x^*(n)=x(-n) & X(f) = -X^*(-f)=X(-f) & \text{Imaginary and even} \\ \hline \text{imaginary} & \text{odd} & x(n)=-x^*(n)=-x(-n) & X(f) = -X^*(-f)=-X(-f) & \text{Real and odd} \\ \hline\hline \end{array} \end{equation}$

Finally, we have $\begin{equation} \fbox{ $ \begin{array}{lll} \text{Real even + imaginary odd } & \flecheTF & \text{ Real}\\ \text{Real odd + imaginary even } & \flecheTF & \text{ Imaginary} \end{array} $} \end{equation}$

2 Table of Fourier transform properties¶

The following table lists the main properties of the Discrete time Fourier transform. The table is adapted from the article on discrete time Fourier transform on Wikipedia.

$\begin{equation} \begin{array}{||l|l|l||} \hline\hline \hline \text{Property} & \text{Time domain } x(n) & \text{Frequency domain } X(f) \\ \hline \text{Linearity} & ax(n) + by(n) & aX(f) + bY(f) \\ \hline \text{Shift in time} & x(n-n_0) & X(f)e^{-j2\pi fn_0} \\ \hline \text{Shift in frequency (modulation)} & x(n)e^{j2\pi f_0 n} & X(f-f_0) \\ \hline \text{Time scaling} & x(n/k) & X( kf) \\ \hline \text{Time reversal} & x(-n) & X(-f) \\ \hline \text{Time conjugation} & x(n)^* & X(-f)^* \\ \hline \text{Time reversal \& conjugation} & x(-n)^* & X(f)^* \\ \hline \text{Sum of }x(n) & \sum_{n=-\infty}^{\infty} x(n) & X(0) \\ \hline \text{Derivative in frequency} & \frac{n}{i} x(n) & \frac{d X(f)}{d f} \\ \hline \text{Integral in frequency} & \frac{i}{n} x(n) & \int_{[1]} X(f) d f \\ \hline \text{Convolve in time} & x(n) * y(n) & X(f) \cdot Y(f) \\ \hline \text{Multiply in time} & x(n) \cdot y(n) & \int_{[1]} X(f_1) \cdot Y(f-f_1) df_1 \\ \hline \text{Area under } X(f) & \displaystyle{x(0)} & \displaystyle{\int_{[1]} X(f)\,df} \\ \hline \text{Parseval's theorem} & \displaystyle{\sum_{n=-\infty}^{\infty} {x(n) \cdot y^*(n)}} & \displaystyle{\int_{[1]}{X(f) \cdot Y^*(f) df}} \\ \hline \text{Parseval's theorem} & \displaystyle{ \sum_{n=-\infty}^{\infty} {|x(n)|^2}} & \displaystyle{\int_{[1]}{|X(f)|^2 df}} \\ \hline\hline \end{array} \end{equation}$

Some examples of Fourier pairs are collected below:

$\begin{equation} \begin{array}{||l|l|l||} \hline\hline \hline \mathbf{\text{Time domain}} & \mathbf{\text{Frequency domain}} \\ \hline x[n] & X(f) \\ \hline \delta[n] & X(f) = 1 \\ \hline \delta[n-M] & X(f) = e^{-j2\pi f M} \\ \hline \sum_{k = -\infty}^{\infty} \delta[n - kM ] & \frac{1}{M}\sum_{k = -\infty}^{\infty} \delta \left( f - \frac{ k}{M} \right) \\ \hline u[n] & X(f) = \frac{1}{1-e^{-j2\pi f}} + \frac{1}{2} \sum_{k=-\infty}^{\infty} \delta (f - k)\!\\ \hline a^n u[n] & X(f) = \frac{1}{1-a e^{-j2\pi f}}\! \\ \hline e^{-j2\pi f_a n} & X(f) = \delta (f +f_a )\\ \hline \cos(2\pi f_a n) & X(f) = \frac{1}{2} [\delta (f+f_a)+\delta (f-f_a)]\\ \hline \sin(2\pi f_a n) & X(f) = \frac{1}{2j} [\delta (f+f_a)-\delta (f-f_a)] \\ \hline \mathrm{rect}_M \left[ { ( n - (M-1)/2 ) } \right] & X(f) = { \sin[ \pi f M ] \over \sin( \pi f ) } \, e^{ -{j\pi f (M-1)} } \! \\ \hline \begin{cases} 0 & n=0 \\\frac{(-1)^n}{n} & \mbox{elsewhere}\\\end{cases} & X(f) = j 2\pi f \\ \hline \begin{cases}0 & n \mbox{ even} \\ \frac{2}{\pi n} & n \mbox{ odd}\\\end{cases} & X(f) = \begin{cases} j & f < 0 \\0 & f = 0 \\-j & f > 0\\\end{cases} \\ \hline\hline \end{array} \end{equation}$

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