Fourier Transform¶

Fourier Series¶

$\begin{align} \text{Fourier series, }x(t)&=\int_{-\infty}^{\infty}c_{k}e^{j2\pi kF_{0}t}\\ \text{where }c_{k} &= \frac{1}{T_{p}}\int_{T_{p}}x(t)e^{-j2\pi kF_{0}t}\ dt \quad (1) \end{align}$

Warning

Equation (1) is only true when $x(t)$ is periodic and satisfies the Direchlet conditions.

Direchlet conditions¶

x(t) has a finite number of discontinuities in any period
x(t) has a finite number of maxima and minima in any period
x(t) is absolutely integrable in any period, i.e.

$\int_{T_{p}}|x(t)| \ dt < \infty\\$

Note

All periodic signals of practical interest satisfy these conditions.

Further Simplification of Fourier Series¶

If x(t) is real, $c_{k}$ and $c_{-k}$ are complex conjugates. $$ c_{k} = |c_{k}|e^{j\theta_{k}}, c_{-k} = |c_{k}|e^{-j\theta_{k}} $$

$\therefore \text{Fourier series, }x(t) = c_{0}+2\sum_{k=1}^{\infty}|c_{k}|\cos(2\pi kF_{0}t + \theta_{k})$

We know that $\cos(2\pi kF_{0}t + \theta_{k})=\cos(2\pi kF_{0}t)\cos\theta_{k} - \sin (2\pi kF_{0}t)\sin\theta_{k}$ ,

$\begin{align} \therefore \text{Fourier series, }x(t) &= a_{0}+\sum^{\infty}_{k=1}(a_{k}\cos 2\pi kF_{0}t - b_{k}\cos 2\pi kF_{0}t)\\ \text{where }a_{0} &= c_{0}, \ a_{k} = 2|c_{k}|\cos\theta_{k}, \ b_{k}=2|c_{k}|\sin\theta_{k} \end{align}$

Info

This is commonly expressed in many beginner texts on Fourier transform as:

$\begin{align} \text{Fourier series, }f(t) = a_{0} + &a_{1}\cos(t)+a_{2}\cos(2t)+a_{3}\cos(3t)+...+a_{n}\cos(nt)\\ +&b_{1}\cos(t)+b_{2}\cos(2t)+b_{3}\cos(3t)+...+b_{n}\cos(nt)\\ \end{align}$

$\text{where }a_{0} = \frac{1}{2\pi}\int_{0}^{2\pi}f(t) \ dt,\ a_{t} = \frac{1}{\pi}\int_{0}^{2\pi}f(t)\cos(nt) \ dt,\ b_{t} = \frac{1}{\pi}\int_{0}^{2\pi}f(t)\sin(nt) \ dt$

Equivalent Forms of Fourier Series¶

Fourier Series, x(t)¶

$x(t)=\int_{-\infty}^{\infty}c_{k}e^{j2\pi kF_{0}t}$
$x(t) = c_{0}+2\sum_{k=1}^{\infty}|c_{k}|\cos(2\pi kF_{0}t + \theta_{k})$
$x(t) = a_{0}+\sum^{\infty}_{k=1}(a_{k}\cos 2\pi kF_{0}t - b_{k}\cos 2\pi kF_{0}t)$ ,
where $a_{0}= c_{0}, \ a_{k} = 2|c_{k}|\cos\theta_{k}, \ b_{k}=2|c_{k}|\sin\theta_{k}$

Fourier coefficient¶

$c_{k} = \frac{1}{T_{p}}\int_{T_{p}}x(t)e^{-j2\pi kF_{0}t}\ dt$
when $x(t)$ is periodic and satisfies the Direchlet conditions

Useful Trigonometrical Identities¶

$\begin{align*} \text{Sine: }\int_{0}^{2\pi} \sin(mt) \ dt &= \ 0 \text{ for any integer }m \\ \text{Cosine: }\int_{0}^{2\pi} \cos(mt) \ dt &= \ 0 \text{ for any non-zero integer }m \\ \\ \text{Product of sine and cosine: }\int_{0}^{2\pi} \sin(mt) \cos(nt) \ dt &= \ 0 \text{ for any integers }m, n \\ \\ \text{Product of sines: }\int_{0}^{2\pi} \sin(mt) \sin(nt) \ dt &= \ 0 \text{ when }m \neq n, m \neq -n \\ \int_{0}^{2\pi} \sin^{2}(mt) \ dt &= \pi \text{ for any non-zero integer }m \\ \\ \text{Product of cosines: }\int_{0}^{2\pi} \cos(mt) \cos(nt) \ dt &= \ 0 \text{ when }m \neq n, m \neq -n \\ \int_{0}^{2\pi} \cos^{2}(mt) \ dt &= \pi \text{ for any non-zero integer }m \\ \end{align*}$

Fourier Transform Theorems and Properties¶

Linearity¶

$a_{1}x_{1}(n)+a_{2}x_{2}(n)\xrightarrow{\mathscr{F}}a_{1}X_{1}(\omega)+a_{2}X_{2}(\omega).$

Time shifting¶

$x(n-k)\xrightarrow{\mathscr{F}}e^{-j\omega k}X(\omega)$ .

Shifting signal in time domain by $k$ changes phase by $-\omega k$ .

Time reversal¶

$x(-n)\xrightarrow{\mathscr{F}}X(-\omega)$

When signal is folded about origin in time, phase spectrum undergoes phase reversal

Convolution¶

$x_{1}(n) \ x_{2}(n) \xrightarrow{\mathscr{F}} X_{1}(\omega)X_{2}(\omega)$

Wiener-Khintchine theorem¶

$r_{xx}(l) \xrightarrow{\mathscr{F}} S_{xx}(\omega)$

Energy spectral density of an energy signal $S$ is the Fourier transform of its autocorrelation sequence $R$

Frequency shifting¶

$e^{j\omega_{0}n}x(n) \xrightarrow{\mathscr{F}} X(\omega-\omega_{0})$

Multiplying $e^{j\omega_{0}n}$ to $x(n)$ is equivalent to translating $X(\omega)$ by $\omega_{0}$

Modulation¶

$x(n)\cos \omega_{0}n \xrightarrow{\mathscr{F}} \frac{1}{2}X(\omega+\omega_{0})+\frac{1}{2}X(\omega-\omega_{0})$

Multiplication (Windowing theorem)¶

$x_{1}(n) \ x_{2}(n) \xrightarrow{\mathscr{F}} \frac{1}{2\pi}\int_{-\pi}^{\pi}X_{1}(\lambda) \ X_{2}(\omega-\lambda) \ d\lambda$

Differentiation in the frequency domain¶

$nx(n) \xrightarrow{\mathscr{F}} j\frac{dX(\omega)}{d\omega}$

Conjugation¶

$x^{*}(n)\xrightarrow{\mathscr{F}}X^{*}(-\omega)$

Parseval's theorem¶

$\int_{-\infty}^{\infty}x_{1}(n) \ x_{2}^{*}(n) = \frac{1}{2\pi}\int_{-\pi}^{\pi}X_{1}(\omega) \ X_{2}^{*}(\omega) \ d\omega$