# 3.4 Continuous-time fourier transform

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In this lab, we learn how to compute the continuous-time Fourier transform (CTFT), normally referred to as Fourier transform, numerically and examine its properties. Also, we explore noise cancellation and amplitude modulation as applications of Fourier transform.

#### Properties of ctft

The continuous-time Fourier transform (CTFT) (commonly known as Fourier transform) of an aperiodic signal $x\left(t\right)$ is given by

$X\left(\omega \right)=\underset{-\infty }{\overset{\infty }{\int }}x\left(t\right){e}^{-\mathrm{j\omega t}}\text{dt}$

The signal $x\left(t\right)$ can be recovered from $X\left(\omega \right)$ via this inverse transform equation

$x\left(t\right)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}X\left(\omega \right){e}^{\mathrm{j\omega t}}\mathrm{d\omega }$

Some of the properties of CTFT are listed in [link] .

 Properties Time domain Frequency domain Time shift $x\left(t-{t}_{0}\right)$ $X\left(\omega \right){e}^{-{\mathrm{j\omega t}}_{0}}$ Time scaling $x\left(\text{at}\right)$ $\frac{1}{\mid a\mid }X\left(\frac{\omega }{a}\right)$ Linearity ${a}_{1}{x}_{1}\left(t\right)+{a}_{2}{x}_{2}\left(t\right)$ ${a}_{1}{X}_{1}\left(\omega \right)+{a}_{2}{X}_{2}\left(\omega \right)$ Time convolution $x\left(t\right)\ast h\left(t\right)$ $X\left(\omega \right)H\left(\omega \right)$ Frequency convolution $x\left(t\right)h\left(t\right)$ $X\left(\omega \right)\ast H\left(\omega \right)$

Refer to signals and systems textbooks [link] - [link] for more theoretical details on this transform.

#### Numerical approximations to ctft

Assuming that the signal $x\left(t\right)$ is zero for $t\text{<0}$ and $t\ge T$ , we can approximate the CTFT integration in Equation (1) as follows:

$\underset{-\infty }{\overset{\infty }{\int }}x\left(t\right){e}^{-\mathrm{j\omega t}}\text{dt}=\underset{0}{\overset{T}{\int }}x\left(t\right){e}^{-\mathrm{j\omega t}}\text{dt}\approx \sum _{n=0}^{N-1}x\left(\mathrm{n\tau }\right){e}^{-\mathrm{j\omega n\tau }}\tau$

where $T=\mathrm{N\tau }$ and N is an integer. For sufficiently small $\tau$ , the above summation provides a close approximation to the CTFT integral. The summation $\sum _{n=0}^{N-1}x\left(\mathrm{n\tau }\right){e}^{-\mathrm{j\omega n\tau }}$ is widely used in digital signal processing (DSP), and both LabVIEW MathScript and LabVIEW have a built-in function for it called fft . In a .m file, if N samples $x\left(\mathrm{n\tau }\right)$ are stored in a vector $x$ , then the function call

>>xw=tau*fft (x)

calculates

$\begin{array}{}{X}_{\omega }\left(k+1\right)=\tau \sum _{n=0}^{N-1}x\left(\mathrm{n\tau }\right){e}^{-{\mathrm{j\omega }}_{k}\mathrm{n\tau }}\begin{array}{cc}& 0\le k\le N-1\end{array}\\ \approx X\left({\omega }_{k}\right)\end{array}$

where

${\omega }_{k}=\left\{\begin{array}{cc}\frac{2\pi k}{\mathrm{N\tau }}& 0\le k\le \frac{N}{2}\\ \frac{2\pi k}{\mathrm{N\tau }}-\frac{2\pi }{\tau }& \frac{N}{2}+1\le k\le N-1\end{array}$

with N assumed to be even. The fft function returns the positive frequency samples before the negative frequency samples. To place the frequency samples in the right order, use the function fftshift as indicated below:

>>xw=fftshift(tau*fft (x ) )

Note that $X\left(\omega \right)$ is a vector (actually, a complex vector) of dimension N. $X\left(\omega \right)$ is complex in general despite the fact that $x\left(t\right)$ is real-valued. The magnitude of $X\left(\omega \right)$ can be computed using the function abs and the phase of $X\left(\omega \right)$ using the function angle .

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