# 0.7 Lab 6a - discrete fourier transform and fft (part 1)

 Page 1 / 2

Questions or comments concerning this laboratory should be directedto Prof. Charles A. Bouman, School of Electrical and Computer Engineering, Purdue University, West Lafayette IN 47907;(765) 494-0340; bouman@ecn.purdue.edu

## Introduction

This is the first week of a two week laboratory that covers the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT)methods. The first week will introduce the DFT and associated samplingand windowing effects, while thesecond week will continue the discussion of the DFT and introduce the FFT.

In previous laboratories, we have used the Discrete-Time Fourier Transform (DTFT) extensively foranalyzing signals and linear time-invariant systems.

$\begin{array}{ccc}\hfill \text{(DTFT)}\phantom{\rule{16.pt}{0ex}}X\left({e}^{j\omega }\right)& =& \sum _{n=-\infty }^{\infty }x\left(n\right){e}^{-j\omega n}\hfill \end{array}$
$\begin{array}{ccc}\hfill \text{(inverse}\phantom{\rule{4.pt}{0ex}}\text{DTFT)}\phantom{\rule{16.pt}{0ex}}x\left(n\right)& =& \frac{1}{2\pi }{\int }_{-\pi }^{\pi }X\left({e}^{j\omega }\right){e}^{j\omega n}d\omega .\hfill \end{array}$

While the DTFT is very useful analytically, it usually cannot be exactlyevaluated on a computer because [link] requires an infinite sum and [link] requires the evaluation of an integral.

The discrete Fourier transform (DFT) is a sampled version of the DTFT, hence it is better suited for numerical evaluation on computers.

$\begin{array}{ccc}\hfill \text{(DFT)}\phantom{\rule{16.pt}{0ex}}{X}_{N}\left(k\right)& =& \sum _{n=0}^{N-1}x\left(n\right){e}^{-j2\pi kn/N}\hfill \end{array}$
$\begin{array}{ccc}\hfill \text{(inverse}\phantom{\rule{4.pt}{0ex}}\text{DFT)}\phantom{\rule{16.pt}{0ex}}x\left(n\right)& =& \frac{1}{N}\sum _{k=0}^{N-1}{X}_{N}\left(k\right){e}^{j2\pi kn/N}\hfill \end{array}$

Here ${X}_{N}\left(k\right)$ is an $N$ point DFT of $x\left(n\right)$ . Note that ${X}_{N}\left(k\right)$ is a function of a discrete integer $k$ , where $k$ ranges from 0 to $N-1$ .

In the following sections, we will study the derivation of the DFT from the DTFT,and several DFT implementations. The fastest and most important implementation is known as the fast Fourier transform (FFT). The FFT algorithmis one of the cornerstones of signal processing.

## Truncating the time-domain signal

The DTFT usually cannot be computed exactly because the sum in [link] is infinite. However, the DTFT may be approximately computedby truncating the sum to a finite window. Let $w\left(n\right)$ be a rectangular window of length $N$ :

$w\left(n\right)=\left\{\begin{array}{cc}\hfill 1& \hfill 0\le n\le N-1\\ \hfill 0& \hfill \text{else}\end{array}\right)\phantom{\rule{0.166667em}{0ex}}.$

Then we may define a truncated signal to be

${x}_{\mathrm{tr}}\left(n\right)=w\left(n\right)x\left(n\right)\phantom{\rule{4pt}{0ex}}.$

The DTFT of ${x}_{\mathrm{tr}}\left(n\right)$ is given by:

$\begin{array}{ccc}\hfill {X}_{\mathrm{tr}}\left({e}^{j\omega }\right)& =& \sum _{n=-\infty }^{\infty }{x}_{\mathrm{tr}}\left(n\right){e}^{-j\omega n}\hfill \\ & =& \sum _{n=0}^{N-1}x\left(n\right){e}^{-j\omega n}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$

We would like to compute $X\left({e}^{j\omega }\right)$ , but as with filter design, the truncation window distortsthe desired frequency characteristics; $X\left({e}^{j\omega }\right)$ and ${X}_{\mathrm{tr}}\left({e}^{j\omega }\right)$ are generally not equal. To understand the relation between these two DTFT's,we need to convolve in the frequency domain (as we did in designing filters with the truncationtechnique):

${X}_{\mathrm{tr}}\left({e}^{j\omega }\right)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }X\left({e}^{j\sigma }\right)W\left({e}^{j\left(\omega -\sigma \right)}\right)d\sigma$

where $W\left({e}^{j\omega }\right)$ is the DTFT of $w\left(n\right)$ . [link] is the periodic convolution of $X\left({e}^{j\omega }\right)$ and $W\left({e}^{j\omega }\right)$ . Hence the true DTFT, $X\left({e}^{j\omega }\right)$ , is smoothed via convolution with $W\left({e}^{j\omega }\right)$ to produce the truncated DTFT, ${X}_{\mathrm{tr}}\left({e}^{j\omega }\right)$ .

We can calculate $W\left({e}^{j\omega }\right)$ :

$\begin{array}{ccc}\hfill W\left({e}^{j\omega }\right)& =& \sum _{n=-\infty }^{\infty }w\left(n\right){e}^{-j\omega n}\hfill \\ & =& \sum _{n=0}^{N-1}{e}^{-j\omega n}\hfill \\ & =& \left\{\begin{array}{cc}\frac{1-{e}^{-j\omega N}}{1-{e}^{-j\omega }},\hfill & \phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\omega \ne 0,±2\pi ,...\hfill \\ N,\hfill & \phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\omega =0,±2\pi ,...\hfill \end{array}\right)\hfill \end{array}$

For $\omega \ne 0,±2\pi ,...$ , we have:

$\begin{array}{ccc}\hfill W\left({e}^{j\omega }\right)& =& \frac{{e}^{-j\omega N/2}}{{e}^{-j\omega /2}}\phantom{\rule{0.166667em}{0ex}}\frac{{e}^{j\omega N/2}-{e}^{-j\omega N/2}}{{e}^{j\omega /2}-{e}^{-j\omega /2}}\hfill \\ & =& {e}^{-j\omega \left(N-1\right)/2}\phantom{\rule{0.166667em}{0ex}}\frac{sin\left(\omega N/2\right)}{sin\left(\omega /2\right)}\phantom{\rule{4pt}{0ex}}.\hfill \end{array}$

Notice that the magnitude of this function is similar to $\text{sinc}\left(\omega N/2\right)$ except that it is periodic in $\omega$ with period $2\pi$ .

## Frequency sampling

[link] contains a summation over a finite number of terms. However, we can only evaluate [link] for a finite set of frequencies, $\omega$ . We must sample in the frequency domain to compute the DTFT on a computer.We can pick any set of frequency points at which to evaluate [link] , but it is particularly useful to uniformly sample $\omega$ at $N$ points, in the range $\left[0,2\pi \right)$ . If we substitute

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers! By Frank Levy By Rhodes By OpenStax By Sandhills MLT By Anonymous User By OpenStax By Edgar Delgado By Richley Crapo By OpenStax By Saylor Foundation