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Numerical Methods. Lecture 9 – Frequency Analysis (using FFT) in Matlab Dr Andy Phillips School of Electrical and Electronic Engineering University of Nottingham. Today’s Topics. More Matlab Frequency Analysis What it is & why we do it Windowing Frequency Analysis in Matlab.

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Numerical methods

Numerical Methods

Lecture 9 – Frequency Analysis (using FFT) in Matlab

Dr Andy Phillips

School of Electrical and Electronic Engineering

University of Nottingham

Today s topics
Today’s Topics

  • More Matlab

  • Frequency Analysis

    • What it is & why we do it

    • Windowing

  • Frequency Analysis in Matlab

More matlab 1 labelling plots
More Matlab 1: Labelling Plots

  • In the last lab we could have provided plot title and x and y axis labels:

    %excerpt from square wave script file


    title(‘Creation of a Square Wave’)



  • Like comments, labels make your work clearer to the reader…

More matlab 2 abs
More Matlab 2: abs

  • To obtain an absolute value or complex magnitude of a scalar or matrix number or variable use abs

    • abs(-2)2

    • if you’ve defined i=sqrt(-1) then


    • if you’ve defined x=-3 then


More matlab 3 functions
More Matlab 3: functions

  • Matlab has lots of familiar maths functions:

    • sin, cos, tan

    • log, log2, log10, exp

      • log is natural log (i.e. base e)

    • cosh, sinh, tanh

    • factorial

  • And less familiar functions e.g gamma, sinc

Frequency analysis
Frequency Analysis

  • One common, and computationally intensive, method we use is Frequency Analysis

  • The most common task is performing Fourier and Inverse Fourier Transforms

    • related to Fourier Series, but don’t need signal to be periodic

Fourier transforms
Fourier Transforms

  • You’ll do these in Signal Processing next year, so don’t worry about the theory. For info:

  • Fourier Transform of h (t) is H (f) where:

  • Inverse Fourier Transform of H (f) is h (t) where:


What does it mean
What does it mean?

  • Fourier Analysis

    • This is a technique which takes a data in the time domain and provides its frequency spectrum

    • The inverse takes us back again

    • We have a ‘Fourier Transform pair’

Dft and fft

  • In engineering functions often represented by finite sets of discrete values e.g. f(t) below could be represented by the set of discrete points (ti ,fi) below

Dft and fft1

  • Using such discrete data a discrete Fourier Transform (DFT) can be defined:

  • Computationally intensive (N2) when calculated straightforwardly

  • Fast Fourier Transform (FFT) computes DFT using very efficient algorithm

    • uses results of previous calcs to reduce number of operations

    • can get to Nlog2N operations, so for N=1024 the FFT is approx. 100 times faster than straight DFT

    • often N required to be an integer power of two (i.e. 2,4,8,16,32,…)


  • Having performed the analysis we can

    • Determine bandwidth

    • Remove Phase information

    • Generally play around with the signal

  • It can be done in C, but it is a lot of work

  • Use a standard library or a package like Matlab

  • Your next lab will involve you using Matlab’s fft function

Simple script file using fft
Simple script file using FFT

%simple example of FFT use

%setting up the signal


x=sin(2*pi*40*t)+sin(2*pi*100*t) %40 Hz and 100 Hz sine waves added

y=x+2*randn(size(t)); %noise added

%randn(size(t)) gives an array of normally distributed random entries

% that is the same size as t

plot(t(1:50),y(1:50)) %plots first 50 points i.e. to 0.05 s

title('40 Hz plus 100 Hz corrupted by noise')

xlabel('time (seconds)')

pause(2) %waits 2 seconds before continuing

Simple script file using fft pt2
Simple script file using FFT pt2

%the fft and output

Y=fft(y,512); %performs the fft. Note 512 elements - power of 2 (2^9)


%Y will have complex elements. The power spectrum is real

%so do an element by element multiplication by complex conjugate


plot(f,Pyy(1:257)) %we are only interested in half the data points

%as the same information is provided in the second half of Pyy

title('Frequency content of corrupted signal')

xlabel('Frequency (Hz)')

Before fft a signal however
Before FFT a signal however

  • We often apply a ‘window’ to the data.

  • This simply means taking the amount we want from the data stream

  • ie

The window is moved along the data; we perform the FFT on this windowed data


  • We extract the data simply by reading data between a start and finish value

    • in effect we apply a rectangular window

  • This however causes problems of discontinuities, as shown in the next slide

The problem
The Problem…

These discontinuities cause errors in our frequency analysis. To avoid this we use a window that ‘tapers’ at the ends

The hann hamming windows
The Hann & Hamming Windows

  • The most common solution is to use either Hann or Hamming Windows

  • Are simply cosine bells, scaled to be the width of the window used (in sample points)

  • They differ only in the choice of 

    • Hann ->  = 0.5 Hamming ->  = 0.54

 - (1-  )cos(2*pi*i/(n-1)) ; 0<i<n

0 ; otherwise

xi =


  • The filter coefficients w(i) of a window of length n are computed according to the following formulae

  • Hamming window

    • w(i) = 0.54 - 0.46*cos(2*pi*i/(n-1))

  • Hann window

    • w(i) = 0.5 - 0.5*cos(2*pi*i/(n-1))

Comparing them
Comparing them





  • Since we often apply this window over and over

    • It might help to put it in a lookup table so we only have to calculate it once

    • We then simply multiply our array of data by the corresponding element in the window’s array

  • i.e.

    • xnew[i] = x[i] * w[i]

And then
And then

  • We perform our FFT to get the frequency spectrum

  • As stated before, we could do this in C, or even manually, but it is simpler to use Matlab

    • (or some standard library e.g. NAG – Numerical Algorithms Group)

  • More FFTs and windowing in your lab on Wednesday…