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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

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

plot(t,sum);

title(‘Creation of a Square Wave’)

xlabel(‘t’)

ylabel(‘x(t)’)

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

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

abs(1+i)1.4142

- if you’ve defined x=-3 then

abs(x)3

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

- 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

- 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:

[j=sqrt(-1)]

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 FFT

- 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,…)

Now…

- 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 example of FFT use

%setting up the signal

t=0:0.001:0.6;

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

%the fft and output

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

Pyy=Y.*conj(Y)/512;

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

%so do an element by element multiplication by complex conjugate

f=1000*(0:256)/512;

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

- 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

However

- 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…

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 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 =

So..

- 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))

Now..

- 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

- 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…

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