The FFT And Spectral Analysis

Spectral Analysis &
The Fourier Transform
Athanasios Anastasiou
Signal Processing and Multimedia
Communications Research Group
University of Plymouth - UK

Learning Objectives
• Familiarise yourselves with the Fourier
Transform and its properties

• Make sense of Fourier spectra

• Carry out basic operations on various signals
in MATLAB
– Forward / Inverse Transform
– Visualisation of spectra

Topics
• Signals In:
– Time Domain
– Frequency Domain

• The Fourier Transform

• The Fourier Spectra

• Practical application in MATLAB

Fourier Transform (?)
• It was named after Jean Baptiste Joseph Fourier (1768 – 1830)

• The Fourier Transform
– Changes the representation of a signal
– Fundamental characteristics of the signal (power, phase) remain the
same, only their representation changes
– Representation (?)
• Time Domain
• Frequency Domain
– Enables us to observe signals and change them far more easily in the
Frequency Domain, than it would be in the Time Domain.

• The same concepts apply similarly to other transforms as well

What is so special about the
Fourier Transform?
• An extremely versatile tool

• Time Series Analysis / Processing / Synthesis
– Tidal observations
– Mechanical Vibration Analysis
– Lossy Compression
– Image Reconstruction
– Signal Features / Classification
• Speech Recognition / Animal Species Recognition
– Electromagnetic Spectrum
• SETI

• …and some special cases:
– Fast Fluid Simulation (for computer graphics)

Time Domain
Frequency Domain
• One Signal
– x(n)
– Amplitude samples of some quantity over time

• Many Different Representations
– X(ω)
– A series of weights to:
• Trigonometric functions
• Other functions

• How do we pass from one domain to the other?

Integral Transforms
Transformed Signal
Signal Kernel (Time Domain)
(Some Domain)
t2

X U t, x(t ) dt Forward Transform
t1

t2
1
xt U ,t X d Inverse Transform
t1

What about the Fourier Transform?

Frequency Domain Kernel Time Domain

i2 t
X e x t dt

ei cos i sin Euler’s Formula

Re-expresses the signal x(t) as a series of weights X(ω)
of sinusoids and co-sinusoids of
different frequencies ω

The Inverse Fourier Transform
Time Domain Kernel Frequency Domain

xt ei 2 t
X d

Re-composes the signal x(t) as a remix of
sinusoids and co-sinusoids at different frequencies (ω) and
different “strengths” X(ω)

The Discrete Fourier Transform
Remember, DSP is all about sampled signals…

N 1 i2 k n
1 N
k e xn
N n 0
k N 1 i2 k n
N
xn e X k
k 0

…and this has some consequences on the properties of the Fourier Transform
(Key properties: Linearity, Periodicity, Shifting, Convolution
Duality, Interpolation, Frequency & Time Scaling)

Please Note:This is the 1-Dimensional case of the Fourier Transform.
It can be generalised to higher dimensions with applications to
Image Processing, Volumetric Data processing and others

Digression: Fast Fourier Transform
The Discrete Fourier Transform (DFT) as a matrix operation
X U x

Re U

Im U

Perform The Unique Operations Only

Fourier Spectra
• “Spectrum” is another name for X(k)
– The strength X(k) by which sinusoids and co-sinusoids of
frequency k contribute to signal x(t)

• “Fourier Spectrum” is synonymous with “Signal
Spectrum”
– Usually, “Spectral Analysis” refers to observing the Fourier
Spectra of a signal

• Hold on….Fourier Spectra?...There’s more than one?
– Oh, yeah!

Fourier Spectra

Ak X k Amplitude
2
Pk X k Power

1 Im X k Phase
k tan
Re X k

Please note:
• X(k) is a complex number
• The power spectrum is not the same as the Amplitude spectrum!

• How do these “spectra” look like and how do we make sense of them?

Fourier Spectra
Amplitude Spectrum
Time Domain Frequency Domain
5
1 10

0.8

0.6
0
10
0.4

0.2

X(k)
x(t)

-5
0 10

-0.2

-0.4
-10
10
-0.6

-0.8

-15
-1 10
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 0.5 1 1.5 2 2.5
time (sec) Frequency x 10
4

What are the units of the Frequency Axis?
Where are the 0, Fs/2 and Fs points on the Frequency Axis?
What can the units of X(k) be?

Fourier Spectra
Amplitude Spectrum
0
1 10

0.8 -2
10

0.6 -4
10

0.4
-6
10
0.2
-8
10

X(k)
x(t)

0
-10
10
-0.2
-12
10
-0.4
-14
10
-0.6

-16
-0.8 10

-18
-1 10
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000
time (sec) Frequency (Hz)

Physical Frequency k
VS f Fs
Normalised Frequency
N

Any general observations on X(k)?
What about the relationship between X(k) and x(t)?

Fourier Spectra
Spectral Resolution
0
1 10

0.8 -2
10

0.6 -4
10

0.4
-6
10
0.2
-8
10

X(k)
x(t)

0
-10
10
-0.2
-12
10
-0.4
-14
10
-0.6

-16
-0.8 10

-18
-1 10
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000

Physical Frequency k
VS f Fs
Normalised Frequency
N

What happens when some (f) falls between two kn,kn+1?

Fourier Spectra
Digression: Windowing Functions
To reduce spectral leakage in x(n):

x* n xn W n
X* k F x n *F W n
Where W(n) is some “Window Function”

2 5
1 10 1 10

0.8 0 0.8
10
0
10
0.6 0.6
-2
A

A
10
A

A

0.4 0.4 -5
10
-4
10
0.2 0.2

-10
-6 0 10
0 10 0 50 100 150 0 0.1 0.2 0.3 0.4 0.5
0 50 100 0 0.1 0.2 0.3 0.4 0.5
n k n k

Hamming Window Blackman Window

Fourier Spectra
Power Spectrum
0
1 10

0.8
-5
10

0.6

-10
0.4 10

0.2 -15
10

P(k)
x(t)

0
-20
10
-0.2

-0.4 10
-25

-0.6
-30
10
-0.8

-35
-1 10
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000

What does P(k) shows and what could be its units?

Fourier Spectra
Power Spectrum (Parseval’s Theorem)
0
1 10

0.8
-5
10

0.6

-10
0.4 10

0.2 -15
10

P(k)
x(t)

0
-20
10
-0.2

-0.4 10
-25

-0.6
-30
10
-0.8

-35
-1 10
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000

N 1 N 1
2 1 2
xn X k
n 0 N k 0

Fourier Spectra
(Phase)
1
4

0.8
3
0.6

2
0.4

0.2 1
x(t)

phi(k)
0
0

-0.2
-1
-0.4

-2
-0.6

-0.8 -3

-1 -4
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0 1000 2000 3000 4000 5000 6000 7000 8000

What does the φ(k) axis shows?

Fourier Spectra
(Phase)
4

1 3

2

1

0.8

phi(k)
0

3 -1

-2

0.6 -3

-4
0 1000 2000 3000 4000 5000 6000 7000 8000
Frequency (Hz)

0.4
2
0.2
x(t)

0
1

-0.2

phi(k)
-0.4
0

-0.6

-0.8 -1
X: 440
Y: -1.571
-1
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016
time (sec)
-2

-3
380 400 420 440 460 480 500 520 540
Frequency (Hz)

What is the phase of x(n)?

Fourier Spectra
Summary
Harmonics
Fundamental 5
10

0
10

|A(k)|
-5
10

1
-10
10

0.5
-15
10
0 2000 4000 6000 8000
x(n)

0 Frequency (Hz)
X(k)=F(x(n)) 4

3
-0.5
2

1
-1

phi(k)
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0
time (sec) x 10
-4

-1

-2

-3

-4
0 2000 4000 6000 8000
Frequency (Hz)

0 Hz Fs/2 Hz FsHz


Digression (1): Phase & Phase Shift
• Positive Phase Shift
– x(n)=x(n+a)
– Phase Is Advancing
– Signal Appears Shifted To The LEFT!

• Negative Phase Shift
– x(n)=x(n-a)
– Phase Is Lagging
– Signal Appears Shifted To The RIGHT!

• This is also important for “Filter Causality”

Digression(2): The Phase Spectrum!
• The Phase Spectrum is extremely important
– It encodes the “structure” of the signal
– Two signals can have the same A(k)=A1(k)=A2(k) but
different φ1(k), φ2(k)
• This implies two completely different x1(n), x2(n)

• Practical Example
– Draw bars on an Organ
– Additive Sound Synthesis
– The importance of phase in speech enhancement

Fourier Transform
• Practical Application

Spectral Analysis &
• We will be using MATLAB

• Create / Acquire Some Signals

• Perform Forward / Inverse Fourier Transform

• Visualise and read the spectra of some basic signals

• Explore some properties of the Fourier Transform
– And their consequences to DSP

• Brief Case Study On Sound Analysis

Creating / Acquiring Some Signals
• Task 1 - For some…
– Sampling Fequency Fs (Hz)
– Signal Duration D (Sec)
– Fs=8000; D=3;
n=0:(1./Fs):D;
p=2*pi*n;
• What are the (n) and (p)?

• …generate some basic signals (for some frequency f of your choice)
– x1=sin(f*p); #Sin
– x2=cos(f*p); #Cos
– x3=square(f*p) #...
– x4=sawtooth(f*p); #...
– x5=sawtooth(f*p,0.5); #Triangle
– x6=zeros(size(t));x6(1)=1; #Delta

Creating / Acquiring Some Signals
• Task 1.1 - Other signals?
– Text Format
• MATLAB can load a list of numbers very easily through the
command “load”
• Start “Notepad”, enter a coma separated list of
numbers, save the file in your current path, load the file in
MATLAB.
– WAV Format
• Sound files can be acquired from “freesound.org” and be
loaded into your workspace through “wavread”
• Locate a .wav file, load it into matlab using
[x7, Fs]=wavread(‘path/to/some/file.wav’);

Digression:
Visualising / Sonifying Some Signals
• Visualisation
– Any time series data
• plot(x*);
– plot(n,x*)
• stem(x*);

• Sonification
– How does a time series of values sounds like?
– sound(x*,Fs)
– soundsc(x*,Fs)

The Forward / Inverse Fourier
Transform
• Select some signal (x*) from your workspace
• Perform Forward Fourier Transform
– Fx*=fft(x*);
• Perform Inverse Fourier Transform
– IFx*=ifft(Fx*)

• Please note: The “*” symbol stands for some signal
created in the previous step

• What can we observe about Fx*?
• Was IFx* the same as x?

Visualising Signal Spectra
• Task 2
– Select some Fx* from your workspace
– Derive its Amplitude spectrum
• A(Fx*) = abs(Fx*);
– Derive its Power spectrum
• P(Fx*) = abs(Fx*)^2;
– Derive its Phase Spectrum
• Phi(Fx*) = angle(Fx*);

• How do they look like?
– How can we improve their presentation?
• How does the spectrum of the “cos” function looks like?
– How is it different to the sin spectrum?
• How does the spectrum of the “Delta” function looks like?
– Why? What does this mean?
• How does the spectrum of the “square” signal looks like?

Exploring Some Properties Of The
Fourier Transform
• Shifting In Time / Frequency
– How would you shift your x(n) by 5 samples in the
time domain?
• x(n)=x(n+5); %Yes…I know…
• But how can it be done through the Fourier Transform?

• After you have shifted your signal
– What was put in the place of these 5 values?
– Why?

Brief Case Study Of
Real World Signal Spectra
200

150

100
Amplitude

50

0

-50

-100

-150
0 0.5 1 1.5 2 2.5 3
Sample (n) x 10
4

6
10

5
10

4
10
Amplitude

3
10

2
10

1
10

0
10
0 20 40 60 80 100 120
Frequency (Hz)

Human EEG: “Reading” The Spectrum
Signals courtesy of Javier Escudero Rodriguez

What Does The Spectrum “Tells” Us
2

Power Line

Amplitude
Hum / 0
Other
E.M.I. -2
0 20 40 60 80 100 120
Sample (n)
200
Actual Amplitude
Signal 0
From the + xn
Brain -200
0 0.5 1 1.5 2 2.5 3
Sample (n) x 10
4

Heart’s
Amplitude

10
Electrical
Activity 0

-10
7000 7500 8000 8500 9000
Sample (n)

Brief Case Study Of
Real World Signal Spectra
0.6

0.4

0.2
Amplitude

0

-0.2

-0.4

-0.6

-0.8
0 500 1000 1500 2000 2500 3000 3500
Sample (n)

5
10

0
10
Amplitude

-5
10

-10
10

-15
10
0 10 20 30 40 50 60 70 80 90 100
Frequency (Hz)

Pendulum Net Acceleration (on Earth): “Reading” The Spectrum

What Does The Spectrum “Tells” Us
• Data Acquisition Fault
– The MCU was not sampling the Accelerometer
while writing data to the SD card.
– Phase Modulation

• Very limited actual bandwidth

• Did not have a huge impact on the actual
application because of the nature of the signal

Questions for further experimentation
• What is the meaning of the DC component? (0Hz)
– As far as sampled signals are concerned what other condition can lead to a sampled signal of
0Hz Frequency?

• What happens when the FT is evaluated over less / more frequency points than
there are samples in a signal?

• How would the spectra change if the FS was increased / decreased?

• If some x(n) expresses calibrated “Sea Level” measurements sampled every day in
a particular location, what is the unit of the Frequency Axis (k) and what is the
interpretation of the spectrum?
– Is the Frequency Axis still reading in Hz? If not, what can it be interpreted in?

• If some x(n) expressed “Hours Of Sleep” measurements sampled every night for a
particular person over the course of 10 years, what is the unit of the Frequency
Axis (k) and what is the interpretation of the spectrum?

Appendix:
Crash Course On Complex Numbers
Real Part Re(Z)
Imaginary Part Im(Z)

The Complex Number (Z): Z a i b
You can think of it as a vector i 1

And because Z is a vector, it has “Magnitude” and “Orientation” (angle)
b
rZ a 2 b2 Z tan
a

And through these, Z can be expressed in “Polar Form”:
z rZ cos Z i sin or z rZ ei Z

The basic operations between two Complex Numbers Z,Y in each form are as follows:
Z a i b, Y c i d
z rZ e i Z , y rY e i Y
Z Y a c i b d
z y rZ rY e i Z Y
Z Y ac bd i bc ad
Z ac bd bc ad
z rZ i
i e Z Y

Y c2 d 2 c2 d 2 y rY

Appendix:
Properties Of The Fourier Transform
Please Note: In all of the following properties, capitalised letters are
the Fourier Transforms of the corresponding small case time series

Periodicit
y X k N X k
Convolutio
n xn yn X k Y k and x n y n X k Y k
i2 i2
nm nm
N N
Shifting X k m xn e and x n m X k e
Linearity a xn b yn a X k b Y k
1 k
Scaling xa n X
a a

The FFT And Spectral Analysis

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