1. The Fourier transform allows signals to be represented and analyzed in the frequency domain by decomposing them into their constituent frequencies. 2. For periodic signals, the Fourier transform results in a discrete spectrum, while for non-periodic signals it yields a continuous spectrum. 3. The continuous time Fourier transform (CTFT) represents a non-periodic signal as an integral that expresses the signal as a sum of complex exponentials oscillating at all possible frequencies.

1. 1
ECNG3025
Lecture 3
Fourier Analysis 1
The CTFT
ECNG3025 © 2014 2
Use of Frequency Domain
Analysis
Telecommunication Channel |A|
f
fc
Frequency Response
 Signal frequencies > fc are attenuated and distorted
Communication channels
Speech recognition
The sound of ‘a’ spoken
by 2 people
|A|
f
SPEECH ANALYSIS
 RECOGNITION
 IDENTIFICATION

2. 2
ECNG3025 © 2014 3
Use of Frequency Domain Analysis
 The tone controls on stereo equipment
 Low freq - Bass
 High freq – Treble
 Harmonic Analysis in power systems
 Stock market fluctuation analysis
 Audiology
 Image processing
 And more…..
ECNG3025 © 2014 4
Spectrum
 The frequency content of a signal is called
 Its spectrum
 Analogous to light
 Information about the frequency content of
a signal is called
 Spectrum analysis
 For continuous signals can be done using a
spectrum analyser

3. 3
ECNG3025 © 2014 5
Fourier Series Highlights
 A periodic signal can be expressed as an
infinite sum of orthogonal functions.
 When these functions are the cosine and
sine, the sum is called the Fourier
Series.
 The frequency of each of the sinusoidal
functions is an integer multiple of the
fundamental frequency.
Jean Baptiste Joseph Fourier
ECNG3025 © 2014 6
Fourier Series Highlights
Tp
Original Signal xp(t) First 4 Terms of Fourier Series
Sum of First 4 Terms of
Fourier Series and xp(t)
Original xp(t)
First 4 Terms of Fourier Series
Periodic signal expressed as infinite sum of sinusoids.

4. 4
ECNG3025 © 2014 7
Fourier Series Highlights
 The Fourier series has a number of
common descriptions
 See ECNG 2011 Notes
 We used the complex exponential form:




k
tjk
kectx 0
)( 



pT
tjk
p
k dtetx
T
c 0
)(
1 
where
Recall that Euler’s relationships express e
in terms of the sine and cosine functions
ECNG3025 © 2014 8
Fourier Series Highlights
 Using the complex exponential form , we
have:
etc2,1,0,where
)(
1 0

 

k
dtetx
T
c
T
tjk
k
Complex
Fourier
coefficients

5. 5
ECNG3025 © 2014 9
Complex Fourier Coefficients
kj
kk ecc 

We can express the complex coefficients in this form
The plot of |ck| versus angular frequency is called
The Amplitude Spectrum of x(t)
The plot of k versus angular frequency is called
The Phase Spectrum of x(t)
dtetx
T
c
T
tjk
k 

 0
)(
1 
The index, k, assumes integer values, so that the
Amplitude and Phase spectra appear at discrete
frequencies, k0.
ECNG3025 © 2014 10
Example
 Recall the following example:
The Fourier series
coefficients for this
function were shown
to be:
2
a
2
a
2
T 2
T
f(t)
t



















T
a
k
T
a
k
ck


sin

6. 6
-3 -2 -1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
A sinc spectrum
A plot of the
magnitude of
ck (normalised ie
magnitude = 1)



















T
a
k
T
a
k
ck


sin
ECNG3025 11© 2014
Line spacing is a/T and
the zero crossing points
are at k= mT/a, m = - to
 (integer values)
ECNG3025 © 2014 12
Observations
-3 -2 -1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
A sinc spectrum
 Periodic signal
has a discrete
spectrum
 Intuitive
 Think of a
sinusoid
 What is its
spectrum?
 What happens as
period increases?

7. 7
ECNG3025 © 2014 13
Observations
As T increases
Line spaces decrease
Spectral lines get closer together
Zero crossing points move further apart
Spectrum flattens
Line spacing is a/T and the zero
crossing points are at k= mT/a, m = -
to  (integer values)
ECNG3025 © 2014 14
Fourier Series Limits
 FS applies to periodic signals
 How can it represent non-periodic
waveforms such as speech?
 For example, a single spoken word is
a non-periodic waveform because it
does not repeat itself.
What is the period of a non-periodic waveform?

8. 8
ECNG3025 © 2014 15
Non-periodic signal
 Solution is to treat it as a periodic
waveform with an infinite period.
 If we assume that T tends towards
infinity, then we can produce
equations for non-periodic signals
 Remember our previous
observations as T increased
 What is the period of an
aperiodic signal?
ECNG3025 © 2014 16
Non-periodic signal
As the fundamental frequency
approaches 0 a number of things
happen





2
)(
0
0
0
0





f
Cc
k
T
k



0
0
)(
1
0
T
tjk
k dtetf
T
c 



T
tj
dtetfC 


 )(
2
)(
Thus
Evolves to

9. 9
ECNG3025 © 2014 17
Non-periodic signal
 The harmonic designator k is dropped from the
equations
 We now have an infinite number of k values
 ie k is now a continuous 
 C() is therefore a continuous variable.
But T is now infinite!
Therefore: 

 dtetfC tj


 )(
2
)(



T
tj
dtetfC 


 )(
2
)(
ECNG3025 © 2014 18
Observations
 Periodic signal has a discrete spectrum
 From before
 Non-periodic signal has a continuous
spectrum

10. 10
ECNG3025 © 2014 19
Fourier Transform
 With a bit of juggling, normalising and
substitution:




 dtetxX tj
 )()(

 dtetfC tj


 )(
2
)(
Called the Continuous Time Fourier Transform
(CTFT) of x(t)
ECNG3025 © 2014 20
Fourier Transform
 Of course there must be an Inverse
Fourier Transform
x t X e dj t
( ) ( )



1
2
 

11. 11
ECNG3025 © 2014 21
Comments
 The Fourier Transform allows you to
compute the frequency domain
representation of a signal from the time
domain signal
 The inverse Fourier Transform allows
you to compute the time domain
representation from the frequency
domain signal
ECNG3025 © 2014 22
Comments
 The CTFT and the inverse CTFT are
known as the Fourier Transform
pair
 They apply to non periodic signals
 The application to periodic signals
will be covered in a later lecture.

12. 12
ECNG3025 © 2014 23
Continuous Time Fourier Transform
(CTFT)
 Standard notation




 dtetxjXXtx tj
 )()(or)()}({F


 
deXtxX tj




 )(
2
1
)()}({1
F
•Note the similarity in form to the Laplace Transform.
•Many of its properties are likewise similar.
•There are several important exceptions.