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o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• CT FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• DTFS & LTI systems
o Summary
ELEC264: Signals And Systems
Topic 3: Fourier Series (FS)
Aishy Amer
Concordia University
Electrical and Computer Engineering
Figures and examples in these course slides are taken from the following sources:
•A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997
•M.J. Roberts, Signals and Systems, McGraw Hill, 2004
•J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003
2
Signal Representation
Time-domain representation x(t)
Waveform versus time
Periodic / non-periodic
Signal value: amplitude
Frequency-domain representation
X(f)
Signal versus frequency
Signal value: contribution of a
frequency
Periodic / non-periodic
Concepts of
frequency, bandwidth,
filtering
** A Frequency (Fourier) representation is unique, i.e., no two same signals in time domain give the same function in frequency domain
3
Signal Representation
Time Representation:
Plot of the signal value vs. time
• Sound amplitude, temperature reading, stock price, ..
Mathematical representation: x(t)
• x: signal value
• t: independent variable
Frequency Representation :
Plot of the magnitude value vs. frequency
• Sound changes per second, …
Mathematical representation: X(f)
• X: magnitude value
• f: independent variable
4
Signal Representation: Sinusoidal Signals
angle Phase :
22
frequency Angular : econd)(radians/s
Period lFundementa: (seconds)
Frequency lFundementa :cond)(cycles/se
at t signal theof value: x(t)
instant time: t
Signal : x
Amplitude Signal :A
)cos()(
000
0
0
0
0
Tf
T
f
tAtx
Src: Wikipedia
f0 = 1000Hz
f0 = 2000Hz
Sinusoidal signals: important because they can be used to synthesize any signal
5
Transforming Signals
Often we transform signals x(t) in a different domain
Frequency analysis allows us to view signals in the frequency domain
In the frequency domain we examine which frequencies are present in the signal
Frequency domain techniques reveal things about the signal that are difficult to see otherwise in the time domain
6
Transforming signals
Frequency: the number of “changes” per unit time (cycles/second)
“amount of change”
Jean Baptiste Joseph Fourier (1768 – 1830):
French mathematician and physicist
Invented the Fourier series and their applications to physic problems (e.g.,
heat transfer)
Fourier analysis: x(t) <=> X(f)
decompose a signal x(t) into its frequency content
• X(f) of a musical chord: the amplitudes of the individual notes that make the chord up
Spectrum: plot of X(f)
9
Illustration of Frequency
x1(t) and x2(t) look similar
Frequency (or spectrum) analyzer reveals the difference:
X2(f) shows two large “spikes” but not X1(f)
x2(t) contains a sinusoidal signal that causes these “spikes”
10
Concept of frequency: Periodic Signals
A CT x(t) signal is periodic if there is a positive value T for which x(t)=x(t+T)
Period T of x(t) : The interval on which x(t) repeats
Fundamental period T0 : the smallest positive value of T for which the equation above holds T0 the smallest such repetition interval T0 =1/f0
x(t) : a sum of sinusoidal signals of different frequencies fk=kf0
Harmonic frequencies of x(t): kf0 , k is integer, )2cos()( 0tkfatx
k
k
12
Concept of frequency:
Constant Signal x(t)=A
•Let the fundamental frequency be zero,
i.e., constant signal (d.c) has zero
rate of oscillation
• If x(t) is periodic with period T for any
positive value of T
fundamental period is undefined0
0
1
fT
f0=0
13
Concept of frequency
A constant signal: only zero frequency component (DC component)
A sinusoid : Contain only a single frequency component
Arbitrary signal x(t): Does not have a unique frequency
Contains the fundamental frequency f0 and harmonics kf0
It can be decomposed into many sinusoidal signals with different frequencies
Slowly varying : contain low frequency only
Fast varying : contain very high frequency
Sharp transition : contain from low to high frequency
16
Fourier representation of
signals
The study of signals using sinusoidal representations is termed Fourier analysis, after Joseph Fourier (1768-1830)
The development of Fourier analysis has a long history involving a great many individuals and the investigation of many different physical phenomena, such as the motion of a vibrating string, the phenomenon of heat propagation and diffusion
Fourier methods have widespread application beyond signals and systems, being used in every branch of engineering and science
The theory of integration, point-set topology, and eigenfunction expansions are just a few examples of topics in mathematics that have their roots in the analysis of Fourier series and integrals
17
Fourier representation of
signals
Spectrum of x(t): the plot of the magnitudes and phases of different
frequency components
Fourier analysis: find spectrum for signals
Spectrum (Fourier representation) is unique, i.e., no two same
signals in time domain give the same function in frequency domain
Bandwidth of x(t): the spread of the frequency components with
significant energy existing in a signal
18
Fourier representation of
signals Any periodic signal can be approximated by a sum of
many sinusoids at harmonic frequencies of the signal
(kf0 ) with appropriate amplitude and phase
The more harmonic components are added, the more accurate the approximation becomes
Instead of using sinusoidal signals, mathematically, we can use the complex exponential functions with both positive and negative harmonic frequencies
tjte oo
tj o sincos
19
Fourier representation:
Introductory example 1
Componentsconstant -NonComponent Constant )(
.4.4
.7.710)(
)2
500cos(8)3
200cos(1410)(
2502225022
1002310023
tx
eeee
eeeetx
tttx
tjj
tjj
tjj
tjj
20
Fourier representation:
Introductory example 2
a Square as a Sum of Sinusoids
1
The Fourier series analysis:
Period T
22
Fourier representation: Example 2
Square as a Sum of Sinusoids
Each line corresponds to
one harmonic frequency.
The line magnitude
(height) indicates the
contribution of that
frequency to the signal
The line magnitude drops
exponentially, which is
not very fast. The very
sharp transition in square
waves calls for very high
frequency sinusoids to
synthesize
1
23
Frequency Representation:
Applications
Shows the frequency composition of the signal
Change the magnitude of any frequency
component arbitrarily by a filtering operation
Lowpass -> smoothing, noise removal
Highpass -> edge/transition detection
High emphasis -> edge enhancement
Shift the central frequency by modulation
A core technique for communication, which uses
modulation to multiplex many signals into a single
composite signal, to be carried over the same
physical medium
24
Frequency Representation:
Applications
Typical Filtering applied to x(t):
Lowpass -> smoothing, noise removal
Highpass -> edge/transition detection
Bandpass -> Retain only a certain frequency range
25
Frequency Representation: Applications
Speech signals
Each signal is a sum of sinusoidal signals
They are periodic within each short time interval
Period depends on the vowel being spoken
Speech has a frequency span up to 4 KHz
Audio (e.g., music) has a much wider spectrum, up to 22KHz
26
Frequency Representation: Applications
Music signals
Music notes are essentially sinusoids at different frequencies
A sum of sinusoidal signals
They contain both slowly varying and fast varying components wide
bandwidth
They have more periodic structure than speech signals
Structure depends on the note being played
Music has a much wide spectrum, up to 22KHz
27
Fourier representation of signals:
Types
A Fourier representation is unique, i.e., no two same signals in time domain give the same function in frequency domain
28
Fourier representation :
Notation
Variable Period Continuous
Frequency
Discrete
Frequency
DT x[n] n N k
CT x(t) t T k
Nkk /2
Tkk /2
)(
• DT-FS: Discrete in time; Periodic in time; Discrete in Frequency; Periodic in Frequency
• CT-FS: Continuous in time; Periodic in time; Discrete in Frequency; Aperiodic in Frequency
• DT-FT: Discrete in time; Aperiodic in time; Continous in Frequency; Periodic in Frequency
• CT-FT: Continuous in time; Aperiodic in time; Continous in Frequency; Aperiodic in Frequency
32
Complex Numbers and Sinusoids
)sin(2)cos(2
:ForumlaEuler
phase) (initialshift phase theis
)sin()cos(||)(
:Signal lExponentiaComplex
)()(:Note
:ConjugateComplex
sincos :tionrepresentaPolar
tan is z of Phase
|| is z of Magnitude
:tionrepresentaCartesian
:NumberComplex
00
)(
***
1
22
0
tjeetee
tzjtzeztx
realarezzandzzjbaz
zjzezz
a
bz
baz
jbaz
tjtjtjtj
tj
j
34
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
35
Fourier Series: Periodic Signalsx(t) is periodic with period T
2 spectrum); (symmetric :signals realFor
numbercomplex a general,in is,
,...2,1,0;)(1
][
) transform(forward Analysis SeriesFourier
complex) and realboth for sided, (double
only) signal realfor sided, (single )cos()(
transform)(inverser Synthesis SeriesFourier
*
0
1
00
0
0
zzR(z)aaaa
a
kdtetxT
akX
ea
ktaatx
kkkk
k
T
ktj
k
ktj
k
k
k
kk
37
Fourier series: CT periodic
signals
Consider the following continuous-time complex
exponentials:
T0 is the period of all of these exponentials and it can
be easily verified that the fundamental period is equal
to
Any linear combination of is also periodic with
period T0
38
Fourier series: CT periodic
signals
fundamental components or
the first harmonic components
The corresponding fundamental frequency
is ω0
Fourier series representation of a
periodic signal x(t):
40
Fourier series: CT periodic signal
tscoefficien current) ng(alternati AC thecalled are
tcoefficien current)(direct DC thecalled is 0
ka
a
44
CT Fourier Series: Summary
T
T
0
0
0
P-CTDF
)( :SeriesFourier Inverse CT
DFP-CT
)(1
; )(1
:SeriesFourier CT
0
0
k
tjk
k
TT
tjk
k
eatx
dttxT
adtetxT
a
o CT FS: Continuous and periodic signal in time
Discrete & aperiodic in frequency
46
Fourier series: CT periodic
signal: Example 1 Note that the Fourier
coefficients are complex numbers, in general
Thus one should use two figures to demonstrate them completely:
real and imaginary parts or
magnitude and angle
48
Fourier series of a CT
periodic signal: Example 2
The Fourier series coefficients are shown next for T=4T1 and T=16T1
Note that the Fourier series coefficients for this particular example are real
51
Fourier series of a CT
periodic signal: Example 3
Problem: Find the Fourier series coefficients of the periodic continuous signal: and is the period of the signal. Plot the spectrum (magnitude and phase) of x(t).
Solution: Using the definition of Fourier coefficients for a periodic continuous signal, the coefficients are:
which would be simplified after some manipulations to:
What is the spectrum of x(t-4)?
Solution: Use FS properties
30),3
cos()( tttx
3T
3
0
3
2333
0
3
2
023
1)
3cos(
3
1)(
10 dte
eedtetdtetx
Ta
tjktjtj
tjkT
tjk
k
)41(
42k
kjak
52
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
60
Effect of Signal Symmetry: Inverse
CT Fourier Series Example:
The CT Fourier Series representation of the above cosine signal X[k] is
is odd
The discontinuities make X[k] have significant higher harmonic content
)(txF
)(txF
)(txF
61
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
62
Properties of CT Fourier
series
The properties are useful in determining the Fourier
series or inverse Fourier series
They help to represent a given signal in term of
operations (e.g., convolution, differentiation, shift) on
another signal for which the Fourier series is known
Operations on {x(t)} Operations on {X[k]}
Help find analytical solutions to Fourier Series
problems of complex signals
Example:
tionmultiplicaanddelaytuatyFS t })5()({
63
Properties of CT Fourier
series
Let x(t): have a fundamental period T0x
Let y(t): have a fundamental period T0y
Let X[k]=ak and Y[k]=bk
In the Fourier series properties which follow:
Assume the two fundamental periods are the same
T= T0x =T0y (unless otherwise stated)
The following properties can easily been shown using the
equation of the Fourier series
79
Steps for computing Fourier
series of x(t)
1. Identify fundamental period of x(t)
2. Write down equation for x(t)
3. Observe if the signal has any symmetry (e.g., even or odd)
4. Observe any trigonometric relation in x(t)
5. Find the FS using either using its equation or by inspect from x(t)
6. Observe if you can use the FS properties to simplify the solution
80
Numerical Calculation of CT
Fourier Series1. The original analog signal x(t) is digitized to x[n]
2. Then a Fast Fourier Transform (FFT) algorithm is
applied, which yields samples of the FT at equally
spaced intervals
3. For a signal that is very long, e.g. a speech signal or
a music piece, spectrogram is used
Fourier transforms over successive overlapping short
intervals
A spectrogram is a time-varying spectral representation
(image-like) that shows how the spectral density of a
signal varies with time
84
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
87
Convergence of the CT Fourier series
The Fourier series representation of a periodic signal x(t)
converges to x(t) if the Dirichlet conditions are satisfied
Three Dirichlet conditions are as follows:
1. Over any period, x(t) must be absolutely integrable.
For example, the following signal does not satisfy this
condition
k
tjk
keatx 0)(
88
Convergence of the CT
Fourier series
2. x(t) must have a finite number of maxima and minima in
one period
For example, the following signal meets Condition 1, but not
Condition 2
89
Convergence of the CT
Fourier series
3. x(t) must have a finite number of discontinuities, all of
finite size, in one period
For example, the following signal violates Condition 3
90
Convergence of the CT
Fourier series
Every continuous periodic signal has an FS representation
Many not continuous signals has an FS representation
If a signal x(t) satisfies the Dirichlet conditions and is not continuous, then the Fourier series converges to the midpoint of the left and right limits of x(t) at each discontinuity
Almost all physical periodic signals encountered in engineering practice, including all of the signals with which we will be concerned, satisfy the Dirichlet conditions
94
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• CT FS & LTI systems: what is y(t) to a periodic x(t)?
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
95
CT FS & LTI systems
Convolution represents:
The input as a linear combination of impulses
The response as a linear combination of impulse responses
Fourier Series represents:
a periodic signal as a linear combination of complex
sinusoids
dtxtx )()()(
dthxty )()()(
022)( fkfeatx kk
k
tj
kk
96
CT FS & LTI systems
If x(t) can be expressed as a sum of complex sinusoids
the response can be expressed as the sum of responses to
complex sinusoids
kk
k
tj
k febty k 2)(
97
CT FS & LTI systems: eigen
functions
Eigen-function of a linear operator S:
a non-zero function that returns from the operator
exactly as-is except for a multiplicator (or a scaling
factor)
Eigen-function of a system S:
characteristic function of S
function-eigen :)(
vector)null-non (a value-eigen :
)()}({:)(function aon applied System
tx
txtxStx
98
CT FS & LTI systems: eigen
functions
Eigen-function of an LTI system is the complex
exponential
Any LTI system S excited by a complex sinusoid
responds with another complex sinusoid of the same
frequency, but generally a different amplitude and
phase
The eigen-values are either real or, if complex, occur
in complex conjugate pairs
tj ke
dtethjHejHtyetx tjtjtj kk
)()( where)()()( If
99
CT FS & LTI systems
CTFS is defined for periodic signals
ejw0t are eigen-functions of CT LTI systems:
y(t)=H(jw0) ejw0
H(jw) is the eigen-value of the LTI system associated with
the eigen-function ejwt
H(jw) is the Frequency representation of the impulse
response h(t)
H(jw) is the frequency response of the LTI system
100
CT FS & LTI systems
)( ;()(
:is x(t), this toh(t) with system LTIan of response they(t),
lsexponentiacomplex of sum a is x(t)
)(
a FS with x(t)signal periodic aConsider
)()(
0)
k
0
0
0
jkHabejkHaty
eatx
dtethjH
kk
tjk
k
k
tjk
k
k
tj
101
CT FS & LTI systems:
Example 1
.....;1
)
0
)
3
3
33221100
3
3
10
0
0
0
0
(Let
)1/(1)(
()(
3/1;2/1;4/1;1;2with
)( );()(
bb
kk
j
tjk
k
k
tjk
k
k
t
jkHab
jdeejH
ejkHaty
aaaaaaa
eatxtueth
102
CT FS & LTI systems:
Generalization with
Let a continuous-time LTI system be excited by a complex exponential of the form,
The response is the convolution of the excitation with the impulse response or
The quantity
is the Laplace transform (or s-transform) of the impulse response h(t)
jseetx tjst ;)( )(
104
CT FS & LTI systems: Generalization
H(s) is called “System function” or “Transfer function”
H(s) is a complex constant whose value depends on s and is given by:
Complex exponential est are eigen-functions of CT LTI systems
H(s) is the eigen-value associated with the eigen-function est
105
Response to CT Complex
Exponential: Example 1
Consider an LTI system whose input x(t) and
output y(t) are related by a time shift as follows:
Find the output of the system to the following
inputs:
109
Outlineo Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
110
Periodic DT Signals
A DT signal is periodic with period
where is a positive integer if
The fundamental period of is the
smallest positive value of for which the
equation holds
Example:
is periodic with fundamental period
118
DT Fourier Series: Summary
NN
1
0
NN
1
0
P-DTP-DF
][1
][ SeriesFourier DT Inverse
P-DFP-DT
][][ SeriesFourier DT
0
0
N
k
knj
N
n
knj
k
ekXN
nx
enxkXa
o Discrete and periodic signal in time Discrete &
periodic FS signal in frequency
123
Example 2: periodic impulse train
DTFS of a periodic impulse train
Since the period of the signal is N
We can represent the signal with these DTFS
coefficients as
else
rNnrNnnx
r 0
1][~
1][][~~ 0/21
0
/21
0
/2
kNjN
n
knNjN
n
knNj eenenxkX
1
0
/21][~
N
k
knNj
r
eN
rNnnx
124
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
128
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• DT FS & LTI systems
o Summary
129
DT FS & LTI systems
What is the response of a LTI system to a periodic x[n]?
DT systems:
rzreznjn ||
;0
130
DT FS & LTI systems: eigen functions
H(z) is “system function”, a complex constant whose value depends on z and is given by:
H(z) is z (frequency+) representation of h(n), the impulse response
Complex exponential zn are eigen-functions of DT LTI systems
H(z) is the eigen-value associated with the eigenfunction zn
132
Outline
o Introduction to frequency analysis of signals
o CT FS
• Fourier series of CT periodic signals
• Signal Symmetry and CT Fourier Series
• Properties of CT Fourier series
• Convergence of the CT Fourier series
• FS & LTI systems
o DT FS
• Fourier Series of DT periodic signals
• Properties of DT Fourier series
• FS & LTI systems
o Summary
134
Fourier series: summary
Sinusoid signals:
Can determine the period, frequency, magnitude and
phase of a sinusoid signal from a given formula or
plot
Fourier series for periodic signals
Understand the meaning of Fourier series
representation
Can calculate the Fourier series coefficients for
simple signals
Can sketch the line spectrum from the Fourier series
coefficients
135
Fourier Series: Summary
T
T
0
P-CTDT ; )( :SeriesFourier Inverse CT
DTP-CT ; )(1
:SeriesFourier CT
0
0
k
tjk
k
T
tjk
k
eatx
dtetxT
a
NN
1
0
NN
1
0
P-DTP-DT ; ][1
][ SeriesFourier DT Inverse
P-DTP-DT ; ][][ SeriesFourier DT
0
0
N
k
knj
N
n
knj
ekXN
nx
enxkX
o CT FS: Continuous and periodic signal in time
Discrete & aperiodic in frequency
o DT FS: Discrete and periodic signal in time
Discrete & periodic in frequency
136
Fourier series: summary
Steps to compute the FS
1. Identify fundamental period of x(t)
2. Write down equation for x(t)
3. Observe if the signal has any symmetry (e.g., even or odd)
4. Observe any trigonometric relation in x(t)
5. Find the FS using either using its equation or by inspect from x(t)
6. Observe if you can use the FS properties to simplify the solution