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Roadmap
Signal System
InputSignal
OutputSignal
characteristics
Given input and systeminformation, solve for
the response
Solvingdifferentialequation
Calculateconvolution
LinearTime-invariant
InvertibleCausal
MemoryBound
classification
transformation
Special properties(even/odd, periodic)
Math. description
Energy & power
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ECE310 – Lecture 13
Fourier Series - CTFS02/26/01
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Why a New Domain? It is often much easier to analyze signals and
systems when they are represented in the frequency domain
The entire subject of signals & systems consists primarily the following concepts: Writing signals as functions of frequency Looking at how systems respond to inputs of
different frequencies Developing tools for switching between time-domain
and frequency-domain representations Learning how to determine which domain is best
suited for a particular problem
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Scenario for Selected Legs
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5two legs selected
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6two legs selected
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Leg1 Analysis (AAV from N to W)
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Leg1 Analysis (DW from W to E)
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Fourier Series & Fourier Transform They both represent signal in the
form of a linear combination of complex sinusoids
FS can only represent periodic signals for all time
FT can represent both periodic and aperiodic signals for all time
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Limitations of FS Dirichlet conditions
The signal must be absolutely integrable over the time, t0 < t < t0 + TF
The signal must have a finite number of maxima and minima in the time, t0 < t < t0 + TF
The signal must have a finite number of discontinuities, all of finite size in the time, t0 < t < t0 + TF
FTt
t
dttx0
0
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The Fourier Series of x(t) over TF Fourier series (xF) represents any function over a finite
interval TF
Outside TF, xF repeats itself periodically with period TF. xF is one period of a periodic function which matches the
function x(t) over the interval TF. If x(t) is periodic with period = T0
if TF=nT0, then the Fourier series representation (TSR) equals to x(t) everywhere;
if TF != nT0, then FSR equals to s(t) only in the time period TF, not anywhere else.
-k
2
00
kX
,
tkfjF
FF
Fetx
Tttttxtx
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Examples
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The Fourier Series
FF
F
FF
F
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t
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n
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sF
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c
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, :formcomplex
2sin2 ,2cos2
,10
2sin2cos0 :form rictrigonomet
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kjXkXkXXX
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Some Parameters TF is the interval of signal x(t) over which
the Fourier series represents fF = 1/TF is the fundamental frequency of
the Fourier series representation n is called the “harmonic number”
2fF is the second harmonic of the fundamental frequency fF.
The Fourier series representation is always periodic and is linear combinations of sinusoids at fF and its harmonics.
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Interpretation The FS coefficient tells us how
much of a sinusoid at the nth harmonic of fF are in the signal x(t)
In another word, how much of one signal is contained within another signal
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Calculation of FS Sinusoidal signal (ex 1,2) Non-sinusoidal signal (ex 3) Periodic signal over a non-integer
number of periods (ex 4) Periodic signal over an integer number
of periods (ex 4) Even and odd periodic signals (ex 5) Random signal (no known mathematical
description) (ex 6)
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Example 1 – Finite Nonzero Coef x(t) = 2cos(400t) over 0<t<10ms Band-limited signals Analytically
Trignometric form and Complex form Graphically (p6-10~6-12)
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Example 2 – Finite Nonzero Coef x(t) = 0.5 - 0.75cos(20t) + 0.5sin(30t)
over -100ms < t < 100ms Band-limited signals (p6-14,6-15)
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Example 3 – Infinite Nonzero Coef x(t) = rect(2t)*comb(t) over –0.5<t<0.5 When we have infinite nonzero coefficients, we
tend to use magnitude and phase of the CTFS versus harmonic to present the CTFS (p6-19)
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Example 4 – Periodic Signal x(t) = 2cos(400pt) over 0<t<7.5ms Over a non-integer number of period p6-20
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Example 5 – Periodic Even/Odd Signals For a periodic even function, X[k] must be real
and Xs[k] must be zero for all k For a periodic odd function, X[k] must be
imaginary and Xc[k] must be zero for all k
2
0
2
2
2
2
22
2cos22sin2cos1
11 0
0
FF
F
F
F
F
F
F
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dttkftjxtkftxT
dtetxT
dtetxT
kX
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Example 6 – Random Signal Is it necessary to know the mathematical
description of the signal in order to derive its CTFS?
No Graphically (p6-24, 6-25)
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Convergence of the CTFS For continuous signal
As N increases, CTFS approaches x(t) in that interval
For signals with discontinuities As N increases, there is an overshoot or
ripple near the discontinuities which does not decrease – Gibbs phenomenon
When N goes to infinity, the height of the overshoot is constant but its width approaches zero, which does not contribute to the average power
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Example P6-35, 6-36
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Exercises 6.1.2
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Response of LTI System with Periodic Excitation Represent the periodic excitation using complex
CTFS Since it’s an LTI system, the response can be
found by finding the response to each complex sinusoid
Example: RC lowpass circuit
Magnitude and phase of Vout[k]/Vin[k] (p6-53)
k
tkfjininoutout ekVtvtvtRCv 02'
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Properties of CTFS
Linearity Time shifting Time reversal Time scaling Time differentiation Time integration Time multiplication Frequency shifting Conjugation Parseval’s theorem
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2
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Parseval’s Theorem Only if the signal is periodic The average power of a periodic
signal is the sum of the average powers in its harmonic components
kT
kXdttxT
22
0 0
1
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Summary - CTFS The essence of CTFS The limitation of CTFS The calculation of CTFS The convergence of CTFS
Continuous signals Signals with discontinuities – Gibbs phenomena
Properties of CTFS Especially Parseval’s theorem
Application in LTI system
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Test 2ECE310 Test 2 Statistics
0123456
90-100
80-89 70-79 60-69 <60
Avg. 69.9, [42, 89]
Nr. o
f Stu
dent
s
Series1