week1 intro system fourier(1)
DESCRIPTION
Week1 Intro System Fourier(1)TRANSCRIPT
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Course Introduction and Review of LinearSystems and Fourier Transform
ELEC3540 - Analog and Digital Communications
Dr. Duy Ngo
School of Electrical Engineering and Computer ScienceUniversity of Newcastle
Feb. 24, 2015
Course Outline, Chapter 1 (1.2, 1.3, 1.4, 1.5, 1.6),Appendices A, B of textbook
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Course Admin Teaching Staff
Lecturer: Dr Duy Ngo Office: EAG19 - EA Building Email: [email protected] Phone: (02) 4921 8947 Consultation: Wednesday 10.00 AM - 12.00 NOON
Lab Demonstrator: Mr. Behzad Asadi Office: ES230 - ES Building Email: [email protected]
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Course Admin Contents
Textbook Martin S. Roden, Analog and Digital CommunicationSystems, 5th edition, 2003. ISBN: 9780964696976.
First half of Semester: Analog Communication Review of Fourier transform, linear systems, random
processes Analog baseband transmission, analog AM, FM, PM
Second half of Semester: Digital Communication Baseband digital transmission, digital AM, FM, PM Multiplexing techniques: TDM, FDM, CDM
References to Chapters/Sections of textbook will be givenat the beginning of lecture slides
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Course Admin Assessments
Assignments (15%) 2 assignment, 7.5% each Due in Weeks 6 and 12 Hard copies; Marked by me
Labs (20%) 5 labs, 4% each Report to be submitted 1 week after your assigned time slot Hard copies; Marked by Lab Demonstrator
Mid-term quiz (15%) To be conducted in lecture time, Week 7; Marked by me Closed-book Review lecture in Week 7
Final exam (50%) Closed-book; Marked by me Review lecture in Week 13
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Course Admin My Advice and Expectations
Lectures: Buy and do read textbook Regular attendance, take notes, no ringing phone, no
private chatting, questions are always welcome (dointerrupt!)
Solutions to example exercises are only given in thelectures
Tutorials: Regular attendance Problem-solving skills needed for assignments, quiz and
exam Solutions to tutorial problems are only given in the tutorials Only solutions to unsolved questions posted on the
Blackboard site
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Course Admin My Advice and Expectations
Labs: Complete online Lab Induction and Access Quiz to get
access Contact Lab Manager (Mr. Peter Turner) if you have any
question regarding access Attend your scheduled sessions
Quiz and Exam: will be based on textbook, lectures notes,tutorial problems
Adverse circumstances: email (from UniMail), see me Check Blackboard site and Email regularly for
announcements Have fun!
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Overview of a Communication System
Transmitter: change raw information into a format that ismatched to channel characteristics
Channel: introduce changes to the signals Linear channels with transfer function: H(f ) = A(f )ej(f ) Amplitude distortion: A(f ) is not a constant Phase distortion: (f ) is not linear in f
Receiver: detect and process signals from channel
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Expanded Block Digram of a Communication System
This course studies the transmission techniques used toreliably and efficiently transmit and receive signals over achannel with uncertainty
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Types of Signals
Analog signals: Waveform takes on a continuum of values for any time
within a range of times E.g., human speech waveform
Analog sampled (or discrete) signals: Sample analog signals at discrete time points Signals can still take an infinite number of values
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Types of Signals
Digital signals: A form of discrete signals Signals can only take a finite number of values E.g., pressing keys at ATM, using computer keyboard, A/D
conversion Advantages of digital over analog transmission:
Improved noise immunity Processing simplicity
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Nyquist Sampling Theorem Suppose a signal s(t) is defined over bandwidth [0, fm]
We sample s(t) at all t = nTs where n is integer to obtainss(t)
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Nyquist Sampling Theorem We can reconstruct s(t) from ss(t) precisely and
accurately if
Ts 2fm (2)
That is, sampling rate must be more than twice the highestfrequency of the signal for a perfect reconstruction
2fm is called Nyquist sampling rate12 / 44
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A/D Conversion Sampling an analog signal gives a discrete signal, which
still takes a continuum of values. Analog-to-digital (A/D) conversion: Code the list of
infinite numbers into a set of finite code words. Uniform quantization:
Divide the continuum of functional values into uniform-widthregions
Assign an integer code to each region n-bit quantizer contains 2n regions.
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D/A Conversion
Digital-to-analog (D/A) conversion An integer code is associated with a region Usually, the analog value is taken as the centre point of the
region.
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Communication Channels
Copper wires, coaxial cables, fiber optic cables, air All channels have a maximum frequency beyond which
input signals are almost entirely attenuated. As frequencies increase, parallel capacitance short out
the signal Series inductance open circuits
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Communication Channels
Bandpass channel: Low-frequency cutoff Lowpass channel: No lower-frequency cutoff (i.e., dc
coupled) Narrowband: f < 300Hz Voiceband: 300Hz < f < 4kHz Wideband: f > 4kHz The higher the frequency, the more the transmission takes
on the characteristics of light 16 / 44
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Classification of Signals
Signal is any function that carries some information. Continuous-time signal x(t) when t takes real numbers.
E.g., x(t) = A cos(2pif0t + ). Discrete-time signal x [n] when n takes integral values.
E.g., x [n] = A cos(2pif0n + ). Real signals take values in the set of real numbers:x(t) R.
Complex signals take values in the set of complexnumbers: x(t) C.
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Classification of Signals
Deterministic signals: x(t) is a real or complex number atany time instant t .
Random signals: x(t) is random variable at any timeinstant t , defined by a probability density function.
Periodic signals: Continuous: x(t + kT0) = s(t), t ,k Z and period T0 > 0 Discrete: x [n + kN0] = s[n], n, k Z and period N0 Z+.
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Some Important Signals
Sinusoidal: x(t) = A cos(2pif0t + ) with amplitude A,frequency f0 and phase .
Complex exponential: x(t) = Aej(2pif0t+). Impulse (or Delta): Precisely, (t) is not a function, but a
distribution or a generalized function.
(t) ={
0, t 6= 0,, t = 0. (3)
(t)(t)dt = (0). (4)
Sinc:
sinc(t) ={
sin(pit)/(pit), t 6= 0,1, t = 0.
(5)
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Energy and Power
For any signal x(t), the energy content or energy Ex is:
Ex = |x(t)|2dt = lim
T
T/2T/2
|x(t)|2dt . (6)
The power content or power Px is:
Px = limT
1T
T/2T/2
|x(t)|2dt . (7)
For real signals, |x(t)|2 = x2(t). A signal x(t) is an energy-type signal iff Ex
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Example Consider signal x(t) = A cos(2pif0t + ) x(t) is not energy-type:
Ex = limT
T/2T/2
A2 cos2(2pif0t + )dt = (8)
The power of x(t):
Px = limT
1T
T/2T/2
A2 cos2(2pif0t + )dt
= limT
1T
T/2T/2
A2
2[1 + cos(4pif0t + 2)]dt
= limT
[A2T2T
+
[A2
8pif0Tsin(4pif0t + 2)
]]=
A2
2
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Exercise
Suppose x(t) is periodic with period T0 Show that x(t) is not an energy-type signal.
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Exercise Suppose x(t) is periodic with period T0 Show that the power content of x(t) is equal to the average
power in one period.
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Systems
System is an interconnection of various elements andbehaves as a whole
A set of rules that associate an output time function toevery input time function
Example: y(t) = ax(t) + bx2(t) defines a system Communication viewpoint:
A system is excited by an input signal x(t) and, as a resultof such an excitation, produces an output signal y(t).
Output y(t) must be uniquely defined for any legitimateinput x(t) (i.e., must not be a one-to-many mapping!).
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Classification of Systems
Continuous-time systems: Continuous-time input x(t)and continuous-time output y(t)
E.g., differentiator:
y(t) =ddtx(t) (10)
Discrete-time systems: Discrete-time input x [n] anddiscrete-time output y [n]
E.g., discrete-time differentiator:
y [n] = x [n] x [n 1] (11)
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Classification of Systems
Linear systems: Superposition property is satisfied.
T [x1(t) + x2(t)] = T [x1(t)] + T [x2(t)], (12)
where x1(t), x2(t) are two legitimate inputs and , aretwo scalars.
E.g., y(t) = x2(t) is nonlinear because its response to2x(t) is:
T [2x(t)] = 4x2(t) 6= 2x2(t) = 2T [x(t)] (13)
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Classification of Systems
Time-invariant systems: The response due to an input isindependent upon the actual time of occurrence of theinput.
T [x(t t0)] = y(t t0). (14)
E.g., y(t) = x(t) cos(2pif0t) is a time-varying systembecause the response to x(t t0) is
x(t t0) cos(2pif0t) 6= y(t t0) (15)
Causal systems: For physical realizability of systems
y(t0) = T [x(t) : t t0]. (16)
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LTI Systems
We will consider linear time-invariant systems (LTI). For this class of systems, the input-output relation is
particularly simple and can be expressed in terms of theconvolution integral.
Instead of requiring that we know the response due toevery possible input, we need only know the output h(t) ofone test input (t).
h(t) = T [(t)]. (17)
For an LTI system, the impulse response h(t) completelycharacterizes the system. It is all the information we needto describe the system behavior.
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Analysis of LTI Systems in Time Domain Response to x(t) is the convolution of x(t) and h(t):
y(t) = T [x(t)] = T[
x()(t )d]
=
x()T [(t )]d
=
x()h(t )d
= x(t)Fh(t). (18) Given an LTI system, the time-domain analysis involves:
Find impulse response h(t) by applying an impulse (t) atthe input.
To obtain output y(t) to other input x(t), perform theconvolution integral:
y(t) = x(t)Fh(t) =
x()h(t )d. (19)
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Motivation of Fourier Series
Analyzing LTI systems requires direct application ofconvolution integral. Although straightforward, evaluatingthe integral is not always easy.
Alternative approach: Expand the input as a linear combination of some basic
signals whose output can be obtained easily; Then, employ the linearity properties of the system to
obtain the corresponding output.
Question: Which basic signals? Complex exponentials ej2pif0t are the eigenfunctions of
LTI systems Response to a complex exponential is a complex
exponential with the same frequency and with a change inamplitude and phase.
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Fourier Series of Periodic Signals Question: Which signals can be expanded in terms of
complex exponentials? And how? Fourier series theorem: Let signal x(t) be a periodic
signal with period T0 = 1/f0. Subject to the Dirichletconditions, x(t) can be expanded as:
x(t) =
n=xnej2pinf0t , (20)
where the Fourier series coefficients are
xn =1T0
+T0
x(t)ej2pinf0tdt , (21)
for some arbitrary . To describe x(t), it is sufficient to specify a countable set of
(in general, complex) numbers.31 / 44
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Fourier Series of Periodic Signals For real signal x(t):
x(t) =a02
+n=1
an cos(2pinf0t) + bn sin(2pinf0t), (22)
where
an =2T0
+T0
x(t) cos(2pinf0t)dt , (23)
bn =2T0
+T0
x(t) sin(2pinf0t)dt . (24)
Parsevals relation: Result of orthogonality of the basis ofcomplex exponentials
1T0
+T0
|x(t)|2dt =
n=|xn|2. (25)
LHS: power content of x(t); |xn|2: power content ofxnej2pinf0t , the nth harmonic. 32 / 44
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Response of LTI Systems to Periodic Signals Given impulse response h(t), response to the exponentialej2pif0t is H(f0)ej2pif0t , where the frequency response H(f ) isdefined as:
H(f ) =
h(t)ej2piftdt . (26)
With Fourier series expansion of x(t), response to x(t) is:
y(t) = T [x(t)] = T[ n=
xnej2pinf0t]
=
n=xnT
[ej2pinf0t
]=
n=
xnH(nf0) yn
ej2pinf0t . (27)
Output y(t) is also periodic with period T0 = 1/f0.33 / 44
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From Fourier Series to Fourier Transform Extend the ideas of Fourier series representation for
periodic signal to non-periodic signals. Suppose x(t) is non-periodic, we define truncated signal
xT0(t) ={
x(t), T02 < t T02 ,0, otherwise.
(28)
Define periodic signal xPT0(t) =
n= xT0(t nT0), then
xPT0(t) =
n=xPn e
j2pinf0t
=
n=
[1T0
T0/2T0/2
x()ej2pinf0d
]ej2pinf0t
=
n=
[f T0/2T0/2
x()ej2pifd
]ej2pift , (29)
where f = nf0 and f = (n + 1)f0 nf0 = 1/T0.34 / 44
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Fourier Transform for Non-Periodic Signals To obtain the result for non-periodic signal x(t), we letT0 . Then, f0 0 and summation over n becomesintegration.
x(t) =
[
x()ej2pifd]ej2piftdf . (30)
Define the Fourier transform of x(t) as:
X (f ) =
x(t)ej2piftdt . (31)
Then, x(t) can be expressed as:
x(t) =
X (f )ej2piftdf . (32)
Notation: X (f ) = F [x(t)] and x(t) = F1[X (f )].35 / 44
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Exercise Fourier Transform
Find the Fourier transform of the rectangular function:
(t) ={
A, T2 < t < T2 ,0, otherwise.
(33)
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Table of Common Fourier TransformsTime Domain Frequency Domain
(t) 1
1 (f )
(t t0) ej2pift0ej2pif0t (f f0)cos(2pif0t) 12(f f0) + 12(f + f0)sin(2pif0t) 12j (f f0) 12j (f + f0)(t) sinc(f )
sinc(t) (f )
(t) sinc2(f )
sinc2(t) (f )n= (t nT0) 1T0
n= (f nf0 )
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Fourier Transform for Periodic Signals Fourier series representation of periodic signal x(t):
x(t) =
n=xnej2pi nT0 t . (34)
Taking Fourier transform of both sides gives:
X (f ) =
n=xn
(f n
T0
). (35)
The spectrum is no longer continuous, but consists ofdiscrete lines at multiples of f0 = 1/T0.
It can be shown that:
X (f ) =1T0
n=
XT0
(nT0
)
(f n
T0
). (36)
A convenient way to compute xn is: xn = 1T0XT0(
nT0
).
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Key Properties of Fourier Transform
Linearity: For any (real or complex) scalars , ,
F [x1(t) + x2(t)] = X1(f ) + X2(f ). (37)
Scaling: For any real a 6= 0,
F [x(at)] = 1|a|X(fa
). (38)
Time shift:
F [x(t t0)] = ej2pift0X (f ). (39)
Modulation:
F [x(t)ej2pif0t ] = X (f f0). (40)
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Key Properties of Fourier Transform
Convolution: If x(t) and h(t) possess Fourier transforms:
F [x(t)Fh(t)] = X (f )H(f ). (41) Much simpler to find response in frequency domain than in
time domain. This result is the basis of frequency domain analysis of LTI
systems.
Parsevals relation: |x(t)|2dt =
|X (f )|2df . (42)
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Fourier Transform of Real, Even and Odd Signals
X (f ) =
x(t)ej2piftdt
=
x(t) cos(2pift)dt j
x(t) sin(2pift)dt
= XR(f ) jXI(f ). (43)
If x(t) is real XR(f ) is real and even; XI(f ) are real and odd. X (f ) = X (f ). |X (f )| = |X (f )|. X (f ) = X (f ).
If x(t) is real and even, XI(f ) = 0. Then X (f ) = XR(f ) isreal and even.
If x(t) is real and odd, XR(f ) = 0. Then X (f ) = jXI(f ) isimaginary and odd.
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Energy and Power Relations The energy content of an energy-type signal x(t):
Ex = |x(t)|2dt =
|X (f )|2df =
Gx(f )df . (44)
Energy spectral density: Gx(f ) = |X (f )|2. We define autocorrelation function Rx() as:
Rx() = x()Fx() =
x(t)x(t )dt . (45)
It can be easily verified thatEx = R(0) (46)
Gx(f ) = F [Rx()]. (47) Input-output relation for y(t) = x(t)Fh(t):
Ry (t) = Rx(t)FRh(t) (48)Gy (f ) = Gx(f )Gh(f ) = |X (f )|2|H(f )|2. (49)
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Energy and Power Relations Time-average autocorrelation of power-type signal x(t):
Rx() = limT
1T
T/2T/2
x(t)x(t )dt . (50)
By setting = 0, we obtain the power content of x(t):
Px = Rx(0) = limT
1T
T/2T/2
|x(t)|2dt . (51)
Define power spectral density: Sx(f ) = F [Rx()], then:
Px = Rx(0) = Sx(f )df . (52)
Input-output relation of y(t) = x(t)Fh(t):Ry (t) = Rx(t)Fh(t)Fh(t) (53)Sy (f ) = Sx(f )|H(f )|2. (54)
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Energy and Power Relations For periodic (i.e., power-type) signal x(t):
Rx() =1T0
T0/2T0/2
x(t)x(t )dt =
n=|xn|2ej2pi
nT0.
Power spectral density of x(t):
Sx(f ) = F [Rx()] =
n=|xn|2
(f n
T0
). (55)
Power content of x(t): Px = Sx(f ) =
n= |xn|2.
Input-output relation for y(t) = x(t)Fh(t):
Sy (f ) = Sx(f )|H(f )|2 =
n=|xn|2
H ( nT0)2 (f nT0
)
Py =
n=|xn|2
H ( nT0)2 . (56)
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