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EEE 330Introduction to
Communication Systems
Lecture # 2Signal and SystemsSignal ComparisonReview of the Fourier SeriesReview of the Fourier Transform
Overview
� The Objectives of Today’s Lecture
� Signal and Systems
� Signal Comparison
� Overview/Motivation for Fourier Theory
� Review of the Fourier Series
� Review of the Fourier Transform
� Reading� B.P. Lathi, Modern Digital and Analog Communication
Systems, 3rd Ed., Oxford University Press, 1998.
�Chapter 2
�Chapter 3
To Study Communication Systemsyou must understand…
� Signals and Systems
� Fourier Analysis
� Modulation Theory
� We will study this in detail
� Detection Theory
� Given that this signal is corrupt at the receiver, how do we determine the original signal?
� Probability Theory
� Since the transmit signal and noise are both unknown to the receiver, we can use probability theory to study communications systems
Signals and Systems
� In this class we will rely on mathematical representations of signals and systems to describe communications
� Relies on background obtained from EEE301
� A system is characterized by inputs and outputs which are mathematically modeled as signals
� We will also mathematically represent the signals at various points within a communications system
� Mathematical representations of the various components of the system can be viewed as subsystems with input-output relationships defined by
� Impulse response in the time domain
� Transfer function in the frequency domain
System Representation
� A system is any process that results in the transformation of signals.
� H is typically used to represent the system
� x(t) is typically used to represent the excitation or input to the system
� y(t) is typically used to represent the response or output of the system
� Systems can have multiple inputs and/or mulitple outputs
� Example of a Single-Input Single Output system:
System Properties
� There are several properties of systems that are important to understand
� Many properties allow us to make simplifications in our analysis
� Specific properties� Time Invariance
� Linearity
� Stability
� Causality
� Memory
� Invertibility
Time-invariance
� A system is time-invariant if a time-shift in the input causes a time shift in the output
� Ex: y(t) = sin(x(t))
� y(t-t0) = sin(x(t-t0))
� If a system is not time-invariant, then it is time-varying.
� Ex: y(t) = t x(t)
� y(t-t0) = t x(t-t0) (t-t0) x(t-t0)≠
Linearity
� A linear system is any system that obeys the properties of scaling (homogeneity) and superposition (additivity)
If y(t) = H(x(t)) then
α y(t) = H(α x(t))and
H(α x1(t)+ β x2(t))= α H(x1(t))+ β H(x2(t))
Stability
� A stable system is one where the output does not diverge as long as the input does not diverge.
� If the input is bounded then the output is also bounded (BIBO system)
� However, this is not always true.
x[n] = u[n] (unit step function – Bounded)
y[n] = H(x[n] ) =(n+1)u[n] (not bounded)
Causality
� A causal system is one that is nonanticipative; that is, the output may depend on current and past inputs, but not future inputs.
Ex: y[n] = H(x[n]) = x[n] – x[n-1]
Memory
� A system is memoryless if its output for each value of independent variable is dependent only on the input at the same time.
Ex: y[n] = H(x[n]) = [x[n] ]2
Invertibility
� A system is called invertible, if distinct inputs lead to distinct outputs.
� By observing output, you can determine its input
Ex: y(t) = 2x(t) � x(t) = 0.5 y(t)
Signals
� A signal is a function representing a physical quantity.
� Signals are represented mathematically as functions of one or more independent variables.� Speech signal is represented by acoustic
pressure as a function of time.
� Picture is represented by brightness function of tqo spatial variables.
� Although functions can operate on any type of variable, we will be most concerned with functions of time
Physically realizable functions
� Have finite time duration (finite energy!)
� Occupy finite frequency spectrum
� Are continuous
� Have finite peak value
� Are real-valued
Mathematical Representations
Classification of Signals
� Signals (or more specifically their mathematical representations) can be categorized according to a few major features� Continuous Time vs. Discrete Time
� Analog vs. Digital
� Deterministic vs. Propabilistic (Random)
� Power vs. Energy
� Periodic vs. Aperiodic
� Even vs. Odd
Contiuous Time vs. Discrete Time
� This classification is determined by whether or not the time axis-independent variable) is discrete (countable) or continuous.
� A continuous-time signal are defined for a continuum of values of time
� A discrete-time signal is only defined at discrete times.
Analog vs. Digital
� Analog signal can take any value for all t
� Digital signal can take only finite number of distinct values
Deterministic vs. Random
� A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. � The future values of the signal can be calculated from past values with complete
confidence.
� If a signal is known only in terms of probabilistic description such as mean value, mean squared value, and so on, it is a random signal.� The future values of a random signal cannot be accurately predicted and can
usually only be guessed based on the averages of sets of signals.
Power vs. Energy
� Energy signals have finite energy
� Every signal in real life is an energy signal
� Power signal have finite and nonzero power.
� Power signal is of infinite duration
Periodic vs. Aperiodic
� Periodic signals repeat with some period T.
� A signal is called aperiodic if it is not periodic.
Even vs. Odd
� An even signal is any signal f such that f(t) =f(−t) .
� Even signals can be easily spotted as they are symmetric around the vertical axis.
� An odd signal is a signal fsuch that f(t) =−f(−t)
Signal Comparison (Orthogonality)
� Orthogonality: Two complex signals are said to be orthogonal over an interval t1≤ t ≤ t2, if
or
� Significance:
� Sum of weighted orthogonal signals are used to represent any signal with minimum error
� We can transmit signals over orthogonal signals
� We can reject undesired signals to select just one that we want, by filtering at the demodulator
� Orthogonal signals are used in CDMA
2
1
*1 2( ) ( ) 0
t
tx t x t dt =∫
2
1
*1 2( ) ( ) 0
t
tx t x t dt =∫
Orthogonality (Approximation of functions)
� xn are N mutually orthogonal signals
� cn are coefficients
Orthogonality (Transmitter)
� Any two sinusoids that are harmonically related are orthogonal over the whole cycle
� All sinusoids are orthogonal over the interval -∞ to ∞.
� This means we can modulate information over separate carriers and “tune in” the channel we want.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
sin(
π t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
sin(
2 π t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1si
n(4 π
t)
t
Orthogonality (Demodulator)
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (kHz)
abs(
Mod
ulat
ed S
igna
l)
f=3khz
Local oscillator
0
∞
∫Performs operation
Signals orthogonal to cos (2∗π∗3*10^3*t) will be cancelled
Orthogonality (CDMA)
� In CDMA the spectrum is used by N users
� Each user is assigned to a unique code
� The N codes are orthogonal to each other
� 3G systems
0 0.5 1 1.5 2-1
-0.5
0
0.5
1
code
1
0 0.5 1 1.5 2-1
-0.5
0
0.5
1
time
code
2
Signal Comparison (Correlation)
� Correlation is related with the information that how much two signals are similar
� cn is the correlation coefficient and
� normalizes the levels of g(t) and
x(t), which are complex signals.
1( ) ( )n
g x
c g t x t dtE E
∞∗
−∞
= ∫
1 1nc− ≤ ≤
1
g xE E
Correlation
� cn = 1 �Two signals are similar
� Two best friends
� cn = 0 �Two signals are orthogonal
� Unrelated
� Complete strangers
� cn =-1 �Two signals are dissimilar
� Worst enemies
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
g(t
)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
x 1(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
x 2(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-1
0
1
time
x 3(t)
Correlation (Contd.)
� x2(t) is a shifted version of x1(t), hence they are “IDENTICAL”
� However cn=0
� Use cross-correlation instead of correlation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
x 1(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
time
x 2(t)
Cross-Correlationfunction
g(t)
z(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
time
z(t)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
g(t)
Cross-Correlation
� The normalization factor is dropped
� It can take any value
� It plots the similarity index for all relative time shifts between two signals
� For complex signals
Autocorrelation Function
� The correlation of signal with itself is called autocorrelation
� The autocorrelation function for a real signal g(t) is
Autocorrelation
� Autocorrelation of a random noiseor signal indicates the “periodicity”structure of the signal
� Truely random (unpredictable) noise has Ψg(τ)=δ(τ).
Review of the Fourier Series and the Fourier Transform
Motivation
� If a system is linear, the response due to a sum of signals is the sum of the responses to each individual signal
� System analysis can be simplified by decomposing an input signal into a sum of simpler signals
� The system output can then be found as the sum of the system responses to these simpler signals
� A physically meaningful way of decomposing signals is to represent them as a sum (or integral) of sinusoids
� Periodic signals – Fourier Series
� Aperiodic signals – Fourier Transform� Periodic signals can also be represented using the
Fourier Transform
� This gives rise to the idea of the frequency domain
Fourier Theory
� Two basic types of signals
� Power signals
� Energy signals
� We can represent a signal in time or in frequency
� Fourier Representations
� Fourier Series� Representation valid for all time if signal is periodic
(i.e., power signals)
� Representation is valid only over a certain interval for aperiodic signals
� Fourier Transform� Applies directly to energy signals
� Requires introduction of the impulse for application to power signals
Fourier Theory (cont.)
� Fourier Theory tells us that signals can be represented as weighted sums (or integrals) of sinusoids.
� The “amount” of each sinusoid is equivalent to the “frequency domain” information of a particular signal
� If the signal is periodic, the signal can be represented as an infinite sum of sinusoids whose frequencies are integer multiples of the fundamental frequency, fo.
� If a signal is aperiodic we can take the limit of the Fourier Series as the period goes to infinity. The result is the Fourier Transform
� The Fourier Transform doesn’t technically apply to periodic signals.
� However we can create a FT through the use of the delta function
Trigonometric Fourier Series
� Trigonometric Fourier Series
where To is the period, and
Compact trigonometric Fourier Series
Example 2.7
Example 2.7 (Cont’d)
Example 2.7 (Cont’d)
Exponential Fourier Series
� We can represent a periodic signal g(t) with period T0 exactly by the sum of complex sinusoids
� where
� The above integral must converge
� This is termed the Exponential Fourier Series
� We can represent the relationship between g(t) and Dnas
( ) FSng t D←→
Exponential Fourier Series (Cont’d)
� Dn are complex numbers in general.
� For any real signal, | Dn | is even function and the phase is always an odd function
� It still represents a signal on one period.
Example 2.10
Example 2.10 (Cont’d)
Example 2.10 (Cont’d)
Negative frequencies ?
� Negative frequencies arise as a necessary implication of the exponential phasor view of the signals, required to be able to represnt purely real signal.
� Frequency is not negative
cos( )2
j t j te et
ω ω
ω−+=
sin( )2
j t j te et
j
ω ω
ω−−=
Energy Signals
� The Fourier Series applies to periodic signals which are also power signals.� They all have a “line spectrum”
� However, we would like to analyze both power signals and energy signals.
� Energy signals have a “continuous spectrum”� There is some energy at every frequency in the
signal spectrum
� Thus, we need a more powerful analysis tool.
� The Fourier Transform is the answer
Fourier Transform
� It is the Fourier series in the limit
or
� G(ω) � has the same amplitude symmetry and phase
anti-symmetry properties of exp. FS
� For a single pulse g(t), it gives the envelope of exp. FS that is obtaines if g(t) is repeated periodically.
FT and Exp. FS
Continuous Spectrum
Line Spectrum
G(ω)
� In general G(ω) is a complex number
� G(ω) = | G(ω) |*exp(jθG(ω))� For real signals g(t)
� G(ω)= G*(-ω) (Conjugate symmetry property)
Hence
� |G(-ω)|= |G(ω)| (even function of ω)� θG(-ω)= -θG(ω) (odd function of ω)
Inverse Fourier Transform
� Inverse Fourier Transform reconstructs the signal from its spectrum
� g(t) and G(ω) forms a FT pair
The Frequency Domain� The original signal g(t) is said to be in the time domain
since its argument represents time� The Fourier Transform G(ω) representation is said to be in
the frequency domain since its argument ω represents frequency
� Notes:� Frequency is the reciprocal of time� The Fourier Transform is referred to as an analysis of the
signal g(t) since it extracts the frequency components of g(t) at each value of ω
� The Inverse Fourier Transform is referred to as synthesis since it recombines the components G(ω) to obtain the original signal g(t)
� The physical meaning of G(ω) depends on the meaning of g(t). If g(t) has units of volts, G(ω) has units volts/Hz. � Thus it represents how much of the voltage signal is present
at each frequency.
The Frequency Domain
� We can think of the Fourier Transform and the Inverse Fourier Transform as means for moving between the time and frequency domains
� Note that no information is lost in the transformation and both are equivalent representations of a signal
This is sometimestermed the“Analysis equation”
This is sometimestermed the“Synthesis equation”
Example 3.1
Existence of FT
� Not all the signals are Fourier transformable
� The existence of FT is assured for any g(t) satisfying the Dirichlet’s conditions, i.e.
FT of Rectangular pulse
-τ/2 τ/2
Plots
Time vs. Frequency
τ =0.1
τ =0.01
Time vs. Frequency
� Time and frequency are reciprocal� If a function speeds up in time, it slows down in
frequency� If a signal changes rapidly it requires more high
frequency components� Signals which change rapidly in time are said to have
a large bandwidth (a measure of the frequency content)
� If a function slows down in time, it speeds up in frequency� If a signal changes slowly in time it requires less high
frequency components and more low-frequency components
� Signals which change slowly in time are said to have a small bandwidth
Definitions of Bandwidth forBaseband Signals
� Bandwidth is a term used to describe a positive frequency range over which the signal has significant content. There are various definitions for bandwidth including:
� Absolute Bandwidth (Babs)
� Defined as B where G(ω)=0 ω>B� 3-dB Bandwidth (half-power bandwidth - (B3dB))
� Defined as B where
� X-dB Bandwidth
� Defined as B where
� First Null Bandwidth (Bfirst null)
� For baseband systems this is equal to the frequency of the first null in the spectrum
( ) ( )10 10 max20log ( ) 20log ( ) -X >G G Bω ω ω<
22 max
( )( ) >
2
GG B
ωω ω<
Bandwidth - Baseband
|G(ω)|2
Bandwidth - Bandpass
|G(ω)|2
Properties of Fourier Transform
� Time-Frequency Duality
� Symmetry
� Linearity
� Scaling
� Time-shifting
� Frequency-shifting
� Convolution and multiplication
� Time-differentiation and Time-Integration
Refer to Table 3.2 on pg 101 for the properties
Time-Frequency Duality
� Due to the similar nature of the Fourier Transform and the Inverse Fourier Transform, there is the duality property.
� Whenever we derive any result, we can be sure that it has a dual
Example 3.8
Symmetry (part of duality)
� If
then
Example:
Linearity
� If g(t)=α g1(t)+β g2(t) then
G(ω)=α G1(ω)+β G2(ω)
α g1(t)+β g2(t) ���� α G1(ω)+β G2(ω)
Scaling
� If
then for a real constant a
Example
Scaling - Interpretation
� Scaling property states that the time compression of signal results in the spectral expansion, and time expansion of signal results in the spectral compression.
� Time Compression: α > 1.� Scaling a signal in time by α speeds the signal up in time.� The resulting transform is scaled by 1/α which slows the
transform down in frequency – this means that more of the larger frequency values are present to accomplish faster changes.
� Time Expansion: α < 1.� Scaling a signal in time by 1/α slows the signal down in
time. � The resulting transform is scaled by α which speeds it up in
frequency – this means that more low frequency values are present to account for slower changes.
Time and Frequency-shifting
� Time-Shifting
If
then
� Frequency-shifting
If
then
Frequency-Shifting and Modulation
� Since
� then
Amplitude modulation
Carrier
Example of Modulation
Example
� g(t) = rect(t) |G(ω)| = |sinc(ω/2)|
Example– cont’d.
z(t)=rect(t)cos(200πt), ωo=200π
Z(ω)=0.5*(sinc((ω−200π)/2)+ sinc((ω+200π)/2))
Bandpass signals
Low pass
Bandwidth: 2πB
Band pass
Bandwidth: 4πBIf a linear combination of these two band pass signals will be a band pass signal
Bandpass signal
Convolution and multiplication
� If
then
and
� Thus, convolution in the time domain results in multiplication in the frequency domain while multiplication in the time domain results in convolution in the frequency domain.
� This can greatly simplify some system analysis
BW= B1 BW= B2
BW= B1+B2
Time-Differentiation and Time-Integration
� If then
time-differentiation
and time-integration
Summary
� In this lecture we have discussed
� Signals and systems
� Fourier series
� Fourier Transform.
� The Fourier Transform is useful for providing a frequency domain representation of periodic and aperiodic signals that is valid for all time.
� Understanding the relationship between time and frequency is perhaps one of the most important concepts in this course.
