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COMM 401: Signals & Systems Theory
Dr. Ahmed El-MahdyAssociate Prof. in Communications Department
E-mail: [email protected]
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COMM 401: Signals & Systems Theory
• Instructor : Dr. Ahmed El-Mahdy• Office : C3.213
• Lecture Time :
MET: Saturday, 1 st slot H13IET : Saturday, 3 rd slots H13
• Office Hours : Tuesday (10 am-5pm)
• Email : [email protected]
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• Teaching Assistants- Eng. Moustafa Adly- Eng. Abdel-Rahman Kamel- Eng. Hatem Mohamed Ayman- Eng. Mohamed Osama El-Shaer
- Eng. Ahmed Taha- Eng. Mohamed Ahmed Abdel Ghany- Eng. Arsany Amir
- Eng. Dalia El-Banna- Eng. Mohamed Essam
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Text Book
Alan V. Oppenheim and Alan S. Willsky,Signals & Systems , 2 nd edition, Prentice-Hall, 1997.
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Grading• Assignments 10%
• Lab. 20%
• Midterm Exam 20%
• Quizzes 10%
• Final Exam 40%
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Course Contents
-Sampling and reconstruction6
- Discrete Time Fourier Transform5
-Communication Systems.7
SubjectNo.
-Signal Classifications.1
-Linear Time Invariant Systems.2
- Fourier Series Representation for Periodic Signals.3
- Continuous Time Fourier Transform.4
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Practice and LabPractice and Lab assignments are performed usingMtalab.
What is MatLab?
MatLab is a programming language and datavisualization software package which is especiallyeffective in signal processing and systems analysis.
website
http://www.mathworks.com
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COMM 401: Signals & Systems
TheoryLecture 1
Signal Classifications
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How Can We Represent the Signal?
• by mathematical formula (function) Ex.: x(t)=A sin(t)
• by computer program• by plot• by sound
• Definition: A signal is a function representing aphysical quantity.
What is the Signal?
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fs = 100000; % sampling frequency Ampl=1; % signal amplitude
f=400; % signal frequencyt = 0:1/fs:0.01; % sample periodx = Ampl*cos(2*pi*f*t); %signal
clf; % clear last figurefigure, plot(t,x) %figure, plotaxis([0 .01 -1.2*Ampl 1.2*Ampl]); %axisxlabel('time (sec)'); %labeling in x-axis
ylabel('x(t)') %labeling in y-axis
Example: Sinusoidal Signal inMatLab
% sinusoidal signal plotfs = 100000;
Ampl=1;f=400;t = 0:1/fs:0.01;x = Ampl*cos(2*pi*f*t);
clffigure, plot(t,x)axis([0 .01 -1.2*Ampl 1.2*Ampl])xlabel('time (sec)')
ylabel('x(t)')
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MatlabPlot of Sinusoidal Signal by
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Sound of Sinusoidal Signal
fs = 100000; Ampl=1;
f=400;
t = 0:1/fs:1.5;x = Ampl*cos(2*pi*f*t);sound(x,fs)wavwrite(x,fs,'sig2');
% sinusoidal signal
%writes data to WAV-file
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Classification of Signals• Mathematically, a signal is represented as a function of time
x(t ).Signals
Continuous-TimeSignals (CT)
Discrete-TimeSignals (DT)
t is a continuous
variable
t is a discrete variable
t = n Ts , Ts : sampling period
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Continuous Versus Discrete Time Signal
Continuous Time Signal ( CT) Discrete Time Signal ( DT)
• the independent variable t is continuous .• the signal values are defined for all t inthe interval of interest.
• notation x(t) .• Example: volt or current.
• the independent variable t is discrete .• it takes only a discrete values n .• n is an integer.• notation x(nT s ) or x(n) , Ts is the sampling time.• it results from sampling of (CT) signal.
2-2
5
-5
n
X(n )
1
2.5
-2.5
- 101
-1
5
-5
t
X(t)
0
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Classifications of Signals (Contd.)Signals
Real Signals ComplexSignals
x(t ) is a real signal ifits value is a realnumber
Example: x(t)=sin(t)
x(t ) is a complexsignal if its value isa complex number
Example :t jet x ω =)(
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Classifications of SignalsSignals
DeterministicSignals
RandomSignals
whose values are completely specified for any given time
Ex: sin, step, ramp,..
Take random values at any given time and must be characterized statistically.
Ex: noise of wireless comm.
channel
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Classifications of SignalsSignals
PeriodicSignals
Non periodicSignals
x ( t) is periodic with period T o ifthere is a +ve nonzero value ofT o for which x(t + T o) = x (t ).
In general : x(t + mT o)=x (t ) (1)
m =1, 2, ..
x[n] is periodic with period N o ifthere is a +ve nonzero integer N ofor which x[n + N o] = x [n].
In general : x[n + mN o] = x [n] (2)
m =1, 2, ..
The fundamental period T o
(or N o) is the smallest positivevalue (or number) for which theequalities (1) or (2) holds.
Cont.Time
Discr.Time
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Examples:• Continuous Time (CT) Signals
CT periodic signal CT non-periodic
signal
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Examples:
DT periodic sequence
DT non-periodic sequence
• Discrete Time (DT) Signals
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• Example: The signal:is periodic because it repeats itself every
)sin()( t t x =
)2sin()sin( π += t t
π 2
• Similarly:
The signal:is periodic because it repeats itself every
)2cos()cos( π += t t
)cos()( t t x =
π 2
0>t for
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• Example:
≥
<=
0)sin(0)cos()(
t if t t if t t x
X(t) has a discontinuity at the time origin anddoes not happen at any other time. Sinceevery feature in the shape of a periodic signalmust occur periodically , then the signal x(t) isnon-periodic.
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Power and Energy Signals• Total Energy:
- for CT signals:
- for DT signals:
joulesdt t x E ∞
∞−
∞=
2)(
Wattsn xP N
N n N N
−=
+∞→
∞ =2
12 1 ][lim
Wattsdt t xPT
T T T
−
∞→∞
=2
21 )(lim
joulesn x E n
∞
−∞=
∞ =2
][
• Average Power:
- for CT signals:
- for DT signals:
-For Non-Periodic Signals:
-For Periodic Signals:
signaltheist x )(
period theisT dt t xP
T
T pT 0
2
2
21
0
00 )( −
=
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Power and Energy Signals
∞ E
∞P
Signals
Periodic
Signals
Non-periodicSignals
Calculate P
P finite
Power
Signal
NeitherPower nor energy
Yes No
Calculate
finite
Energysignal
Yes No
NeitherPower orenergy
Calculate
Finite
NoYes
Power
Signal
-Any Periodic Signal hasinfinite energy.
∞P
∞P
∞ E
∞ E
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Example:• S(t)=A cos(t) it is periodic signal then it has infinite energy.
Watt A
dt t A
dt t A
dt t sT
PT
T
2
)]2cos(1[4
)(cos21
)(1
2
2
22
2 /
2 /
2
0
0
0
=
+=
=
=
−
−
−
π
π
π
π
π
π
Then s(t) is a power signal
∞==
=
∞
∞−
∞
∞−
∞
dt t A
dt t x E
)(cos
)(
22
2
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Transformation of the independent variable: Time Scaling
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Reflection and Time Shift
Reflection
Time Shift
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Scaling with Time Shift (CT signals)-TimeThe proper order in which the operations of time scaling and time shiftingshould be applied in the case of the continuous-time signal
1 2
1 2
The incorrect way of applying the precedence rule .
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Scaling with Time Shift (DT signals)-Time
The proper order of applying the operations of time scaling andtime shifting for the case of a discrete-time signal.
2
1
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Time Reversal
Reflection
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Even and Odd Signals
Even signal: symmetric around y-axis
Odd signal: symmetric around the origin
(must be zero at the origin)
t
x(t)
n
x[n]
][][);()( n xn xt xt x −−=−−=
][][);()( n xn xt xt x −=−=
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Signal Decomposition