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COMM 401: Signals & Systems Theory

Dr. Ahmed El-MahdyAssociate Prof. in Communications Department

E-mail: [email protected]



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]



• 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



Text Book

Alan V. Oppenheim and Alan S. Willsky,Signals & Systems , 2 nd edition, Prentice-Hall, 1997.



Grading• Assignments 10%

• Lab. 20%

• Midterm Exam 20%

• Quizzes 10%

• Final Exam 40%



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



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



COMM 401: Signals & Systems

TheoryLecture 1

Signal Classifications



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?



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)')



MatlabPlot of Sinusoidal Signal by



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



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



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



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 =)(



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



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



Examples:• Continuous Time (CT) Signals

CT periodic signal CT non-periodic

signal



Examples:

DT periodic sequence

DT non-periodic sequence

• Discrete Time (DT) Signals



• 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



• 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.



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 )( −

=



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



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



Transformation of the independent variable: Time Scaling



Reflection and Time Shift

Reflection

Time Shift



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 .



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



Time Reversal

Reflection



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 −=−=



Signal Decomposition

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