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Signals and Systems(Part one: Continuous)
Dr. Yasmine FahmyDr. Yasmine Fahmy
General Information
LecturerLecturer– Dr. Yasmine FahmyDr. Yasmine Fahmy
ReferenceReference– Signals and systems, Oppenheim Signals and systems, Oppenheim
AA..VV.., Wilski A, Wilski A..SS.., Prentice Hall, 1997, Prentice Hall, 1997
Dr. Yasmine FahmyDr. Yasmine Fahmy
Course Contents Signals (definition, properties , important Signals (definition, properties , important
signals)signals) Systems (definition, properties)Systems (definition, properties) Linear Time Invariant (LTI) SystemsLinear Time Invariant (LTI) Systems Fourier SeriesFourier Series Fourier TransformFourier Transform LTI Systems in Frequency DomainLTI Systems in Frequency Domain Applications (Filters, Sampling, Modulation)Applications (Filters, Sampling, Modulation)
22 11 11 11 22 11 11 99
Dr. Yasmine FahmyDr. Yasmine Fahmy
Signals
Electrical signalsElectrical signals– Voltages and currents in a circuitVoltages and currents in a circuit
Acoustic signalsAcoustic signals– Acoustic pressure (sound) over timeAcoustic pressure (sound) over time
Mechanical signalsMechanical signals– Velocity of a car over timeVelocity of a car over time
Video signalsVideo signals– Intensity level of a pixel (camera, video) Intensity level of a pixel (camera, video)
over timeover time
variables carrying informationvariables carrying information
Dr. Yasmine FahmyDr. Yasmine Fahmy
Continuous / Discrete
VelocityVelocity VoltageVoltage
x[n]
n
x(t)
t -3 -2 -1 0 1 2 3 4
PixelsPixels Daily stock Daily stock
priceprice
Dr. Yasmine FahmyDr. Yasmine Fahmy
Analog / Digital Continuous Continuous AnalogAnalog Signal Signal
Discrete Discrete AnalogAnalog Signal Signal
Continuous Quantized Continuous Quantized AnalogAnalog Signal Signal
t
n-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4
Dr. Yasmine FahmyDr. Yasmine Fahmy
Analog / Digital
SamplingSampling QuantizationQuantization CodingCoding
n-3 -2 -1 0 1 2 3 4
-3 -2 -1 0 1 2 3 4
01001111010011010
n-3 -2 -1 0 1 2 3 4
t
Dr. Yasmine FahmyDr. Yasmine Fahmy
Properties of Signals
1.1. Signal Energy and PowerSignal Energy and Power2.2. Transformation in Time Transformation in Time
(Shift, Reverse, Scaling)(Shift, Reverse, Scaling)3.3. Periodic SignalsPeriodic Signals4.4. Even and Odd SignalsEven and Odd Signals
Dr. Yasmine FahmyDr. Yasmine Fahmy
Signal Energy and Power Energy over time intervalEnergy over time interval
Average Power over time intervalAverage Power over time interval
dttxEt
ttt
22
1
21)(
12
2
12
212
1
21)(1
ttE
dttxtt
P ttt
ttt
Dr. Yasmine FahmyDr. Yasmine Fahmy
Signal Energy and Power Total EnergyTotal Energy
Average PowerAverage Power
dttxET
TT
2
)(lim
TEdttx
TP
T
T
TT 2)(
21 limlim
2
Dr. Yasmine FahmyDr. Yasmine Fahmy
Transformation in Time
Time ShiftTime Shiftx(t)
t
x(t+to)
t
x(t-to)
t
+to Advance -to Delay
-to +to
Dr. Yasmine FahmyDr. Yasmine Fahmy
Transformation in Time
Time ReverseTime Reversex(t)
t
x(-t)
t
Dr. Yasmine FahmyDr. Yasmine Fahmy
Transformation in Time
Time ScalingTime Scalingx(t)
t
x(׀α׀ t)
t t
x(׀α׀ t)
1 > ׀ α׀ Compressed 1׀ < α׀ Stretched
Dr. Yasmine FahmyDr. Yasmine Fahmy
Example 1
20E 20P E P
Find:1. The equation of x(t)2. The values of 3. x(t+1)4. x(-t+1)5. x(-3/2t+1)6. x(-3/2t-1)
,
,
,
0 1 2 t
X(t)
1
Dr. Yasmine FahmyDr. Yasmine Fahmy
NOTENOTE
Energy signals: Energy signals: – Finite EnergyFinite Energy– Zero PowerZero Power
Power signals: Power signals: – Infinite Infinite
EnergyEnergy– Finite PowerFinite Power
Dr. Yasmine FahmyDr. Yasmine Fahmy
Periodic Signals
x(t) = x(t+x(t) = x(t+TT))
Where Period := T Fundamental Period := To
(is the minimum value of T)
t
Dr. Yasmine FahmyDr. Yasmine Fahmy
Example 2 Find the period of the following Find the period of the following
signals:signals:
–
–
12( ) 5 cos sin
3 9t tx t
2 ( ) cos 2 1 sin 5 2x t t t
Dr. Yasmine FahmyDr. Yasmine Fahmy
Even & Odd SignalsEvenEven
x(t) = x(-t)x(t) = x(-t)
Symmetric around the Symmetric around the axisaxis
tt
OddOddx(t) = -x(-t)x(t) = -x(-t)
Symmetric around the Symmetric around the originorigin
Dr. Yasmine FahmyDr. Yasmine Fahmy
Even & Odd SignalsFor any signal For any signal x(t)x(t)
x(t) = xx(t) = xe(t)+ x(t)+ xo(t)(t)
WhereWhere xxe(t)=(t)=1/2 [1/2 [x(t)x(t)++x(-t)x(-t)]] xxo(t)=(t)=1/2 [1/2 [x(t) x(t) --x(-t)x(-t)]]
Dr. Yasmine FahmyDr. Yasmine Fahmy
Example 3
Find and SketchThe Even and Odd components of x(t)
,
,
,
-1 0 1
X(t)
1
Dr. Yasmine FahmyDr. Yasmine Fahmy
Example 3
Dr. Yasmine FahmyDr. Yasmine Fahmy
Lecture Overview Signal
(continuous/discrete/analog/digital)
Signal Properties1. Signal Energy and Power2. Transformation in Time (Shift, Reverse,
Scaling)
3. Periodic Signals4. Even and Odd Signals