(part one: continuous)

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

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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

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Continuous / Discrete

VelocityVelocity VoltageVoltage

x[n]

n

x(t)

t -3 -2 -1 0 1 2 3 4

PixelsPixels Daily stock Daily stock

priceprice

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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

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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

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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

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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

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Signal Energy and Power Total EnergyTotal Energy

Average PowerAverage Power

dttxET

TT

2

)(lim

TEdttx

TP

T

T

TT 2)(

21 limlim

2

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Transformation in Time

Time ShiftTime Shiftx(t)

t

x(t+to)

t

x(t-to)

t

+to Advance -to Delay

-to +to

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Transformation in Time

Time ReverseTime Reversex(t)

t

x(-t)

t

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Transformation in Time

Time ScalingTime Scalingx(t)

t

x(׀α׀ t)

t t

x(׀α׀ t)

1 > ׀ α׀ Compressed 1׀ < α׀ Stretched

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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

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NOTENOTE

Energy signals: Energy signals: – Finite EnergyFinite Energy– Zero PowerZero Power

Power signals: Power signals: – Infinite Infinite

EnergyEnergy– Finite PowerFinite Power

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Periodic Signals

x(t) = x(t+x(t) = x(t+TT))

Where Period := T Fundamental Period := To

(is the minimum value of T)

t

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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

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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

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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)]]

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Example 3

Find and SketchThe Even and Odd components of x(t)

,

,

,

-1 0 1

X(t)

1

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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