topic 01: signals’ properties, classi cations, and useful operations · 2020. 9. 28. · periodic...

Topic 01: Signals’ Properties, Classifications, and

Useful Operations

Dr. Hilal M. El Misilmani

COME 480 − Discrete-time Signals & Systems

Dr. Hilal M. El MisilmaniNot For

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Useful Operations: Shifting, Scaling, InversionClassification of Signals

Some Useful ModelsEven and Odd Functions

Lectures Schedule

Topic Nb Topics

Topic 00 Course description, evaluation and regulations

Topic 01 Signals’ properties, classifications, and useful operations

Topic 02 DSP scheme, sampling, signal reconstruction, anti-aliasing filter, quantization

Topic 03Digital sequences, LTI systems, causal systems, difference equation and impulseresponses, BIBO stability, digital convolution

Topic 04Fourier series & Fourier transform, Discrete Fourier transform, Fast Fourier transform(decimation in time and frequency methods)

Topic 05Z-transform, properties, convolution, inverse Z-transform, solution of differenceequation using Z-transform

Topic 06Difference equations and digital filters, Z-plane pole-zero plot stability, digital filterfrequency response, basic filtering and realization of digital filters

Topic 07Finite impulse response (FIR) filter design, Fourier transform design, window method,design for customer specifications

Topic 08Infinite impulse response (IIR) filter design, Bilinear transformation (BLT) design method,digital Butterworth and Chebyshev filter designs

Topic 09 Realization of digital filters, FIR filter realization, IIR filter realization

Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 2/60

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Outline of Topics1 Useful Operations: Shifting, Scaling, Inversion

Time ShiftingTime ScalingTime ReversalCombined Operations

2 Classification of SignalsContinuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals

3 Some Useful ModelsUnit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est

4 Even and Odd FunctionsSome Properties of Even and Odd FunctionsEven and Odd Components of a Signal


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Current Section1 Useful Operations: Shifting, Scaling, Inversion






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Time Shifting

Consider a signal x(t) and the same signal delayed by Tseconds φ(t)

Whatever happens in x(t) at some instant t also happens inφ(t) T seconds later at the instant t + T .

φ(t + T ) = x(t)

φ(t) = x(t − T )

To time-shift a signal by T , we replace t with t − T ⇒ x(t − T )represents x(t) time-shifted by T seconds

1 If T > 0, the shift is to the right → delay

2 If T < 0, the shift is to the left → advance

Example: x(t − 2) is x(t) delayed (right-shifted) by 2 seconds,and x(t + 2) is x(t) advanced (left-shifted) by 2 seconds


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Example 1


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Example 2


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Time Scaling

The compression or expansion of a signal in time is known astime scaling

If the signal φ(t) is x(t) compressed in time by a factor of 2

whatever happens in x(t) at some instant t also happens toφ(t) at the instant t/2

φ(t/2) = x(t)

φ(t) = x(2t)

Application: if x(t) was recorded on a tape and played backat twice the normal recording speed, we would obtain x(2t)


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Time Scaling

1 if x(t) is compressed in time by a factor a, (a > 1), theresulting signal φ(t) is given by

φ(t) = x(at)

2 if x(t) is expanded in time (slowed down) by a factor a,(a > 1), the resulting signal

φ(t) = x(t/a)

To time-scale a signal by a factor a, we replace t with at:

1 If a > 1, the scaling results in compression

2 if a < 1, the scaling results in expansion


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Example 3


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Example 4

The figure below shows a signal x(t):1 Sketch and describe mathematically this signal

time-compressed by factor 32 Repeat for the same signal time-expanded by factor 2


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Example 4


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Time Reversal

To time-reverse x(t), we rotate it by 180◦ about the verticalaxis ⇒ time reversal is the reflection of x(t) about thevertical axis

Whatever happens to x(t) at some instant t also happens tothe time reverse x(t) at the instant −t, and vice versa

To time-reverse a signal we replace t with −t, and the timereversal of signal x(t) results in a signal x(−t)

Note: the reversal of x(t) about the horizontal axis results in−x(t)


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Example 5


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Example 6

For the signal x(t) illustrated in Figure, sketch x(−t), whichis time-reversed x(t)


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Combined Operations

Certain complex operations require simultaneous use of morethan one of the operations just described

The most general operation involving all the three operationsis x(at − b), realized in two possible sequences

1 Time-shift x(t) by b to obtain x(t − b). Then time-scale the shiftedsignal x(t − b) by a (i.e., replace t with at) to obtain x(at − b)

2 Time-scale x(t) by a to obtain x(at). Then time-shift x(at) by b/a(i.e., replace t with t − (b/a)) to obtain x [a(t − b/a)] = x(at − b)

In either case, if a is negative, time scaling involves timereversal


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Example 7

Example: x(2t − 6) can be obtained in two ways:1 Delay x(t) by 6 → x(t − 6), then time-compress this signal by

factor 2 → x(2t − 6)2 Time-compress x(t) by factor 2 → x(2t), then delay this

signal by 3 → x(2t − 6)


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Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals







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Continuous-time and discrete-time signals

1 Continuous-time signal: a signal that is specified for acontinuum of values of time t

2 Discrete-time signal: a signal that is specified only at discretevalues of t

(a) (b)


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Analog and Digital Signals

→ Continuous & analog; discrete & digital: are not the same!1 Analog signal: its amplitude can take on any value2 Digital signal: its amplitude can take on only a finite number

of values

Figure: (a) analog, continuous time, (b) digital, continuous time, (c)analog, discrete time, and (d) digital, discrete time


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Periodic and Aperiodic signals

1 Periodic signal: a signal x(t) is said to be periodic if for somepositive constant T0

x(t) = x(t + T0)

The smallest value of T0 that satisfies this periodicitycondition is the fundamental period of x(t)

By definition, a periodic signal x(t) remains unchanged whentime-shifted by one (or more) period(s):

x(t) = x(t + nT0)

2 Aperiodic signal: a signal that does not repeats its patternover a period is called aperiodic signal or non periodic


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A periodic signal, by definition, must start at t = −∞if it started at some finite instant, say t = 0, the time-shiftedsignal x(t + T0) would start at t = −T0 and x(t + T0) wouldnot be the same as x(t)

Another important property of a periodic signal x(t) is thatx(t) can be generated by periodic extension of any segment ofx(t) of duration T0 (the period)


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An additional useful property of a periodic signal x(t) ofperiod T0 is that the area under x(t) over any interval ofduration T0 is the same:→ For any real numbers a and b:∫ a+T0

ax(t)dt =

∫ b+T0

bx(t)dt


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More Signal Definitions

Some definitions:1 Everlasting signal: exists over the entire interval −∞ < t <∞.

A periodic signal, by definition, is an everlasting signal2 Causal signal: exists only for t > 0, it does not start before

t = 03 Noncausal signal: starts before t = 0:

An everlasting signal is always noncausal but a noncausalsignal is not necessarily everlasting

4 Anticausal signal: is zero for all t > 0

Question: do we really have an everlasting signal? if not, whydo we study it?


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Size of a Signal

How can a signal that exists over a certain time interval withvarying amplitude be measured by one number that willindicate the signal size or signal strength?

Such a measure must consider not only the signal amplitude,but also its duration

The area under a signal x(t) can be considered as a possiblemeasure of its size, however, this is a defective measure:

For a large signal x(t), its positive and negative areas couldcancel each other, indicating a signal of small size!

→ What is the solution?


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Signal Energy

This difficulty can be corrected by defining the signal size asthe area under x2(t), which is always positive → We call thismeasure the signal energy Ex , defined (for a real signal) as

Ex =

∫ +∞

−∞x2(t)dt

This definition can be generalized to a complex valued signalx(t) as

Ex =

∫ +∞

−∞|x(t)|2dt

There are also other possible measures of signal size, such asthe area under |x(t)|


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Signal Power

The signal energy must be finite for it to be a meaningfulmeasure of the signal size

A necessary condition for this is that the signal amplitude → 0as |t| → ∞If not, the integral will not converge → the signal energy isinfinite

A more meaningful measure of the signal size when the energyis infinite would be the time average of the energy, if it exists.→ This measure is called the power of the signal


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Signal Power Condition

Generally, the mean of an entity averaged over a large timeinterval approaching ∞ exists if the entity either is periodic orhas a statistical regularity

When x(t) is periodic, |x(t)|2 is also periodic → the power ofx(t) can be computed by averaging |x(t)|2 over one period

If such a condition is not satisfied, the average may not exist


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Signal Power

Examples:A ramp signal x(t) = t increases indefinitely as |t| → ∞ ⇒neither the energy nor the power exists for this signalThe unit step function, which is not periodic nor has statisticalregularity, does have a finite power!

(a) Ramp Function (b) Unit Step

Units of Energy and Power: depend on the nature of thesignal x(t)


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Signal Power

For a signal x(t), we define its power Px as:

Px = limT→∞

1

T

∫ T/2

−T/2x2(t)dt

We can generalize this definition for a complex signal x(t) as:

Px = limT→∞

1

T

∫ T/2

−T/2|x(t)|2dt

Px is the time average (mean) of the signal amplitude squared→ the mean-squared value of x(t)→ The square root of Px (

√Px) is the familiar rms

(root-mean-square) value of x(t)


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Energy and Power Signals

1 Energy signal: a signal with finite energy (zero power)

2 Power signal: a signal with finite and nonzero power (infiniteenergy)

Since the power is the time average of energy:

a signal with finite energy has zero powerand a signal with finite power has infinite energy→ a signal cannot both be an energy signal and a power signal


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Example 8

Determine the suitable measures of the signals shown

(a)

(b)


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Example 9

Determine the power and the rms vale of:

1 x(t) = C cos(ω0t + θ)

2 x(t) = C1 cos(ω1t + θ1) + C2 cos(ω2t + θ2); ω1 6= ω2

3 x(t) = De jω0t


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Deterministic and Probabilistic Signals

1 Deterministic signal: a signal whose physical description isknown completely, either in a mathematical form or agraphical form

2 Random signal: a signal whose values cannot be predictedprecisely but are known only in terms of probabilisticdescription, such as mean value or mean-squared value

In this course we shall exclusively deal with deterministicsignals. Random signals are beyond the scope of this study


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Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est







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Unit Step Function u(t)

Causal signals can be conveniently described in terms of unitstep function u(t)

A unit step function is defined as:

u(t) =

{1, t > 0

0, t < 0


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Unit Step Function u(t)

The unit step can also be shifted in time:

u(t − t0) =

{1, t > t0

0, t < t0


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Unit Step Function u(t): Some Applications

If we want a signal to start at t = 0 (so that it has a value ofzero for t < 0), we need only to multiply the signal by u(t)

It is also very useful in specifying a function with differentmathematical descriptions over different intervals

Example:

→ x(t) = u(t − 2)− u(t − 4)


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Example on the Use of Unit Step u(t)

Describe the signal shown as a function of unit step function


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The Unit Impulse Function δ(t)

Defined by Paul Dirac, δ(t) is one of the most importantfunctions in the study of signals and systems

Regarded as a rectangular pulse with:

a width that has become infinitesimally smalla height that has become infinitely largean overall area that has been maintained at unity


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The Unit Impulse Function δ(t) Properties

δ(t) = 0, for t 6= 0∫∞−∞ δ(t)dt = 1

kδ(t) is an impulse function whose area is k, and kδ(t) = 0,for t 6= 0

An exact impulse function cannot be generated in practice; itcan only be approached

As α→∞ the pulse height →∞ and the width → 0


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Multiplication of a Function by an Impulse

Multiplication of a continuous-time function φ(t) by δ resultsin:

an impulse located at where the δ is definedand has strength φ(t where δ is defined)

Generalizing this result, provided that φ(t) is continuous att = T :

φ(t)δ(t − T ) = φ(T )δ(t − T )


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Sampling Property of the Unit Impulse Function

Provided that δ(t) is continuous at t = 0:

The area under the product of a function with an impulse δ(t) isequal to the value of that function at the instant at which the unitimpulse is located

→ Property known as sampling property of the unit impulse→ We say that the impulse sampled the function at t = 0∫ ∞

−∞φ(t)δ(t)dt =

∫ ∞−∞

φ(0)δ(t)dt = φ(0)

∫ ∞−∞

φ(t)δ(t − T )dt = φ(T )


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Unit Step and Impulse Functions

An interesting application of the generalized functiondefinition of an impulse:

du(t)

dt= δ(t)

and ∫ t

−∞δ(τ)dτ =

{0, t < 0

1, t > 0

Similarly, the delta dirac δ(t) can be delayed:∫ t

−∞δ(τ − a)dτ =

{0, t < a

1, t > a


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Example on δ(t) Calculations

Calculate the following equations as a function of δ(t)

1(t3 + 3

)δ(t) = 3δ(t)

2[sin(t2 − π

2 )]δ(t) = −δ(t)

3 e−2tδ(t) = δ(t)

4 ω2+1ω2+9δ(ω − 1) = 1

5δ(ω − 1)


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Example: Unite Step and Delta Dirac

Compute and graph the derivative of the rectangular pulseshown below


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Example: Unite Step and Delta Dirac

Solution:

f (t) = Au(t)− Au(t − t0)

df (t)

d(t)= Aδ(t)− Aδ(t − t0)


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The Exponential Function est

Another important function in the area of signals and systemsis the exponential signal est , where s is complex in general,given by:

s = σ + jω

→ est = e(σ+jω)t = eσte jωt = eσt(cosωt + jsinωt)

If we take the conjugate of s → s∗ = σ − jω:

es∗t = e(σ−jω)t = eσte−jωt = eσt(cosωt − jsinωt)

→eσtcos(ωt) =1

2

(est + es

∗t)


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The Complex Frequency Plane (s plane)

s = σ + jω

|ω|: the radian frequency, indicatesthe frequency of oscillation of est

σ: the neper frequency, givesinformation about the rate ofincrease or decrease of theamplitude of est


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For signals whose complex frequencies lie on the real axis (σaxis, where ω = 0), the frequency of oscillation is zero.→ These signals are monotonically increasing or decreasingexponentially (Fig. a)For signals whose frequencies lie on the imaginary axis (jωaxis where σ = 0), eσt = 1.→ These signals are conventional sinusoids with constantamplitude (Fig. b)


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For the signals in Fig. c & d, both σ and ω are nonzero; thefrequency s is complex and does not lie on either axis.

If σ is negative the signal decays exponentially (Fig. c), and slies to the left of the imaginary axisIf σ is positive the signal increase exponentially (Fig. d), and slies to the right of the imaginary axis

The case s = 0⇒ σ = ω = 0 corresponds to a constant (dc)signal because e0t = 1


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Some Properties of Even and Odd FunctionsEven and Odd Components of a Signal







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Even and Odd Functions

Even function: a function xe(t) is said to be an even functionof t if

xe(t) = xe(−t)

→ symmetrical about the vertical axisOdd function: a function xo(t) is said to be an odd functionof t if

xo(t) = −xo(−t)

→ antisymmetrical about the vertical axis


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Some Properties of Even and Odd Functions

Even function × odd function = odd function

Odd function × odd function = even function

Even function × even function = even function

Since xe(t) is even, symmetrical about the vertical axis:∫ a

−axe(t)dt = 2

∫ a

0xe(t)dt

For xo(t): ∫ a

−axo(t)dt = 0


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Even and Odd Components of a Signal

Every signal x(t) can be expressed as a sum of even and oddcomponents:

x(t) =1

2[x(t) + x(−t)]︸︷︷︸

even

+1

2[x(t)− x(−t)]︸︷︷︸

odd

Example: Consider the following function, get its even andodd components:

x(t) = e−atu(t)


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Example Cont.

Example: Consider the following function, get its even andodd components:

x(t) = e−atu(t)


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topic 01: signals’ properties, classi cations, and useful operations · 2020. 9. 28. · periodic...

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