Post on 23-Mar-2021
transcript
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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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 3/60
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Public
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Current Section1 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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 4/60
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Public
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 5/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 1
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 6/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 2
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 7/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 8/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 9/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 3
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 10/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 11/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 4
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 12/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 4
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 13/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 14/60
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Public
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 5
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 15/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
Example 6
For the signal x(t) illustrated in Figure, sketch x(−t), whichis time-reversed x(t)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 16/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 17/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Time ShiftingTime ScalingTime ReversalCombined Operations
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 18/60
Not For
Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
Current Section1 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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 19/60
Not For
Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 20/60
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Public
Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 21/60
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Public
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 22/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
Periodic and Aperiodic signals
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 23/60
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Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
Periodic and Aperiodic signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 24/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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?
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 25/60
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Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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?
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 26/60
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Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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)|
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 27/60
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Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 28/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 29/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 30/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 31/60
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Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 32/60
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Public
Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
Example 8
Determine the suitable measures of the signals shown
(a)
(b)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 33/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 34/60
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Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Continuous-time and discrete-time signalsAnalog and Digital SignalsPeriodic and Aperiodic SignalsEnergy and Power SignalsDeterministic and Probabilistic Signals
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 35/60
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e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Current Section1 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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 36/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 37/60
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Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Unit Step Function u(t)
The unit step can also be shifted in time:
u(t − t0) =
{1, t > t0
0, t < t0
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 38/60
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Releas
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 39/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Example on the Use of Unit Step u(t)
Describe the signal shown as a function of unit step function
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 40/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Example on the Use of Unit Step u(t)
Describe the signal shown as a function of unit step function
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 41/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Example on the Use of Unit Step u(t)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 42/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Example on the Use of Unit Step u(t)
Describe the signal shown as a function of unit step function
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 43/60
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Public
Releas
e
Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 44/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 45/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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 )
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 46/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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 )
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 47/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 48/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 49/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Example: Unite Step and Delta Dirac
Compute and graph the derivative of the rectangular pulseshown below
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 50/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
Example: Unite Step and Delta Dirac
Solution:
f (t) = Au(t)− Au(t − t0)
df (t)
d(t)= Aδ(t)− Aδ(t − t0)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 51/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 52/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 53/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
The Complex Frequency Plane (s plane)
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 54/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Unit Step Function u(t)The Unit Impulse Function δ(t)The Exponential Function est
The Complex Frequency Plane (s plane)
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 55/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Some Properties of Even and Odd FunctionsEven and Odd Components of a Signal
Current Section1 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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 56/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Some Properties of Even and Odd FunctionsEven and Odd Components of a Signal
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 57/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Some Properties of Even and Odd FunctionsEven and Odd Components of a Signal
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
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 58/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Some Properties of Even and Odd FunctionsEven and Odd Components of a Signal
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)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 59/60
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Useful Operations: Shifting, Scaling, InversionClassification of Signals
Some Useful ModelsEven and Odd Functions
Some Properties of Even and Odd FunctionsEven and Odd Components of a Signal
Example Cont.
Example: Consider the following function, get its even andodd components:
x(t) = e−atu(t)
Dr. Hilal M. El Misilmani Topic 01: Signals’ Properties & Operations 60/60
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