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Introduction to Signals & Systems
Signals & Systems
Examples of practicalCommunication andControl systems
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 1
Notions
SignalSystem
Noise
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Scope
Study of mathematical concepts and
techni ues, useful for anal sis of
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 2
communication andcontrolsystems
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Introduction to Si nal Theor
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 3
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Signal ConceptsWhat is a Signal?What is a Signal?What is a Signal?What is a Signal?
Variation of aphysical quantity,containing someinformation
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 4
Signal DescriptionSignal DescriptionSignal DescriptionSignal Description
Mathematical Model:Single-valued function
ConceptsConceptsConceptsConcepts
DomainDimension
Energy/ PowerCross Energy/PowerNorm
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Signal ClassificationDimensionDimensionDimensionDimension
Single/ multi-dimensional
DomainDomainDomainDomain
Time/ frequency / spatial
ValueValueValueValue
Real/ Complex
Extent in magnitudeExtent in magnitudeExtent in magnitudeExtent in magnitude
Bounded/ unbounded
PredictabilityPredictabilityPredictabilityPredictability
Deterministic/ Random
Relative to time originRelative to time originRelative to time originRelative to time origin
Causal/ non-causal
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 5
Extent in timeExtent in timeExtent in timeExtent in time
Finite/ eternal/ semi- infinite
Finiteness of Energy/ PowerFiniteness of Energy/ PowerFiniteness of Energy/ PowerFiniteness of Energy/ Power
Energy/ power
Continuity/ quantizationContinuity/ quantizationContinuity/ quantizationContinuity/ quantization
Continuous/ discrete
Symmetry about originSymmetry about originSymmetry about originSymmetry about origin
Even/ odd
PeriodicityPeriodicityPeriodicityPeriodicity
Aperiodic/ periodic
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Some Properties of Periodic SignalsA periodic function is also periodic with integer multiples of
the fundamental period.
Sum of two periodic signals has a fundamental period = LCMof the two periods
Product of two periodic signals has a period = LCM of the
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 6
two perio s not necessari y un amenta perio .
Sum or product of periodic signals would not be periodic ifratio of the periods is irrational.
A periodic signal with period T is said to have half-wave orrotational symmetry if f(tT/2)= f(t)
All periodic signals bounded in amplitude are power signals.
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Some Notes of Even & Odd Signalseven + even = evenodd + odd = oddodd + even = neither odd nor even
An even function expressed as a sum cannot have any odd components; and, an oddfunction can not have any even components.
Any arbitrary signal can be split into an even
An odd function alwayspasses through the origin.Integral of an odd functionequals zero.
Constant is an evenfunction.
Multiplication by an odd
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 7
( ) ( ) ( )
( ) ( )( )
2( ) ( )
( )2
even odd
even
odd
f t f t f t
f t f t f t
f t f t f t
= +
+ =
=
even x even = even
odd x odd = eveneven x odd = odd
part and an odd part in a unique manner:function alters evenness tooddness and vice versa,while multiplication by aneven function does not.
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Some Elementary/ Standard Signalsexponential sinusoidal
step u(t) impulseramp r(t)
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 8
rectangular signum sgn(x)
sinc(t)
sin( )
=
t
t
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Elementary Signals contd..Unit Impulse:
(t) = 0 for t 0
(t) dt = 1
Unit Step:
u(t) = 0 for t 0
1 for t > 0
Unit Ramp:
r(t) = 0 for t 0
t for t > 0
( )0as a Some functions which approach (t) in the limit
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 9
1 11. Rectangular pulse for 2. Triangular pulse 1 for2 2
1 13. Exponential pulse 4. Double exponential pulse
2
5. Sinc pulse
tt
aa
ta at ta a a
e ea a
<
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Elementary Signals contd..Ex. Classify u(t) and exp(-2t)u(t) as energy/ power signals
Ex. Find even and odd components of exponential and step functions
Ex. Show that derivative of u(t) equals (t); also verify that running integralof(t) equals u(t).
Ex. Find the area under sinc(t) by integration
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 10
Some notesSignal value at a discontinuity (such as in a step) for practical purposes maybe equated to the left limit, right limit or the mid-value.
Integral value of a continuous function equals area under its graph (positivearea - negative area)
Random waveforms produced by sources of finite average power, are powersignals.
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Mathematical Operations on Signals
Operations ona signal
Operations ontwo signals
Amplitude Shift
Convolution
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 11
me
Amplitude Scaling
Time Scaling
Correlation
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Transformation of Independent Variable
A function once defined, is invariant under transformation of independentvariable. That is, the function value for any value of argument remains thesame even after transformation.
f(t-1) is f(t) shifted right by one, and f(t+1) is f(t) shifted left by one.
f(at) is compression of f(t) for a>0, expansion for a
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Reversal about
t = 0.5
Transformation of Independent Variable:
Example
0
1 1
1
1 2
t
t t
f(t)
f(2t-1) f(-2t+1)
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 13
0
0
0
1
0.5 1.5 - 0.5 0.5
0- 0.5 0.5 - 0.5-1.5
1
t t
f(2t+1)
f(-2t-1)Reversal about
t = - 0.5
Reversal about t= 0
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Signal Operations contd..
Ex: For the signals x(t) and h(t) sketched below, find and sketcha) x(t)h(t+1) b)x(2- 0.5t) c) x(t-1) h(1-t) d) x(t)h(-t)
Ex.: For the f(t) given below sketch f(2t-1), f(2t+1), f(-2t+1) and f(-2t-1).Also verify.
0 1 2
1
t
f(t)
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 14
32-1 1
1
2
210-1-2
1
-1
t t
x(t) h(t)
Verify your answers by substituting suitable values for t.
0
Ex: Express f(t), x(t) and h(t) given above, as linear combinations of other
basic signals with suitable transformations.
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Convolution of Signals:
Example of Graphical Method
Procedure
Change to
dummy variable
Time-reverse
either signal
y(t) = x(t) h(t) = x( )h(t- )d
Presented by APN Rao, Dept ECE, GRIET, Hyderabad. Jul 2011 15
Shift
Multiply
Find area
Repeat
1 3
product function
for t