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SIGNALS & SYSTEMS
Unit:I Mr. M. S. PatilVIIT, Pune-48
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Elementary Signals
Exponential signals
Sinusoidal signals
Exponentially damped sinusoidal signals
( ) atx t Be
( ) cos( )x t A t
( ) cos( )atx t Ae t
Sinusoidal & Exponential Signals• Sinusoids and exponentials are important in signal
and system analysis because they arise naturally inthe solutions of the differential equations.
• Sinusoidal Signals can expressed in either of twoways :cyclic frequency form- A sin 2Пfot = A sin(2П/To)tradian frequency form- A sin ωot
ωo = 2Пfo = 2П/To
To = Time Period of the Sinusoidal Wave
Sinusoidal & Exponential Signals Contd.
x(t) = A sin (2Пfot+ θ)= A sin (ωot+ θ)
x(t) = Aeat Real Exponential
= Aejω̥t = A[cos (ωot) +j sin (ωot)] Complex Exponential
θ = Phase of sinusoidal waveA = amplitude of a sinusoidal or exponential signalfo = fundamental cyclic frequency of sinusoidal signalωo = radian frequency
Sinusoidal signal
x(t) = e-at
x(t) = eαt
Real Exponential Signals and damped Sinusoidal
Discrete Time Exponential and Sinusoidal Signals
• DT signals can be defined in a manner analogous to their continuous-time counter partx[n] = A sin (2Пn/No+θ)
= A sin (2ПFon+ θ)
x[n] = an
n = the discrete timeA = amplitudeθ = phase shifting radians, No = Discrete Period of the wave
1/N0 = Fo = Ωo/2 П = Discrete Frequency
Discrete Time Sinusoidal Signal
Discrete Time Exponential Signal
Discrete Time Sinusoidal Signals
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Real exponential sequence
nanx )(
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
x(n)
. . .. . .
Unit Step Function
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Step function ( ) ( )x t u t
Unit Ramp Function
, 0ramp u u
0 , 0
tt tt d t t
t
•The unit ramp function is the integral of the unit step function.•It is called the unit ramp function because for positive t, its slope is one amplitude unit per time.
Discrete Time Unit Ramp Function
, 0ramp u 1
0 , 0
n
m
n nn m
n
Unit Impulse Function
As approaches zero, g approaches a unit
step andg approaches a unit impulse
a t
t
So unit impulse function is the derivative of the unit step function or unit step is the integral of the unit impulse function
Functions that approach unit step and unit impulse
Representation of Impulse Function
The area under an impulse is called its strength or weight. It is represented graphically by a vertical arrow. An impulse with a strength of one is called a unit impulse.
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Impulse function ( ) ( )x t t
(a) Evolution of a rectangular pulse of unit area into an impulse of unit strength (i.e., unit impulse). (b) Graphical symbol for unit impulse. (c) Representation of an impulse of strength a that results from allowing the duration Δ of a rectangular pulse of area a to approach zero.
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Unit impulse – the shape doesn’t matter
There is nothing special about rectangular pulse approx.to the unit impulse. A triangular pulse approximation is just as good. As far as our definition is concerned both the rectangular and triangular pulse are equally good approximations. Both can act as impulses
1/
t 2
1/
t
Area = 1
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3 important properties of unit impulse
Sampling properties
000 tttxtttx
Scaling properties
abt
abat 1
The Replication Property
g(t)⊗ δ(t) = g (t)
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Exercise:
dttt )10(2
5
0
2 )10( dttt
20
0
2 )10( dttt
Evaluate the following signals:
a)
b)
c)
= 100
= 0
= 100
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Integration of the unit impulse yields the unit step function
dtut
Conversely, the impulse is the derivative of the step function
t t
Unit impulse, (t) Unit step, u(t)
1 1integration
differentiation
)()( tudtdt
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Unit impulse – what do we need it for?
The unit impulse is a valuable idealization and is used widely in science and engineering. Impulses in time are useful idealizations
impulse of current in time delivers a unit charge instantaneously to a network.
impulse of force in time delivers an instantaneous momentum to a mechanical system.
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Unit-sample sequence (n)
0001
)(nn
n
1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8
n
(n)1
Sometime call (n) a discrete-time impulse; or an impulse )1()()( nunun
Fact:
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(a) Rectangular pulse x(t) of amplitude A and duration of 1 s, symmetric about the origin. (b) Representation of x(t) as the difference of two step functions of amplitude A, with one step function shifted to the left by ½ and the other shifted to the right by ½; the two shifted signals are denoted by x1(t) and x2(t), respectively. Note that x(t) = x1(t) – x2(t).
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Why it’s called singularity?
A set of functions relating through integrals and derivatives to mathematically describe signals
• using combination of singularity functions, we can form more complicated function
Singularity functions
Unit step Unit rampintegration
differentiation
Unit ramp Unit parabolicdifferentiation
integration
Unit impulse Unit stepintegration
differentiation
Signum Function
1 , 0
sgn 0 , 0 2u 11 , 0
tt t t
t
Precise Graph Commonly-Used Graph
The signum function, is closely related to the unit-step function.
Sinc Function
sinsinc
tt
t
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Basic Operations on Signals
•Operations performed on dependent signals
•Operations performed on the independent signals
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Operations performed on dependent signals
Amplitude scaling
Addition
Multiplication
Differentiation
Integration
( ) ( )y t cx t
1 2( ) ( ) ( )y t x t x t
1 2( ) ( ) ( )y t x t x t
( ) ( )dy t x tdx
( ) ( )t
y t x d
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Operations performed on the independent signals
Time scalinga>1 : compressed0<a<1 : expanded
( ) ( )y t x at
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Operations performed on the independent signals
Reflection or folding
( ) ( )y t x t
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Operations performed on the independent signals
Time shifting/ Origin Shifting
Precedence Rule for time shifting & time scaling
0( ) ( )y t x t t
( ) ( ) ( ( ))by t x at b x a ta
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The incorrect way of applying the precedence rule. (a) Signal x(t). (b) Time-scaled signal v(t) = x(2t). (c) Signal y(t) obtained by shifting v(t) = x(2t) by 3 time units, which yields y(t) = x(2(t + 3)).
The proper order in which the operations of time scaling and time shifting (a) Rectangular pulse x(t) of amplitude 1.0 and duration 2.0, symmetric about the origin. (b) Intermediate pulse v(t), representing a time-shifted version of x(t). (c) Desired signal y(t), resulting from the compression of v(t) by a factor of 2.
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• All these singularity signals can be combined to produce pulse signals
u(t) - u(t-1) r(t) - 2r(t-1)+r(t-2)
1
1
t t1 2
1
To do the above operation, you need to understand the operation on signals !
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Classification of Signals
Continuous and discrete-time signals Continuous and discrete-valued signals Even and odd signals Periodic signals, non-periodic signals Deterministic signals, random signals Causal and anticausal signals Right-handed and left-handed signals Finite and infinite length
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Classification of signals
CT or DT
Deterministic Random/stochastic
Periodic Aperiodic
Complex Transient Almostperiodic
Sinusoid
Power, Energy
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Deterministic and Random Signals
Deterministic signal: Signal which can be completely modeled as function of time.
Random signal: Signal which is processed using probability theory.
e.g. x(t) = sin(20πt) for all t
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Deterministic signals, random signals
Deterministic signals -There is no uncertainty with respect to its value
at any time. (ex) sin(3t)
Random signals - There is uncertainty before its actual
occurrence.
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Even and odd signals
Even signals : x(-t)=x(t)Odd signals : x(-t)=-x(t)
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Even and Odd CT Functions
Even Functions Odd Functions
g t g t g t g t
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Even and Odd Parts of Functions
g gThe of a function is g
2e
t tt
even part
g gThe of a function is g
2o
t tt
odd part
A function whose even part is zero is odd and a functionwhose odd part is zero is even.
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Combination of even and odd function
Function type
Sum Difference Product Quotient
Both even Even Even Even Even
Both odd Odd Odd Even Even
Even and odd
Neither Neither Odd Odd
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Two Even Functions
Products of Even and Odd Functions
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Cont…An Even Function and an Odd Function
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An Even Function and an Odd Function
Cont…
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Two Odd Functions
Cont…
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Integrals of Even and Odd Functions
0
g 2 ga a
a
t dt t dt
g 0a
a
t dt
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Periodic signals, non-periodic signals
Periodic signals - A function that satisfies the condition
x(t)=x(t+T) for all t
- Fundamental frequency : f=1/T
- Angular frequency : = 2/T
Non-periodic signals
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Periodicity Condition for DT Signal
A sequence x(n) is defined to be periodic with period N if
NNnxnx allfor )()( Example: consider njenx 0)(
)()( 0000 )( Nnxeeeenx njNjNnjnj
kN 20 0
2
k
N0
2 must be a rational
number
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How to prove Periodic or Aperiodic
To prove periodic: - for a simple sinusoid signal, use
x(t+To)=x(t)- for a sum or a product of 2 signals, find the
ratio of their periods that can be expressed as rational number. Use least common multiple method if possible.
Try these:1. x(t)=10 sin(12πt) + 4 cos (18πt)2. x(t)=10 sin(12πt) + 4 cos (18t)3. x(t)=10 cos(πt) + sin (4πt)
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Signal Energy and Power
How to express size of a signal by as single value??
Integration : Area under the signalPower: Time Average of Energy and MS
value of a signalUse: Comm-SNR, Image: MSE
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Signal Energy and Power
2x xE t dt
The signal energy of a signal x(t) is
All physical activity is mediated by a transfer of energy.
No real physical system can respond to an excitation unless it has energy.Signal energy of a signal is defined as the area under the square of the magnitude of the signal.
The units of signal energy depends on the unit of the signal.
If the signal unit is volt (V), the energy of that signal is expressed in V2.s.
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Signal Energy and PowerSome signals have infinite signal energy. In that caseit is more convenient to deal with average signal power.
/ 2
2x
/ 2
1lim xT
TT
P t dtT
The average signal power of a signal x(t) is
For a periodic signal x(t) the average signal power is
2x
1 xT
P t dtT
where T is any period of the signal.
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Signal Energy and Power
A signal with finite signal energy is called an energy signal.
A signal with infinite signal energy and finite average signal power is called a power signal.
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Energy and Power Signals
• x(t) : energy signal iif 0 < E < , so that P = 0• x(t) : power signal iif 0 < P < , so that E = • If cannot satisfy both properties: neither energy nor power signal
00
0
2
0
1 Tt
tave dttx
TPFor periodic signal :
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Energy of a Sequence Energy of a sequence is defined by
|)(| 2
n
nnxE