Signal Operation

7/30/2019 Signal Operation

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Signal Operations


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Basic Operation of the Signals.Basic Operation of the Signals.1.3.1.1.3.1. Time ShiftingTime Shifting

1.3.2 Reflection and Folding.1.3.2 Reflection and Folding.

1.3.3.1.3.3. Time ScalingTime Scaling

1.3.4 Precedence Rule for Time Shifting and Time Scaling.1.3.4 Precedence Rule for Time Shifting and Time Scaling.

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Time Shifting

Time shifting is, as the name suggests, the

shifting of a signal in time. This is done byadding or subtracting the amount of the shiftto the time variable in the function.

Subtracting a fixed amount from the timevariable will shift the signal to the right (delay)that amount,

while adding to the time variable will shift thesignal to the left (advance).


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A time shift delaydelay or advancesadvances the signal in time by a time interval +t0 ort0, withoutchanging its shape.

y(t) = x(t - t0)

If t0 is positive the waveform ofy(t) is obtained by shiftingx(t) toward the rightright,relative to the tie axis. (Delay)

1.3.3 Time Shifting.1.3.3 Time Shifting.

4

0 , .


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ExampleExample 1:1: Continuous Signal.Continuous Signal.A CT signal is shown in FigureA CT signal is shown in Figure belowbelow, sketch and label each of this signal;, sketch and label each of this signal;

a)a) x(tx(t - -1)1)

2

x(t)

5

-1 3

t


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Solution:Solution:(a) x(t -1)

0 4

t

x(t-1)

2

6


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Quiz 1:Quiz 1:

Time Shifting.Time Shifting.Given the rectangular pulseGiven the rectangular pulse xx((tt) of unit amplitude and unit duration.) of unit amplitude and unit duration.

FindFindyy((tt)=)=x (tx (t -- 2)2)


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AnsAns Quiz 1:Quiz 1: Time Shifting.Time Shifting.Given the rectangular pulseGiven the rectangular pulse xx((tt) of unit amplitude and unit duration. Find) of unit amplitude and unit duration. Findyy((tt)=)=x (tx (t -- 2)2)

Solution:Solution:t0 is equal to 2 time units. Shift x(t) to the right by 2 time units.

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Figure 1.16: Time-shifting operation:

(a) continuous-time signal in the form of a rectangular pulse of amplitude 1.0 andduration 1.0, symmetric about the origin; and

(b) time-shifted version ofx(t) by 2 time shifts.


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Discrete Time Signal.Discrete Time Signal.

A discreteA discrete--time signal x[n] is shown below,time signal x[n] is shown below,Sketch and label each of the following signal.Sketch and label each of the following signal.

(a) x[n(a) x[n 2]2]

x[n]

9

n

5

3

0 1 2 3


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(a) A discrete-time signal,x[n-2].

A delay by 2

x[n-2]

ContdContd

10

5

3

0 1 2 3 4 5 n


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QuizQuiz 22Discrete Time Signal.Discrete Time Signal.

A discreteA discrete--time signal x[n] is shown below,time signal x[n] is shown below, SketchSketch and label each of the following signaland label each of the following signal..

(a)(a) x[nx[n 3]3] x[n]

n

5

3

0 1 2 3


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Quiz 2Quiz 2

Ans x[n-2]

5

3

0 1 2 3 4 5 n6


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Let x(t) denote a continuous-time signal andy(t) is the signal obtained byreplacingreplacing timetwith t;

y(t) is the signal represents a refracted version ofx(t) about t= 0.

Two special casesspecial cases for continuous and discrete-time signal;

(i) Even signal; x(-t) = x(t) an even signal is same as reflected version.

(ii) Odd signal; x(-t) = -x(t) an odd signal is the negative of its reflected version.

( ) ( )txty =

1.3.2 Reflection and Folding.1.3.2 Reflection and Folding.

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ExampleExample ::..A CT signal is shown in Figure 1.17 below, sketch and label each of this signal;A CT signal is shown in Figure 1.17 below, sketch and label each of this signal;

a)a) x(x(--t)t)

2

x(t)

14

-1 3

t


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Solution:Solution:

(c)x(-t)

2

t

15

-3 1


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The continuous-timeversion of the unit-step function is defined by,

( )0

0

,0

,1

=t

ttu

ContdContd

16

The discontinuity exhibit at t= 0 and the value of u(t) changes instantaneously from 0 to 1when t= 0. That is the reason whyu(0) is undefined.


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STEPS To Remember

If you have Time shifting and Reversal

(Reflection)Reflection) together Do 1st Shifting

T en Re ection


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Question 1Question 1


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Solution


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Question 2:Question 2: Discrete Time Signal.Discrete Time Signal.

A discreteA discrete--time signal x[n] is shown below,time signal x[n] is shown below,Sketch and label each of the following signal.Sketch and label each of the following signal.

(a) x[(a) x[--n+2]n+2] (b) x[ (b) x[--n]n]

x[n]

20

n

6

4

0 1 2 3


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(a) A discrete-time signal,x[-n+2].

Time shifting and reversal

x(-n+2)

ContdContd

21

6

4

-1 0 1 2 n


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(b) A discrete-time signal,x[-n].

Time reversal

x(-n)

ContdContd

22

4

-3 -2 -1 0 1 n


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Question 2 ContinuousQuestion 2 Continuous SignalSignalAA continuouscontinuous signalsignal xx((tt)) isis shownshown inin FigureFigure.. SketchSketch andand labellabel eacheach ofof thethe followingfollowing signalssignals..

a)a) x(t)=x(t)= u(tu(t - -11))

b)b) x(t)= [u(t)x(t)= [u(t)--u(tu(t--1)]1)]c)c) x(t)=x(t)= (t(t -- 3/2)3/2)


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Solution:Solution:

(a) x(t)= u(t(a) x(t)= u(t --1)1) ((b)b) x(t)= [u(t)x(t)= [u(t)--u(tu(t--1)] (1)] (c)c) x(t)=d(tx(t)=d(t -- 3/2)3/2)

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Time scaling refers to the multiplicationmultiplication of the variable by a real positive constant.

Ifaa > 1 the signaly(t) is a compressedcompressedversion ofx(t).

If 0 < aa < 1 the signal y(t) is an expandedexpandedversion ofx(t).

( ) ( )atxty =

TimeTime Scaling.Scaling.

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Example

Theimage cannotbe displayed.Your computer may nothave enough memory to open theimage,or theimage may havebeen corrupted. Restartyour computer,and then open thefileagain. Ifthe red x stillappears,yo u may haveto deletetheimage and then insertit again.

C tdC td


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In the discrete time,

It is defined for integer value of k, k > 1. Figure below for k = 2, sample for n = +-1,

[ ] [ ],knxny =

ContdContd

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Figure 1.12: Effect of time scaling on a discrete-time signal:(a) discrete-time signalx[n] and (b) version ofx[n] compressed by a factor of 2, with

some values of the originalx[n] lost as a result of the compression.


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The discrete-time version of the unit-step function is defined by,

[ ]0

0

,0

,1

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Signal Operation