Signal Operations

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 by adding or subtracting the amount of the shift to the time variable in the function.

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

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

� A time shift delaydelay or advancesadvances the signal in time by a time interval +t0 or –t0, without changing its shape.

y(t) = x(t - t0)

� If t0 is positive the waveform of y(t) is obtained by shifting x(t) toward the rightright,

relative to the tie axis. (Delay)

� If t0 is negative, x(t) is shifted to the leftleft. (Advances)

1.3.3 Time Shifting.1.3.3 Time Shifting.

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� If t0 is negative, x(t) is shifted to the leftleft. (Advances)

Example Example 1:1: Continuous Signal. Continuous Signal. A CT signal is shown in Figure A 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

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

0 4

t

x(t-1)

2

6

Quiz 1: Quiz 1:

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

Find Find yy((tt)=)=x (t x (t -- 2)2)

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

duration 1.0, symmetric about the origin; and

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

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

x[n]

n

5

3

0 1 2 3

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

�A delay by 2

x[n-2]

Cont’d…Cont’d…

10

5

3

0 1 2 3 4 5 n

•• Quiz Quiz 22Discrete Time Signal. Discrete Time Signal.

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

(a)(a) x[n x[n –– 3]3] x[n]

n

5

3

0 1 2 3

•• Quiz 2Quiz 2

• Ansx[n-2]

5

3

0 1 2 3 4 5 n6

� Let x(t) denote a continuous-time signal and y(t) is the signal obtained by replacingreplacing time

t with –t;

� y(t) is the signal represents a refracted version of x(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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Example Example ::. . 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

Solution:Solution:

(c) x(-t)

2

t

15

-3 1

t

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

( )0

0

,0

,1

<

>

=t

ttu

Cont’d…Cont’d…

16

� The discontinuity exhibit at t = 0 and the value of u(t) changes instantaneously from 0 to 1

when t = 0. That is the reason why u(0) is undefined.

STEPS To Remember

• If you have Time shifting and Reversal

(Reflection)Reflection) together

• Do 1st Shifting

• Then Reflection• Then Reflection

•• Question 1Question 1

• Solution

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

x[n]

n

6

4

0 1 2 3

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

Time shifting and reversal

x(-n+2)

Cont’d…Cont’d…

21

6

4

-1 0 1 2 n

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

�Time reversal

6

x(-n)

Cont’d…Cont’d…

22

6

4

-3 -2 -1 0 1 n

Question 2 Continuous Question 2 Continuous SignalSignal

AA 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)= δδ(t (t -- 3/2)3/2)

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(t x(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.

� If aa > 1 the signal y(t) is a compressedcompressed version of x(t).

� If 0 < aa < 1 the signal y(t) is an expandedexpanded version of x(t).

( ) ( )atxty =

Time Time Scaling.Scaling.

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Example

The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

� In the discrete time,

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

[ ] [ ],knxny =

Cont’d…Cont’d…

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Figure 1.12: Effect of time scaling on a discrete-time signal:

(a) discrete-time signal x[n] and (b) version of x[n] compressed by a factor of 2, with

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

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

[ ]0

0

,0

,1

<

≥

=n

nnu

1.4.5 Step Function.1.4.5 Step Function.

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Tutorial 1Tutorial 1

Q 1Q 1A continuousA continuous--time signal time signal x(t)x(t) is shown below, Sketch and label is shown below, Sketch and label

each of the following signaleach of the following signal

(a) (a) x(t x(t –– 2)2) (b) (b) x(2t)x(2t) (c.) (c.) x(t/2)x(t/2) (d) (d) x(x(--t)t)

x(t)

4

29

t0 4

Tutorial 1Tutorial 1

•• Q3Q3

Tutorial 1Tutorial 1

� The discretediscrete--timetime version of the unit impulse is defined by,

[ ]

≠

==

0,0

0,1

n

nnδ

1.4.6 Impulse Function.1.4.6 Impulse Function.

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