Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is...
Transcript of Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is...
![Page 1: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/1.jpg)
Signal Operations
![Page 2: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/2.jpg)
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.
2
![Page 3: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/3.jpg)
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).
![Page 4: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/4.jpg)
� 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.
4
� If t0 is negative, x(t) is shifted to the leftleft. (Advances)
![Page 5: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/5.jpg)
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
![Page 6: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/6.jpg)
Solution:Solution:(a) x(t -1)
0 4
t
x(t-1)
2
6
![Page 7: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/7.jpg)
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)
![Page 8: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/8.jpg)
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.
8
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.
![Page 9: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/9.jpg)
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
![Page 10: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/10.jpg)
(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
![Page 11: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/11.jpg)
•• 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
![Page 12: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/12.jpg)
•• Quiz 2Quiz 2
• Ansx[n-2]
5
3
0 1 2 3 4 5 n6
![Page 13: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/13.jpg)
� 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.
13
![Page 14: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/14.jpg)
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
![Page 15: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/15.jpg)
Solution:Solution:
(c) x(-t)
2
t
15
-3 1
t
![Page 16: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/16.jpg)
� 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.
![Page 17: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/17.jpg)
STEPS To Remember
• If you have Time shifting and Reversal
(Reflection)Reflection) together
• Do 1st Shifting
• Then Reflection• Then Reflection
![Page 18: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/18.jpg)
•• Question 1Question 1
![Page 19: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/19.jpg)
• Solution
![Page 20: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/20.jpg)
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
![Page 21: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/21.jpg)
(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
![Page 22: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/22.jpg)
(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
![Page 23: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/23.jpg)
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)
![Page 24: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/24.jpg)
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)
24
![Page 25: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/25.jpg)
� 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.
25
![Page 26: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/26.jpg)
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.
![Page 27: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/27.jpg)
� 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…
27
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.
![Page 28: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/28.jpg)
� 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.
28
![Page 29: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/29.jpg)
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
![Page 30: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/30.jpg)
Tutorial 1Tutorial 1
•• Q3Q3
![Page 31: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/31.jpg)
Tutorial 1Tutorial 1
![Page 32: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/32.jpg)
� 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.
32
![Page 33: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/33.jpg)
![Page 34: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/34.jpg)
![Page 35: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/35.jpg)
![Page 36: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/36.jpg)
![Page 37: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/37.jpg)
![Page 38: Signal Operations - WordPress.com€¦ · · 2014-03-20A discreteA discrete--time signal x[n] is shown below, ... Sketch Sketch and label each of the following signaland label each](https://reader033.fdocuments.us/reader033/viewer/2022052712/5aeca2ec7f8b9a3b2e8f692c/html5/thumbnails/38.jpg)