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signals and systems_isooperations.pptx
1. SECA 1301 –
Signals and Systems
Dr. S. Emalda Roslin, M.E., Ph.D.,/ Dr. Grace Kanmani, M.E., Ph.D.
Professor,
Department of ECE,
Sathyabama Institute of Science and Technology
SATHYABAMA
INSTITUTE OF SCIENCE AND TECHNOLOGY
1
3. 3
INTRODUCTION TO SIGNALS
Signals- Continuous time signals (CT signals) and Discrete time
signals (DT signals) -Step, Ramp, Pulse, Impulse, Exponential and
Sinusoidal Signal – Basic Operations on signals -Amplitude Scaling,
Time Scaling, Time Reversal, Time Shifting, Signal Addition,
Subtraction-classification of CT and DT signals-Deterministic and
Non-deterministic Signals, Even and Odd Signals, Periodic and
Aperiodic Signals, Energy and Power Signals, Real and Imaginary
Signals
UNIT 1
4. COURSE OUTCOMES
CO1: Perform various operations on different types of signals and
systems
CO2: Analyze suitable transforms for continuous time and discrete
time signals
CO3: Develop Input output relationship for Linear Time Invariant
systems
CO4: Evaluate various responses of LTI systems
CO5: Apply Mathematical tools for analysis of various real time
signals
CO6: Build and analyze real time systems for various applications
4
5. A signal is any physical phenomenon which conveys information
or
A signal is defined as a physical quantity that varies with time, space or
any other independent variables.
• Signals are represented mathematically as functions of
one or more independent variables that contain
information
SIGNAL
Continuous Time Signals
• The signals that are defined for every instant
of time
• Functions whose amplitude value
continuously varies with time
Discrete Time Signals
• The signals that are defined for discrete
instant of time
• Functions whose amplitude value does not
continuously varies with time
5
6. REPRESENTATION OF DISCRETE TIME SIGNALS
• Graphical Representation
• Functional Representation
• Tabular Representation
• Sequence Representation
x(n) = {3,5,5,3}
n 0 1 2 3
x(n) 3 5 5 3
3
3
2
5
1
5
0
3
)
(
n
for
n
for
n
for
n
for
n
x
Graphical Representation Functional Representation Tabular Representation
Sequence Representation
6
7. REPRESENTATION OF DISCRETE TIME SIGNALS
0 1 2 3 4 5 6 …. n
5
3
x(n)
At n= 0, x(0)=3
At n= 1, x(1)=5
At n= 2, x(2)=5
At n= 3, x(3)=3
x(n) = {3,5,5,3}
n=0 x(0)=3
n=1 x(1)=5
n=2 x(2)=5
n=3 x(3)=3
n 0 1 2 3
x(n) 3 5 5 3
7
8. x(n)={0,1,2,3,4,3,2,1,0}
At n=-4 , x(-4)=0
At n=-3 , x(-3)=1
At n=-2 , x(-2)=2
At n=-1 , x(-1)=3
At n=0 , x(0)=4
At n=1 , x(1)=3
At n=2 , x(2)=2
At n=3 , x(3)=1
At n=4 , x(4)=0
x(0)=4
-4 -3 -2 -1 0 1 2 3 4 … n
x(-1)=3
x(-2)=2
x(-3)=1
x(-4)=0
x(1)=3
x(2)=2
x(3)=1
x(4)=0
REPRESENTATION OF DISCRETE TIME SIGNALS
8
10. OPERATIONS ON DT SIGNALS
• Time Shifting y(n) = x(n-T)
• Time Reversal y(n) = x(-n)
• Amplitude Scaling y(n) = ax(n)
• Time Scaling y(n) = x(an)
• Addition y(n) =x1(n) +
x2(n)
10
11. 1. TIME SHIFTING
Time shifting (also called translation) maps the input signal x to the output signal y as
given by
y ( t) = x ( t − T ), where T is a real number.
For a continuous time signal x(t), time shifting may occur at a
delay or advance the signal in time
Such a transformation shifts the signal (to the left or right) along the time axis.
• If T > 0, y is shifted to the right by |T|, relative to x (i.e., delayed in time or shifting
delay)
• If T < 0, y is shifted to the left by |T|, relative to x (i.e., advanced in time or shifting
advance).
11
12. Problem : A discrete-time signal x[n] is shown below,
Sketch and label x(n-2).
0 1 2 3 4 5 6 …. n
5
3
x(n)
At n= 0, x(0)=3
At n= 1, x(1)=5
At n= 2, x(2)=5
At n= 3, x(3)=3
0 1 2 3 4 5 6 …. n
x(n-2)
5
3
At n= 2, x(2)=3
At n= 3, x(3)=5
At n= 4, x(4)=5
At n= 5, x(5)=3
x(n) = {3,5,5,3}
n=0 x(0)=3
n=1 x(1)=5
n=2 x(2)=5
n=3 x(3)=3
n 0 1 2 3
x(n) 3 5 5 3
12
13. Problem : A discrete-time signal x[n] is shown below,
Sketch and label x(n+2).
0 1 2 3 4 5 6 …. n
5
3
x(n)
At n= 0, x(0)=3
At n= 1, x(1)=5
At n= 2, x(2)=5
At n= 3, x(3)=3
-2 -1 0 1 2 3 4 5 6 …. n
x(n+2)
5
3
At n= -2, x(-2)=3
At n= -1, x(-1)=5
At n= 0, x(0)=5
At n= 1, x(1)=3
x(n) = {3,5,5,3}
13
14. Problem : A DT signal x(n)={0,1,1,0.5}, sketch and label x(n-
3) and x(n+3) n=0 x(0)=0
n=1 x(1)=1
n=2 x(2)=1
n=3 x(3)=0.5
14
15. Problem : A DT signal x(n) shown in Figure, sketch and label
x(n-2) and x(n+3)
15
16. Problem : A CT signal is shown in Figure , sketch and label
time delayed signal y(t);
a) y(t) = x(t -1)
b) y(t)=x (t - 2)
-1 0 1 2 3 4 …. t
y(t)=x(t-1)
16
17. 2. TIME REVERSAL / FOLDING
• Time reversal of the signal is done by folding the signal about at t= 0
• Let x(t) denote a continuous-time signal and y(t) is the signal obtained by
replacing time t with –t; y(t) = x(− t)
x(-t)
a)
b)
-3 -2 -1 0 1 2 3 4 5 6 …. n
x(-n)
6
4
17
18. Problem: A discrete-time signal x[n] is shown below,
Sketch and label y(n)=x(-n+3).
(a) x[-n+2]
Step 1: folding or reversal x(n) to get x(-n)
-3 -2 -1 0 1 2 3 4 5 6 …. n
x(-n)
6
4
Step 2: Time shifting of x(-n) to get x(-n+3)
x(-n+3)
6
4
-3 -2 -1 0 1 2 3 4 5 6 …. n
18
19. 19
Problem: Draw the waveform x(-t) and x(2-t) of the
signal
x(t) = t ; 0≤ t ≤3
0 ; t > 3
t=0 x(0)=3
t=1 x(1)=3
t=2 x(2)=3
t=3 x(3)=3
20. Problem: A CT signal x2(t) and x3(t) is shown below,
Sketch and label the folded signals.
20
22. Problem: A continuous-time signal x[t] is shown below,
Sketch and label the signal y(t) = 4 x(t) for the given signal
x(t).
x(0)=1 ; t= 0 to 1
x(1)=1.5; t= 1 to 2
x(2)=2 ; t= 2 to 3
y(0)=4*x(0)=4*1 =4; t= 0 to 1
y(1)=4*x(1)=4*1.5 =6; t= 1 to 2
y(0)=4*x(2)=4*2 =8; t= 2 to 3
y(t)=4 x(t)
22
23. Problem: A continuous-time signal x[t] is shown below,
Sketch and label the signal y(t) = 2 x(t) and y(t) = 0.5 x(t)
for the given signal x(t).
-1 -1 -1
1 1
1
At t= -1 x(-1)=2
At t= 0 x(0)=2
At t= 1 x(1)=2
t t
t
At t=-1 y(-1)=2*x(-1)=2*2 =4
At t=0 y(0)=2*x(0)=2*2 =4
At t=1 y(1)=2*x(1)=2*2 =4
At t=-1 y(-1)=0.5*x(-1)=0.5*2 =1
At t=0 y(0)=0.5*x(0)=0.5*2 =1
At t=1 y(1)=0.5*x(1)=0.5*2 =1
23
y(t)=2*x(t)
y(t)=0.5*x(t)
24. 4. TIME SCALING
• Time scaling is obtained by replacing t by a*t in the signal
x(t)
y(t) = x(at)
• Time scaling refers to the multiplication of the variable by a
real positive constant.
• If a > 1 the signal y(t) is a compressed version of x(t).
If 0 < a < 1 the signal y(t) is an expanded version of x(t).
24
25. Problem: A Discrete-time sequence x[n] is given below,
Sketch and label the signal y(n) = x(2n) for the given signal
x(n).
x(n)={0,1,2,3,4,3,2,1,0}
At n=-4 , x(-4)=0
At n=-3 , x(-3)=1
At n=-2 , x(-2)=2
At n=-1 , x(-1)=3
At n=0 , x(0)=4
At n=1 , x(1)=3
At n=2 , x(2)=2
At n=3 , x(3)=1
At n=4 , x(4)=0
y(n)=x(2n)
n= -3 y(-3)=x(2*-3)= x(-6)=0
n= -2 y(-2)=x(2*-2)= x(-4)=0
n= -1 y(-1)=x(2*-1)= x(-2)=2
n= 0 y(0)=x(2*0)= x(0)=4
n= 1 y(1)=x(2*1)= x(2)=2
n= 2 y(2)=x(2*2)= x(4)=0
n= 3 y(3)=x(2*3)= x(6)=0
x(0)=4
-4 -3 -2 -1 0 1 2 3 4 … n
x(-1)=3
x(-2)=2
x(-3)=1
x(-4)=0
x(1)=3
x(2)=2
x(3)=1
x(4)=0
-4 -3 -2 -1 0 1 2 3 4 … n
y(n)=x(2n)
y(-1)=2
y(0)=4
y(1)=2
25
26. 5. SIGNALADDITION /SUBTRACTION
26
• The sum of two continuous time signals can be obtained by adding their
values at every instant.
• Similarly the subtraction of two continuous time signal can be obtained by
subtracting their values at every instant.
+
+
x1(t)
x2(t)
y(t)=x1(t)+x2(t) +
-
x1(t)
x2(t)
y(t)=x1(t)-x2(t)
y(t) = x1(t) ± x2(t)
27. 27
• The sum of two SEQUENCES can be obtained by adding sequence values at
the same index ‘n’ for which the sequences are identified.
• The subtraction of two SEQUENCES can be obtained by subtracting the
sequence values at the same index ‘n’ for which the sequences are
identified.
+
+
X1(n)
x2(n)
y(n)=x1(n)+x2(n) +
-
X1(n)
X2(n)
y(n)=x1(n)-x2(n)
y(n) = x1(n) ± x2(n)
FOR DISCRETE TIME SIGNAL
28. 28
Problem: Sketch and label the summation of the CT signals
x1(t) and x2(t)
At t=0 x1(0) = 1 x2(0) = 0.2 y(0)=x1(0)+x2(0)=1+0.2=1.2
At t=1 x1(1) = 2 x2(1) = 1 y(1)=x1(1)+x2(1)=2+1=3
At t=2 x1(2) = 0.5 x2(2) = 0.5 y(2)=x2(2)+x2(2)=0.5+0.5=1
29. 29
Problem: Sketch and label the difference of the sequences
x1(n) and x2(n)
The difference between the sequences is given by y(n) = x1(n) – x2(n)
30. 30
6. SIGNAL MULTIPLICATION
The multiplication of two signals can be obtained by
multiplying their values at every instant.
y(t) = x1(t) * x2(t)
x1(t)
x2(t)
y(t)=x1(t)*x2(t)
x1(n)
x2(n)
y(n)=x1(n)*x2(n)
31. 31
Problem: Sketch and label z(t), the multiplication of the
given signals x1(t) and x2(t).
x1(t)*x2(t)=0*2=0
