Signals and Systems
Introduction
Instructor Details
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Name: Dr Ramesh V
Cabin: TT240
Email: vramesh@vit.ac.in
Mobile: 9944730510
Syllabus and Textbook: Download from Course
Page
What is a signal?
• Class discussion
What is a system?
• Class discussion
Signal power
• Many of the electrically transduced signals are
voltages.
• By Ohms law power=v2/R
v2
1 2
• Energy=  dt   v dt
R
R
• Average Power for t1≤t≤t2
t2
1
v2
dt

t 2  t1 t1 R
Signal Energy and Power
• The total energy
for any signal x(t) for t1≤t≤t2
t
2
x
(
t
)
dt . (x may be complex).
is given by 
t
• Dividing by the time average t2-t1 the average
power is obtained
• Similarly for a discrete signal
the total energy
n
2
for n1≤n≤n2 is given by x[n] .
nn
• Dividing by the number of points n2-n1+1 the
average power between n1 and n2 is obtained
2
1
2
1
Power and Energy in the infinite range
• t1 can be -∞ and t2 can be + ∞. Similarly for n1
and n2. In that case we conclude as follows
• Signals with finite energy will have zero
average power.
• Signal with finite average power will have
infinite energy. (Sinusoidal Signal)
• A signal with neither finite power nor finite
energy in the infinite range is x(t)=t
Example
• Energy Signal x(t)=e-t for t>=0, 0 for t<0


2 t
e
E   | e t |2 dt   e  2t dt  
 1/ 2
2

0
• Average Power (1/2)/∞=0
• Power Signal x(t)=Acos(ωt+ϴ)
1
P  lim
t  2T
A2
 lim
t  2T
T
2




|
A
cos

t


|
dt

T
2
1  cos2t  2 
A
dt 
T
2
2
T
Signal Transformation-Time Shift
• f(t+a) is f(t) shifted leftwards by a distance a if
a is positive and rightwards if a is negative.
• Examples
Signal Transformation-Scaling
• For f(αt) the t coordinate is divided by 1/|α|
while the vertical ordinate is unchanged. Thus
if |α|>1 there is compression and |α|<1
there is stretching
• Examples
Signal Transformation-Time Reversal
• f(-t) is obtained by reflecting the right half
plane to the left about the vertical axis and
the left half plane to the right.
• Examples
Sketch the function below
1  t
f (t )  
1

t

1  t  0
0  t 1
f (t )  0 elsewhere
X(t)
-1
1
X(2t)
-0.5
0.5
X(0.5t)
-2
2
X(3t+2)
-1
1
X(t+2)
-3
-1
X(3t+2)
-1
-1/3
Further Exercises
• Find x(2t+4); x(2t-4); x(-2t-1)
• x(3t)+x(3t+2)
Discrete Time Signal
• Sketch the signal described by
n  1,2
1

x[n]   1
n  1,2
 0 n  0 & | n | 2

• Find y[n]=x[2n+3]
Sketch x[n]
Sketch x[2n+3]
X[n+3]
X[2n+3]
Exercise
• Consider a discrete time signal
1  2  n  2
x[n]  
| n | 2
0
• Find y[n]=x[3n-2]
• Answer
1
n  0,1
y[n]  
0 otherwise