it is all about convolution which is used in signal and system

1. G H PATEL COLLEGE OF
ENGINEERING AND
TECHNOLOGY
SUB : SIGNAL &SYSTEM
BATCH : 1A09
BRANCH : ELECTRICAL
PREPERED BY:-
YASH KOTHADIA - 150110109019
ABHISHEK LALKIYA -150110109020
MAULIK VASOYA - 150110109060

2. CONVOLUTION

3.  Consider the DT system:
 If the input signal is and the system has no
energy at , the output
is called the impulse response of the system
[ ]y n[ ]x n
[ ]h n[ ]n
[ ] [ ]x n n
[ ] [ ]y n h n
System
System
0n 
DT Unit-Impulse Response

4.  Consider the DT system described by
 Its impulse response can be found to be
[ ] [ 1] [ ]y n ay n bx n  
( ) , 0,1,2,
[ ]
0, 1, 2, 3,
n
a b n
h n
n
 
 
   
EXAMPLE

5.  Let x[n] be an arbitrary input signal to a DT LTI system
 Suppose that for
 This signal can be represented as
0
[ ] [0] [ ] [1] [ 1] [2] [ 2]
[ ] [ ], 0,1,2,
i
x n x n x n x n
x i n i n
  



     
  
1, 2,n   [ ] 0x n 
PRESENTING SYSTEM IN TERM OF
SHIFTED AND SCALED IMPULS

6. 0
[ ] [ ] [ ], 0
i
y n x i h n i n


  
EXPLOTING TIME -
INVERIANCE AND LINEARITY

7.  This particular summation is called the convolution sum
 Equation is called the convolution
representation of the system.
 A DT LTI system is completely described by its impulse
response h[n].
0
[ ] [ ] [ ]
i
y n x i h n i


 
[ ] [ ]x n h n
[ ] [ ] [ ]y n x n h n 
CONVOLUTION SUM
(LINEAR CONVOLUTION)

8.  Since the impulse response h[n] provides the
complete description of a DT LTI system, we write
[ ]y n[ ]x n [ ]h n
BLOCK DIAGRAM
REPRESENTATION OF DT LTI
SYSTEM

9.  Suppose that we have two signals x[n] and v[n] that
are not zero for negative times .
 Then, their convolution is expressed by the two-
sided series
[ ] [ ] [ ]
i
y n x i v n i


 
THE CONVOLUTION SUM FOR
NONCAUSAL SIGNALS

10.  Suppose that both x[n] and v[n] are equal to the
rectangular pulse p[n] (causal signal) represent
below
EXAMPLE: CONVOLUTION OF TWO
RECTANGULAR PULSES

11.  The signal is equal to the pulse p[i] folded about
the vertical axis
[ ]v i
THE FOLDED PULS

12. SHIFTING V[N- I] OVER X[I]

13. PLOT OF X[N]*V[N]

14.  Associatively
 Commutatively
 Distributive w.r.t. addition
[ ] ( [ ] [ ]) ( [ ] [ ]) [ ]x n v n w n x n v n w n    
[ ] [ ] [ ] [ ]x n v n v n x n  
[ ] ( [ ] [ ]) [ ] [ ] [ ] [ ]x n v n w n x n v n x n w n     
PROPERTIES OF CONVOLUTION SUM

15. CONVOLUTION INTEGRAL
 This particular integration is called the convolution
integral
 Equation is called the convolution
representation of the system
 A CT LTI system is completely described by its impulse
response h(t)
( ) ( )x t h t
( ) ( ) ( )y t x t h t 
0
( ) ( ) ( ) , 0y t x h t d t  

  

16.  Since the impulse response h(t) provides the
complete description of a CT LTI system, we write
( )y t( )x t ( )h t
BLOCK DIAGRAM REPRESENTATION OF CT
LIT SYSTEM

17.  Suppose that where p(t) is the
rectangular pulse depicted in figure
( ) ( ) ( ),x t h t p t 
( )p t
tT0
Example: Analytical Computation
of the Convolution Integral

18.  In order to compute the convolution integral
we have to consider four cases:
0
( ) ( ) ( ) , 0y t x h t d t  

  
Example

19.  Case 1: 0t 
( )x 
T0
( )h t 
t T t
( ) 0y t 
Example

20.  Case 2: 0 t T 
( )x 
T0
( )h t 
t T t
0
( )
t
y t d t 
Example

21.  Case 3:
( )x 
T0
( )h t 
t T t
( ) ( ) 2
T
t T
y t d T t T T t

     
0 2t T T T t T     
Example

22.  Case 4:
( ) 0y t 
( )x 
T0
( )h t 
t T t
2T t T T t   
Example

23.  Associativity
 Commutativity
 Distributivity w.r.t. addition
( ) ( ( ) ( )) ( ( ) ( )) ( )x t v t w t x t v t w t    
( ) ( ) ( ) ( )x t v t v t x t  
( ) ( ( ) ( )) ( ) ( ) ( ) ( )x t v t w t x t v t x t w t     
Properties of the Convolution
Integral