convolution

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 System System DT Unit-Impulse Response

4. Consider the DT system described by Its impulse response can be found to be EXAMPLE

5. Let x [ n ] be an arbitrary input signal to a DT LTI system Suppose that for This signal can be represented as PRESENTING SYSTEM IN TERM OF SHIFTED AND SCALED IMPULS

6. 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 ]. CONVOLUTION SUM (Linear convolution)

8. Since the impulse response h [ n ] provides the complete description of a DT LTI system, we write 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 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 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 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 )

16. Since the impulse response h (t) provides the complete description of a CT LTI system, we write Block diagram representation of ct lit system

17. Suppose that where p ( t ) is the rectangular pulse depicted in figure Example: Analytical Computation of the Convolution Integral

18. In order to compute the convolution integral we have to consider four cases: Example

19. Case 1: Example

20. Case 2: Example

21. Case 3: Example

22. Case 4: Example

23. Associativity Commutativity Distributivity w.r.t . addition Properties of the Convolution Integral

24. THANK YOU