Convolution sum using graphical and matrix method

Signals and Systems

1. Convolution Sum using Graphical and Matrix Method Dr.K.G.SHANTHI Professor/ECE R.M.K College of Engineering and Technology

2. Convolution Sum 2

3. Convolution sum using Matrix method Steps : Let T 1 be the length of x[n] Let T 2 be the length of h[n] Convolution sum can be obtained as Y=X.H X is the matrix formed from x[n] with the dimension of (T 1 + T 2 -1)x T 2 H is the matrix formed from h[n] with the dimension of T 2 x1 3

4. Matrix Method( contd ) Formation X Matrix: The first column is formed from x[n] .The other elements are made to zero . Second column is formed by shifting the first column from top to bottom. The remaining columns are formed in the same way until the dimension of the matrix is achieved. 4

5. Obtain the Convolution sum using Matrix method x[n]={1,2,3,4} and h[n]={1,2} Solution: T 1 =4, T 2 =2 X is the matrix formed from x[n] with Dimension (T 1 + T 2 -1)x T 2 =(4+2-1)x2=5x2 H is the matrix formed from h[n] with the Dimension T 2 x1 =2x1 Convolution sum Y=X.H Y[n]={1,4,7,10,8} 5

6. Obtain the Convolution sum using Matrix method x[n]={1,2,3,4} and h[n]={1,1,1,1} Solution: T 1 =4, T 2 =4 X is the matrix formed from x[n] with Dimension (T 1 + T 2 -1)x T 2 =(4+4-1)x4=7x4 H is the matrix formed from h[n] with the Dimension T 2 x1 =4x1 Convolution sum Y=X.H 6

7. Convolution sum Y=X.H 7 y[n]={1,3,6,10,9,7,4}

8. Convolution sum using Graphical method Steps : Represent x[k] Plot h[k] Obtain h[-k] Obtain h[n-k]. Shift h[n-k]to the extreme left and start moving towards right so that x[k] and h[n-k] overlap with each other. Calculate y[n] at the instant of overlap –Multiply x[k]and h[n-k] Repeat this procedure until there is no overlap. 8

9. Obtain the Convolution sum using Graphical method x[n]={1,2,3,4} and h[n]={1,2} 9 k x(k) 1 2 3 4 1 2 3 k h(k) 1 2 1 h(-k) -k 1 2 -1 1) y(n)=x[k]h[n-k] y(0)= x[k]h[-k]=0x2+1x1=1 h(1-k) -k 1 2 1 k 2) y(n)=x[k]h[n-k] y(1)= x[k]h[1-k]=1x2+2x1=4 h(2-k) -k 1 2 1 k 2 3) y(n)=x[k]h[n-k] y(2)= x[k]h[2-k]=1x0+2x2+3x1=7

10. Contd 10 k x(k) 1 2 3 4 1 2 3 h(3-k) -k 1 2 1 k 2 3 4) y(n)=x[k]h[n-k] y(3)= x[k]h[3-k]=1x0+2x0+3x2+4x1 =10 h(4-k) -k 1 2 1 k 2 3 4 5) y(n)=x[k]h[n-k] y(4)= x[k]h[4-k]=4x2+0x1=8 For n=5 there is no overlap and hence the process can be stopped. Y[n]={1,4,7,10,8}

11. Obtain the Convolution sum using Graphical method x[n]={1,2,1,1} and h[n]={2,3,1} 11 1) y(n)=x[k]h[n-k] y(-1)= x[k]h[-1-k]=0x1+0x3+1x2=2 3) y(n)=x[k]h[n-k] y(1)= x[k]h[1-k]=1x1+2x3+1x2=9 k x(k) 1 2 -1 2 3 1 1 1 1 k h(k) 2 1 3 2 h(-k) -k 1 2 -1 3 -2 h(1-k) -k 1 k 1 2 3 -1 h(-1-k) -k 1 k 1 2 3 -1 -2 -3 2) y(n)=x[k]h[n-k] y(0)= x[k]h[-k]=1x3+2x2=7

12. Contd 12 4) y(n)=x[k]h[n-k] y(2)= x[k]h[2-k]=2x1+1x3+1x2=7 k x(k) 1 2 -1 2 3 1 1 1 h(2-k) -k 1 k 1 2 3 -1 2 5) y(n)=x[k]h[n-k] y(3)= x[k]h[3-k]=1x1+1x3=4 h(3-k) -k 1 k 1 2 3 -1 2 3 h(4-k) -k 1 k 1 2 3 -1 2 3 4 6) y(n)=x[k]h[n-k] y(3)= x[k]h[3-k]=1x1=1 For n=5 there is no overlap and hence the process can be stopped. y[n]={2,7,9,7,4,1}

13. 13 13 THANK YOU ALL