The document discusses convolution sums and provides examples of calculating convolution sums for different signals. It also includes block diagram representations of direct form I and direct form II structures. Specifically: 1. It gives the mathematical expression for a convolution sum and calculates the convolution sum of two signals x[n]=anu[n] and h[n]=bnu[n]. 2. It calculates the convolution sum of x[n]=r[n] and h[n]=u[n], where r[n]=n and u[n]=1, showing the overlapping interval is from 0 to n. 3. It briefly mentions block diagram representations of direct form I and direct form II structures at the end

1. Convolution Sum
&
Block Diagram representation
Dr.K.G.SHANTHI
Professor/ECE
R.M.K College of Engineering
and Technology

2. Convolution Sum
2
][][][][][ nhnxknhkxny
k
 



3. Convolution sum using Mathematical expression for
infinite sequence
1.Find the convolution sum of x[n]=anu[n] and h[n]= bnu[n]
3
][][][ knubkuany kn
k
k
 



Where x[k]=aku[k] and h[k]= bku[k]][][][][][ nhnxknhkxny
k
 



4. ✘ The common range of x[k] and h[n-k] is 0 to n
4
][][][ knubkuany kn
k
k
 



kn
k
n
n
k
k
k
nk
n
k
knkn
n
k
k
b
a
b
b
a
bbabbany  











0000
][
x
x
x
NN
n
n





 1
1 1
0
Using the formula:











 






















































 


ab
b
b
ab
b
b
a
b
a
b
b
a
b
a
bny n
nn
n
n
n
n
n
n
1
111
11
1
1
1
1
][

5. Problem(contd)
5
][][
11
nu
ab
ab
ny
nn





























 











 




ab
ab
ab
b
bb
ab
b
ab
b
b
ab
bny
nn
n
nn
n
n
nn
n
1111
1
11
.
][

6. 2.Find the convolution sum of
x[n]=r[n] and h[n]= u[n]
6
][][][][][ nhnxknhkxny
k
 


Here x[k]=r[k] and h[k]= u[k]
][][][
0
knukrny
n
k
 
Here both x[k] and h[k] are causal signals and hence the overlapping interval is from 0 to n
W.K.T r[k]=k u[k] =k and u[n-k]= 1
][][][
0
knukrny
n
k
  2
)1(
][
0

 
nn
kny
n
k
][
2
][
2
][
2
nu
n
nu
n
ny 
k
x(k)
1
2
3
0 1 2 3
k
h(k)
1
0 1
1 11
h(n-k)
-k
0
k
1
n
1 11
-3 -2 -1
1 1 1 1 11

7. 7

8. 8
April –May 2015

9. 9

10. 10

11. 11
April –May 2019

12. Direct Form II 12

13. Direct Form I 13

14. 1414
THANK YOU ALL