494
Practical MATLAB
® Applications for Engineers
R.5.107 Let the notation f(n − 1) 7 denote that the fi nite sequence f(n) consisting of the elements [1 2 3 4 5 6 7] is shifted by one element to the right, returning the sequence
f(n − 1) 7 = [7 1 2 3 4 5 6]; and the expression f(n + 1) 7 indicate that the elements
of f(n) are rotated (or shifted) to the left by one element, returning the sequence
f(n + 1) 7 = [2 3 4 5 6 7 1].
R.5.108 Let us next compare the linear convolution with the circular convolution. Recall
that the linear discrete convolution of the sequences f 1 (n) with f 2 (n) is indicated
by ⊗ and defi ned by the following equation:
f n
f n
f n
f m f n m
L
m
( )
( )
( )
( ) (
)
ϭ
ϭ
Ϫ
ϭϪϱ
ϩϱ
1
2
1
2
⊗
∑
which consists of a sequence of length 2N − 1.
The assumption is that each of the sequences f 1 (n) and f 2 (n) are of equal length N.
However, the circular convolution of the two sequences given by f 1 (n) and f 2 (n) is
indicated by the character ° and it is defi ned by the following equation:
f n
f n f n
f m f n m
C
m
N
( )
( )
( )
( ) (
)
ϭ
ϭ
Ϫ
ϭ
Ϫ
1
2
1
0
1
2
∑
returning the sequence f C (n) which consists of (length) N elements.
R.5.109 Let us gain some insight of the circular process by evaluating the circular convolution given by the sequence
f 1 (n) = [1 0 2 4 2] with f 2 (n) =[2 1 2 3 5]
ANALYTICAL Solution
The elements of the convolution f C (n) = f 1 (n) ° f 2 (n) can be computed as follows:
f C (0) = f 1 (0)f 2 (0) + f 1 (1)f 2 (4) + f 1 (2)f 2 (3) + f 1 (3)f(2) + f 1 (4)f 2 (1)
f C (0) = 1(2) + (0)(5) + 2(3) + 4(2)+ 2(1)
f C (0) = 2 + 0 + 6 + 8 + 2 = 18
f C (1) = f 1 (0)f 2 (1) + f 1 (1)f 2 (0) + f 1 (2)f 2 (4)+ f 1 (3)f 2 (3) + f 1 (4)f 2 (2)
f C (1) = 1(1) + 0(2) + 2(5) + 4(3) + 2(2)
f C (1) = 1 + 0 + 10 +12 + 4 = 27
f C (2) = f 1 (0)f 2 (2) + f 1 (1)f 2 (1) + f 1 (2)f 2 (0) + f 1 (3)f 2 (4) + f 1 (4)f 2 (3)
f C (2) = 1(2) + 0(1) + 2(2) + 4(5) + 2(3)
f C (2) = 2 + 0 + 4 + 20 + 6 = 32
f C (3) = f 1 (0)f 2 (3) + f 1 (1)f 2 (2) + f 1 (2)f 2 (1) + f 1 (3)f 2 (0) + f 1 (4)f 2 (4)
CRC_47760_CH005.indd 494
CRC_47760_CH005.indd 494
7/23/2008 1:47:27 PM
7/23/2008 1:47:27 PM
Practical MATLAB
® Applications for Engineers
R.5.107 Let the notation f(n − 1) 7 denote that the fi nite sequence f(n) consisting of the elements [1 2 3 4 5 6 7] is shifted by one element to the right, returning the sequence
f(n − 1) 7 = [7 1 2 3 4 5 6]; and the expression f(n + 1) 7 indicate that the elements
of f(n) are rotated (or shifted) to the left by one element, returning the sequence
f(n + 1) 7 = [2 3 4 5 6 7 1].
R.5.108 Let us next compare the linear convolution with the circular convolution. Recall
that the linear discrete convolution of the sequences f 1 (n) with f 2 (n) is indicated
by ⊗ and defi ned by the following equation:
f n
f n
f n
f m f n m
L
m
( )
( )
( )
( ) (
)
ϭ
ϭ
Ϫ
ϭϪϱ
ϩϱ
1
2
1
2
⊗
∑
which consists of a sequence of length 2N − 1.
The assumption is that each of the sequences f 1 (n) and f 2 (n) are of equal length N.
However, the circular convolution of the two sequences given by f 1 (n) and f 2 (n) is
indicated by the character ° and it is defi ned by the following equation:
f n
f n f n
f m f n m
C
m
N
( )
( )
( )
( ) (
)
ϭ
ϭ
Ϫ
ϭ
Ϫ
1
2
1
0
1
2
∑
returning the sequence f C (n) which consists of (length) N elements.
R.5.109 Let us gain some insight of the circular process by evaluating the circular convolution given by the sequence
f 1 (n) = [1 0 2 4 2] with f 2 (n) =[2 1 2 3 5]
ANALYTICAL Solution
The elements of the convolution f C (n) = f 1 (n) ° f 2 (n) can be computed as follows:
f C (0) = f 1 (0)f 2 (0) + f 1 (1)f 2 (4) + f 1 (2)f 2 (3) + f 1 (3)f(2) + f 1 (4)f 2 (1)
f C (0) = 1(2) + (0)(5) + 2(3) + 4(2)+ 2(1)
f C (0) = 2 + 0 + 6 + 8 + 2 = 18
f C (1) = f 1 (0)f 2 (1) + f 1 (1)f 2 (0) + f 1 (2)f 2 (4)+ f 1 (3)f 2 (3) + f 1 (4)f 2 (2)
f C (1) = 1(1) + 0(2) + 2(5) + 4(3) + 2(2)
f C (1) = 1 + 0 + 10 +12 + 4 = 27
f C (2) = f 1 (0)f 2 (2) + f 1 (1)f 2 (1) + f 1 (2)f 2 (0) + f 1 (3)f 2 (4) + f 1 (4)f 2 (3)
f C (2) = 1(2) + 0(1) + 2(2) + 4(5) + 2(3)
f C (2) = 2 + 0 + 4 + 20 + 6 = 32
f C (3) = f 1 (0)f 2 (3) + f 1 (1)f 2 (2) + f 1 (2)f 2 (1) + f 1 (3)f 2 (0) + f 1 (4)f 2 (4)
CRC_47760_CH005.indd 494
CRC_47760_CH005.indd 494
7/23/2008 1:47:27 PM
7/23/2008 1:47:27 PM