Circular conv_7767038.ppt

Chapter 5
Finite-Length
Discrete Transform

§5.2.1 The Discrete Fourier
Transform
 Definition - The simplest relation between a
length-N sequence x[n], defined for 0≤n≤N-1,
and its DTFT X(ejω) is obtained by uniformly
sampling X(ejω) on the ω-axis between
0≤ω≤2π at ωk= 2πk/N, 0≤k≤N-1
 From the definition of the DTFT we thus
have
1
0
,
]
[
)
(
]
[
1
0
/
2
/
2




 




N
k
e
n
x
e
X
k
X
N
n
N
k
j
N
k
j 




Transform
 Note: X[k] is also a length-N sequence in the
frequency domain
 The sequence X[k] is called the discrete
Fourier transform (DFT) of the sequence x[n]
 Using the notation WN= e-j2π/N the DFT is
usually expressed as:
1
0
,
]
[
]
[
1
0







N
k
W
n
x
k
X
N
n
n
k
N

Transform
 The inverse discrete Fourier transform
(IDFT) is given by
 To verify the above expression we multiply
both sides of the above equation by and
sum the result from n = 0 to n=N-1
W
n
N

1
0
,
]
[
1
]
[
1
0








N
n
W
k
X
N
n
x
N
k
kn
N

Transform
 resulting in


 





























1
0
1
0
)
(
1
0
1
0
)
(
1
0
1
0
1
0
]
[
1
]
[
1
]
[
1
]
[
N
k
N
k
n
K
N
N
n
N
k
n
K
N
N
n
n
N
N
k
kn
N
N
n
n
N
W
k
X
N
W
k
X
N
W
W
k
X
N
W
n
x





Transform
 Making use of the identity
we observe that the RHS of the last
equation is equal to X[l]
]
[
]
[ 

X
W
n
x
N
n
n
N 



1
0


 







otherwise
,
0
integer
an
,
for
,
1
0
)
( r
rN
k
N
W
N
n
n
k
N


 Hence

Transform
 Example – Consider the length-N sequence
1
1
0
0
1




N
n
n
,
,




]
[n
x
1
0
1
]
0
[
]
[
]
[ 0
1
0





 


N
k
W
x
W
n
x
k
X N
N
n
kn
N ，
 Its N-point DFT is given by

Transform












1
1
,
1
0
,
0
,
1
]
[
N
n
m
m
n
m
n
n
y
1
0
]
[
]
[
]
[
1
0





 


N
k
W
W
m
y
W
n
y
k
Y km
N
km
N
N
n
kn
N ，
 Its N-point DFT is given by

Transform
defined for 0≤n≤N-1
1
0
),
/
2
cos(
]
[ 



 N
r
N
rn
n
g
 
)
(
2
1
2
1
]
[ /
2
/
2
rn
N
rn
N
N
rn
j
N
rn
j
W
W
e
e
n
g





 

 Using a trigonometric identity we can write

Transform
 The N-point DFT of g[n] is thus given by
1
0
2
1
]
[
]
[
1
0
1
0
)
(
)
(
1
0












 










N
k
W
W
W
n
g
k
G
N
n
N
n
n
k
r
N
n
k
r
N
N
n
kn
N

Transform


 







otherwise
,
0
integer
an
,
for
,
1
0
)
( r
rN
k
N
W
N
n
n
k
N


we get
1
0
otherwis
0
for
,
2
/
for
,
2
/
]
[












N
k
r
N
k
N
r
k
N
k
G
，

§5.2.2 Matrix Relations
 The DFT samples defined by
1
0
,
]
[
]
[
1
0



 


N
k
W
n
x
k
X
N
n
kn
N
T
T
N
x
x
x
x
N
X
X
X
X
]
1
[
]
1
[
]]
0
[
[
]]
1
[
]
1
[
]
0
[
[






can be expressed in matrix form as
X=Dnx
where

and DN is the N×N DFT matrix given by






















2
)
1
(
)
1
(
2
)
1
(
)
1
(
2
4
2
)
1
(
2
1
1
1
1
1
1
1
1
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
W
W
W
W
W
W
W
W
W
D










 Likewise, the IDFT relation given by
where is the N×N DFT matrix
DN
1

1
0
,
]
[
1
]
[
1
0



 



N
n
W
K
X
N
n
x
N
k
kn
N
X
D
x N
1


can be expressed in matrix form as

where
 Note:
D
D N
N
N
*
1 1


































2
)
1
(
)
1
(
2
)
1
(
)
1
(
2
4
2
)
1
(
2
1
1
1
1
1
1
1
1
1
N
N
N
N
N
N
N
N
N
N
N
N
N
N
N
W
W
W
W
W
W
W
W
W
D










§5.2.3 DFT Computation Using
MATLAB
 The functions to compute the DFT and the
IDFT are fft and ifft
 These functions make use of FFT
algorithms which are computationally highly
efficient compared to the direct computation
 Programs 5_1.m and 5_2.m illustrate the use
of these functions

§5.2.3 DFT Computation Using
MATLAB
 Example –a Program 5_3.m can be used to
compute the DFT and the DTFT of the
sequence x[n]=cos(6πn/16), 0≤n≤15
as shown below
indicates DFT samples

§5.3.3 DTFT from DFT by
Interpolation
 The N-point DFT X[k] of a length-N
sequence x[n] is simply the frequency
samples of its DTFT X(ejω) evaluated at N
uniformly spaced frequency points
1
0
,
/
2 



 N
k
N
k
k 


 Given the N-point DFT X[k] of a length-N
sequence x[n], its DTFT X(ejω) can be
uniquely determined from X[k]

Interpolation
 Thus

 

 

S
N
n
n
N
k
j
N
k
n
j
N
n
kn
N
N
k
N
n
n
j
j
e
e
W
e
e
k
X
N
k
X
N
n
x
X


 



















1
0
)
/
2
(
1
0
1
0
1
0
1
0
]
[
1
]
]
[
1
[
]
[
)
(






Interpolation
 To develop a compact expression for the
sum S, let
)
/
2
( N
k
j
e
r 



 Then
 From the above




1
0
N
n
n
r
S
1
1
1
1
1
0
1
1
1












 





N
N
N
n
n
N
N
n
N
n
n
n
r
S
r
r
r
r
r
rS

Interpolation
 Or, equivalently,
 Hence
]
2
/
)
1
)][(
/
2
[
)]
/
2
(
[
)
2
(
2
2
sin
2
2
sin
1
1
1
1













 





 







N
N
k
j
N
k
j
k
N
j
N
e
N
k
N
k
N
e
e
r
r
S










N
r
S
r
rS
S 



 1
)
1
(

Interpolation
 therefore
]
2
/
)
1
)][(
/
2
[
1
0
2
2
sin
2
2
sin
]
[
1
)
(











 





 
  N
N
k
j
N
k
j
e
N
k
N
k
N
k
X
N
e
X








§5.3.4 Sampling the DTFT
 Consider a sequence x[n] with a DTFT
X(ejω)
 We sample X(ejω) at N equally spaced
points ωk=2πk/N, 0≤k≤N-1 deleloping the N
frequency samples
 These N frequency samples can be
considered as an N-point DFT Y[k] whose
N- point IDFT is a length-N sequence y[n]
)}
(
{ e
k
j
X


 Now 




＝－


 
 j
j
e
x
e
X ]
[
)
(











＝－
＝－ 




 k
N
N
k
j
N
k
j
j
W
x
e
x
e
X
e
X
k
Y k
]
[
]
[
)
(
)
(
]
[
/
2
/
2



kn
N
N
k
W
k
Y
N
n
y 



1
0
]
[
1
]
[
＝
 Thus
 An IDFT of Y[k] yields

 i.e.










 








 


1
0
)
(
1
0
1
]
[
]
[
1
]
[
N
k
n
k
N
kn
N
k
N
N
k
W
N
x
W
W
x
N
n
y






＝


 






otherwise
,
0
for
,
1
1 1
0
)
( mN
n
r
W
N
N
n
r
n
k
N
＝

 we arrive at the desired relation









m
N
n
mN
n
x
n
y 1
0
,
]
[
]
[
 Thus y[n] is obtained from x[n] by adding an
infinite number of shifted replicas of x[n],
with each replica shifted by an integer
multiple of N sampling instants, and
observing the sum only for the interval
0≤n≤N-1

 To apply
1
0
,
]
[
]
[ 



 



N
n
mN
n
x
n
y
m
to finite-length sequences, we assume that
the samples outside the specified range are
zeros
 Thus if x[n] is a length-M sequence with
M≤N, then y[n]= x[n] , for 0≤n≤N-1

 If M > N, there is a time-domain aliasing of
samples of x[n] in generating y[n], and x[n]
cannot be recovered from y[n]
 Example – Let {x[n]}={0 1 2 3 4 5}
↑
 By sampling its DTFT X(ejω) at ωk=2πk/4,
0≤k≤3 and then applying a 4-point IDFT to
these samples, we arrive at the sequence
y[n] given by

y[n]= x[n]+ x[n+4]+x[n-4] 0≤n≤3
 i.e. {y[n]}={4 6 2 3}
↑
{x[n]} cannot be recovered from {y[n]}

Numerical Computation of the
DTFT Using the DFT
 A practical approach to the numerical
computation of the DTFT of a finite-length
sequence
 Let X(ejω) be the DTFT of a length-N
sequence x[n]
 We wish to evaluate X(ejω) at a dense grid of
frequencies ωk=2πk/M, 0≤k≤M-1,where M >>
N:

DTFT Using the DFT
 Define a new sequence
M
kn
j
N
n
n
j
N
n
j
e
n
x
e
n
x
e
X k
k /
2
1
0
1
0
]
[
]
[
)
( 

 






 

M
kn
j
M
n
e
j
e
n
x
e
X k /
2
1
0
]
[
)
( 
 














1
,
0
1
0
,
]
[
]
[
M
n
N
N
n
n
x
n
xe
 Then

DTFT Using the DFT
 The DFT Xe[k] can be computed very
efficiently using the FFT algorithm if M is an
integer power of 2
 The function freqz employs this approach to
evaluate the frequency response at a
prescribed set of frequencies of a DTFT
expressed as a rational function in e-jω
 Thus is essentially an M-point DFT
Xe[k] of the length-M sequence xe[n]
)
(e
k
j
X


DFT Properties
 Like the DTFT, the DFT also satisfies a
number of properties that are useful in
signal processing applications
 Some of these properties are essentially
identical to those of the DTFT, while some
others are somewhat different
 A summary of the DFT properties are given
in tables in the following slides

Table 5.1: DFT Properties: Symmetry
Relations
x[n] is a complex sequence

Table 5.2: DFT Properties: Symmetry
Relations
x[n] is a real sequence

Table 5.3: General Properties of DFT

§5.4.1 Circular Shift of a Sequence
 This property is analogous to the time-
shifting property of the DTFT as given in
Table 3.4, but with a subtle difference
 Consider length-N sequences defined for
0≤n≤N-1
 Sample values of such sequences are equal
to zero for values of n<0 and n≥N

 If x[n] is such a sequence, then for any
arbitrary integer n0, the shifted sequence
x1[n]= x(n-n0)
is no longer defined for the range 0≤n≤N-1
 We thus need to define another type of a
shift that will always keep the shifted
sequence in the range 0≤n≤N-1

 The desired shift, called the circular shift,
is defined using a modulo operation:
]
[
]
[ 0 N
c n
n
x
n
x 
















0
0
0
0
0
,
]
[
1
,
]
[
]
[
n
n
for
n
n
N
x
N
n
n
for
n
n
x
n
xc
 For n0>0 (right circular shift), the above
equation implies

 Illustration of the concept of a circular shift
]
5
[
]
1
[
6
6






n
x
n
x －
]
2
[
]
4
[
6
6






n
x
n
x －
]
[n
x

 As can be seen from the previous figure, a
right circular shift by n0 is equivalent to a left
circular shift by N-n0 sample periods
 A circular shift by an integer number n0
greater than N is equivalent to a circular shift
by N
n0

§5.4.2 Circular Convolution
 This operation is analogous to linear
convolution, but with a subtle difference
 Consider two length-N sequences, g[n] and
h[n], respectively
 Their linear convolution results in a length-
(2N-1) sequence yL[n] given by
2
2
0
1
0




 


N
n
m
n
h
m
g
n
y
N
m
L ],
[
]
[
]
[

 In computing yL[n] we have assumed that
both length-N sequence have been zero-
padded to extend their lengths to 2N-1
 The longer form of yL[n] results from the
time-reversal of the sequence h[n] and its
linear shift to the right
 The first nonzero value of yL[n] is
yL[0]=g[0]h[0] ,and the last nonzero value is
yL[2N-2]=g[N-1]h[N-1]

 To develop a convolution-like operation
resulting in a length-N sequence yC[n] , we
need to define a circular time-reversal, and
then apply a circular time-shift
 Resulting operation, called a circular
convolution, is defined by
1
0
],
[
]
[
]
[
1
0



 





N
n
m
n
h
m
g
n
y
N
m
N
C

 Since the operation defined involves two
length-N sequences, it is often referred to as
an N-point circular convolution, denoted as
 The circular convolution is commutative, i.e.
]
[
]
[
]
[
]
[ n
g
n
h
n
h
n
g  N
N
]
[
]
[
]
[ n
h
n
g
n
y  N

 Example – Determine the 4-point circular
convolution of the two length-4 sequences:
as sketched below



 }
1
1
2
2
{
]}
[
{
},
1
0
2
1
{
]}
[
{ n
h
n
g

 The result is a length-4 sequence yC[n]
given by
6
)
2
1
(
)
1
0
(
)
1
2
(
)
2
1
(
]
1
[
]
3
[
]
2
[
]
2
[
]
3
[
]
1
[
]
0
[
]
0
[
]
[
]
[
]
0
[ 4
3
0















 

h
g
h
g
h
g
h
g
m
h
m
g
m
C
y
,
]
[
]
[
]
[
]
[
]
[
3
0
4
 





m
C m
n
h
m
g
n
h
n
g
n
y 4
3
0 
 n
 From the above we observe

 Likewise
7
)
1
1
(
)
1
0
(
)
2
2
(
)
2
1
(
]
2
[
]
3
[
]
3
[
]
2
[
]
0
[
]
1
[
]
1
[
]
0
[
]
1
[
]
[
]
1
[ 4
3
0


















h
g
h
g
h
g
h
g
m
h
m
g
m
C
y
6
)
1
1
(
)
2
0
(
)
2
2
(
)
1
1
(
]
3
[
]
3
[
]
0
[
]
2
[
]
1
[
]
1
[
]
2
[
]
0
[
]
2
[
]
[
]
2
[ 4
3
0


















h
g
h
g
h
g
h
g
m
h
m
g
m
C
y

 The circular convolution can also be
computed using a DFT-based approach as
indicated in Table 5.3
5
)
2
1
(
)
2
0
(
)
1
2
(
)
1
1
(
]
0
[
]
3
[
]
1
[
]
2
[
]
2
[
]
1
[
]
3
[
]
0
[
]
3
[
]
[
]
3
[ 4
3
0
















 

h
g
h
g
h
g
h
g
m
h
m
g
m
C
y
yC[n]
and

 Example – Consider the two length-4
sequences repeated below for convenience:
 The 4-point DFT G[k] of g[n] is given by
G[k]= g[0]+g[1]e-j2πk/4 +g[2]e-j4πk/4 +g[3]e-j6πk/4
=1+2e-jπk/2 +e-j3πk/2 , 0≤k≤3

 Therefore G[0]=1+2+1=4,
G[1]=1-j2+j=1-j,
G[2]=1-2-1=-2,
G[3]= 1+j2-j=1+j
 Likewise,
H[k]=h[0]+h[1]e-j2πk/4 +h[2]e-j4πk/4
+h[3]e-j6πk/4
=2+2e-jπk/2 +e-jπk +e-j3πk/2 , 0≤k≤3

 Hence， H[0]=2+2+1+1=6,
H [1]=2-j2-1+j=1-j,
H [2]=2-2+1-1=0,
H [3]=2+j2-1-j=1+j
 The two 4-point DFTs can also be computed
using the matrix relation given earlier









































































j
j
j
j
j
j
g
g
g
g
D
G
G
G
G
1
2
1
4
1
0
2
1
1
1
1
1
1
1
1
1
1
1
1
1
]
3
[
]
2
[
]
1
[
]
0
[
]
3
[
]
2
[
]
1
[
]
0
[
4







































































j
j
j
j
j
j
h
h
h
h
D
H
H
H
H
1
0
1
6
1
1
2
2
1
1
1
1
1
1
1
1
1
1
1
1
]
3
[
]
2
[
]
1
[
]
0
[
]
3
[
]
2
[
]
1
[
]
0
[
4
D4 is the 4-point DFT matrix

 If YC[k] denotes the 4-point DFT of yC[n]
then from Table 3.5 we observe
YC[k]= G[k] H[k] , 0≤k≤3









































2
0
2
24
]
3
[
]
3
[
]
2
[
]
2
[
]
1
[
]
1
[
]
0
[
]
0
[
]
3
[
]
2
[
]
1
[
]
0
[
j
j
H
G
H
G
H
G
H
G
Y
Y
Y
Y
C
C
C
C
 Thus

 A 4-point IDFT of YC[k] yields













































































5
6
7
6
2
0
2
24
1
1
1
1
1
1
1
1
1
1
1
1
4
1
]
3
[
]
2
[
]
1
[
]
0
[
4
1
]
3
[
]
2
[
]
1
[
]
0
[
*
4
j
j
j
j
j
j
Y
Y
Y
Y
D
y
y
y
y
C
C
C
C
C
C
C
C

 Example – Now let us extended the two
length-4 sequences to length 7 by appending
each with three zero-valued samples, i.e.
















6
4
,
0
3
0
,
]
[
]
[
6
4
,
0
3
0
,
]
[
]
[
n
n
n
h
n
h
n
n
n
g
n
g
e
e

 We next determine the 7-point circular
convolution of ge[n] and he[n] :
 From the above
y[0]=ge[0]he[0]+ge[1]he[6]+ge[3]he[4]
+ge[4]he[3]+ge[5]he[2]+ge[6]he[1]
=ge[0]he[0]=1×2=2
6
0
,
]
[
]
[
]
[
6
0
7





n
m
n
h
n
g
n
y
m
e
e

 Continuing the process we arrive
y[1]=g[0]h[1]+g[1]h[0]=(1×2)+(2×2)=6，
y[2]=g[0]h[2]+g[2]h[0]
=(1×2)+(2×2)+(0×2)=5，
y[3]=g[0]h[3]+g[1]h[2]+g[2]h[1]+g[3]h[0]
=(1×1)+(2×1)+(0×2)+(1×2)=5，
y[4]=g[1]h[3]+g[2]h[2]+g[3]h[1]
=(2×1)+(0×1)+(1×2)=4，

y[5]=g[2]h[3]+g[3]h[2]=(0×1)+(1×1)=1,
y[6]=g[3]h[3]=(1×1)=1
 As can be seen from the above that y[n] is
precisely the sequence yL[n] obtained by a
linear convolution of g[n] and h[n]

 The N-point circular convolution can be
written in matrix form as
























































 ]
1
[
]
2
[
]
1
[
]
0
[
]
0
[
]
3
[
]
2
[
]
1
[
]
3
[
]
0
[
]
1
[
]
2
[
]
2
[
]
1
[
]
0
[
]
1
[
]
1
[
]
2
[
]
1
[
]
0
[
]
1
[
]
2
[
]
1
[
]
0
[
N
g
g
g
g
h
N
h
N
h
N
h
h
h
h
h
h
N
h
h
h
h
N
h
N
h
h
N
y
y
y
y
C
C
C
C











 Note: The elements of each diagonal of the
N×N matrix are equal
 Such a matrix is called a circulant matrix

 First, the samples of the two sequences are
multiplied using the conventional
multiplication method as shown on the next
slide
 Consider the evaluation of y[n]=h[n] g[n]
where {g[n]} and {h[n]} are length-4
sequences
＊
 Tabular Method
 We illustrate the method by an example

 The partial products generated in the 2nd, 3rd, and
4th rows are circularly shifted to the left as
indicated above

 The modified table after circular shifting is
shown below
 The samples of the sequence { yc[n]} are
obtained by adding the 4 partial products in
the column above of each sample

 Thus
yc[0]=g[0]h[0]+ g[3]h[1]+ g[2]h[2]+ g[1]h[3]
yc[1]=g[1]h[0]+ g[0]h[1]+ g[3]h[2]+ g[2]h[3]
yc[2]=g[2]h[0]+ g[1]h[1]+ g[0]h[2]+ g[3]h[3]
yc[3]=g[3]h[0]+ g[2]h[1]+ g[1]h[2]+ g[0]h[3]

§5.9 Computation of the DFT of
Real Sequences
 In most practical applications, sequences of
interest are real
 In such cases, the symmetry properties of
the DFT given in Table 5.2 can be exploited
to make the DFT computations more
efficient

§5.9.1 N-Point DFTs of Two Length-
N Real Sequences
 Let g[n] and h[n] be two length-N real
sequences with G[k] and H[k] denoting their
respective N-point DFTs
 These two N-point DFTs can be computed
efficiently using a single N-point DFT
 Define a complex length-N sequence
x[n]=g[n]+jh[n]
 Hence, g[n]=Re{x[n]} and h[n]=Im{x[n]}

N Real Sequences
 Let X[k] denote the N-point DFT of x[n]
 Then, from Table 5.1 we arrive at
 
 
]
[
]
[
2
1
]
[
]
[
]
[
2
1
]
[
*
*
N
N
k
X
k
X
j
k
H
k
X
k
X
k
G








]
[
]
[ *
*
N
N k
N
X
k
X 





 Note that for 0≤k≤N-1,

N Real Sequences
 Example – We compute the 4-point DFTs of
the two real sequences g[n] and h[n] given
below
       



 1
1
2
2
]
[
,
1
0
2
1
]
[ n
h
n
g
 Then {x[n]}={g[n]}+j{h[n]} is given by
   




 j
j
j
j
n
x 1
2
2
2
1
]
[

N Real Sequences
 Its DTF X[k] is





























































2
2
2
6
4
1
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
]
3
[
]
2
[
]
1
[
]
0
[
j
j
j
j
j
j
j
j
j
j
X
X
X
X
 From the above
X*[k]=[4-j6 2 -2 -j2]
 Hence
X*[<4-k>4]=[4-j6 -j2 -2 2]

N Real Sequences
 Therefore
{G[k]}=[4 1-j -2 1+j]
{G[k]}=[6 1-j 0 1+j]
verifying the results derived earlier

§5.9.2 2N-Point DFT of a Real
Sequence Using an N-point DFT
 Let v[n] be a length-2N real sequence with
an 2N-point DFT V[k]
 Define two length-N real sequences g[n]
and h[n] as follows:
N
n
n
v
n
h
n
v
n
g 



 0
,
]
1
2
[
]
[
,
]
2
[
]
[
 Let G[k] and H[k] denote their respective N-
point DFTs

 Define a length-N complex sequence
{x[n]}={g[n]}+j{h[n]}
with an N-point DFT X[k]
 Then as shown earlier
]}
[
]
[
{
2
1
]
[
]}
[
]
[
{
2
1
]
[
N
N
k
X
k
X
j
k
H
k
X
k
X
k
G











 Now
1
2
0
,
]
[
]
[
]
[
]
[
]
1
2
[
]
2
[
]
[
]
[
1
0
2
1
0
2
1
0
1
0
)
1
2
(
2
1
0
2
2
1
0
2
1
2
0

































N
k
n
h
n
g
n
h
n
g
n
v
n
v
n
v
k
V
W
W
W
W
W
W
W
W
W
nk
N
N
n
k
N
nk
N
N
n
k
N
nk
N
N
n
nk
N
N
n
k
n
N
N
n
nk
N
N
n
nk
N
N
n

 i.e.,
1
2
0
,
]
[
]
[
]
[ 2 







 N
k
k
H
W
k
G
k
V N
k
N
N

 }
1
1
1
0
2
2
2
1
{
]}
[
{ n
v
 We form two length-4 real sequences as
follows
 Example – Let us determine the 8-point
 DFT V[k] of the length-8 real sequence

 Now
7
0
,
]
[
]
[
]
[ 4
8
4 






 k
k
H
W
k
G
k
V k
 Substituting the values of the 4-point DFTs
G[k] and H[k] computed earlier we get







}
1
1
2
2
{
]}
1
2
[
{
]}
[
{
}
1
0
2
1
{
]}
2
[
{
]}
[
{
n
v
n
h
n
v
n
g

2
6
4
]
0
[
]
0
[
]
4
[
4142
.
0
1
)
1
(
)
1
(
]
3
[
]
3
[
]
3
[
2
0
2
]
2
[
]
2
[
]
2
[
4142
.
2
1
)
1
(
)
1
(
]
1
[
]
1
[
]
1
[
10
6
4
]
0
[
]
0
[
]
0
[
4
8
4
/
3
3
8
2
/
2
8
4
/
1
8












































j
j
j
j
e
H
W
G
V
j
j
e
j
H
W
G
V
e
H
W
G
V
j
j
e
j
H
W
G
V
H
G
V

4142
.
2
1
)
1
(
)
1
(
]
3
[
]
3
[
]
7
[
2
0
2
]
2
[
]
2
[
]
6
[
4142
.
0
1
)
1
(
)
1
(
]
1
[
]
1
[
]
5
[
4
/
7
7
8
2
/
3
6
8
4
/
5
5
8
j
j
e
j
H
W
G
V
e
H
W
G
V
j
j
e
j
H
W
G
V
j
j
j































§5.10 Linear Convolution Using
the DFT
 Linear convolution is a key operation in
many signal processing applications
 Since a DFT can be efficiently implemented
using FFT algorithms, it is of interest to
develop methods for the implementation of
linear convolution using the DFT

§5.10.1 Linear Convolution of Two
Finite-Length Sequences
 Let g[n] and h[n] be two finite-length
sequences of length N and M, respectively
 Denote L=N+M-1
 Define two length-L sequences




















1
,
0
1
0
,
]
[
]
[
1
,
0
1
0
,
]
[
]
[
L
n
M
N
n
n
h
n
h
L
n
N
N
n
n
g
n
g
e
e

§5.10.1 Linear Convolution of Two
Finite-Length Sequences
 The corresponding implementation scheme
is illustrated below
Then
Zero-padding
with
(M-1) zeros
Zero-padding
with
(N-1) zeros
(N+M-1)
point DFT
(N+M-1)
point DFT
(N+M-1)
point IDFT
g[n]
h[n]
ge[n]
he[n]
yL[n]
Length-N
Length-M Length-(N+M-1)
L
yL[n]=g[n] h[n]=yC[n]=ge[n] he[n]
﹡

§5.10.2 Linear Convolution of a
Finite- Length Sequence with an
Infinite-Length Sequence
 We next consider the DFT-based
implementation of
]
[
]
[
]
[
]
[
]
[
1
0
n
x
n
h
n
x
h
n
y
M



 





where h[n] is a finite-length sequence of
length M and x[n] is an infinite length (or a
finite length sequence of length much
greater than M)

§5.10.2 Overlap-Add Method
 We first segment x[n], assumed to be a
causal sequence here without any loss of
generality, into a set of contiguous finite-
length subsequences xm [n] of length N each:





0
]
[
]
[
m
m mN
n
x
n
x
where



 




otherwise
,
0
1
0
,
]
[
]
[
N
n
mN
n
x
n
xm

Thus we can write
where






0
]
[
]
[
]
[
]
[
m
m mN
n
y
n
x
n
h
n
y ﹡
Since h[n] is of length M and xm [n] is of
length N, the linear convolution
is of length N+M-1
﹡ ]
[
]
[ n
x
n
h m
]
[
]
[
]
[ n
x
n
h
n
y m
m  ﹡

 As a result, the desired linear convolution
has been broken up into a
sum of infinite number of short-length linear
convolutions of length N+M-1
each: ]
[
]
[
]
[ n
h
n
x
n
y m
m  ﹡
]
[
]
[
]
[ n
x
n
h
n
y  ﹡
 Each of these short convolutions can be
implemented using the DFT-based method
discussed earlier, where now the DFTs (and
the IDFT) are computed on the basis of
N+M-1 points

 There is one more subtlety to take care of
before we can implement





0
]
[
]
[
m
m mN
n
y
n
y
 Now the first convolution in the above sum,
, is of length N+M-1 and
is defined for 0≤ N+M-2
﹡ ]
[
]
[
]
[ 0
0 n
x
n
h
n
y 
using the DFT-based approach

 There is an overlap of M-1samples
between these two short linear convolutions
 The second short convolution
, is also of length N+M-1
but is defined for N≤n≤2N+M-2
]
[
]
[
]
[ 1
1 n
x
n
h
n
y  ﹡
 Likewise, the third short convolution
, is also of length N+M-1 but
is defined for 2N≤n≤3N+M-2
﹡ ]
[
]
[
]
[ 2
2 n
x
n
h
n
y 

 This process is illustrated in the figure on
the next slide for M=5 and N=70
 In general, there will be an overlap of M-1
samples between the samples of the short
convolutions h[n] xr-1[n] and h[n] xr [n] for
(r-1)N≤n≤rN+M-2
﹡
﹡
 Thus there is an overlap of M-1 samples
between h[n] x1[n] and h[n] x2[n]
﹡
﹡

 Therefore, y[n] obtained by a linear
convolution of x[n] and h[n] is given by
y[n]=y0[n], 0≤n≤6
y[n]=y0[n]+ y1[n-7], 7≤n≤10
y[n]=y1[n-7], 11≤n≤13
y[n]= y1[n-7]+ y2[n-14], 14≤n≤17
y[n]= y2[n-14], 18≤n≤20


 The above procedure is called the overlap-
add method since the results of the short
linear convolutions overlap and the
overlapped portions are added to get the
correct final result
 The function fftfilt can be used to implement
the above method

 Program 5_5 illustrates the use of fftfilt in the
filtering of a noise-corrupted signal using a
length-3 moving average filter
 The plots generated by running this program
is shown below

§5.10.2 Overlap-Save Method
 In implementing the overlap-add method
using the DFT, we need to compute two
(N+M-1)-point DFTs and one (N+M-1)-point
IDFT since the overall linear convolution
was expressed as a sum of short-length
linear convolutions of length (N+M-1) each
 It is possible to implement the overall linear
convolution by performing instead circular
convolution of length shorter than (N+M-1)

 To this end, it is necessary to segment x[n]
into overlapping blocks xm[n], keep the terms
of the circular convolution of h[n] with xm[n]
that corresponds to the terms obtained by a
linear convolution of h[n] and xm[n], and
throw the other parts of the circular
convoluition

 To understand the correspondence between
the linear and circular convolutions, consider
a length-4 sequence x[n] and a length-3
sequence h[n]
 Let yL[n] denote the result of a linear
convolution of x[n] with h[n]
 The six samples of yL[n] are given by

yL[0]=h[0]x[0]
yL[1]=h[0]x[1]+h[1]x[0]
yL[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]
yL[3]=h[0]x[3]+h[1]x[2]+h[2]x[1]
yL[4]=h[1]x[3]+h[2]x[2]
yL[5]=h[2]x[3]

 If we append h[n] with a single zero-valued
sample and convert it into a length-4
sequence he[n], the 4-point circular
convolution yC[n] of he[n] and x[n] is given by
yC[0]=h[0]x[0]+h[1]x[3]+h[2]x[2]
yC[1]=h[0]x[1]+h[1]x[0]+h[2]x[3]
yC[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]
yC[3]=h[0]x[3]+h[1]x[2]+h[2]x[1]

 If we compare the expressions for the
samples of yL[n] with the samples of yC[n],
we observe that the first 2 terms of yC[n] do
not correspond to the first 2 terms of yL[n],
whereas the last 2 terms of yC[n] are
precisely the same as the 3rd and 4th terms
of yL[n], i.e.,
yL[0]≠yC[0], yL[1]≠yC[1]
yL[2]≠yC[2], yL[3]≠yC[3]

 General case: N-point circular convolution of
a length-M sequence h[n] with a length-N
sequence x[n] with N > M
 First M-1samples of the circular convolution
are incorrect and are rejected
 Remaining N - M+1 samples correspond to
the correct samples of the linear convolution
of h[n] with x[n]

 Now, consider an infinitely long or very long
sequence x[n]
 Break it up as a collection of smaller length
(length-4) overlapping sequences xm[n] as
xm[n]= x [n+2m], 0≤n≤3, 0≤m≤∞
 Next,form
wm[n]=h[n] xm[n]
4

 Or, equivalently,
wm[0]=h[0]xm[0]+h[1]xm[3]+h[2]xm[2]
wm[1]=h[0]xm[1]+h[1]xm[0]+h[2]xm[3]
wm[2]=h[0]xm[2]+h[1]xm[1]+h[2]xm[0]
wm[3]=h[0]xm[3]+h[1]xm[2]+h[2]xm[1]
 Computing the above for m = 0, 1, 2, 3, . . . ,
and substituting the values of xm[n] we arrive
at

w0[0]=h[0]x[0]+h[1]x[3]+h[2]x[2] ← Reject
w0[1]=h[0]x[1]+h[1]x[0]+h[2]x[3] ← Reject
w0[2]=h[0]x[2]+h[1]x[1]+h[2]x[0]=y[2]← Save
w0[3]=h[0]x[3]+h[1]x[2]+h[2]x[1]=y[2]← Save
w1[0]=h[0]x[2]+h[1]x[5]+h[2]x[4] ← Reject
w1[1]=h[0]x[3]+h[1]x[2]+h[2]x[5] ← Reject
w1[2]=h[0]x[4]+h[1]x[3]+h[2]x[2]=y[4]← Save
w1[3]=h[0]x[5]+h[1]x[4]+h[2]x[3]=y[5]← Save

w2[0]=h[0]x[4]+h[1]x[5]+h[2]x[6] ← Reject
w2[1]=h[0]x[5]+h[1]x[4]+h[2]x[7] ← Reject
w2[2]=h[0]x[6]+h[1]x[5]+h[2]x[4]=y[6]← Save
w2[3]=h[0]x[7]+h[1]x[6]+h[2]x[5]=y[7]← Save

 It should be noted that to determine y[0] and
y[1], we need to form x-1[n] :
x-1[0]=0, x-1[1]=0,
x-1[2]=x[0], x-1[3]=x[1]
]
[
]
[
]
[ 1
1 n
x
n
h
n
w 
  4
and compute for 0≤n≤3
reject w-1[0] and w-1[1] ,and save w-1[2]= y[0]
and w-1[3]= y[1]

 General Case: Let h[n] be a length-N
sequence
 Let xm[n] denote the m-th section of an
infinitely long sequence x[n] of length N and
defined by
xm[n]= x [n+m(N-m+1)], 0≤n≤N-1
with M<N

 Then, we reject the first M-1 samples of
wm[n] and “abut” the remaining N-M+1
samples of wm[n] to form yL[n], the linear
convolution of h[n] and x[n]
 If ym[n] denotes the saved portion of wm[n],
i.e.
 Let wm[n]=h[n] xm[n]
N











2
1
,
]
[
2
0
,
0
]
[
N
n
M
n
w
M
n
n
y
m
m

 Then
yL [n+m(N-M+1)]=ym[n] , M-1≤n≤N-1
 The approach is called overlap-save method
since the input is segmented into
overlapping sections and parts of the results
of the circular convolutions are saved and
abutted to determine the linear convolution
result

 Process is illustrated next

Circular conv_7767038.ppt

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Circular conv_7767038.ppt