The document discusses efficient methods for computing the discrete Fourier transform (DFT) of real sequences by exploiting symmetry properties. It describes how the DFTs of two real sequences can be computed using a single DFT, and how the DFT of a real sequence can be obtained from the DFT of an extended complex sequence. The document also covers using the DFT to compute linear convolution, including breaking long convolutions into overlapping shorter convolutions that can each use the efficient DFT method.

1. 1
Copyright © 2001, S. K. Mitra
Computation of the DFT of
Computation of the DFT of
Real Sequences
Real Sequences
• In most practical applications, sequences of
interest are real
• In such cases, the symmetry properties of
the DFT given in Table 3.7 can be exploited
to make the DFT computations more
efficient

2. 2
Copyright © 2001, S. K. Mitra
N
N-Point
-Point DFTs
DFTs of Two
of Two Length
Length-
-N
N
Real Sequences
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
• Hence, g[n] = Re{x[n]} and h[n] = Im{x[n]}
]
[
]
[
]
[ n
h
j
n
g
n
x +
=

3. 3
Copyright © 2001, S. K. Mitra
N
N-Point
-Point DFTs
DFTs of Two Length-
of Two Length-N
N
Real Sequences
Real Sequences
• Let X[k] denote the N-point DFT of x[n]
• Then, from Table 3.6 we arrive at
• Note that
]}
[
*
]
[
{
]
[
2
1
N
k
X
k
X
k
G 〉
〈−
+
=
]}
[
*
]
[
{
]
[
2
1
N
j
k
X
k
X
k
H 〉
〈−
−
=
]
[
*
]
[
* N
N k
N
X
k
X 〉
−
〈
=
〉
〈−

4. 4
Copyright © 2001, S. K. Mitra
N
N-Point
-Point DFTs
DFTs of Two Length-
of Two Length-N
N
Real Sequences
Real Sequences
• Example - We compute the 4-point DFTs of
the two real sequences g[n] and h[n] given
below
• Then is given by
}
{
]}
[
{
},
{
]}
[
{ 1
1
2
2
1
0
2
1 =
= n
h
n
g
↑ ↑
]}
[
{
]}
[
{
]}
[
{ n
h
j
n
g
n
x +
=
}
{
]}
[
{ j
j
j
j
n
x +
+
+
= 1
2
2
2
1
↑

5. 5
Copyright © 2001, S. K. Mitra
N
N-Point
-Point DFTs
DFTs of Two Length-
of Two Length-N
N
Real Sequences
Real Sequences
• Its DFT X[k] is
• From the above
• Hence










−
+
=










+
+
+










−
−
−
−
−
−
=










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
]
[
]
[
]
[
]
[
]
[
]
[
* 2
2
2
6
4 j
j
k
X −
−
−
=
]
2
2
2
6
4
[
]
4
[
* 4 −
−
−
=
〉
−
〈 j
j
k
X

6. 6
Copyright © 2001, S. K. Mitra
N
N-Point
-Point DFTs
DFTs of Two Length-
of Two Length-N
N
Real Sequences
Real Sequences
• Therefore
verifying the results derived earlier
}
{
]}
[
{ j
j
k
G +
−
−
= 1
2
1
4
}
{
]}
[
{ j
j
k
H +
−
= 1
0
1
6

7. 7
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-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:
• Let G[k] and H[k] denote their respective N-
point DFTs
N
n
n
v
n
h
n
v
n
g ≤
≤
+
=
= 0
1
2
2 ],
[
]
[
],
[
]
[

8. 8
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-point DFT
• Define a length-N complex sequence
with an N-point DFT X[k]
• Then as shown earlier
]}
[
{
]}
[
{
]}
[
{ n
h
j
n
g
n
x +
=
]}
[
*
]
[
{
]
[
2
1
N
k
X
k
X
k
G 〉
〈−
+
=
]}
[
*
]
[
{
]
[
2
1
N
j
k
X
k
X
k
H 〉
〈−
−
=

9. 9
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-point DFT
• Now ∑
−
=
=
1
2
0
2
N
n
nk
N
W
n
v
k
V ]
[
]
[
∑ ∑
−
=
−
=
+
+
+
=
1
0
1
0
1
2
2
2
2 1
2
2
N
n
N
n
k
n
N
nk
N W
n
v
W
n
v )
(
]
[
]
[
∑ ∑
−
=
−
=
+
=
1
0
1
0
2
N
n
N
n
k
N
nk
N
nk
N W
W
n
h
W
n
g ]
[
]
[
∑ ∑
−
=
−
=
−
≤
≤
+
=
1
0
1
0
2 1
2
0
N
n
N
n
nk
N
k
N
nk
N N
k
W
n
h
W
W
n
g ,
]
[
]
[

10. 10
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-point DFT
• i.e.,
• Example - Let us determine the 8-point
DFT V[k] of the length-8 real sequence
• We form two length-4 real sequences as
follows
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
↑

11. 11
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-point DFT
• Now
• Substituting the values of the 4-point DFTs
G[k] and H[k] computed earlier we get
}
{
]}
[
{
]}
[
{ 1
0
2
1
2 =
= n
v
n
g
↑
}
{
]}
[
{
]}
[
{ 1
1
2
2
1
2 =
+
= n
v
n
h
↑
7
0
],
[
]
[
]
[ 4
8
4 ≤
≤
〉
〈
+
〉
〈
= k
k
H
W
k
G
k
V k

12. 12
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-point DFT
10
6
4
0
0
0 =
+
=
+
= ]
[
]
[
]
[ H
G
V
]
[
]
[
]
[ 1
1
1 1
8 H
W
G
V +
=
4142
2
1
1
1 4
.
)
(
)
( /
j
j
e
j j
−
=
−
+
−
= − π
2
0
2
2
2
2 2
2
8 −
=
⋅
+
−
=
+
= − /
]
[
]
[
]
[ π
j
e
H
W
G
V
]
[
]
[
]
[ 3
3
3 3
8 H
W
G
V +
=
4142
0
1
1
1 4
3
.
)
(
)
( /
j
j
e
j j
−
=
+
+
+
= − π
2
6
4
0
0
4 4
8 −
=
⋅
+
=
+
= − π
j
e
H
W
G
V ]
[
]
[
]
[

13. 13
Copyright © 2001, S. K. Mitra
2
2N
N-Point DFT of a Real
-Point DFT of a Real
Sequence Using an
Sequence Using an N
N-point DFT
-point DFT
]
[
]
[
]
[ 1
1
5 5
8 H
W
G
V +
=
4142
0
1
1
1 4
5
.
)
(
)
( /
j
j
e
j j
+
=
−
+
−
= − π
2
0
2
2
2
6 2
3
6
8 −
=
⋅
+
−
=
+
= − /
]
[
]
[
]
[ π
j
e
H
W
G
V
]
[
]
[
]
[ 3
3
7 7
8 H
W
G
V +
=
4142
2
1
1
1 4
7
.
)
(
)
( /
j
j
e
j j
+
=
+
+
+
= − π

14. 14
Copyright © 2001, S. K. Mitra
Linear Convolution Using the
Linear Convolution Using the
DFT
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

15. 15
Copyright © 2001, S. K. Mitra
Linear Convolution of Two
Linear Convolution of Two
Finite-Length Sequences
Finite-Length Sequences
• Let g[n] and h[n] be two finite-length
sequences of length N and M, respectively
• Denote
• Define two length-L sequences
1
−
+
= M
N
L



−
≤
≤
−
≤
≤
=
1
0
1
0
L
n
N
N
n
n
g
n
ge ,
],
[
]
[



−
≤
≤
−
≤
≤
=
1
0
1
0
L
n
M
M
n
n
h
n
he ,
],
[
]
[

16. 16
Copyright © 2001, S. K. Mitra
Linear Convolution of Two
Linear Convolution of Two
Finite-Length Sequences
Finite-Length Sequences
• Then
• The corresponding implementation scheme
is illustrated below
]
[
]
[
]
[
]
[
]
[
]
[ n
h
n
g
n
y
n
h
n
g
n
y C
L =
=
= * L
Zero-padding
with
zeros
1)
( −
N point DFT
Zero-padding
with
zeros
1)
( −
M
−
−
+ )
( 1
M
N
point DFT
×
g[n]
h[n]
Length-N
]
[n
ge
]
[n
he −
−
+ )
( 1
M
N
−
−
+ )
( 1
M
N
point IDFT
]
[n
yL
Length-M Length- )
( 1
−
+ M
N

17. 17
Copyright © 2001, S. K. Mitra
Linear Convolution of a Finite-
Linear Convolution of a Finite-
Length Sequence with an
Length Sequence with an
Infinite-Length Sequence
Infinite-Length Sequence
• We next consider the DFT-based
implementation of
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)
]
[
]
[
]
[
]
[
]
[ n
x
n
h
n
x
h
n
y
M
∑
−
=
=
−
=
1
0
l
l
l *

18. 18
Copyright © 2001, S. K. Mitra
Overlap-Add Method
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 of length N each:
where
]
[n
xm
∑
∞
=
−
=
0
m
m mN
n
x
n
x ]
[
]
[


 −
≤
≤
+
=
otherwise
0
1
0
,
],
[
]
[
N
n
mN
n
x
n
xm

19. 19
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• Thus we can write
where
• Since h[n] is of length M and is of
length N, the linear convolution
is of length
]
[n
xm
]
[
]
[ n
x
n
h m
*
]
[
]
[
]
[ n
x
n
h
n
y m
m = *
∑
∞
=
−
=
=
0
m
m mN
n
y
n
x
n
h
n
y ]
[
]
[
]
[
]
[ *
1
−
+ M
N

20. 20
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• As a result, the desired linear convolution
has been broken up into a
sum of infinite number of short-length
linear convolutions of length
each:
• 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
points
]
[
]
[
]
[ n
x
n
h
n
y = *
1
−
+ M
N
)
( 1
−
+ M
N
]
[
]
[
]
[ n
h
n
x
n
y m
m = L

21. 21
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• There is one more subtlety to take care of
before we can implement
using the DFT-based approach
• Now the first convolution in the above sum,
, is of length
and is defined for
∑
∞
=
−
=
0
m
m mN
n
y
n
y ]
[
]
[
1
−
+ M
N
2
0 −
+
≤
≤ M
N
n
]
[
]
[
]
[ 0
0 n
x
n
h
n
y = *

22. 22
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• The second short convolution
, is also of length
but is defined for
• There is an overlap of samples
between these two short linear convolutions
• Likewise, the third short convolution
, is also of length
but is defined for
1
−
+ M
N
2
2 −
+
≤
≤ M
N
n
N
2
0 −
+
≤
≤ M
N
n
1
−
M
]
[
]
[ n
x
n
h 2
*
]
[
]
[ n
x
n
h 1
*
=
]
[
2 n
y
=
]
[
1 n
y
1
−
+ M
N

23. 23
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• Thus there is an overlap of samples
between and
• In general, there will be an overlap of
samples between the samples of the short
convolutions and
for
• This process is illustrated in the figure on
the next slide for M = 5 and N = 7
]
[
]
[ n
x
n
h r 1
−
* ]
[
]
[ n
x
n
h r
*
1
−
M
1
−
M
]
[
]
[ n
x
n
h 1
* ]
[
]
[ n
x
n
h 2
*

24. 24
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method

25. 25
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
Add
Add

26. 26
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• Therefore, y[n] obtained by a linear
convolution of x[n] and h[n] is given by
],
[
]
[ n
y
n
y 0
=
],
[
]
[
]
[ 7
1
0 −
+
= n
y
n
y
n
y
],
[
]
[ 7
1 −
= n
y
n
y
],
[
]
[
]
[ 14
7 2
1 −
+
−
= n
y
n
y
n
y
],
[
]
[ 14
2 −
= n
y
n
y
•
•
•
6
0 ≤
≤ n
10
7 ≤
≤ n
13
11 ≤
≤ n
17
14 ≤
≤ n
20
18 ≤
≤ n

27. 27
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• 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

28. 28
Copyright © 2001, S. K. Mitra
Overlap-Add Method
Overlap-Add Method
• Program 3_6 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
0 10 20 30 40 50 60
-2
0
2
4
6
8
Time index n
Amplitude
s[n]
y[n]

29. 29
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• In implementing the overlap-add method
using the DFT, we need to compute two
-point DFTs and one -
point IDFT since the overall linear
convolution was expressed as a sum of
short-length linear convolutions of length
each
• It is possible to implement the overall linear
convolution by performing instead circular
convolution of length shorter than
)
( 1
−
+ M
N )
( 1
−
+ M
N
)
( 1
−
+ M
N
)
( 1
−
+ M
N

30. 30
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• To this end, it is necessary to segment x[n]
into overlapping blocks , keep the
terms of the circular convolution of h[n]
with that corresponds to the terms
obtained by a linear convolution of h[n] and
, and throw away the other parts of
the circular convolution
]
[n
xm
]
[n
xm
]
[n
xm

31. 31
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• 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 denote the result of a linear
convolution of x[n] with h[n]
• The six samples of are given by
]
[n
yL
]
[n
yL

32. 32
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
]
[
]
[
]
[ 0
0
0 x
h
yL =
]
[
]
[
]
[
]
[
]
[ 0
1
1
0
1 x
h
x
h
yL +
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 0
2
1
1
2
0
2 x
h
x
h
x
h
yL +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 1
2
2
1
3
0
3 x
h
x
h
x
h
yL +
+
=
]
[
]
[
]
[
]
[
]
[ 2
2
3
1
4 x
h
x
h
yL +
=
]
[
]
[
]
[ 3
2
5 x
h
yL =

33. 33
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• If we append h[n] with a single zero-valued
sample and convert it into a length-4
sequence , the 4-point circular
convolution of and x[n] is given
by
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 2
2
3
1
0
0
0 x
h
x
h
x
h
yC +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 3
2
0
1
1
0
1 x
h
x
h
x
h
yC +
+
=
]
0
[
]
2
[
]
1
[
]
1
[
]
2
[
]
0
[
]
2
[ x
h
x
h
x
h
yC +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 1
2
2
1
3
0
3 x
h
x
h
x
h
yC +
+
=
]
[n
he
]
[n
he
]
[n
yC

34. 34
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• If we compare the expressions for the
samples of with the samples of ,
we observe that the first 2 terms of do
not correspond to the first 2 terms of ,
whereas the last 2 terms of are
precisely the same as the 3rd and 4th terms
of , i.e.,
]
[n
yL
]
[n
yL
]
[n
yL
]
[n
yC
]
[n
yC
]
[n
yC
],
[
]
[ 0
0 C
L y
y ≠ ]
[
]
[ 1
1 C
L y
y ≠
],
[
]
[ 2
2 C
L y
y = ]
[
]
[ 3
3 C
L y
y =

35. 35
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• General case: N-point circular convolution
of a length-M sequence h[n] with a length-N
sequence x[n] with N > M
• First samples of the circular
convolution are incorrect and are rejected
• Remaining samples correspond
to the correct samples of the linear
convolution of h[n] with x[n]
1
−
M
1
+
− M
N

36. 36
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• Now, consider an infinitely long or very
long sequence x[n]
• Break it up as a collection of smaller length
(length-4) overlapping sequences as
• Next, form
]
[n
xm
∞
≤
≤
≤
≤
+
= m
n
m
n
x
n
xm 0
3
0
2 ,
],
[
]
[
]
[
]
[
]
[ n
x
n
h
n
w m
m = 4

37. 37
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• Or, equivalently,
• Computing the above for m = 0, 1, 2, 3, . . . ,
and substituting the values of we
arrive at
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 2
2
3
1
0
0
0 m
m
m
m x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 3
2
0
1
1
0
1 m
m
m
m x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 0
2
1
1
2
0
2 m
m
m
m x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 1
2
2
1
3
0
3 m
m
m
m x
h
x
h
x
h
w +
+
=
]
[n
xm

38. 38
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 2
2
3
1
0
0
0
0 x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 3
2
0
1
1
0
1
0 x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 2
0
2
1
1
2
0
2
0 y
x
h
x
h
x
h
w =
+
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 3
1
2
2
1
3
0
3
0 y
x
h
x
h
x
h
w =
+
+
=
←Reject
←Reject
←Save
←Save
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 4
2
5
1
2
0
0
1 x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 5
2
2
1
3
0
1
1 x
h
x
h
x
h
w +
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 4
2
2
3
1
4
0
2
1 y
x
h
x
h
x
h
w =
+
+
=
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 5
3
2
4
1
5
0
3
1 y
x
h
x
h
x
h
w =
+
+
=
←Reject
←Reject
←Save
←Save

39. 39
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 6
2
5
1
4
0
0
2 x
h
x
h
x
h
w +
+
= ←Reject
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 7
2
4
1
5
0
1
2 x
h
x
h
x
h
w +
+
= ←Reject
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 6
4
2
5
1
6
0
2
2 y
x
h
x
h
x
h
w =
+
+
= ←Save
]
[
]
[
]
[
]
[
]
[
]
[
]
[
]
[ 7
5
2
6
1
7
0
3
2 y
x
h
x
h
x
h
w =
+
+
= ←Save

40. 40
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• It should be noted that to determine y[0] and
y[1], we need to form :
and compute for
reject and , and save
and
]
[n
x 1
−
,
]
[
,
]
[ 0
1
0
0 1
1 =
= −
− x
x
]
[
]
[
]
[ n
x
n
h
n
w 1
1 −
− = 4 3
0 ≤
≤ n
]
[0
1
−
w ]
[1
1
−
w ]
[
]
[ 0
2
1 y
w =
−
]
[
]
[ 1
3
1 y
w =
−
]
[
]
[
],
[
]
[ 1
3
0
2 1
1 x
x
x
x =
= −
−

41. 41
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• General Case: Let h[n] be a length-N
sequence
• Let denote the m-th section of an
infinitely long sequence x[n] of length N
and defined by
with M < N
1
0
)],
1
(
[
]
[ −
≤
≤
+
−
+
= N
n
m
N
m
n
x
n
xm
]
[n
xm

42. 42
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• Let
• Then, we reject the first samples of
and “abut” the remaining samples of
to form , the linear convolution of
h[n] and x[n]
• If denotes the saved portion of ,
i.e.
]
[n
wm
]
[n
wm
]
[
]
[
]
[ n
x
n
h
n
w m
m = N
1
−
M
1
+
− M
N
]
[n
yL
]
[n
ym ]
[n
wm



−
≤
≤
−
−
≤
≤
=
2
1
],
[
2
0
,
0
]
[
N
n
M
n
w
M
n
n
y
m
m

43. 43
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• Then
• 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
1
1
],
[
)]
1
(
[ −
≤
≤
−
=
+
−
+ N
n
M
n
y
M
N
m
n
y m
L

44. 44
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method
• Process is illustrated next

45. 45
Copyright © 2001, S. K. Mitra
Overlap-Save Method
Overlap-Save Method