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Digital Signal Processing
Chapter 3
Discrete transforms
2
2.1 Introduction
• Up to now, we have converted analog signals into discrete-time sequences. So now we can process these sequences by computers. But we cannot process the spectra, i.e., Fourier transforms, in digital way, because they are continuous.
• In order for that the signals can be processed by computers in the discrete-frequency domain, we should sample the continuous-frequency Fourier transforms, to obtain the discrete-frequency Fourier transforms.
3
3.2 Discrete Fourier transform
• Sampling X(z) at |z| =1, the discrete Fourier transform can be obtained.
N
2
]Re[z
]Im[zj
4
3.2 Discrete Fourier transform
The continuous-frequency Fourier transform X(ejω) is sampled by an impulse train as follows
This sampling process can be described as
According to the convolution theorem, we get that
k
kN
2
k
jj kN
eXeX 2
)()(
k
j kN
nxeXnx 2
)()()( 11 FF
5
3.2 Discrete Fourier transform
p
N
k
N
N
nj
N
N
njN
k
N
N
nj
kk
pNnN
pNnN
dekN
dekN
dekN
kN
)(2
otherwise,0
,2
2
2
1
2
2
1
2
2
12
1
0
2
2 1
0
21
F
6
3.2 Discrete Fourier transform
This expression shows that, after sampling the continuous-frequency Fourier transform, the sequence becomes periodic. is a precise periodic shifted repetition of x(n).
p
p
k
j
pNnxN
pNnN
nx
kN
nxeXnx
)(2
)(2
)(
2)()()( 11
FF
)(nx
7
3.2 Discrete Fourier transform
• For a finite length sequence, its Fourier transform can be represented by the samples in frequency.
• Only if the number of samples, N, of the Fourier transform in one period is larger than or equal to the length of the sequence, can represent x(n) utterly.)(nx
10for ),(2
)( NnnxN
nx
8
3.2 Discrete Fourier transform
0 2
)( jeH
0 2
0 2
)( jeH
0 n
)(nx
0
0 n
)(nx
n
9
3.2 Discrete Fourier transform
10for )(1
)(2
)(
)(2
1
2)(
2
1
)(2
1)(
2)(
2)()(
1
0
22
1
0
22
22
2
2
NneeXN
nxN
nx
eeX
dekN
eX
deeXnx
kN
eXkN
eXeX
N
k
knN
jkN
j
N
k
knN
jkN
j
N
N
nj
k
kN
j
N
N
njj
k
kN
j
k
jj
10
3.2 Discrete Fourier transform
• The expression
shows that a finite-length sequence can be obtained from the samples of its Fourier transform.
This expression is referred to as the inverse discrete Fourier transform, IDFT for short.
10for ,)(1
)(2
)(1
0
22
NneeXN
nxN
nxN
k
knN
jkN
j
11
3.2 Discrete Fourier transform
0 2
)( jeH
0 2
)( jeH
0 n
)(nx
0 n
)(nx
0 2
)( jeH
0 n
)(nx
12
3.2 Discrete Fourier transform
• The Fourier transform X(ejω), after sampled, is defined as the discrete Fourier transform,that is
10for ),(
)(
)()(
2
1
0
2
2
1
02
NkeX
enx
enxeX
kN
j
N
n
knN
j
kN
N
n
nj
kN
j
13
3.2 Discrete Fourier transform
1
0
)1(2
2
1
02
2
1
0
1
0
22
1
0
1
0
221
0
2sin
2sin
)(1
1
1)(
1
)(1
)(1
)()(
N
k
NN
kjk
Nj
N
k kN
j
NjkN
j
N
k
N
n
nkN
jkN
j
N
n
njN
k
knN
jkN
jN
n
njj
e
Nk
kN
eXN
e
eeX
N
eeXN
eeeXN
enxeX
14
3.2 Discrete Fourier transform
• Usually, the simplified notation is used, that is
10for ,)(1
)(
10for ,)()(
1
0
2
1
0
2
NnekXN
nx
NkenxkX
N
n
knN
j
N
n
knN
j
15
3.2 Discrete Fourier transform
• Furthermore, let
we get the canonical form of discrete Fourier transforms, that is
Nj
N eW2
10for ,)(1
)(
10for ,)()(
1
0
1
0
NnWkXN
nx
NkWnxkX
N
n
knN
N
n
knN
16
3.2 Discrete Fourier transform
a finite-length sequence with length N: x(n)
a periodic sequence with period N: )(nx
the Fourier transform of x(n): X(ejω)
the Fourier transform of : )(nx )( jeX
sampling
FT
IFTtaking one period
DFStaking one period
a finite-length sequence with length N: x(n)
taking one period
DFT
The discrete Fourier transform of x(n): X(k)
17
Example 1
Compute the DFT of the following sequence
Solution
96,0
50,1)(
n
nnx
k
ke
e
eekX
kj
kj
kj
n
knj
10sin
53
sin
1
1)( 2
10
2
610
25
0
10
2
18
Example 1 (cont.)
0
6
0 k
|H(k)|
0
–π
–π/2
π/2
π
4
2
1 2 3 4 5 6 7 8 9
k0 1 2 3 4 5 6 7 8 9
arg[H(k)]
19
Example 2
Compute the DFT of the following sequence
Solution
95,1
40,1)(2 n
nnx
0)0(
1
)1(2
11
1)(
2
10
510
10
1010
510
10
510
9
510
4
0102
X
W
W
W
WW
W
WWWkX
k
k
k
kk
k
k
n
kn
n
kn
20
Example 3
Compute the DFT of the following sequence
Solution
95,0
40,2)(3 n
nnx
10)0(
1
)1(2
1
122)(
3
10
510
10
510
4
0103
X
W
W
W
WWkX
k
k
k
k
n
kn
21
Comparison between Ex. 2 and Ex. 3
)(2jeX
)(3jeX
0 2
0 1 2 3 4 5 6 7 8 9k
22
Matrix notation of discrete Fourier transform
)1(
)2(
)1(
)0(
1
)1(
)2(
)1(
)0(
)1(
)2(
)1(
)0(
)1(
)2(
)1(
)0(
2
2
)1()1(2)1(0
)1(2420
)1(210
0000
)1()1(2)1(0
)1(2420
)1(210
0000
NX
X
X
X
WWWW
WWWW
WWWW
WWWW
N
Nx
x
x
x
Nx
x
x
x
WWWW
WWWW
WWWW
WWWW
NX
X
X
X
NN
NN
NNN
NNNNN
NNNNN
NNNN
NN
NN
NNN
NNNNN
NNNNN
NNNN
23
3.3 Properties of the DFT
• Linearity
If x(n) = k1x1(n) + k2x2(n), then
X(k) = k1X1(k) + k2X2(k)
Note that the two sequences and two DFTs must have the same length.
24
3.3 Properties of the DFT
• Time-reversal
If x(n) ↔ X(k), then
x((– n) mod N) ↔ X ((–k) mod N)
x(n)
n
x(– n)
n
x((– n) mod N)
n
25
3.3 Properties of the DFT
• Time-shift theorem
Note that the sequence x(n+l) should be considered as a circular shift of x(n).
)()mod)(( kXWNlnx lkN
x(n)
n
x((n–2) mod N)
n
26
3.3 Properties of the DFT
• Circular frequency-shift theorem (modulation theorem)
Note that the DFT X(k+l) should also be considered as a circular shift of X(k).
From
we get that
)mod)(()( NlkXnxW nlN
nl
Njnl
NW nl
N
2sin
2cos
)mod)(()mod)(((2
12sin)(
)mod)(()mod)(((2
12cos)(
NlkXNlkXnlN
nx
NlkXNlkXnlN
nx
27
3.3 Properties of the DFT
• Circular convolution in time
)()()()()()()()(
)()(),()(
10for ,)mod()mod)((
)mod)(()mod()(
1
0
1
0
kYkHkXnxnhnhnxny
kHnhkXnx
NnNlhNlnx
NlnhNlxny
N
l
N
l
28
3.3 Properties of the DFT
Proof
1
0
1
0
1
0
1
0
1
0
1
0
1
0
)mod)(()(
)(1
)(
)()(1
)()(1
)(1
)(
N
l
N
l
N
k
klN
knN
N
k
knN
N
l
klN
N
k
knN
N
k
knN
Nlnhlx
WWkHN
lx
WkHWlxN
WkHkXN
WkYN
ny
29
Example 1
Compute the 6 point circular convolution.
Solution
}2,3,4,1{)(},1,3,2,1{)(
)()()(
nhnx
nhnxny
9)5(143200)6mod)5((
17)4(014320)6mod)4((
21)3(001432)6mod)3((
14)2(200143)6mod)2((
6)1(320014)6mod)1((
3)0(432001)6mod)0((
001321)(
ylh
ylh
ylh
ylh
ylh
ylh
lx
30
Example 1 (cont.)
The 6 point circular convolution is
Now, let’s see the linear convolution.
}9,17,21,14,6,3{)()()( nhnxny
2)6(1432)6(
9)5(1432)5(
17)4(1432)4(
21)3(1432)3(
14)2(1432)2(
6)1(1432)1(
1)0(1432)0(
1321)(
ylh
ylh
ylh
ylh
ylh
ylh
ylh
lx
31
Example 1 (cont.)
The 6 point circular convolution is
and the linear convolution is
Comparison between the circular and the linear.– Usually, two convolutions are different from each other;
– The circular one is the periodic repetition of the linear one.
– In a certain condition, two convolutions could be the same, that is, the period N is greater than or equal to the length of the linear convolution.
}9,17,21,14,6,3{)()()( nhnxny
}2,9,17,21,14,6,1{)()()( nhnxny
32
3.3 Properties of the DFT
• Circular correlation
The DFT of the correlation in time between two sequences is such that
• Circular autocorrelation
)mod)(()()mod)(()(1
0
NkHkXNlnhnxN
n
)mod)(()()mod)(()(1
0
NkXkXNlnxnxN
n
33
3.3 Properties of the DFT
• Real and imaginary sequences– If x(n) is a real sequence, then
Re[X(k)] = Re[X((–k) mod N)]
Im[X(k)] = – Im[X((–k) mod N)]
– If x(n) is an imaginary sequence, then
Re[X(k)] = – Re[X((–k) mod N)]
Im[X(k)] = Im[X((–k) mod N)]
– If x(n) is a real or an imaginary sequence, then
|X(k)| = |X((–k) mod N)|
arg[X(k)] = – arg[X((–k) mod N)]
34
3.3 Properties of the DFT
• Symmetric and antisymmetric sequences– If x(n) = x*((–n) mod N), it is a periodically conjugate symmet
ric sequence;
– If x(n) = – x*((–n) mod N), it is a periodically conjugate antisymmetric.
Periodically conjugate symmetric and antisymmetric sequences have the following properties:– If x(n) is conjugate symmetric, X(k) is real;
– If x(n) is conjugate antisymmetric, X(k) is imaginary;
35
3.3 Properties of the DFT
• Parseval’s theorem
1
0
21
0
2)(
1)(
N
n
N
n
kXN
nx
36
3.3 Properties of the DFT
• Relation between the DFT and z transform
– Sampling the z transform at , we can obtain the DFT.
– The z transform can be obtained from the DFT.
kN
2
1
01
1
01
1
0
1
0
1
0
1
0
1
0
1
)(1
1
1)(
1)(
1
)(1
)()(
N
kk
N
N
N
kk
N
NkNN
N
k
N
n
nknN
N
n
nN
k
knN
N
n
n
zW
kX
N
z
zW
zWkX
NzWkX
N
zWkXN
znxzX
37
3.4 Digital filtering using the DFT
• Linear and circular convolutions
Assume that x(n) has duration L and h(n) has duration K. For the linear convolution , the duration of y(n) is L + K –1.
If the convolution size N satisfies the condition
N ≥ L + K –1
then the circular convolution is equivalent to the linear convolution.
)()()( nhnxny
38
3.4 Digital filtering using the DFT
• Overlap-and-add method
…x(n)
L L Lh(n)
K
L+K–1
L+K–1
L+K–1
y(n)
…
39
3.4 Digital filtering using the DFT
• Overlap-and-save method
…
x(n)
h(n)
L+K–1
y(n)
…
K
L
saveddiscarded
K–1
40
3.5 Fast Fourier transform (FFT)
• Algorithm analysis of DFT
When computing the DFT of an N-point sequence, we need about N2 complex multiplications and N2 complex additions.
In early 1960s, Cooley and Tukey proposed the fast Fourier transform, which requires the order of Nlog2N.
10for ,)(1
)(
10for ,)()(
1
0
1
0
NnWkXN
nx
NkWnxkX
N
n
knN
N
n
knN
41
3.5.1 Radix-2 decimation-in-time algorithm
Separating x(n) into its even- and odd-numbered points, we obtain that
or with n=2r for n even and n=2r+1 for n odd,
1
odd0
1
even0
1
0
)()()()(N
nn
knN
N
nn
knN
N
n
knN WnxWnxWnxkX
1)2(
0
21)2(
0
2
1)2(
0
)12(1)2(
0
2
))(12())(2(
)12()2()(
N
r
rkN
kN
N
r
rkN
N
r
krN
N
r
rkN
WrxWWrx
WrxWrxkX
42
3.5.1 Radix-2 decimation-in-time algorithm
If we note that
for N even, then we have that
An N-point DFT can be broken into two (N/2)-point computations.
2)2(
2222
NN
jN
j
N WeeW
1)2(
02
1)2(
02
1)2(
0
21)2(
0
2
)12()2(
))(12())(2()(
N
r
rkN
kN
N
r
rkN
N
r
rkN
kN
N
r
rkN
WrxWWrx
WrxWWrxkX
43
3.5.1 Radix-2 decimation-in-time algorithm
• Algorithm analysis
Computing this DFT, we need
complex multiplications and
complex additions.
1)2(
02
1)2(
02 )12()2()(
N
r
rkN
kN
N
r
rkN WrxWWrxkX
NN
NN
222
22
222
22NN
44
3.5.1 Radix-2 decimation-in-time algorithm
• If N is equal to a power of 2, we can go on to break two (N/2)-point DFTs into four (N/4)-point DFTs.
• And repeat this procedure till the computation is reduced to N one point DFTs.
• For N=2r, where r is an integer, the procedure of decomposition can be done at most r =log2N times.
45
3.5.1 Radix-2 decimation-in-time algorithm
• Algorithm analysis once more– When an N-point DFT is broken into two (N/2)-point DFTs, N
2 multiplications are decreased to 2(N/2)2+N multiplications;
– When an (N/2)-point DFT is decomposed into two (N/4)-point DFTs, (N/2)2 decreases to 2(N/4)2+ N/2, so the overall computation needs 2[2(N/4)2+ N/2]+N = 4(N/4)2+2N complex multiplications ;
– After carrying out this procedure to the last, we actually accomplish the decompositions r =log2N times. At this time, the computation requires Nlog2N multiplications.
46
3.5.1 Radix-2 decimation-in-time algorithm
x(0)
x(2)
x(4)
x(6)
x(1)
x(3)
x(5)
x(7)
X(1)
X(2)
X(3)
X(4)
X(5)
X(6)
X(7)
4-pointDFT
4-pointDFT
0NW
1NW
2NW
3NW
4NW
5NW
6NW
7NW
X(0)
47
3.5.1 Radix-2 decimation-in-time algorithm
x(0)
x(6)
x(2)
x(4)
x(1)
x(5)
x(3)
x(7)
X(1)
X(2)
X(3)
X(4)
X(5)
X(6)
X(7)
0NW
1NW
2NW
3NW
4NW
5NW
6NW
7NW
X(0)2-pointDFT
0NW
2NW
4NW
6NW
0NW
2NW
4NW
6NW
2-pointDFT
2-pointDFT
2-pointDFT
48
3.5.1 Radix-2 decimation-in-time algorithm
x(0)
x(6)
x(2)
x(4)
x(1)
x(5)
x(3)
x(7)
X(1)
X(2)
X(3)
X(4)
X(5)
X(6)
X(7)
0NW
1NW
2NW
3NW
4NW
5NW
6NW
7NW
X(0)0
NW
2NW
4NW
6NW
0NW
2NW
4NW
6NW
0NW
4NW
0NW
4NW
0NW
4NW
0NW
4NW
49
3.5.1 Radix-2 decimation-in-time algorithm
• Basic butterfly computation
lNW
2
Nl
NW
)( pX m
)(qX m
)(1 pX m
)(1 qX m
)()()(
)()()(
21
1
qXWpXqX
qXWpXpX
m
Nl
Nmm
ml
Nmm
)()()(
)()()(
1
1
qXWpXqX
qXWpXpX
ml
Nmm
ml
Nmm
lNW 1
)( pX m
)(qX m
)(1 pX m
)(1 qX m
50
3.5.1 Radix-2 decimation-in-time algorithm
x(0)
x(2)
x(4)
x(6)
x(1)
x(5)
x(3)
x(7)
X(1)
X(2)
X(3)
X(4)
X(5)
X(6)
X(7)
1NW
3NW
X(0)
0NW 1
1
1
1
0NW
2NW
0NW
2NW
1
1
1
1
0NW
2NW
1
1
1
1
0NW
0NW
0NW
51
3.5.1 Radix-2 decimation-in-time algorithm
• Several characteristics of decimation-in-time algorithm– In-place
– Bit-reversed ordering
lNW 1
)( pX m
)(qX m
)(1 pX m
)(1 qX m
)7()111()111()7(
)6()110()011()3(
)5()101()101()5(
)4()100()001()1(
)3()011()110()6(
)2()010()010()2(
)1()001()100()4(
)0()000()000()0(
XXxx
XXxx
XXxx
XXxx
XXxx
XXxx
XXxx
XXxx
)(
)(
0
01
11
11
)(
)(
1
1
)(
)(
1
1
qX
pX
W
qX
pX
W
W
qX
pX
m
ml
N
m
m
lN
lN
m
m
52
3.5.2 Decimation-in-frequency algorithm
Dividing X(k) into smaller parts, we can get another class of FFT algorithms.
1)2(
0
2
1)2(
0
21)2(
0
1)2(
0
21)2(
0
1
2
1)2(
0
1
0
2)(
2)(
2)(
)()()()(
N
n
nkN
kN
N
N
n
nkN
kN
N
N
n
nkN
N
n
kN
n
N
N
n
nkN
N
Nn
nkN
N
n
nkN
N
n
nkN
WN
nxWnx
WN
nxWWnx
WN
nxWnx
WnxWnxWnxkX
53
3.5.2 Decimation-in-frequency algorithm
Now consider k even and k odd separately, that is
1)2(
0
1)2(
0
2
2)1()(
2)()(
N
n
nkN
k
N
n
nkN
kN
N
WN
nxnx
WN
nxWnxkX
1)2(
0
)12(
1)2(
0
2
2)()12(
2)()2(
N
n
nrN
N
n
nrN
WN
nxnxrX
WN
nxnxrX
54
3.5.2 Decimation-in-frequency algorithm
An N-point DFT X(k) can be divided into two (N/2)-point DFTs.
Repeat this procedure till an N-point DFT X(k) is divided into N one-point DFTs.
1)2(
0
2
1)2(
0
2
2)()12(
2)()2(
N
n
rnN
nN
N
n
rnN
WWN
nxnxrX
WN
nxnxrX
55
3.5.2 Decimation-in-frequency algorithm
X(2)
X(4)
X(6)
X(1)
X(3)
X(5)
X(7)
0NW
1NW
2NW
3NW
X(0)x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
1
1
1
1
4-pointDFT
4-pointDFT
56
3.5.2 Decimation-in-frequency algorithm
X(4)
X(2)
X(6)
X(1)
X(5)
X(3)
X(7)
X(0)x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
2-pointDFT
2-pointDFT
2-pointDFT
2-pointDFT
0NW
1NW
2NW
3NW
1
1
1
1
1
1
1
1
0NW
2NW
0NW
2NW
57
3.5.2 Decimation-in-frequency algorithm
X(4)
X(2)
X(6)
X(1)
X(5)
X(3)
X(7)
X(0)x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
x(7)
0NW
1NW
2NW
3NW
1
1
1
1
1
1
1
1
0NW
2NW
0NW
2NW
1
1
1
1
0NW
0NW
0NW
0NW
58
3.5.3 Radix-4 algorithm
When N=4l, we can divide an N-point DFT into four (N/4)-point DFTs.
1)4(
0
431)4(
0
42
1)4(
0
41)4(
0
4
1)4(
0
)34(1)4(
0
)24(
1)4(
0
)14(1)4(
0
4
)34()24(
)14()4(
)34()24(
)14()4()(
N
r
rkN
kN
N
r
rkN
kN
N
r
rkN
kN
N
r
rkN
N
r
krN
N
r
krN
N
r
krN
N
r
rkN
WrxWWrxW
WrxWWrx
WrxWrx
WrxWrxkX
59
3.5.3 Radix-4 algorithm
0NW
rNW
rNW 2
rNW 3
j1 j
11 1
j1
1
60
3.5.4 Algorithms for arbitrary values of N
• If N is not a power of 2 but a composite number, that is
N = N1N2…Nl = N1 N2l
then we can initially divide an N-point DFT into N1 DFTs with length N2l.
• We can continue this decomposition, until the lengths of all DFTs are Nl.
1
0
1
01
1
0
1 2
2)()()(
N
r
N
m
mkN
rkN
N
n
nkN
l
lWrmNxWWnxkX
61
3.6 Other discrete transforms (skipped)
62
3.7 Signal representations
• Laplace transform– Time domain: nonperiodic function of a continuous and real ti
me variable.
– Frequency domain: nonperiodic function of a continuous and complex frequency variable.
• z transform– Time domain: nonperiodic function of a discrete and integer ti
me variable.
– Frequency domain: nonperiodic function of a continuous and complex frequency variable.
63
3.7 Signal representations
• Fourier transform of continuous signals– Time domain: nonperiodic function of a continuous and real ti
me variable.
– Frequency domain: nonperiodic function of a continuous and imaginary frequency variable.
• Fourier transform of discrete signals– Time domain: nonperiodic function of a discrete and integer ti
me variable.
– Frequency domain: periodic function of a continuous and imaginary frequency variable.
64
3.7 Signal representations
• Fourier series of periodic continuous signals– Time domain: periodic function of a continuous and real time
variable.
– Frequency domain: nonperiodic function of a discrete and integer frequency variable.
• Fourier series of periodic discrete signals (DFS)– Time domain: periodic function of a discrete and integer time
variable.
– Frequency domain: periodic function of a discrete and integer frequency variable.
65
3.7 Signal representations
• Discrete Fourier transform (DFT)– Time domain: finite-length function of a discrete and integer
time variable.
– Frequency domain: finite-length function of a discrete and integer frequency variable.
• Fast Fourier transform (FFT)– FFT is no transform
– FFT is a kind of fast algorithm of DFT
66
Exercises
• 3.3 (d)• 3.7• 3.8 (b)(c)• 3.17
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