γλώσσες
Σελίδες
Νομικός
CE 40763Digital Signal Processing
Fall 1992
Discrete Fourier Transform (DFT)Hossein Sameti
Department of Computer Engineering Sharif University of Technology
2
Motivation The DTFT is defined using an infinite sum over a
discrete time signal and yields a continuous function X(ω)◦ not very useful because the outcome cannot be stored on
a PC. Now introduce the Discrete Fourier Transform
(DFT), which is discrete and can be stored on a PC.
We will show that the DFT yields a sampled version of the DTFT.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Review of Transforms
3
dtetxtxCTFTX tj)()}({)(
n
njenxnxDTFTX )()}({)(
)(txC.
)(nxD.
C.
C.
n
nznxnxzzX )()}({)()(nxD. Complex
Inf. orFinite
1
0
2~~~)()}({)(
N
n
Nknj
enxnxDFSkX
)(~
nxD. Int.
periodicperiodic
1
0
2
)()}({)(N
n
Nknj
enxnxDFTkX
)(nxD.
Int.finite finite
Review of Transforms
4
dtetxtxCTFTX tj)()}({)(
n
njenxnxDTFTX )()}({)(
)(tx
)(nx
n
nznxnxzzX )()}({)()(nx
1
0
2~~~)()}({)(
N
n
Nknj
enxnxDFSkX
)(~
nx
1
0
2
)()}({)(N
n
Nknj
enxnxDFTkX
)(nx
Discrete Fourier Series (DFS)
5
periodNkNnxnx :)()(~~
• Decompose in terms of complex exponentials that are periodic with period N.
)(~
nx
• How many exponentials? N
1,...,1,0)(2
Nkene Nknj
k
1
0
2~)(1)(
N
k
Nknj
ekXN
nx
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Discrete Fourier Series (DFS)
6
1,...,1,0)(2
Nkene Nknj
k
Exponentials that are periodic with period N.
)()( nene rNkk arbitrary integer
* Proof:
NnrNkj
rNk ene)(2
)(
NrnNj
Nknj
rNk eene 22
.)(
)()(2
neene kN
knj
rNk
1
....)()()( 20 nenene NN....)()()( 1211 nenene NN
7
Discrete Fourier Series (DFS)
1
0
2~)(1)(
N
k
Nknj
ekXN
nx
How to find X(k)?
Answer:
1
0
2~)()(
N
n
Nknj
enxkX
Proof : substitute X(k) in the first equation.
• It can also easily be shown that X(k) is periodic with period N:
)()( rNkXkX arbitrary integer
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
DFS Pairs
8
Analysis:
1
0
2~~)()(
N
n
Nknj
enxkX
Synthesis:
1
0
2~~)(1)(
N
k
Nknj
ekXN
nx
Periodic N pt.seq. in time domain
Periodic N pt.seq. in freq. domain
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example
9
n0
1
rrNnnx )()(
~
1 1
N
1
0
2~~)()(
N
nN
knj
enxkX
1
0
2
)(N
nN
knj
en
Nkj
e)0(2
1
n0
1 1
N
…1
…
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example (cont.)
10
rrNnnx )()(
~
1
0
2~~)()(
N
nN
knj
enxkX
1
0
2
)(N
nN
knj
en
Nkj
e)0(2
1
1
0
2~~)(1)(
N
k
Nknj
ekXN
nx
1
0
21 N
kN
knj
eN
(eq.1)
(eq.2)
(eq.1) & (eq.2)
rrNn )(
1
0
21 N
kN
knj
eN
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Shift property:
Periodic convolution:
Properties of DFS
11
)(~
kX)(~
nx
)(~2 0
kXe Nkn
j
)( 0
~nnx
3
~
2
~
1
~xxx
Period N
1
02
~
1
~)()(
N
mmnxmx
)()()( 3~
2~
1~
kXkXkX Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Periodic convolution - Example
12Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
13
In the list of properties:
Properties of DFS
1,...,1,0)(2
Nkene Nknj
k
=
=
Where:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Properties of DFS
14Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Properties of DFS
15Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Discrete Fourier Transform
16
)(kX)(nx
N pt. N pt.
DFTN pt. N pt.
DFT
DTFT
DFSHossein Sameti, Dept. of Computer Eng., Sharif University of Technology
1) Start with a finite-length seq. x(n) with N points (n=0,1,…, N-1).
2) Make x(n) periodic with period N to get
Deriving DFT from DFS
17
)(~
nx
)()()(~
nRnxnx N Extracts one period of )(~
nx
otherwise
NnnRN 0
1,...,1,01)(
rrNnxnx )()(
~
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
3) Take DFS of 4) Take one period of to get DFT of x(n):
18
Deriving DFT from DFS (cont.))(
~nx )(
~kX
)(~
kX )(kX
)()()(~
kRkXkX N
)(nx )(~
nx )(~
kX )(kX
N pt. N pt.N pt.periodic
N pt.periodic
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example
19
n0 1 2 3
12
3
1
)(nx
n0 1 2 3
12
3
1
)(~
nx
k0 1 2 3
k0 1 2 3
)(~
kX
)(kX
12
3
1 12
3
1
Definition of DFT:
Discrete Fourier Transform
20
otherwise
NkenxkXN
n
Nknj
0
0)()(1
0
2
N pt. DFT of x(n)
otherwise
NnekXNnx
N
k
Nknj
0
0)(1)(
1
0
2
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example
21
otherwise
nnx
0
501)( ?)( kX
otherwise
kenxkX n
knj
0
50)()(4
0
52
otherwise
ke
ekX kj
kj
0
501
1)( 5
2
2
otherwise
kkX
005
)(
Example, cont’d
Mehrdad Fatourechi, Electrical and Computer Engineering, University of
British Columbia, Summer 201122
Relationship between DFT and DTFT
23
otherwise
NkenxkXN
n
Nknj
0
0)()(1
0
2
n
njenxX )()(
otherwise
NkXkX N
k
0
0)()(
2
DFT thus consists of equally-spaced samples of DTFT.
1
0)(
N
n
njenx
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
24
Relationship between DFT and DTFT
)(X
3
8 pt. sequence 8 pt. DFT
7,...,1,08
2 kk
3 2
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
M pt. DFT of N pt. Signal
25
otherwise
NkXkX N
k
0
0)()(
2
So far we calculated the N pt. DFT of a seq. x(n) with N non-zero values:
Suppose we pad this N pt. seq. with (M-N) zeros to get a sequence with length M.
n0 1 2 3
12
3
1
)(nx We can now take an M-pt. DFT of the signal x(n)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
26
M pt. DFT of N pt. Signal
otherwise
MkenxkXM
n
Mknj
0
0)()(1
0
2
DFTN pt. M pt.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example
27
4 pt. DFT:
otherwise
kenxkX n
knj
0
40)()(3
0
42
6 pt. DFT:
otherwise
kenxkX n
knj
0
60)()(5
0
62
n0 1 2 3
12
3
1
)(nx
8 pt. DFT:
otherwise
kenxkX n
knj
0
80)()(7
0
82
100 pt. DFT:
otherwise
kenxkX n
knj
0
1000)()(99
0
1002
How are these related to each other?
28
M pt. DFT of N pt. Signal
)(X
20
Going from N pt. to 2N pt. DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
29
M pt. DFT of N pt. Signal
N pt.DFT
N pt. seq. N pt.
2N pt.DFTN pt. seq.
padded with N zeros
2N pt.
What is the minimum number of N needed to recover x(n)?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Assume y(n) is a signal of finite or infinite extent.
Problem Statement
30
)}({)( nyDTFTY • Sample at N equally-spaced points.)(Y
)()( 2 kXYN
k
N pt. sequence.
)()}({ nxkXIDFT N pt. sequence.
What is the relationship between x(n) and y(n)?
What happens if N is larger , equal or less than the length of y(n)?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
We start with x(n) and find its relationship with y(n):
Solution to the Problem Statement
31
otherwise
NnekXNnx
N
k
Nknj
0
0)(1)(
1
0
2
otherwise
NneemyNnx
N
k
Nknj
m
Nkmj
0
0))((1)(
1
0
22
)()( 2 kXYN
k
Change the order of summation:
otherwise
NneN
mynxN
k
Nnmkj
m0
01)()(1
0
)(2
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
32
Solution to the Problem Statement
However, we have shown that:
otherwise
NneN
mynxN
k
Nnmkj
m0
01)()(1
0
)(2
1
0
2~)(.11)(
N
k r
Nknj
rNneN
nx
rrNnnf )()(
rrNmnmnf )()(
otherwise
Nnmnfmynx m0
0)()()(Convolution with train of delta functions
101)(~
NkkX
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
33
Solution to the Problem Statement
otherwise
NnrNnynx r0
0)()(
One period of the replicated version of y(n)
• Examples
If we sample at a sampling rate that is higher than the number of points in y(n), we should be able to recover y(n).
)(Y
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Shift property:
Properties of DFT
34
)()( kXnx DFT N pt. seq.
Nkn
jDFT ekXnnx02
0 )()(
Nk 0
Nk 0
The above relationship is not correct, because of the definition of DFT.
The signal should only be non-zero for the first N points.
Nkn
jDFTN ekXnRnnx
02
0
~)()()(
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
35
In the list of properties:
Properties of DFT
1,...,1,0)(2
Nkene Nknj
k
= , =
where:
)()(]))[(( 0
~
0 nRnnxnnx NN
otherwise
NnnRN 0
1,...,1,01)(
and
where:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
36
Summery of Properties of DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
37
Summery of Properties of DFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Using DFT to calculate linear convolution
38Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
We are familiar with, “linear convolution”. Question: Can DFT be used for calculating the linear
convolution? The answer is: NO! (at least not in its current format) We now examine how DFT can be applied in order to
calculate linear convolution.
Convolution
39Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Definitions of convolution
40
3
~
2
~
1
~xxx
1
02
~
1
~)()(
N
mmnxmx
• Linear convolution:
)()()( 321 nxnxnx k
knxkxnx )()()( 213
Application in the analysis of LTI systems
• Periodic convolution:
A seq. with period N
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Definitions of convolution(cont.)
41
321 xxx • Circular convolution:
N pt. seq.
• Circular convolution is closely related to periodic convolution.
)()].()([)()()( 2
~
1
~
213 nRnxnxnxnxnx N
)().()( 213 kXkXkX
N pt. DFT of x1N pt. DFT of x2N pt. DFT of x3
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example: Circular Convolution
42
)1()(1 nnx
seq.pt.:)(2 NnxCircular convolution?
Make an N pt. seq.)(1 nx
Nn
nn
nx10
11100
)(1
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
43
Example: Circular Convolution
Circular Convolution & DFT
44
)()].()([)()()( 2
~
1
~
213 nRnxnxnxnxnx N
We know from DFS properties:
3
~
2
~
1
~xxx )()()( 3
~2
~1
~kXkXkX
)}({)( 1
~
1
~nxDFSkX )}({)( 2
~
2
~nxDFSkX )}({)( 3
~
3
~nxDFSkX
)()()()()( 3~
2~
1~
kRkXkRkXkX NN
)()().( 321 kXkXkX Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
45
Circular Convolution & DFT
)()].()([)()()( 2
~
1
~
213 nRnxnxnxnxnx N
)()().( 321 kXkXkX
If we multiply the DFTs of two N pt. sequences, we get the DFT of their circular convolution and not the DFT of their linear convolution.
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Example : Circular Convolution
46
otherwise
Lnnxnx
0
01)()( 21
Calculate N pt. circular convolution of x1 and x2 for the following two cases of N:1) N=L2) N=2L
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
47
Case 1: N=L
otherwise
Lnnx
0
101)(1
N pt. DFT of x1
otherwise
LkenxkXL
n
Lknj
0
10)()(1
0
2
otherwise
kLkXkX
00
)()( 21
)().()( 213 kXkXkX
otherwise
kL0
02
otherwise
LnLnx
010
)(3
IDFT
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
48
Case 1: N=L
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
49
Case 2: N=2L• Pad each signal with L extra zeros to get an 2L pt. seq.:
N=2L pt. DFT of x1
otherwise
LkenxkXL
n
Lknj
0
120)()(12
0
22
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
50
Case 2: N=2L
Same as linear convolution!!
Our hypothesis is that if we pad two DT signals with enough zeros so that its length becomes N, we can use DFT to calculate linear convolution.
Using DFT to Calculate linear Convolution
51
)(1 nx L pt. seq. LN
)(2 nx P pt. seq. PN
)(*)()( 213 nxnxnx Using DFTGoal: calculate
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
52
n
njenxnxDTFTX )()}({)(
Using DFT to Calculate linear Convolution
)(*)()( 213 nxnxnx )()()( 213 XXX
To get DFT, we have to sample the above DTFT at N equally-spaced points:
otherwise
NnrNnxr0
0)(3
NkXkY
23 )()(
)}({ kYIDFTSolution to the problem statement
(Eq.1)
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
53
Using DFT to Calculate linear Convolution
Nk
Nk
Nk XXXkY
222123 )()()()(
N pt. DFT of x1 N pt. DFT of x2
otherwise
NnrNnxr0
0)(3
(Eq.2))()()}({ 21 nxnxkYIDFT
Circular convolution
(Eq.1)
(Eq.2))()( 21 nxnx
Replicated version of the linear convolution
On the other hand, we know that:
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
In other words, the N pt. circular convolution of two DT signals is the same as their linear convolution, if we make the result of linear convolution periodic with period N and extract one period.
54
Using DFT to Calculate linear Convolution
1)(3 PLnx
1 PLNTo avoid aliasing:
We can thus use DFT in order to calculate the linear convolution of two sequences!
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
1) Start with
Algorithm for calculating linear convolution using DFT
55
)(1 nx L pt. seq.
)(2 nx P pt. seq.
3) Pad with N-L zeros to get N points.4) Pad with N-P zeros to get N points.
)(1 nx)(2 nx
5) Calculate the N pt. DFTs of the above two sequences and multiply them together.6) Calculate IDFT of the resulting N pt. sequence.
2) Choose 1 PLN
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Using DFT for linear filtering when one sequence is very long
56
h(n))(nx )(ny
A very long sequence
An FIR filter with a limited number of taps (P)
• Examples of this situation?
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
57
Using DFT for linear filtering when one sequence is very long
Solution to the problem
Overlap - add
Main idea: using the following property:)(*)()(*)()(*)]()([ 2121 nhnxnhnxnhnxnx
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Segment the long sequence into non-overlapping chunks of data with the length of L.
Convolve each chunk with h(n) to get (L+P-1) new points.
Add the results of the convolution of all chunks to get the final answer.
Overlap- add method
58Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
59
Overlap- add method- Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
60
Overlap- add method- Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
61
Overlap- add method- Example
Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Discussed DFS and DFT and examined the relationship between DFT and DTFT
Showed how DFT can be used for calculating convolution sum.
Next: Fast Fourier Transform
Summary
62Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology
Top Related