Remote Sensing LaboratoryDept. of Information Engineering and Computer Science
University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy
Remote Sensing LaboratoryDept. of Information Engineering and Computer Science
University of TrentoVia Sommarive, 14, I-38123 Povo, Trento, Italy
Digital Signal ProcessingLecture 7
Digital Signal ProcessingLecture 7
E-mail: [email protected] page: http://rslab.disi.unitn.it
Quote of the DayWhoever despises the high wisdom of mathematics
nourishes himself on delusion.
Leonardo da Vinci
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Discrete Time Fourier Transform
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Discrete Fourier Transform
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• Consider a sequence with a Fourier transform
• Assume that a sequence is obtained by sampling the DTFT
• Since the DTFT is periodic resulting sequence is also periodic.
• is the Discrete Fourier Series of a sequence.
• Write the corresponding sequence:
jDTFT eXnx ][
kNj
kN
j eXeXkX /2
/2
~
kX~
1N
0k
knN/2jekX~N1]n[x~
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Discrete Fourier Transform
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• The only assumption made on the sequence is that DTFT exist.
• Combine equation to get:
• Term in the parenthesis is
• So we get
m
mjj emxeX kNjeXkX /2~
1
0
/2~1][~ N
k
knNjekXN
nx
mm
N
k
mnkNj
N
k
knNj
m
kmNj
mnpmxeN
mx
eemxN
nx
~1
1][~
1
0
/2
1
0
/2/2
r
N
k
mnkNj rNmneN
mnp 1
0
/21~
rr
rNnxrNnnxnx ][~
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Discrete Fourier Transform
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Discrete Fourier Transform
Sampling in Frequency Domain-Example
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Discrete Fourier Transform
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If the original sequence• is of finite length• and we take sufficient number of samples of its DTFT• the original sequence can be recovered by
else
Nnnxnx
010~
Discrete Fourier Transform• Representing a finite length sequence by samples of DTFT
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Discrete Fourier Transform
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• Consider a finite length sequence x[n] of length N
• For given length-N sequence associate a periodic sequence
• The Discrete Fourier Series (DFS) coefficients of the periodic sequence are samples of the DTFT of x[n]. Since x[n] is of length N there is no overlap between terms of x[n-rN] and we can write the periodic sequence as:
• To maintain duality between time and frequency– We choose one period of as the Fourier transform of x[n]
10 of outside 0 Nnnx
r
rNnxnx~
Nnxnxnx N mod ~
kX~
else
NkkXkX0
10~ NkXkXkX N mod ~
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Discrete Fourier Transform
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• The DFS pair:
• The equations involve only on period so we can write
• The Discrete Fourier Transform (DFT):
• The DFT pair can also be written as
1
0
/2][~~ N
n
knNjenxkX
1
0
/2~1][~ N
k
knNjekXN
nx
else
NkenxkXN
n
knNj
0
10][~1
0
/2
else
NkekXNnx
N
k
knNj
0
10~1][
1
0
/2
1
0
/2][N
n
knNjenxkX
1
0
/21][N
k
knNjekXN
nx
][kXnx DFT
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Implications of Inherent DFT Periodicity
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Use DFS for interpretations
Use DFT for computations
User Before
User After
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Discrete Fourier TransformSampling in Frequency Domain-Example
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Discrete Fourier Transform
2 /j NNW e
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DFT versus DTFT
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Discrete Fourier Transform
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Discrete Fourier Transform
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• The DFT of a rectangular pulse• x[n] is of length 5• We can consider x[n] of any length greater
than 5• Let’s pick N=5• Calculate the DFS of the periodic form of x[n]
elsek
ee
ekX
kj
kjn
nkj
0,...10,5,05
11
~
5/2
2
4
0
5/2
Example
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Discrete Fourier Transform
How we can obtain other samples of DTFT?
We should increase N. This we can achieve by treating x[n] as an N point sequence by appending zeros.
This is a very important operation called zero padding operation.
This operation is necessary in practice to obtain a dense spectrum ofsignals.
Zero padding is an operation in which more zeros are appended to theoriginal sequence. The resulting longer DFT provides closely spacedsamples of the DTFT of the original sequence (i.e., a high densityspectrum).
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Example-Cont
• If we consider x[n] of length 10• We get a different set of DFT coefficients• Still samples of the DTFT but in different
places
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Discrete Fourier Transform
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Time-domain aliasing
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Illustration of Time-Domain Aliasing
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Illustration of Time-Domain Aliasing
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DFT: Matrix form
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IDFT: Matrix form
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Example
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Properties of Discrete Fourier Transform
1 1
2 2
3 1 2 3 1 2[ ] [ ]
DFT
DFT
DFT
x n X kx n X k
x n ax n bx n X k aX k bX k
• Linearity
If x1[n] and x2[n] have different durations-that is, they are N1 point andN2 point sequences, respectively-then the duration N3 of x3[n] should begreater or equal to max(N1, N2).
Then N3≥ max(N1, N2) point DFTs should be estimated both for x1[n]and x2[n].
If the duration (i.e., length) of any of the signals is smaller than N3, zeropadding should be applied.
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Properties of Discrete Fourier Transform: Linearity
n
1[ ]x n1[ ]x n
0 N11
Duration N1
n2[ ]x n2[ ]x n
0 N21
Duration N2
1 1[ ] [ ]x n X kDFT1 1[ ] [ ]x n X kDFT
2 2[ ] [ ]x n X kDFT2 2[ ] [ ]x n X kDFT
1 2 1 2[ ] [ ] [ ] [ ]ax n bx n aX k bX k DFT1 2 1 2[ ] [ ] [ ] [ ]ax n bx n aX k bX k DFT
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Periodic, circular and modulo-N operations
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a) finite length sequence x[n] b) periodic extension sequence formed byreplicated x[n] c) wrapping the sequence around a cylinder withcircumference N and using modulo-N d) representation of a circularbuffer with modulo-N indexing.
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The n Modulo N Operation
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Circular folding (or reversal)
[0] n=0[ ]
[ ] 1 -1N
xz n x n
x N n n N
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Circular folding (or reversal)
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[ ] [ ]x n X kDFT[ ] [ ]x n X kDFT
[ ] [ ]N N
x n X k DFT[ ] [ ]N N
x n X k DFT
[0] n=0
[ ] 1 -1N
XX k
X N k k N
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Circular Time Shifting of a Sequence
2 / 0 -1
DFT
j k N mDFTN
x n X k
x n m n N X k e
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Circular Shift of a Sequence
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Circular Shift of a Sequence
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Properties of Discrete Fourier Transform-Shifting
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0n
[ ]x n[ ]x n
N
n
[ ]x n[ ]x n
0 N
n
1[ ] [ 2]x n x n 1[ ] [ 2]x n x n
0 N
11
[ ] [ ] 0 1[ ]
0N
x n x n m n Nx n
otherwise
11
[ ] [ ] 0 1[ ]
0N
x n x n m n Nx n
otherwise
Circular Time Shifting of a Sequence
n m
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Circular Shift of a Sequence
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Circular Frequency Shifting Teorem:
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02 /
0 0 -1
DFT
j k N n DFTN
x n X k
x n e X k k k N
Circular Frequency Shifting of a Sequence
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1
3 1 20
[ ] [ ] [ ]N
Nm
x n x m x n m
1
3 1 20
[ ] [ ] [ ]N
Nm
x n x m x n m
1 1[ ] [ ]x n X kDFT1 1[ ] [ ]x n X kDFT
2 2[ ] [ ]x n X kDFT2 2[ ] [ ]x n X kDFT
1
3 1 2 3 1 20
[ ] [ ] [ ] [ ] [ ] [ ]N
Nm
x n x m x n m X k X k X k
DFT1
3 1 2 3 1 20
[ ] [ ] [ ] [ ] [ ] [ ]N
Nm
x n x m x n m X k X k X k
DFT
both of length N
Circular Convolution
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Multiplication of Two N-Point DFTs: Circular Convolution
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Circular Convolution
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Circular Convolution-Example
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Circular Convolution-Graphical Interpretation
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Circular Convolution-Matrix Form
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Circular Convolution-Matrix Form
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Example
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3 1 2 1 2[ ] [ ] [ ] [ ] [ ]m
x n x m x n m x n x n
3 1 2 1 2[ ] [ ] [ ] [ ] [ ]m
x n x m x n m x n x n
1 1[ ] ( )jx n X e DTFT1 1[ ] ( )jx n X e DTFT
2 2[ ] ( )jx n X e DTFT2 2[ ] ( )jx n X e DTFT
3 1 2 3 1 2[ ] [ ] [ ] ( ) ( ) ( )j j jx n x n x n X e X e X e DTFT3 1 2 3 1 2[ ] [ ] [ ] ( ) ( ) ( )j j jx n x n x n X e X e X e DTFT
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Linear Convolution (Review)
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Linear Convolution Using DFT
Let x1[n] be an N1 point sequence and let x2[n] be an N2 point sequence.
Define the linear convolution of x1[n] and x2[n] by x3[n], that is:
Then x3[n] is a N1+N2-1 point sequence.
If we choose N=max(N1, N2) and compute an N point circular convolution we
get an N point sequence which is obviously different from x3[n].
3 1 2
1 2
[ ] [ ] [ ]
[ ] [ ]m
x n x n x n
x m x n m
3 1 2
1 2
[ ] [ ] [ ]
[ ] [ ]m
x n x n x n
x m x n m
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Linear Convolution Using DFT
To obtain the same results with linear convolution, we should make both x1[n]
and x2[n] N1+N2-1 point sequences by padding an appropriate number of
zeros.
By this way the circular convolution is identical to the linear convolution.
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Linear Convolution Using DFT
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Example
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Windowing Property:
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3 1 2[ ] [ ] [ ]x n x n x n3 1 2[ ] [ ] [ ]x n x n x n
1 1[ ] [ ]x n X kDFT1 1[ ] [ ]x n X kDFT
2 2[ ] [ ]x n X kDFT2 2[ ] [ ]x n X kDFT
1
3 1 2 3 1 20
1[ ] [ ] [ ] [ ] [ ] [ ]N
Nl
x n x n x n X k X l X k lN
DFT1
3 1 2 3 1 20
1[ ] [ ] [ ] [ ] [ ] [ ]N
Nl
x n x n x n X k X l X k lN
DFT
both of length N
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[ ] [ ]x n X kDFT[ ] [ ]x n X kDFT
[ ] [ ], 0 1N
X n Nx k k N DFT[ ] [ ], 0 1N
X n Nx k k N DFT
Duality Property:
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Properties of DFT
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1
2 /
0
1[ ]N
j N kn
kx n X k e
N
12 /
0
12 /
01
2 /
0
[ ]
[ ]
[ ]
Nj N kn
n
Nj N kn
kN
j N kn
n
X k x n e
Nx n X k e
Nx k X n e
Proof for the Duality:
DFT
DFT
x n X kX n Nx k
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Parseval's Relation (Related to DFT):
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1 12 2
0 0
1[ ] [ ]N N
n k
E g n G kN
The total energy of a length N sequence g[n] can be computed by summing the square of the absolute values of the DFT samples G[k] divided by N:
1 1* *
0 0
1[ ] [ ] [ ] [ ]N N
n k
g n h n G k H kN
This theorem expresses the sum of sample by sample product of two complex sequences in terms of sum of the product of their discrete Fourier transforms. Specifically, the most general form of this theorem is:
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Symmetry properties of the DFT
x[n] is a real sequence:
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x[n] is a real sequence:
* *[ ] [ ] [ ]DFTN
x n x n X k X k
Sequence x[n] Discrete Fourier Transform X[k]Any real x[n] X[k]=X*[(-k)N]Any real x[n] XRe[k]=XRe[(-k)N]Any real x[n] XIm[k]=-XIm[(-k)N]Any real x[n] |X[k]|=|X[(-k)N]|Any real x[n] X[k]=-X[(-k)N]xe[n] XRe[k]xo[n] jXIm[k]
Symmetry relation of the discrete Fourier transform for a real sequence
Symmetry properties of the DFT
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Symmetry properties of the DFT
Symmetries of real and imaginary parts of the DFT of a real valuedsequence with even and odd number of samples. Note that we check thevalues for k=1,2,…,N-1 for even and odd symmetry about the point N/2;for odd symmetry the value at k=0 should be zero.
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DFT Symmetry Properties-Conjugate Symmetry
Let’s consider a complex sequence x[n]:
*N
x n x n
*N
x n x n
, 0 1cs cax n x n x n n N
*1 , 0 12cs N
x n x n x n n N
*1 , 0 12ca N
x n x n x n n N
Circular conjugate symmetric Circular conjugate antisymmetric
x[n] is said to be circular conjugate symmetric if
x[n] is said to be circular conjugate antisymmetric if
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*N
X k X k
*N
X k X k
, 0 1cs caX k X k X k k N
*1 , 0 12cs N
X k X k X k k N
*1 , 0 12ca N
X k X k X k k N
Circular conjugate symmetric Circular conjugate antisymmetric
X[k] is said to be circular conjugate symmetric if
X[k] is said to be circular conjugate antisymmetric if
Similarly, symmetry properties can be extended to DFT of a signal:
DFT Symmetry Properties-Conjugate Symmetry
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Sequence x[n] Discrete Fourier Transform X[k]x*[n] X*[(-k)N]x*[(-n) N] X*[k]Re{x[n]} Xcs[k] ( circular conjugate-symmetric part)jIm{x[n]} Xca[k] (circular conjugate-antisymmetric part)xcs[n] XRe[k]= Re{X[k]}xca[n] jXIm[k]= jIm{X[k]}
Symmetry relation of the discrete Fourier transform for a complex sequence
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DFT Symmetry Properties-Conjugate Symmetry