the discrete fourier transform (dsp) 4
TRANSCRIPT
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The Discrete Fourier Transform
Mr. HIMANSHU DIWAKAR
Mr. HIMANSHU DIWAKARAssistant Professor
JETGI
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Sampling the Fourier Transform• Consider an aperiodic 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• We can also write it in terms of the z-transform
• The sampling points are shown in figure• could be the DFS of a sequence• Write the corresponding sequence
kN/2jkN/2
j eXeXkX~
jDTFT eX]n[x
kN/2j
ez eXzXkX~ kN/2
kX~
1N
0k
knN/2jekX~N1]n[x~
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Sampling the Fourier Transform Cont’d• 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
1N
0k
knN/2jekX~N1]n[x~ kN/2jeXkX~
mm
1N
0k
mnkN/2j
1N
0k
knN/2j
m
kmN/2j
mnp~mxeN1mx
eemxN1]n[x~
r
1N
0k
mnkN/2j rNmneN1mnp~
rr
rNnxrNnnx]n[x~
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Sampling the Fourier Transform Cont’d
Mr. HIMANSHU DIWAKAR
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Sampling the Fourier Transform Cont’d• Samples of the DTFT of an aperiodic sequence
• can be thought of as DFS coefficients • of a periodic sequence • obtained through summing periodic replicas of original sequence
• 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
• It is not necessary to know the DTFT at all frequencies• To recover the discrete-time sequence in time domain
• Discrete Fourier Transform• Representing a finite length sequence by samples of DTFT
else0
1Nn0nx~nx
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The Discrete Fourier Transform• Consider a finite length sequence x[n] of length N
• For given length-N sequence associate a periodic sequence
• The 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]
1Nn0 of outside 0nx
r
rNnxnx~
NkXN mod kXkX~
kX~
else01Nk0kX~kX
NnxN mod nxnx~
Mr. HIMANSHU DIWAKAR
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The Discrete Fourier Transform Cont’d• The DFS pair
• The equations involve only on period so we can write
• The Discrete Fourier Transform
• The DFT pair can also be written as
1N
0k
knN/2jekX~N1]n[x~
1N
0n
knN/2je]n[x~kX~
else01Nk0e]n[x~kX
1N
0n
knN/2j
else01Nk0ekX~N
1]n[x
1N
0k
knN/2j
1N
0n
knN/2je]n[xkX
1N
0k
knN/2jekXN1]n[x
]n[xkX DFT
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Example• 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]
else0,...10,5,0k5
e1e1
ekX~
5/k2j
k2j
4
0n
n5/k2j
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Example Cont’d
• 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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Properties of DFT• Linearity
• Duality
• Circular Shift of a Sequence
kbXkaXnbxnaxkXnxkXnx
21DFT
21
2DFT
2
1DFT
1
mN/k2jDFT
N
DFT
ekX1-Nn0 mnxkXnx
N
DFT
DFT
kNxnXkXnx
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Example: Duality
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Symmetry Properties
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Circular Convolution
• Circular convolution of two finite length sequences
1N
0mN213 mnxmxnx
1N
0mN123 mnxmxnx
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Example• Circular convolution of two rectangular pulses L=N=6
• DFT of each sequence
• Multiplication of DFTs
• And the inverse DFT
else01Ln01nxnx 21
else00kNekXkX
1N
0n
knN2j
21
else0
0kNkXkXkX2
213
else01Nn0Nnx3
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Example
• We can augment zeros to each sequence L=2N=12• The DFT of each sequence
• Multiplication of DFTs
N
k2j
NLk2j
21e1e1kXkX
2
Nk2j
NLk2j
3e1e1kX
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THANK YOU
Mr. HIMANSHU DIWAKAR