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• 7.1 Discrete Fourier Transform (DFT)

• 7.2 DFT Properties

• 7.3 Cyclic Convolution

• 7.4 Linear Convolution via DFT

Chapter 7 Discrete Fourier Transform

Section 8.1-8.7

2

• Definition: The Discrete Fourier Transform (DFT) of the finite

length sequence is

• Definition: The Inverse Discrete Fourier Transform (IDFT) of

is given by

• The following notation will be used:

7.1.1 Discrete Fourier Transform (DFT)

1 ,..., 1, 0 ]; [ N n n x

1 ,..., 0 for ] [ ] [ DFT ] [1

0

2

N k e n x n x k X

N

n

N

nk j

N

1 ,..., 1, 0 ]; [ N k k X

1 ,..., 0 for ] [1

] [ IDFT ] [1

0

2

N n e k XN

k X n xN

k

N

nk j

N

] [ ] [DFT

k X n x

3

7.1.2 Discrete Fourier Series (DFS)

• Periodic Extension: Given a finite-length sequence

define the periodic sequence by

• The sequence with period N is called the periodic extension

of x[n]. It has a fundamental frequency .

• does not have a Z-transform or a convergent Fourier sum

(why?). But it does have a DFS representation.

• It is actually the DFS that is the true frequency representation of

discrete periodic signals. The DFT is just one period of the DFS.

1 ,..., 0 ]; [ N n n x

n n x]; [ ~

] mod [ ] [ ] [ ~N n x n x n xN

][~ nx

N

2

][~ nx

4

7.1.3 DFT and DFS

• DFS analysis and synthesis pair is expressed as follows:

• Practical significance:

– The length-N DFT of the length-N signal contains all the information about . It is convenient to work with.

– Whenever the DFT is used, actually the DFS is being used – computations involving are affected by the true periodicity of the coefficients.

1 ,..., 0 for ] [ ~ ] [~

1

0

2

N k e n x k X

N

n

N

nk j

1 ,..., 0 for ] [~ 1

] [ ~1

0

2

N n e k XN

n xN

k

N

nk j

][kX ][nx][nx

][kX ][~

kX

][kX

][~

kX

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7.1.4 Relation with Other Transforms

• The DFT samples the Z-transform at evenly spaced samples of

the unit circle over one revolution:

• In other words, the DFT samples on period of the Fourier

Transform at N evenly spaced frequencies

1 ,..., 0 for | ) ( ] [/ 2 N k z X k XN k je z

N

kk

2

1 ,..., 0 for | ) ( ] [/ 2 N k e X k XN kj

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7.1.5 DFT Transformation Matrix

• The DFT can be represented in this way

• This introduces the widely-used and convenient notations:

whence

2) 1 ( ) 1 ( 3 ) 1 ( 2 1

1 3 2 1

... 1

: : : : : :

... 1

1 1 1 1 1 1

NN

NN

NN

NN

NN N N N

N

W W W W

W W W WW T

N jNe W

/ 2

) (symmetric matrix ation transform DFT

Tx X

nkN

N

n

Wn x k X1

0

] [ ] [

X T x1

1

0

] [1

] [

nkN

N

n

Wk XN

n x

]1 [

:

]1[

] 0[

N x

x

x

x

]1 [

:

]1[

] 0[

N X

X

X

X

7

7.1.6 DFT Transformation Matrix: Example

The DFT matrices of dimension 2, 3, 4 are as follows:

If we compute X by as follows:

Where we observe that the real part of X[k] is even-symmetric, and the imaginary part is odd-symmetric – the DFT of the real signal.

1 1

1 12 T

, ] 2 0 3 1[T

x

2

3 1

2

3 11

2

3 1

2

3 11

1 1 1

3

j j

j jT

j j

j j

1 1

1 1 1 1

1 1

1 1 1 1

4 T

Tx X

] 3[

] 2[

]1[

] 0[

5 1

0

5 1

2

2

0

3

1

1 1

1 1 1 1

1 1

1 1 1 1

4

X

X

X

X

j

j

j j

j jx T X

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• Let

be length-N sequences indexed n=0,…,N-1.

• DFT Properties:

– Linearity: For constant a, b:

– Even Sequences: If x[n] is even:

– Odd Sequences: If x[n] is odd:

– Real Sequences: If x[n] is real:

7.2.1 DFT Properties - I

] [ ] [ ], [ ] [ ], [ ] [k H n h k Y n y k X n xDFT DFT DFT

] [ ] [ ] [ ] [k Y k aX n by n axDFT

. 1 ,..., 0 , ; ] [ ] [ ] [ ] [ N k n k N X k X n N x n x

. 1 ,..., 0 , ; ] [ ] [ ] [ ] [ N k n k N X k X n N x n x

] [ ] [ and ] [ ] [k N X k X k N X k X

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7.2.2 DFT Properties - II

– Circular Shift:

– Duality:

– Parseval’s Theorem

(DFT conserves energy)

– Cyclic (circular convolution): If

] [ ] )) [((/ 2

k X e m n xN m j DFT

N

] [ ] [ DFTn N Nx k X N

1

0

2 21

0

2] [ ] [

N

n

N

k

n x N k X

. 1 ,..., 1, 0 ]; [ ] [ ] [ N k k H k X k Y

1

0

] [ ] [ ] [N

mN m n h mx n ythen

1

0

] [ ] [N

mNmh m n x

] [ ] [n h n xN

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7.2.3 DFT Properties – Circular Shift Example

). 2 ( ] [ ] )) [((/ 2

m k X e m n xN m j DFT

N

] [ ] [k X n xDFT ] [ ] [1 1k X n x

DFT

] [ ] [) / 2(

1k X e k Xm N k j

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• The cyclic convolution is not the same as the

linear convolution of linear system theory. It

is a by-product of the periodicity of DFS/DFT.

• When the DFT X[k] is used, the periodic interpretation of the

signal x[n] is implicit: if

then for any integer m:

• Thus, just as the DFT X[k] is implicitly period-N (i.e., is the

DFS), the inverse DFT is also implicitly period-N — the periodic

extension of x[n].

7.3.1 Cyclic Convolution – What?

] [ ] [ ] [n h n x n yN

] [ *] [ ] [n h n x n y

1 ,..., 0 for ] [1

] [ IDFT ] [1

0

2

N n e k XN

k X n xN

k

N

nk j

N

1

0

22 1

0

) ( 2

] [ ] [1

] [1

N

k

km j N

kn j N

k

N

mN n k j

n x e e k XN

e k XN

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7.3.2 Cyclic Convolution – Why?

• Why is cyclic convolution not true linear convolution?

• Because a wraparound effect occurs at the “ends”:

• The procedure of each pair are summed around the circle.

• In a while, it will be seen that can be computed

using

] [ *] [ ] [n h n x n y]. [ ] [ ] [n h n x n y

N

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7.3.3 Cyclic Convolution – Example 1

0] [ ] [ ] [1 0 1

knN

DFTW n X n n n x

] [ ] [ ] [ ] [2 2 1 30

k X W k X k X k Xkn

N

] )) [(( ] [ ] [ ] [0 2 2 1 3N n n x n x n x n x

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7.3.4 Cyclic Convolution – Example 2

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7.4.1 Linear Convolution by DFT

• Of course, linear convolution is desired. Fortunately, the linear convolution can be computed via the DFT, with a minor modification.

• Method: To compute the linear convolution

of a sequence x[n] of length- N1 and a sequence h[n] of length- N2.

via the DFT, form the length N1 + N2 -1 zero-padded sequences

and then] [ *] [ ] [n h n x n y

2 0

1 0 ]; [] [ˆ

2 1 1

1

N N n N

N n n xn x

2 0

1 0 ]; [] [ˆ

2 1 2

2

N N n N

N n n hn h

] [ˆ ] [ˆ ] [ˆ1 2 1

n h n x n yN N

1

0

] [ˆ ] [ˆN

mN m n h mx

2 0 ]; [ *] [2 1 N N n n h n x

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7.4.2 Cyclic Convolution Example -1

The linear convolution is computed as the time instants (in this example) 0 n 4. This can be regarded as a form of time-aliasing – resulting from the sampling of the Fourier Transform.

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7.4.2 Cyclic Convolution Example -1

The linear convolution is computed as the time instants (in this example) 0 n 9. Aliasing is eliminated, so the result is the same as the linear convolution of the non-extended sequence.

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7.4.3 Cyclic Convolution Example -2k

N

LkN

W

Wk X k X

1

1] [ ] [2 1

2

31

1] [

kN

LkN

W

Wk X

). 2 (L N