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Digital Signal Processing (3)

Discrete Fourier Transforms (DFT)

Discrete Fourier Transforms (DFT) ................................................................................ 1

1. Definition ............................................................................................................ 2

2. The Properties of DFT ........................................................................................ 3

2.1. Linearity ................................................................................................... 3

2.2. Circular Shift Theorems ........................................................................... 3

2.3. Circular Convolution Theorem ................................................................ 3

2.4. Symmetry Properties of DFT ................................................................... 4

2.4.1. Conjugate Symmetry of a Finite-Duration Sequence .................... 4

2.4.2. Conjugate Antisymmetry of a Finite-Duration Sequence .............. 5

2.4.3. Conjugate Symmetric/Antisymmetric Part of a Finite-Duration

Sequence ................................................................................................. 5

2.4.4. Conjugate Symmetry of DFT of a Finite-Duration Sequence ....... 6

2.4.5. Conjugate Antisymmetry of DFT of a Finite-Duration Sequence . 6

2.4.6. Conjugate Symmetric/Antisymmetry Part of DFT of a Finite-

Duration Sequence .................................................................................. 6

3. Sample Theorem for the Fourier Transform of a Finite-Duration Sequence ...... 8

4. DFT ApplicationLinear Convolution by Using DFT ...................................... 9

4.1. Linear Convolution of Two Finite-Duration Sequences .......................... 9

4.2. Linear Convolution of an Infinite-Duration Sequence with a Finite-Duration Sequence .......................................................................................... 9

5. DFT Application: Chirp Z-Transform Algorithm ............................................. 11

Appendix: Fourier Transforms of Signals ............................................................. 14

1. Fourier Transforms of Continuous and Non-Periodic Signals .................. 14

2. Fourier Transforms of Continuous and Periodic Signals .......................... 14

3. Fourier Transforms of Discrete and Non-Periodic Signals ....................... 15

4. Fourier Transforms of Discrete and Periodic Signals ............................... 15

Problems ................................................................................................................ 18

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1. Definition

Suppose nx is a finite-duration of length N , the N-point DFT of nx is

defined as follows:

1N

0n

nk

NWnxnxDFTkX , whereN

2j

N eW

.

Remark:Since

1N

0n

njj enxeX , then

k

N

2j

1N

0n

nkN

2j1N

0n

nk

N eXenxWnxkX

The above formula shows that the DFT of a finite-duration sequence can be

derived by periodically sampling its Fourier transform.

TheoremSuppose kX is the N-point DFT of nx , then

1N

0k

nk

NWkXN

1kXIDFTnx

Proof:

nxpxWN

1WWpx

N

1WkX

N

1 1N

0p

1N

0k

knp

N

1N

0k

nk

N

1N

0p

pk

N

1N

0k

nk

N

#

Remark:N 1

nk NkN N

n 0 k

N

N k 0

W 1 W0 k 0

1 W

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2. The Properties of DFT

2.1. Linearity

nyDFTnxDFTnynxDFT

kYDFTkXIDFTkYkXIDFT

2.2. Circular Shift Theorems

Circular Shift Nmnx is referred to as circular shift of nx with

samples m , where Nn denotes n modulo N (the remainder of n after

divided by N ).

Theorem Suppose nxDFTkX , then

kXWmnxDFT mkNN

Proof:

1 1

0 0

N Nn m knk mk

N N NN N Nn n

DFT x n m x n m W W x n m W

1 1

0 0

N

N

N Nn m kmk mk pk mk

N N N N NN p n mn p

W x n m W W x p W W X k

#

Theorem Suppose kXIDFTnx , then

nxWqkXIDFT qnNN

2.3. Circular Convolution Theorem

Circular Convolution Suppose nx and ny are tow finite-duration

sequences of length N , then the N-point circular convolution of nx and

ny is defined as follows:

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1N

0p

Npnypxnynx

Theorem Suppose nxDFTkX , nyDFTkY , then

kYkXnynxDFT

Proof:

1 1 1

0 0 0

N N Nnk nk

N NN

n n p

DFT x n y n x n y n W x p y n p W

1N

0p

1N

0n

kp-n

NN

pk

N

1N

0p

1N

0n

nk

NN WpnyWpxWpnypx

kYkXWpnyWpx1N

0p

1N

0n

kp-n

NN

pk

NN

#

Theorem Suppose nxDFTkX , nyDFTkY , then

kYkXN

1nynxDFT

Proof:

1N

0n

nk

N

N

0p

np

N

1N

0n

nk

N WWpY

N

1nxWnynxnynxDFT

1N

0n

pkn

N

1N

0p

1N

0n

pkn

N

1N

0p

NWnxpYN

1WnxpY

N

1

kYkXN

1pkXpY

N

1 1N

0p

N

#

2.4. Symmetry Properties of DFT

2.4.1. Conjugate Symmetry of a Finite-Duration Sequence

A sequence nx of length N is said to be conjugate symmetric if

Nnxnx . It is easily verified that, if nx is conjugate symmetric,

its DFT kX is a real sequence. In fact,

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1N

0n

nk

NN

1N

0n

nk

N

*1N

0n

nk

N WnxWnxWnxkX

NN 1 N 1

n k nkN NN

n 0 n 0

x n W x n W X k

2.4.2. Conjugate Antisymmetry of a Finite-Duration

Sequence

A sequence nx of length N is said to be conjugate antisymmetric if

Nnxnx

. It is easily verified that, if nx is conjugate

antisymmetric, its DFT kX is an imaginary sequence. In fact,

1N

0n

nk

NN

1N

0n

nk

N

*1N

0n

nk

N WnxWnxWnxkX

kXWpxWnx1N

0p

pk

Nnp

1N

0n

kn

NNN

N

2.4.3. Conjugate Symmetric/Antisymmetric Part of a Finite-

Duration Sequence

For any sequence nx of length N , let

nxnx2

nxnx

2

nxnxnx oe

NN

where

2

nxnxnx Ne

,

2

nxnxnx No

it is clear that nxe is conjugate symmetric and often called the conjugate

symmetric part of nx . Also, nxo is conjugate antisymmetric and often

called the conjugate antisymmetric part of nx .

From the linearity of DFT, we obtain that

kXkXnxDFTnxDFTnxnxDFTkX iroeoe

where

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kXRe2

kXkXnxDFTkX er

kXImj2kXkX

nxDFTkX oi

The above formulae illustrate that the DFT of the conjugate

symmetric/antisymmetric part of nx is nothing but the real/imaginary part

of kX . It is also true that the IDFT of real/imaginary part of kX is

nothing but the conjugate symmetric/antisymmetric part of nx .

2.4.4. Conjugate Symmetry of DFT of a Finite-Duration

Sequence

A N-point DFT kX is said to be conjugate symmetric if

NkXkX . It is easily verified that, if kX is conjugate symmetric,

its IDFT nx is a real sequence. In fact,

1N

0k

nk

NN

1N

0k

nk

N

*1N

0k

nk

N WkXN

1WkX

N

1WkX

N

1nx

nxWpXN

1WkX

N

1 1N

0p

np

Nkp

1N

0k

kn

NNN

N

2.4.5. Conjugate Antisymmetry of DFT of a Finite-Duration

Sequence

An N-point DFT kX is said to be conjugate antisymmetric if

NkXkX . It is easily verified that, if kX is conjugate

antisymmetric, its IDFT nx is an imaginary sequence. In fact,

1N

0k

nk

NN

1N

0k

nk

N

*1N

0k

nk

N

WkXN

1WkX

N

1WkX

N

1nx

nxWpXN

1WkX

N

1 1N

0p

np

Nkp

1N

0k

kn

NNN

N

2.4.6. Conjugate Symmetric/Antisymmetry Part of DFT of a

Finite-Duration Sequence

For any N-point DFT kX , let

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kXkX2

kXkX

2

kXkXkX oe

NN

where

2

kXkXkX Ne

,

2

kXkXkX No

it is clear that kXe is conjugate symmetric and often called the conjugate

symmetric part of kX . Also, kXo is conjugate antisymmetric and called

the conjugate antisymmetric part of kX .

From the linearity of IDFT, we obtain that

nxnxkXIDFTkXIDFTkXkXIDFTkXIDFTnx iroeoe

where

nxRe2

nxnxkXIDFTnx er

nxImj2

nxnxkXIDFTnx oi

The above formulae illustrate that the IDFT of the conjugate

symmetric/antisymmetric part of kX is nothing but the real/imaginary part

of nx . It is also true that the DFT of the real/imaginary part of nx is

nothing but the conjugate symmetric/antisymmetric part of kX .

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3. Sample Theorem for the Fourier Transform

of a Finite-Duration Sequence

TheoremLet nx is a sequence of length N , then

2N 1 N 1j k

j NN N

k 0 k 0

2 2X e X e k X k k

N N

where N 1

j 2N

Nsin

2 e

N sin2

.

Proof:

At first, we have

kN

2

2

1jk

N

2

2

1jk

N

2

2

1j

kN

2

2

Njk

N

2

2

Njk

N

2

2

Nj

kN

2j

NkN

2j

1N

0n

nkN

2j

eee

eee

N1

e1

e1N1e

N1

kN

2e

kN

2

2

1sin

kN

2

2

Nsin

N

1N

kN

2

2

1Nj

Based on this, we further have

1N

0n

nj1N

0k

nk

N

1N

0n

njj eWkXN

1enxeX

2N 1 N 1 N 1j k nN

N

k 0 n 0 k 0

1 2X k e X k k

N N

#

Remark:The theorem illustrates that j

eX can be entirely determined by

its periodic samples.

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4. DFT ApplicationLinear Convolution by

Using DFT

4.1. Linear Convolution of Two Finite-Duration

Sequences

Suppose nx is a sequence of length N and nh a sequence of length M ,

if we let nhnxny , ny is then a sequence of length 1NM . In

order to perform the linear convolution involved in the computation of ny

by using DFT, we now construct two finite-duration sequences of the same

length 1NM :

2MNnN0

1Nn0nx

nx~

,

2MNnM0

1Mn0nh

nh

~

Its easily verified that

nh~

nx~nhnxny

Therefore, from the circular convolution theorem, we obtain that

kH~kX~IDFTnh~nx~nhnxny

where kX~

and kH~

are the 1NM -point DFTs of nx~ and nh~

,

respectively.

4.2. Linear Convolution of an Infinite-Duration

Sequence with a Finite-Duration Sequence

Suppose nx is an infinite-duration sequence and nh a finite-duration

sequence of length M , if we let nhnxny , ny is also an infinite-

duration sequence. In order to perform the linear convolution involved in the

computation of ny by using DFT, we first express nx as a sum of finite-

duration sequences, each of length N :

ii nxnx

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where

others0

1N1iniNnxnxi . Thus ny can be also expressed

as a sum of finite-duration sequences:

i

i

i

i nynhnxnhnxny

where nhnxny ii . nyi is now a linear convolution of two finite-

duration sequences and, as stated in the previous section, can be

implemented by employing 1NM -point DFT. In fact, let

iNnxnx~ ii

and

kHkX~IDFTnhnx~ny~ iii

where kX~i are kH the 1NM -point DFTs of nx~

i and nh ,

respectively. nyi is then seen to be

iNny~ny ii Remark:Note that

y n x n h n x n m h n y n m

Then

nhnx~ny~ ii

nhiNiNnxnhnxny iii iNny~nhiNnx~ ii

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5. DFT Application: Chirp Z-Transform

Algorithm

Suppose nx is a finite-duration sequence of length N and zX the Z-

transform of nx , the chirp Z-transform algorithm is directed toward the

computation of the values of zX at some points of a spiral in the z-plane:

pzX , 1M,,1,0p

where pp AWz , 0j0eAA

, 0j0eWW . The parameter

0W controls the

type, rate and direction of the spiral. If 1W0 , the spiral spirals toward the

origin of the z-plane as p increases, if 1W0 , the spiral spirals outward

from the origin of the z-plane as p increases, and if 1W0 , the spiral

becomes a circular arc.

By using the identity 222 nppn2

1np , pzX can be expressed as

1N

0n

2

np

2

n

n2

p1N

0n

npn1N

0n

n

pp

222

WWAnxWWAnxznxzX

1N

0n

2

p

nphngW

2

where 2n

n

2

WAnxng , 2n 2

Wnh

. The above formula implies that the

computation of pzX can be implemented by linear convolution of two

finite-duration sequences, which, as shown before, can be implemented by

using DFT.

Remark: In radar system, the signals of the form 2n 2

W

are often called

chirpsignals, from which the algorithm we are discussing gets its name.

In order to employ DFT in the computation of pzX , 1M,,1,0p

,

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lets construct two finite-duration sequences

2

n

n 2g n x n A W 0 n N 1g n

0 N n N M 2

,

2n

2h n W 0 n M 1h n

h n N M 1 M n N M 2

the (N+M-1)-point circular convolution of )n(g~ and )n(h~

is then

expressed as

1N

0n

1MN

2MN

0n

1MN nph~

ngnph~

ng~ph~

pg~

TheoremIt can be proven that, when p varies between 0 and M-1,

1N

0n

1N

0n

1MN nphngnph~

ngph~

pg~

Proof:

(1) Since 1Mp0 and 1Nn0 , then 1Mnp1N .

(2) If 1Mnp0 , then

npnp 1MN

Thus, from the definition of nh~

, we obtain that

1N

0n

1N

0n

1N

0n

1MN nphngnph~

ngnph~

ngph~

pg~

(3) If N 1 p n 1 , then

2MNnp1MNnpM 1MN

Thus, from the definition of h n , we obtain that

1N

0n

1N

0n

1MN np1MNh~

ngnph~

ngph~

pg~

1N

0n

1N

0n

nphng1MNnp1MNhng #

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From the theorem, pzX is then given by

kH~kG~IDFTWph~pg~WnphngWzX 2p

2

p1N

0n

2

p

p

222

where kG~

and kH~ are the (N+M-1)-point DFTs of ng~ and nh~

,

respectively.

We may try another way to employ DFT to compute pzX . Let

3MN2nN0

1Nn0ngng~

3MN2n1MN0

2MNn01Nnhnh

~

and

nh~

ng~ny~

where the circular convolution involving ny~ is a (2N+M-2)-point circular

convolution, much larger than the one we recommended above. It is easily

verified that

kH~kG~WIDFT1Npy~zX k1N 2MN2p , 1M,,1,0p

where kG~

and kH~ are the (2N+M-2)-point DFTs of ng~ and nh~

,

respectively.

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Appendix: Fourier Transforms of Signals

Definition

dtetxX tj

According to whether the signal tx is continuous or discrete, periodic or

non-periodic, its Fourier transform X will present different forms

correspondingly. More concrete, If tx is continuous, then its Fourier transform X is non-

periodic

If tx is discrete, then its Fourier transform X is periodic

1. Fourier Transforms of Continuous and Non-

Periodic Signals

If tx is a continuous and non-periodic signal, its Fourier Transform

dtetxX tj

is clearly non-periodic and continuous.

2. Fourier Transforms of Continuous and Periodic

Signals

If tx is a continuous and periodic signal with T as its period, tx can be

then expanded into a Fourier series:

n

ntT

2j

neatx

, where

2

T

2

T

ntT

2j

n dtetxT

1a

Based on the series-form expression of tx , the Fourier transform of tx is

then given by

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n

tnT

2j

n

tj

n

ntT

2j

n

tj dte2

1a2dteeadtetxX

n

n nT

2a2

This shows that X is non-periodic and discrete. By the way, the

coefficientsna of Fourier series of tx are just the samples

T

2nX

of

X , except for a scalar/factor 2 .

3. Fourier Transforms of Discrete and Non-Periodic

Signals

If tx is a discrete and non-periodic signal, tx can be then expressed as:

n

nTtnTxtx

Based on this expression of tx , the Fourier transform of tx is then given

by

n

nTjtj

n

tj enTxdtenTtnTxdtetxX

This shows that X is periodic and continuous. The period of X is

T

2.

Remark:If we let 1T , X is then simplified as

jn

jn eXenxX

This is just the so-called Fourier transform of sequence nx .

4. Fourier Transforms of Discrete and Periodic

Signals

If tx is discrete at the points nTt , where ,2,1,0n , and periodic

with NT as its period, tx can be then expressed as

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k

1N

0n

TkNntnTxtx

Based on this expression of tx , the Fourier transform of tx is then given

by

dteTkNntnTxdtetxX tj

k

1N

0n

tj

k

jNTk1N

0n

TnjTkNnj

k

1N

0n

eenTxenTx

k

1N

0n

Tnj

NT2kenTx

NT2

k

1N

0n

nkN

2j

NT

2kenTx

NT

2

k

1N

0n

nk

NNT

2kWnTx

NT

2

k NT2kkX

NT2

where

1N

0n

nk

NWnTxkX ,N

2j

N eW

Note that kX is periodic with N as its period, X can be further

expressed as

j

1N

0k NT

2jNkkX

NT

2X

Remark 1: X is discrete at the points kNT

2 , ,2,1,0k , and

periodic withT

2as its period and discrete at the points k

NT

2 ,

,2,1,0k .

Remark 2:If we let 1T , kX can be simplified as

1N

0n

nk

NWnxkX

which is just the so-called DFT of the finite-duration sequence nx .

Remark 3:In some literature, the DFT, i.e.,

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1N

0n

nk

NWnxkX

is often called the discrete Fourier series(DFS) of periodic sequence

nx ,

while, the IDFT, i.e.,

1N

0k

nk

NWkXN

1nx

is often called the inverse discrete Fourier series(IDFS) of periodic sequence

nx .

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Problems

0. x n 5 n 4 n 1 2 n 2 n 3 3 n 4 2 n 5 ,

X k ?

0. 1 n 0,1,2,3

x n0 n 4,5

, X k ?

1.Find the N-point DFT for x n

(1) 0 Nx n cos n R n

(2) 2 Nx n n R n

1.Find the N-point DFT for x n

(1)

mn

Nnx

2cos , where Nm0

(2) nnRnx N

6.Suppose X k is the N-point DFT of x n , let nRnxnh rNN , find the rN-

point DFT of h n .

9.Suppose X k is the N-point DFT of x n , let

10

10

rNnN

Nnnxny

Find the rN-point DFT of y n .

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10. Suppose nx1

and nx2

are real sequences of length N and

1

0

21

N

p

Nnpxpxnx ,

find the N-point DFT of

x n.