Chapter 5 Part A - 西安电子科技大学个人主页系统 | 我...

2017/11/15

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Chapter 5

Finite Length Discrete Transforms

Part A

The Discrete Fourier Transform (DFT)

3

Finite-Length Discrete Transforms

It is convenient to map a finite-length sequence from the time domain into a finite-length sequence of the same length in a different domain, and vice-versa.

Such transformations are usually collectively called finite-length transforms.

4

Discrete Fourier Transform

Orthogonal Transforms The Definition of DFT Relation between DTFT and DFT and

their inverses Operations on Finite-Length Sequences

Circular Time-Reversal Circular Shifting Circular Convolution

Classifications of Finite-Length Sequences

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1. Orthogonal TransformsDefinition: with basis sequences

For length-N sequence x[n], 0≤n≤N－1, with X[k] denoting the coefficients of its N-point orthogonal transform :

1*

0

1

0

[ ] [ , ] 0 1

1 , 0 1

N

n

N

k

X k x n k n k N

x n X k k n n NN

y

y

1

*

0

1,1 , ,0,

N

n

l kk n l n

l kNy y

,k ny

6

1. Orthogonal Transforms

Proof: Important consequence--Parseval’s relation

Transforms with good energy compaction properties: most of the signal energy is concentrated in a subset of

the transform coefficients remaining coefficients with very low energy to be set to

zero values

1 12 2

0 0

1N N

n kx n X k

N

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2.1 Definition

Definition

DFT X[k] is obtained by uniformly sampling the DTFT X(ejω) over one principal value interval 0≤ω≤ 2π at ωk= 2π k/N, 0≤k≤ N－1 in the frequency domain.

Sampling the DTFT X(ejω) of x[n], 0≤n≤N－1

Nk

jeXkX

2)(][

8

2.1 Definition

Length-N sequence X[k] : discrete Fourier transform (DFT) of the sequence x[n] in the frequency domain

2

21

0

[ ] ( )

[ ] , 0 1

jk

NkN j n

N

n

X k X e

x n e k N

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2.1 Definition

Using the notation the DFT is usually expressed as:

Inverse discrete Fourier transform (IDFT)

NjN eW /2

10][][1

0

NkWnxkXN

n

knN

10][1][1

0

NkWkXN

nxN

k

knN

Proof 10

2.1 Definition

WN=e- j2π /N : twiddle factor |WN|=1

One of the N N-th roots of unity

NkN

kN WW

10 NNN WW

12/ NNW

kN

NkN WW 2/ 0

1

0

N

k

kNW

otherwiseintergeran is ,for

0,1

0

)( rrNlkNW

N

k

nlkN

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2.1 Definition

Example 1 Consider the length-N sequence

Its N-point DFT is given by1

0

0[ ] [ ] [0] 1,

0 1

Nkn

N Nn

X k x n W x W

k N

1, 0[ ]

0, 1 1n

x nn N

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2.1 Definition

Example 2 Consider the length-N sequence defined for

where r is an integer in the range Using the Euler’s function we can write

10)/2cos(][ NnNrnnx 0 1r N

2 / 2 /1[ ] ( )21 ( )2

j rn N j rn N

rn rnN N

x n e e

W W

p p

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2.1 Definition

The N-point DFT of g[n] is thus given by

1

0

)(1

0

)(

21][

N

n

nkrN

N

n

nkrN WWkX

/ 2, for ,/ 2, for ,

0, otherwise.

N k rN k N r

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2.1 Definition

Example 3 Rectangular Pulse RN [n], width N

N-point DFT is given by1 1

0 0/2 /2 /2

/2 /2 /2

1

1[ ] [ ]

1

sin( )sin( / )

kNN Nkn kn N

N N kn n N

kN kN kNN N N

k k kN N N

Nj kN

WX k x n W W

WW W WW W W

k ek N

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2.1 Definition

2N-point DFT is given by

Length of DFT plays a very important role in DFT

2 1 1

2 20 0

12 2

2

[ ] [ ]

1 sin( / 2)1 sin( / 2 )

N Nkn knN N

n nNkN j kN N

kN

X k x n W W

W keW k N

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2.1 Definition

0 1 2 30

1

2

3

44-Point DFT of R4(n)

n

|X(k

)|

0 1 2 3 4 5 6 7 80

1

2

3

48-point of R4(n)

n

|X(k

)|

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4DTFT of R4(n)

/

Ampl

itude

0 1 2 3 4 5 60

1

2

3

4Relation between DFT and DTFT

Ampl

itude

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2.1 Definition

Mapping Relations between time-domain and frequency-domain transforms

(Time-domain) (Frequency-domain)Continuous AperiodicalDiscrete Periodical

Periodical DiscreteAperiodical Continuous

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2.1 Definition

Type 1: Continuous-Time Fourier Transform(CTFT)

( )ax t ( )aX jContinuousAperiodical

AperiodicalContinuous

( ) ( ) j ta aX j x t e dt

1( ) ( )2

j ta ax t X j e d

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2.1 Definition

Type 2: Continuous-Time Fourier Series(CTFS)

( )ax t 0( )aX jkContinuousPeriodical

AperiodicalDiscrete

0/ 2

0 / 2

1( ) ( )p

p

T jk ta aT

p

X jk x t e dtT

00( ) ( ) jk t

a ak

x t X jk e

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2.1 Definition

Type 3: Discrete-Time Fourier Transform(DTFT)

[ ]x n ( )jX e DiscreteAperiodical

PeriodicalContinuous

( ) ( )j j n

n

X e x n e

-

1[ ] ( )2

j j nx n X e e d

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2.1 Definition

Type 4: Discrete Fourier Transform (DFT)

[ ]x n [ ]X kDiscretePeriodical

PeriodicalDiscrete

1

0

[ ] ( ) , 0 1N

knN

n

X k x n W k N

1

0

1[ ] ( ) , 0 1N

knN

kx n X k W n N

N

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2.1 Definition

The computation of the DFT and the IDFT requires, respectively, approximately N2

complex multiplications and N(N-1) complex additions.

However, elegant methods have been developed to reduce the computational complexity to about N(log2N) operations.

These techniques are usually called fast Fourier transform (FFT) algorithms .

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2.2 Matrix Relations

Since MATLAB stands for MAtrixLABoratory, we represent DFT definition in terms of matrix form

can be expressed in matrix form as

1

0

[ ] [ ] , 0 1N

knN

n

X k x n W k N

NX D x24


Where

And is the DFT matrix given by

[0] [1] [ 1] TX X X N X

[0] [1] [ 1] Tx x x N x

1 2 1

2 4 2( 1)

1 2( 1) ( 1)( 1)

1 1 1 111

1

NN N N

NN N N N

N N N NN N N N N

W W WW W W

W W W

D

ND N N

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Likewise, the IDFT relations can be expressed in

can be expressed in matrix form as

Where is the IDFT matrix

1Nx D X

1

0

[ ] [ ] , 0 1N

knN

k

x n X k W n N

1ND N N

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where

Note:

1 2 ( 1)

1 2 4 2( 1)

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

1 1 1 11

1 1

1

NN N N

NN N N N

N N N NN N N N N

W W WW W W

N

W W W

D

*1 1NN D

ND

27


Obviously, the relation between the two coefficient matrices can be expressed as follows

Therefore, the algorithms designed for DFT are applicable to IDFT

1 *1N NN D D

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2.3 DFT Computation Using MATLAB

Built-in Functions to compute the DFT and the IDFT are fft and ifft

fft(x) ifft(X)fft(x,M) ifft(x,M)

These functions make use of FFT algorithms which are computationally highly efficient compared to the direct computation

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0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

/

Mag

nitu

deDFTDTFT

cos(6 /16) 0 15n n Sequence

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N-point sequence μ[n]

Determine the M-point DFT.

1, 0 1[ ]

0, otherwisen N

u n

0 2 4 6 80

0.2

0.4

0.6

0.8

1Original time-domain sequence

Time index n

Am

plitu

de

0 5 10 150

2

4

6

8Magnitude of the DFT samples

Frequency index k

Mag

nitu

de

0 5 10 15-1.5

-1

-0.5

0

0.5

1

1.5Phase of the DFT samples

Frequency index k

Pha

se

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N-point sequence μ[n]

Determine the M-point DFT.

/ , 0 1[ ]

0, otherwisek K k N

V k

0 2 4 6 80

0.2

0.4

0.6

0.8

1

Frequency index k

Am

plitu

de

Original DFT samples

0 5 10 15-0.2

-0.1

0

0.1

0.2

0.3Real part of the time-domain samples

Time index n

Am

plitu

de

0 5 10 15-0.2

-0.1

0

0.1

0.2Imaginary part of the time-domain samples

Time index n

Am

plitu

de

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3. Relations between DTFT and DFT and their inverses

Relations: (for finite x[n] of length N)

is obtained by uniformly sampling on the ω-axis between

( )jX e[ ]X k ?

sampling

12 /

2 /0

[ ] ( ) [ ] , 0 1N

j j kn N

k Nn

X k X e x n e k Nw p

w p

1

0][][)(

N

n

nj

n

njj enxenxeX

[ ]X k

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3.1 Numerical Computation of theDTFT Using the DFT

A practical approach to the numerical computation of the DTFT of a finite-length sequence.

Let be the DTFT of a length-Nsequence x[n]. We wish to evaluate at a dense grid of frequencies , where M >> N:

2 / , 0 1k k M k M

( )jX e

( )jX e

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Define a new sequence

Then

1 2

0

( )k

M j knj M

e en

X e x n e

[ ], 0 1[ ]

0, 1e

x n n NX n

N n M

1 1

2 /

0 0( )k k

M Nj j n j kn M

n nX e x n e x n e

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Thus is essentially an M-point DFTof the length-M sequence

The DFT can be computed very efficiently using the FFT algorithm if M is an integer power of 2.

( )kjeX e

[ ]eX k ex n[ ]eX k

36


Example Compute the DFT and the DTFT of the

sequence, as shown belowcos(6 /16) 0 15n n

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9

/

Mag

nitu

de

DFTDTFT

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3.1 Numerical Computation of theDTFT using DFT

The function freqz employs this approach to evaluate the frequency response at a prescribed set of frequencies of a DTFT expressed as a rational function in

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3.2 DTFT from DFT by interpolation

1 1 1

0 0 01 1

[ 2 / ]

0 0

1( ) [ ] [ ]

1 [ ]

N N Nj j n kn j n

Nn n k

N Nj k N n

k n

X e x n e X k W eN

X k eN

x[n]IDFT

Exchange of the order of summations

( )jX e [ ]X k ?

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Let and Thus

1

2 /

0

Nj k N n

n

S e

2 /j k Nr e

( 2 )1

2 /0

2 / ( 1) / 2

1 11 1

2sin2

2sin2

N j N kNn

j k Nn

j k N N

r eS rr e

N k

eN k

N

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1

0

2( ) [ ] ( )N

j

k

kX e X kN

2/)1(

2sin

2sin

NjeN

N

]2/)1)][(/2([1

0

22sin

22sin

][1)(

NNkjN

k

j e

NkN

kN

kXN

eX

interpolation formula

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DTFT can be possibly determined by the following interpolation formula

1

0

2( ) [ ] ( )N

j

k

kX e X kN

( )jX e( )X kinterpolation

sampling

2 /( ) [ ]j

NX e X

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3.3 Sampling the DTFT

Sequence x[n] , with a DTFT X(ejω)

Uniformly sample X(ejω) at N equally spaced points ωk=2π k/N, 0≤k≤ N－1 developing the N frequency samples {X(ejωk)}

Let

IDFT of

[ ] ( ), 0 1kjY k X e k N

( ) [ ]j jX e x e

10 Nk

2 /[ ] ( ) | [ ] , 0 1k

k

j kk N NY k X e x W k N

][kY 10,][1][1

0

NnWkYN

nyN

k

knN

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i.e.

Making use of the identity

1

0

1( )

0

1[ ] [ ]

1[ ]

Nk kn

N Nk

Nk n

Nk

y n x W WN

x WN

1( )

0

1, for10, otherwise

Nk n

Nk

n mNW

N

44


We arrive at the desired relation

Thus y[n]is obtained from x[n] by adding an infinite number of shifted replicas of x[n] , with each replica shifted by an integer multiple of N sampling instants, and observing the sum only for the interval 0≤n≤N－1

[ ] [ ], 0 1m

y n x n mN n N

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For finite length-M sequences x(n)

assume that the samples outside the specified range are zeros. If M ≤ N, then y[n]=x[n] for 0≤n≤N－1 If M > N, there is a time-domain aliasing of samples of

x[n] in generating y[n] , and x[n] cannot be recovered from y[n]

[ ] [ ], 0 1m

y n x n mN n N

Sampling Theorem in Frequency-Domain 46


Example Let x[n]={0 1 2 3 4 5}

Sampling 4 point at its DTFT.Can we recover x[n] from the sampling?

47


By sampling its DTFT X(ejω) at ωk=2π k/4, 0≤k≤3, and then applying a 4-point IDFT to these samples, we arrive at the sequence y[n] given by

y[n]= x[n] + x[n+4] + x[n－4] , 0≤k≤3i.e. y[n]= {4 6 2 3}

{x[n] } cannot be recovered from {y[n]}48

4. Operations on Finite-length Sequences

Let x[n] be a sequence of lengthN defined for 0≤n≤N－1, the time-reversal and time-shift of the sequence is no longer defined in 0≤n≤N－1.

We thus need to define another type of operations that will keep the reversed and shifted sequences in the range 0≤n≤N－1.

Similarly, another type of convolution needs to be defined that ensure the convoluted sequence is in the range 0≤n≤N－1.

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4.1 Circular Time-Reversal Operation

The time-reversal operation on a finite-length sequence that develops a sequence also defined for the same range of the time index n, is obtained by using the modulo operation.

Let 0, 1,…, N-1 be a set of N positive integers, and let m be any integer. The integer robtained by evaluating m modulo N is called the residue and is an integer with a value between 0 and N-1.

r =<m>N= m modulo N r=m+N50

4.1 Circular Time-Reversal Operation

Thus, the time-reversal version {y[n]} of the length-N sequence{x[n]} defined for 0≤n≤N－1 is given by

{ [ ]} [ ], 0 1N

y n x n n N

[ ], 0,[ ], otherwise.

x n nx N n

[ ] [ ]NNx n N R n

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4.2 Circular Time-ShiftingOperation

The time-shifting operation on a finite-length sequence that results in another sequence of the same length and defined for the same range of value of n, is referred to as the circular time-shifting operation.

Such a shifting operation is achieved by using the modulo operation.

52


The circular time-shifting operation of a length-N sequence x[n] by an arbitrary amount n0 sample period is defined by the equation

where is also a length-N sequence. If (right circular shift)

0[ ]c Nx n x n n

[ ]cx n

0 0n

0 0

0 0

[ ], for 1,[ ]

[ ], for 0 .c

x n n n n Nx n

x N n n n n

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Given a length-6 sequence x[n], its circularly shifted versions are shown

[ ]x n 6 61 5x n x n 6 6

4 2x n x n

54


As can be seen from the figures, a right circular shift by n0 is equivalent to a left circular shift by N-n0 sample periods.

A circular shift by an integer number n0greater than N is equivalent to a circular shift by .

N0n

55


In the frequency domain, the circular shifting operation by k0 samples on the length-N DFT sequence X[k] is defined by

where Xc[k] is also a length-N DFT.

0[ ]c NX k X k k

56


Steps to get a circular shift of an M-point sequence x[n] Periodize

Time-shifting

Principal value

[ ] [ ]N

y n x n

1 0 0 Ny n y n n x n n

1[ ]C Nx n y n R n

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DFT of the circular shift sequence

1

0

1

0

[ ] [ ] [ ]

[ ] DFT [ ]

[ ] [ ]

[ ]

NN N

Nkn

N NNn

Nkn

NNn

y n x n m R n m

Y k y n

x n m R n W

x n m W

58


'

1'

'1

'

'1 1 1

' 0 ' 0 '1

'

' 01

'

0

[ ] [ ' ]

[ ' ]

(.) (.) (.)

[ ' ]

[ '] [ ]

N mk n m

NNn m

N mkm kn

N NNn m

N m N mkm

Nn n n N

Nkm kn

N NNnN

km kn kmN N N

n

Y k x n W

W x n W

W

W x n W

W x n W W X k

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4.3 Circular Convolution

Analogous to linear convolution, but with a subtle difference

Comparison of linear convolution with circular convolution Consider two length-N sequences, g[n] and h[n]

respectively. Their linear convolution results in a length-(2N-1) sequence given by Ly n

1

0[ ] [ ] [ ], 0 2 2

N

Lm

y n g m h n m n N

60


linear convolution circular convolutionLength of

convolution 2N－1 to be specified

Convolution Formulas

Convolution Signs or

Condition of equivalence ?

1

0

( ) ( )N

C Nm

y n g m h n m

1

0

( ) ( ) ( )N

Lm

y n g m h n m

N

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To develop a convolution-like operation resulting in a length-N sequence yC[n], we need to utilize a circular time-reversal, and then apply a circular time-shift.

Resulting operation, called a circular convolution, is defined by

1

0[ ] [ ] [ ], 0 1

N

C Nm

y n g m h n m n N

62


Since the operation defined involves two length-N sequences, it is often referred to as an N-point circular convolution, denoted as

yC[n]=g[n] h[n] The circular convolution is commutative, i.e.

g[n] h[n]=h[n] g[n]

N

N N

63


Example 1 Length of Circular Convolution is 4

g[n] h[n]

Step 1: Perform Circular time-reversal operation on h[m] (or g[m])

4[ ]h m

These seven samples are enough to calculate the circular convolution because of the periodicity of DFT 64


Step 2: Perform Circular time-shift operation

Red {2 1 1 2}44[ ] [ ]h m R m

Blue {2 2 1 1}44[ 1 ] ( )h m R m

Green {1 2 2 1} 44[ 2 ] ( )h m R m

Purple {1 1 2 2}44[ 3 ] ( )h m R m

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Step 3: Perform multiplication and summation of sequences over the region of 0≤m≤3 for n=0,n=1,n=2and n=3 respectively

y(0)=

1 2 0 12 1 1 22+2+0+2= 6 y(1)=

1 2 0 12 2 1 12+4+0+1= 7

y(2)=

1 2 0 11 2 2 11+4+0+1= 6 y(3)=

1 2 0 11 1 2 21+2+0+2= 5

66


Example 2 Length of Circular Convolution is 7 In order to develop the 7-point circular convolution

on the sequences in the former example, we extended these two sequences to length 7 by appending each with 3 zero-valued samples, i.e.

[ ], 0 3[ ]

0, 4 6e

g n ng n

n

[ ], 0 3[ ]

0, 4 6e

h n nh n

n

67


ge[n] he[n]

Perform Circular time-reversal operation on he[m]

7[ ]eh m

ge[m]

These three samples are not involved in the circular convolution operation

68


In this case, the procedure of circular convolution is equivalent to that of linear convolution over the region of principle value. Obviously, this conclusion always holds when the length of Circular Convolution is not less than 7

SummaryProvided that the length of Circular Convolution is not less than N+M－1 where N and M are the lengths of the two sequences involved, the procedure of circular convolution is equivalent to that of linear convolution

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5.1 Classification Based on Conjugate Symmetry

Based on Conjugate SymmetryIt has been discussed in Ch.2 of 4th edition. Circular Conjugate Symmetry

A length-N circular conjugate-symmetricsequence x[n]

A length-N circular conjugate-antisymmetricsequence

* *[ ] [ ] [ ], 0 1N

x n x n x N n n N

* *[ ] [ ] [ ], 0 1N

x n x n x N n n N 70


A length-N sequence x[n] can be expressed as

wherecircular (periodic) conjugate-symmetric part

circular (periodic) conjugate-antisymmetric part

[ ] [ ] [ ] 0 1pcs pcax n x n x n n N

*1[ ] [ ] [ ] , 0 12pcs N

x n x n x n n N

*1[ ] [ ] [ ] , 0 12pca N

x n x n x n n N

[ ] [ ] [ ], 0 1pcs pcaX k X k X k k N

71


*[ ]Nx n *[ ]x N n

x[n]Conjugating

x*[n]Folding

x*[－ n] Periodical Extension

× RN[n](0 1)n N

x*(n)

n

x*[—n]

n

n

*[ ]Nx n

72


Example Consider the length-4 sequence defined for

Conjugate sequence

Circular conjugate sequence

*4

[ ] 1 4, 5 6, 4 2, 2 3u n j j j j

*[ ] 1 4, 2 3, 4 2, 5 6u n j j j j

[ ] 1 4, 2 3, 4 2, 5 6u n j j j j 0 3n

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Conjugate-symmetric part

Circular conjugate-antisymmetric part

*4

1[ ] [ ] [ ]2

1, 3.5 4.5, 4, 3.5 4.5

PCSu n u n u n

j j

*4

1[ ] [ ] [ ]2

4, 1.5 1.5, 2, 1.5 1.5

pcau n u n u n

j j j j

74

5.2 Classification Based on Geometric Symmetry

Based on Geometric SymmetryA length-N symmetry sequence x[n] satisfies the condition

A length-N antisymmetry sequence x[n]satisfies the condition

[ ] [ 1 ]x n x N n

[ ] [ 1 ]x n x N n

75


Center of symmetry

Center of symmetry

Center of symmetry

Center of symmetry

[ ]h n

n0 1

2

74 5

6

83

Type 1N=9

[ ]h n

n0 1

2 745

63

Type 4N=8

[ ]h n

n0 1

2 7

4

56

83

Type 3N=9

[ ]h n

n0 1

274

563

Type 2N=8

76


[ ]h n

n0 1

2

74 5

6

83

Center of symmetry

Type 1N=9

12

( 1) /2

1

1 1( ) 2 cos( )2 2

N

j j N

n

N NX e e x x n nw w w

( 1)/2( 1) /

1

1 1 2[ ] 2 cos2 2

Nj N k N

n

N N knX k e x x nN

p p

Symmetric Sequence with Odd Length

1( ) , =0 or 2

N

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[ ]h n

n0 1

274

563

Center of symmetry

Type 2N=8

Symmetric Sequence with Even Length

/2

( 1) /2

1

12 cos2 2

Nj j N

n

NX e e x n nw w w

/2( 1) /

1

2 1[ ] 2 cos

2

Nj N k N

n

k nNX k e x nN

p p

1( ) , =0 or 2

N

78


[ ]h n

n0 1

2 7

4

56

83

Center of symmetry

Type 3N=9

Antisymmetric Sequence with Odd Length

)sin(2

122/)1(

1

2/)1( nnNxjeeXN

n

Njj

2/)1(

1

/)1( 2sin2

12][N

n

NkNj

NknnNxjekX

1( ) , =0 or 2 2

N

79


[ ]h n

n0 1

2 745

63

Center of symmetry

Type 4N=8

2/

1

2/)1(

21sin

22

N

n

Njj nnNxjeeX

2/

1

/)1( )12(sin2

2][N

n

NkNj

NnknNxjekX

Antisymmetric Sequence with Even Length

1( ) , =0 or 2 2

N

Chapter 5 Part A - 西安电子科技大学个人主页系统 | 我...

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