Digital Signal Processing Lecture 7

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

University of Trento, Italy B. Demir

University of Trento, Italy

Discrete Time Fourier Transform

B. Demir


Discrete Fourier Transform

B. Demir

• 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~



B. Demir

• 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 ][~



Sampling in Frequency Domain-Example



B. Demir

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



B. Demir

• 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 ~



B. Demir

• 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


Implications of Inherent DFT Periodicity

B. Demir

Use DFS for interpretations

Use DFT for computations

User Before

User After


Discrete Fourier TransformSampling in Frequency Domain-Example

B. Demir



2 /j NNW e


DFT versus DTFT

B. Demir



B. Demir


• 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

B. Demir



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).


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

B. Demir


Time-domain aliasing

B. Demir


Illustration of Time-Domain Aliasing

B. Demir


DFT: Matrix form

B. Demir


IDFT: Matrix form

B. Demir


Example

B. Demir


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.


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


Periodic, circular and modulo-N operations

B. Demir

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.


The n Modulo N Operation

B. Demir


Circular folding (or reversal)

[0] n=0[ ]

[ ] 1 -1N

xz n x n

x N n n N

B. Demir


Circular folding (or reversal)

B. Demir

[ ] [ ]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


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

B. Demir

Circular Shift of a Sequence



B. Demir


Properties of Discrete Fourier Transform-Shifting

B. Demir


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



B. Demir


Circular Frequency Shifting Teorem:

B. Demir

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


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

B. Demir


Multiplication of Two N-Point DFTs: Circular Convolution


Circular Convolution

B. Demir


Circular Convolution-Example

B. Demir


Circular Convolution-Graphical Interpretation

B. Demir


Circular Convolution-Matrix Form

B. Demir


Example

B. Demir


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

B. Demir

Linear Convolution (Review)


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



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.


Example


Windowing Property:

B. Demir

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


[ ] [ ]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:

B. Demir


Properties of DFT

B. Demir

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


Parseval's Relation (Related to DFT):

B. Demir

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:


Symmetry properties of the DFT

x[n] is a real sequence:


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




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.

B. Demir


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


*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:



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

B. Demir


Digital Signal Processing Lecture 7

Documents

Transcript of Digital Signal Processing Lecture 7