Discrete Fourier Transform - VLSI Signal Processing …twins.ee.nctu.edu.tw/courses/dsp_16/Class...

DSP (2015 Spring) Discrete Fourier Transform

NCTU EE 1

Discrete Fourier Transform

What is Discrete Fourier Transform (DFT)?

(Note: It’s not DTFT – discrete-time Fourier transform)

A linear transformation (matrix)

Samples of the Fourier transform (DTFT) of an aperiodic (with finite duration) sequence

Extension of Discrete Fourier Series (DFS)

Review: FT, DTFT, FS, DFS Time signal Transform Coeffs.

(periodic/aperiodic) Coeffs. (con-ti./discrete)

Analog aperiodic FT Aperiodic Continuous Analog periodic FT

FS Aperiodic Aperiodic

Continuous (impulse)Discrete

Discrete aperiodic DTFT Periodic Continuous Discrete periodic DFS Periodic Discrete Discrete finite-duration DFT

The Discrete Fourier Series

Properties of NW

kN

jk

NNj

N eWeW

2

2 thus,

-- NW is periodic with period N. (It is essentially cos and sin) : Nk

NNk

Nk

N WWW 2

--

1

0 if ,0

if ,N

k

lkN mNl

mNlNW

(Pf) (i) 1 , If 0 N

NmkN

lkN WWWNml

NWN

k

N

k

lkN

1

0

1

0

1

(ii) 1 , If lNWNml

0

1

11

1

1

1

0

l

Nl

N

NlN

N

k

lkN

WW

WW

--

1

0

1 N

k m

lkN mNlW

NlY


NCTU EE 2

DFS for periodic sequences

][~ nx

n

NperiodrNnxnx ,~~

Its DFS representation is defined as follows:

Synthesis equation:

1

0

1 2~1~1~

N

k

knN

N

ok

knN

jWkX

NekX

Nnx

Analysis equation:

1

0

~~ N

n

knnWnxkX

Note: The tilde in x~ indicates a periodic signal.

][~

kX is periodic of period N.

Pf)

1

0

~1~N

k

knNWkX

Nnx

Pick an r ( Nr 0 )

rnNW rn

N

N

k

knN

rnN WWkX

NWnx

1

0

~1~

1

0

N

n

rX

rkXXX

WkXN

WWkXN

Wnx

N

n

nkrN

N

k

rnN

N

k

knN

N

n

N

n

rnN

~

1~

01~

00~

)~(1

~1~

1

0

)(1

0

1

0

1

0

1

0

That is,

1

0

~~ N

n

rnNWnxrX . QED


NCTU EE 3

Example: Periodic Rectangular Pulse Train

4

0

10

4

10

510

10

10sin

2sin

1

1~

n

kj

k

kKn

k

k

eW

WWkX

Sampling the Fourier Transform Compare two cases:

(1) Periodic sequence ][~~ kXnx

(2) Finite duration sequence nx = one period of nx~

An aperiodic sequence:

k

N

j

j

eXkXIDFSnx

eXFTnx

2|~~

samples ?

Compare:

jeXDTFTnx

jXFTtx

? samples

DSP The Discrete Fourier Transform

NCTU EE 4

Example:

otherwiase

nnx

,0

40,1

integer

109105,0

10410,1~

r

rnr

rnrnx

r r

m

N

k

knN

kmN

N

k

knN

m

kmN

j

knN

kN

k m

mj

knN

kk

N

j

N

k

KnN

rNnxrNnnx

WWmxN

WemxN

WemxN

WeXN

WkXN

nx

) ge(Interchan 1

1

(FT) 1

(Sampling) 1

(IDFS) ~1~

1

0

1

0

2

2

2

1

0


NCTU EE 5

If x[n] has finite length and we take a sufficient number of equally spaced samples of its

Fourier Transform ( a number greater than or equal to the length of x[n]), then x[n] is re-

coverable from [n]x~ .

Two ways (equivalently) to define DFT: (1) N samples of the DTFT of a finite duration sequence nx

(2) Make the periodic replica of nx nx~

Take the DFS of nx~

Pick up one segment of ][~

kX

kXDFSnx

kXDFTnx

~~segment one periodic

][


NCTU EE 6

Properties of the Discrete Fourier Series

-- Similar to those of FT and z-transform

Linearity

kXbkXanxbnxakXnx

kXnx2121

22

11 ~~~~~~

~~

Shift

kXnx~~ ==>

lkXnxW

kXWmnxnl

N

kmN

~~

~~

Duality

Def:

1

0

1

0

)(#~~

(*)~1~

N

n

nkN

N

k

knN

WnxkX

WkXN

nx

kxNkX

kXnx~~

~~

Symmetry kXnx ~~

kXkXkXnxj

kXkXkXnx

o

e

*

*

~~2

1~~Im

~~2

1~~Re

kXjnxnxnx

kXnxnxnx

o

e

~Im~~

2

1~

~Re~~

2

1~

*

*

kXkXnx *~~ real, is ~ If .

kXkX

kXkX

kXkX

kXkX

~Im~Im

~Re

~Re

~~

~~


NCTU EE 7

Periodic Convolution

nxnx 21~,~ are periodic sequences with period N

1

021213

1

02121

~~1~~~

~~~~

N

l

N

m

lkXlXN

nxnxnx

kXkXmnxmx

Discrete Fourier Transform Definition nx : length N, 10 Nn

Making the periodic replica:

N

r

nx

Nnx

rNnxnx

modulo

~

1

0

~~ N

n

knNWnxkX

Keep one segment (finite duration)

otherwise ,0

10 ,~

NkkXkX That is, ]))[((~

NkXkX

This finite duration sequence ][kX is the discrete Fourier transform (DFT) of nx


NCTU EE 8

10,1

:

10, :

1

0

1

0

NnWkXN

nxeqnSynthesis

NkWnxkXeqnAnalysis

N

k

knN

N

n

knN

Remark: DFT formula is the same as DFS formula. Indeed, many properties of DFT are de-

rived from those of DFS.


NCTU EE 9

Properties of Discrete Fourier Transform Linearity

21

212122

11

,max

NNlength

kbXkaXnbxnaxkXnx

kXnx

Circular Shift

kXnx ==>

NN

kmNN

lkXnxW

kXWmnx

ln

(Pf) From the right side of the 2nd eqn.

mnxmnxmnx

kXekXekXW

NN

kmN

jkmN

jKm

N

~IDFS DFT

~22

QED

Remark: This is circular shift not linear shift. (Linear shift of a periodic sequence = circular

shift of a finite sequence.)


NCTU EE 10

Duality

1-Nk0 ,

NkNxnX

kXnx

Symmetry Properties

0 ,0Re

11 ,2

1

10 ,2

1

~

symmetric-conjugate periodic

*

*

nx

N-n nNxnx

N-n nxnx

nx

nx

NN

e

ep


NCTU EE 11

0 ,0Im

11 ,2

1

ricantisymmet-conjugate periodic

*

nx

N-nnNxnx

nxop

kXjnxkXnx opep Im Re

N

N

N

N

N

kXkX

kXkX

kXkX

kXkX

NkkXkX nx

ImIm

ReRe

10, real, If *

NNop

NNep

kXkXkXnx

kXkXkXnx

*

*

2

1Im

2

1Re

Circular Convolution

kXkXnxnx

mnxmx

nxnxnxN

mN

2121

1

0

213

N-point circular convolution


NCTU EE 12

Example: N-point circular convolution of two constant sequences of length N

2L-point circular convolution of two constant sequences of length L


NCTU EE 13

Linear Convolution Using DFT Why using DFT? There are fast DFT algorithms (FFT)

How to do it?

(1) Compute the N-point DFT of nx1 and nx2 separately

kX1 and kX 2

(2) Compute the product kXkXkX 213

(3) Compute the N-point IDFT of kX 3 nx3

Problems: (a) Aliasing

(b) Very long sequence


NCTU EE 14

Aliasing

nx1 , length L (nonzero values)

nx2 , length P

In order to avoid aliasing, 1 PLN

(What do we mean avoid aliasing? The preceding procedure is circular convolution but we

want linear convolution. That is, nx3 equals to the linear convolution of nx1 and nx2 )


NCTU EE 15

nx1 pad with zeros length N

nx2 pad with zeros length N

Interpretation: (Why call it aliasing?)

kX 3 has a (time domain) bandwidth of size 1 PL

(That is, the nonzero values of nx3 can be at most 1 PL )

Therefore, kX3 should have at least 1 PL samples. If the sampling rate is insuf-

ficient, aliasing occurs on nx3 .


NCTU EE 16

Very long sequence (FIR filtering)

Block convolution

Method 1 – overlap and add

Partition the long sequence into sections of shorter length.

For example, the filter impulse response nh has finite length P and the input data nx

is nearly “infinite”.

otherwise 0,

10 , whereLet

0

L-nrLnxnxrLnxnx r

rr

The system (filter) output is a linear convolution:

0

wherer

rrr nhnxnyrLnynhnxny

Remark: The convolution length is 1 PL . That is, the 1 PL point DFT is used. nyr

has 1 PL data points; among them, (P-1) points should be added

to the next section.

This is called overlap-add method.

(Key: The input data are partitioned into nonoverlapping sections the section outputs

are overlapped and added together.)


NCTU EE 17

Method 2 – overlap and save

Partition the long sequence into overlapping sections.

After computing DFT and IDFT, throw away some (incorrect) outputs.

For each section (length L, which is also the DFT size), we want to retain the correct data of length ))1(( PL points

Let nh , length P

nxr , length L (L>P)

Then, nyr contains (P-1) incorrect points at the beginning.

Therefore, we divide into sections of length L but each section overlaps the preceding

section by (P-1) points. 1-Ln0 ,11 PPLrnxnxr

This is called overlap-save method.

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