elen 5346/4304 dsp and filter design fall 2008 1 lecture 4: frequency domain representation, dtft,...

ELEN 5346/4304 DSP and Filter Design Fall 2008

1

Lecture 4: Frequency domain representation, DTFT, IDTFT, DFT, IDFT

Instructor: Dr. Gleb V. Tcheslavski

Contact: [email protected]

Office Hours: Room 2030

Class web site: http://ee.lamar.edu/gleb/dsp/index.htm


2

Some history

Jean Baptiste Joseph Fourier was born in France in 1768. He attended the Ecole Royale Militaire and in 1790 became a teacher there. Fourier continued his studies at the Ecole Normale in Paris, having as his teachers Lagrange, Laplace, and Monge. Later on, he, together with Monge and Malus, joined Napoleon as scientific advisors to his expedition to Egypt where Fourier established the Cairo Institute.

In 1822 Fourier has published his most famous work: The Analytical Theory of Heat. Fourier showed how the conduction of heat in solid bodies may be analyzed in terms of infinite mathematical series now called by his name, the Fourier series.


3

Frequency domain representation

0cos( ) a (co)sinusoidal inputnLet x A n

frequency

0 00 0( ) ( )

22 22j n jj njn j n

njA A

e e e eA A

x e e complex exponent of 0

complex exponent of -0

A sinusoidal signal is represented by TWO complex exponents of opposite frequencies in the frequency domain.

(4.3.1)

(4.3.2)


4

Frequency domain representation (cont)

0 0 0 00 ( )j n j n k j k j nn k k

kn

k

jnLet x e h e hy H xe ee

: phase

jj H e

j j k jk

kmagnitude

The frequency respon H e h e H e ese

Iff it exists!

* *

*

:

j j

j n j n j nn n n

n n n

j H e j H e

j j

j j

j j

j j

real H e HFor a system h e h e h e

H e e H e

e

H e H e

H e H e

e

has a period of 2 jH e

(4.4.1)

(4.4.2)

(4.4.3)

(4.4.4)


5

Frequency domain representation (cont 2)

For an arbitrary real LTI system:

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-100

-50

0

50

100

Normalized Frequency ( rad/sample)

Pha

se (

degr

ees)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-15

-10

-5

0

5

10

Normalized Frequency ( rad/sample)

Mag

nitu

de (

dB)

Symmetric with respect to

Anti-symmetric with respect to


6


Combining (4.3.2) and (4.4.4) – back to our sinusoid!

0 0 0 0

0 00 0 0 0

0 00 0

0

Stady-state response:2 2

2 2

2

j j

j j

j j n j j nj jn

j H e j H ej j n j j nj j

j n H e j n H ej

A Ay e H e e e H e e real system

A Ae H e e e e H e e e

AH e e e

LTI filtering: 0 00cosj j

ny A H e n H e

due to the input

change due to the system

same as the input

from the input

phase change due to the system

Via design, we manipulate H(ej), therefore, hn, and, finally, manipulate the coefficients in the Linear Constant Coefficient Difference Equation (LCCDE)

(4.6.1)

(4.6.2)

(4.6.3)

(4.6.4)


7


LCCDE:0

0,

, 0

0

j nj nnn

tot n i ii

x e ny c ke

initial conditions

if the system is BIBO( ) , the frequency response existsj j nn n

n n

H e h e h

for large enough n:0

,j n

tot ny ke

0 0 00 0

0

( )n

j n k j k j nn k n k k

k k

j

n

j ny h e u e ee Hh e

(4.7.1)

(4.7.4)

(4.7.2)

(4.7.3)


8


for an LTI: 0, ,

j nn n n tr n ss nx e u y y y

We don’t need systems of order higher than 2: can always make cascades.

0 0, ( ) ,j j n

ss ny H e e exists only for LTI BIBO systems

for a real, LTI, BIBO system:

0 00 , 0cos( ) ( ) cos( ( )j j

n ss nx A n y A H e n H e

effects of filtering

We cannot observe ANY frequency components in the output that are not present in the input (in steady state). We may see less when ( ) 0rjH e

(4.8.1)

(4.8.2)


9


In continuous time:

{we want}t t t t tx s n y c s distortion less transmission signal noise const delay

( )( ) ( ) ( ) ( ) ( )j t j t j jt tY y e dt c s e d t e ce S H S

We need a constant magnitude and linear phase for the frequencies of interest.

Ideal filters:

0( )jH e 0( )jH e

LPF

0( )jH e

HPF

0( )jH e

BPF

0( )jH e

BSF

Ideal filters are non-realizable!

(4.9.1)

(4.9.2)


10

CTFT and ICTFT

(4.10.1)

(4.10.2)

( ) ( )

1( ) ( )

2

j t

j t

X x t e dt

x t X e d

CTFT:

ICTFT:

Examples:


11

DTFT

, ,j j nn

n

a complex funX e x ction periodic continuous one

if exists

(4.11.1)

What’s about convergence???

1. Absolute convergence:

1 i.e. converges absolutely or uniformlyjn n

n

if x l x X e

.

( ) lim 0k

j j nk n

n kabs er

j jkk

ror

Le X et X e x e X e

(4.11.4)

(4.11.5)

2

2 2

cos , sinj j j j j jre im

j j jre im

X e X e X e X e X e X e

X e X e X e

(4.11.2)

(4.11.3)


12

DTFT (cont)

2

1 2

0

. , :

cos( )

n

nn

n

u

a are not l In fact they are not l x

A n

must be

2. Mean-square convergence:

lim 0

energy of the

j j

err

k

r

k

o

X e X e

The total energy of the error must approach zero, not an error itself!

(4.12.1)

Absolutely summable sequences always have finite energy. However, finite energy sequences are not necessary absolutely summable.


13

IDTFT

2

2 ,j n

j pn

n

Let p is a period X e x e

11( ) ( )

2j j n

p

j j nnx XX e e d

pe e d

IDTFT:

(4.13.1)

(4.13.2)

Combining (4.11.1) and (4.12.2)1

2j l j n

n ll

x x e e d

( ) sin ( )1

2 ( )j n l

l l l n l nl l

n lx e d x x x

n l

(4.13.3)

(4.13.4)

jX e shows where xn “lives” in the frequency domain.


14

Back to ideal filters

Ideal LPF:

0( )jH e

20

1

c

1, 0

0,

cjLP

c

H e

Using IDTFT:

,

s n1,

i1

2 2

c c c

c

j n j nj n c

LP n

e ee d

nh n

jn jn n

1. The response in (4.14.2) is not absolutely summable, therefore, the filter is not BIBO stable!

2. The response in (4.14.2) is not causal and is of an infinite length.

(4.14.1)

(4.14.2)

As a result, the filter in (4.14.1) is not realizable.Similar derivations show that none of the ideal filters in slide 9 is realizable.


15

DTFT properties

0

0

0 0( )

:

:

:

:

j jn n

DTFT

j n jn n

DTFT

j n jn

DTFT

nD

Linearity ag bx aG e bX e

Shift in time g e G e

Shift in frequency e g G e

Differentiation n g

( )

* *

:

1:

2

1' :

2

j

TFT

j jn n

DTFT

j jn n

DTFT

j jn n

n

dG ej

d

Linear convolution g x G e X e

Periodic convolution g w G e W e d

Parseval s theorem g w G e W e d

(4.15.1)

(4.15.2)

(4.15.3)

(4.15.4)

(4.15.5)

(4.15.6)

continuous, periodic functions

(4.15.6)


16

DTFTs of commonly used sequences


17

DTFT examples

002 ( 2 )j n j

nDTFT

k

Let x e X e k 0 1 2 ( 2 )j

nDTFT

k

x X e k 1

11 1 2 ( 2 )

1j j j

n n n jDTFTl

u u U e e U e c le

½ of DC value of un1( 2 )

1n jDTFTl

u le

*

* *j n j n jn n

n n

h e h e H e

( )

( )

1 1

2 2

1

2

j n j j n j n j j nn n n n

n n n

j j

h g e H e e g e d H e g e d

H e G e d

(4.17.1)

(4.17.2)

(4.17.3)

(4.17.4)

(4.17.5)

(4.17.6)


18

DTFT examples (cont)

We can re-work the Parseval’s theorem (4.15.6) as follows:

(

2

)

2 * 1

2

1

2j

hS

jn n

j

n

j

e

Energy HE h de ed HH e

energy density (spectrum)

Autocorrelation function:

*( ),

* * * j jn n l n l n l

jgg l g

DTFTl

n n

g g g g g gr G e G Se e

(4.18.1)

(4.18.2)


19

DTFT examples (cont 2)

One obvious problem with DTFT is that we can never compute it since xn needs to be known everywhere! which is impossible! Therefore, DTFT is not practical to compute.

Often, a finite dimension LTI system is described by LCCDE:

0 0 0

0

0

0

N M N Mj i j j m j

i n i m n m i mDTFT

i m i m

j

j

Mj m

mjm

Nj i

ii

b eH e

a

a y b x a e Y e b e X e

Y e

X e e

(4.19.2)

practical (finite dimensions)

Prediction of steady-state behavior of LCCDE

(4.19.1)


20

How to measure frequency response of an actual (unknown) filter?

1. Perform two I/O experiments:

0 0

0 0

( ) ( ) ( )0 0 ,

( ) ( ) ( )0 0 ,

1) cos( ) cos

2) sin( ) sin

j jc c cn n tr n

LTI

j js s sn n tr n

LTI

x A n y A H e n H e y

x A n y A H e n H e y

2. Analyze these measurements and form:

0

00

0

0

0

( )( ) ( )

( ) (

,

),

j

j

j nc sn n n

j n H ejc s

j H e

n

jntr n

n

n n tr n

x x jx Ae

y y jy A H e e y

Finallyy

H e e zx

That’s a good way to measure/estimate a frequency response for every .

(4.20.1)

(4.20.2)

(4.20.3)

(4.20.4)

(4.20.5)


21

DFT and IDFT

Consider an N-sequence xn (at most N non-zero values for 0 n N-1)

1

20

, 0,1,..., 1k

Nj n j

n kk

n

n

j j

n Nn x e X eX e x e kX N

uniformly spaced frequency samples

(4.21.1)

21

0

, 0,1,..., 1N j kn

Nk n

n

X x e k N

(4.21.2)DFT:

Finite sum! Therefore, it’s computable.

2

Using notation:jN

NW e

(4.21.1) can be rewritten as:1

0

Nkn

k n Nn

X x W

1

0

: 0,1,., ..,1

1N

knn k N

n

IDFT n Nx X WN

(4.21.3)

(4.21.4)

(4.21.5)

Btw, DFT is a sampled version of DTFT.


22

DFT and IDFT (cont)

Let us verify (4.21.5). We multiply both sides by l nNW

1 1 1 1 1( )

0 0 0 0 0

1 1N N N N Nl n kn l n k l n

n N k N N k Nn n k n k

x W X W W X WN N

1 1 1( )

0 0 0

1N N Nl n k l n

n N k Nn k n

x W X WN

1( )

0

, ,

0,

Nk l n

Nn

N for k l rN r is IntegerSince W

otherwise

1

0

Nl n

n N ln

x W X

(4.22.1)

(4.22.2)

(4.22.3)

(4.22.4)


23

FFT

In the matrix form: N D TD x FX

where: 0 1 1[ ... ]TNx x x x time domain signal

0 1 1[ ... ]TNX X X X N DFT samples

1 2 1

2 4 2 ( 1)

1 2 ( 1) ( 1) ( 1)

1 1 1 1

1

1

1

NN N N

NN N N N

N N N NN N N

W W W

D W DFT mW W

W W W

atrix

1Nx D X IDFT

1 2 ( 1)

1 2 4 2 ( 1)

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

1 1 1 1

11

1

1

NN N N

NN N N N

N N N NN N N

W W W

D W W WN

W W W

IDFT matrix

1 *1

N ND DN

(4.23.1)

(4.23.2)

(4.23.3)

(4.23.4)

(4.23.5)

(4.23.6)

(4.23.7)

This is actually FFT…


24

Relation between DTFT and DFT

1. Sampling of DTFT

jn

DTFTLet x X e

2 , 0,1,... 1 point DFTkjjk k

kSample X e at k N N frequency samples X e Y N

N

, 0,1,..., 1k nIDFT

Y y n N

2k

kjj klN

k l Nl

Y X e X e xW

1 1 1( )

0 0 0

1 1 1N N Nkn kl kn k n l

n k N l N N l Nk k l l k

y Y W xW W x WN N N

, 0,1,..., 1n n Nm

mConsidering y x n N

(4.24.1)

(4.24.2)

(4.24.3)

yn is an infinite sum of shifted replicas of xn. Iff xn is a length M sequence (M N) than yn = xn. Otherwise, time-domain aliasing xn cannot be recovered!


25

Relation between DTFT and DFT (cont)

2. DTFT from DFT by Interpolation

Let xn be a length N sequence:point

n kN DFT

x X

It’s possible to determine DTFT X(ej) from its uniformly sampled version uniquely!

Let us try to recover DTFT from DFT (its sampled version).

1 1 1 1 1 2

0 0 0 0 0

1 1 knN N N N N jj j n kn j n j nNn k N k

n n k k n

X e x e X W e X e eN N

2 1( 2 ) ( 2 )/21 22

( 2 / ) ( 2 )/20

2 2sin sin

1 2 22 21 sin sin

2 2

k k Nj N k j N kN j n jN N

j k N j k Nn

N k N ke e

Since e eN k N ke e

N N

2 11

2

0

112

0

sin1 2

1s

2sin

1 22

sin2

in2

k NN j NN jjk

k

Nk

k

N N

X e X e

k

e

NN

XN kN

(4.25.1)

(4.25.2)

(4.25.3)


26

Relation between DTFT and DFT (cont 2)

3. Numerical computation of DTFT from DFT

Let xn is a length N sequence: jn

DTFTx X e

defined by N uniformly spaced samples

We wish to evaluate at more dense frequency scale. jX e

2 / , 0,1,..., 1,k k M k M where M N

1 1 2

0 0

k k

nN N j kj j n Mn n

n n

X e x e x e

1 12

, ,0 0

k

nM Mj kj knMl n l n M

n n

X e x e x W

No change in information, no change in DTFT… just a better “plot resolution”.

(4.26.1)

,

, 0 1

0, 1n

l n

x n Nx

N n M

Define: zero-padding (4.26.2)

(4.26.3)


27

A note on WN

2

Recall that:jN

NW e

WN is also called an Nth root of unity, since

2

1jN

NW e

Re

Im

2

N

Re

Im

2nN

(4.27.1)

(4.27.2)


28

DFT properties

1. Circular shiftxn is a length N sequence defined for n = 0,1,…N-1.

An arbitrary shift applied to xn will knock it out of the 0…N-1 range.Therefore, a circular shift that always keeps the shifted sequence in the range 0…N-1 is defined using a modulo operation:

0

0 0

0

0

, ( modulo0

, 1

, 0N

n n

c n n n N n nN n n

x n n Nx x x

x n n

0

0

0 0

1( )

0

1

0

1

1

N

Nk n n

k Nn nk

Nkn knkn

N k N N kDFT

k

x X WN

W X W W XN

(4.28.1)

(4.28.2)

nx 6 61 5n nx x

6 64 2n nx x


29

DFT properties (cont)

2. Circular convolution

A linear convolution for two length N sequences xn and gn has a length 2N-1:1

,0

, 0 2 2N

L n m n mm

y g x n N

A circular convolution is a length-N sequence defined as:

1

,0

N

N

C n m n n n nn mm

y g x g x x g

N N

1 1 1 1 1( )

0 0 0 0 0

1

0

1 1

1

N

n m N

N N N N Nk n m km kn

m m k N l N k Nn mm m k k m

x

Nkn

k k N k kDFT

k

g x g X W gW X WN N

G X W G XN

(4.29.1)

(4.29.2)

(4.29.3)

Procedure: take two sequences of the same length (zero-pad if needed), DFT of them, multiply, IDFT: a circular convolution.


30

DFT properties (cont 2)

Example:

j j nn

n

X e x e Take N frequency samples of (4.30.1) and then IDFT:

(4.22.

2 21 1

0 0

1( )

0

3)

1 1

1

n l rNr

N N j kl jkn knN Nk N l N N

k k l

Nk n

n

nl

l N rl

Nrk

X W x e W W eN N

x WN

x

x

aliased version of xn

(4.30.1)

(4.30.2)

The results of circular convolution differ from the linear convolution “on the edges” – caused by aliasing.To avoid aliasing, we need to use zero-padding…


31

Linear filtering via DFT

Often, we need to process long data sequences; therefore, the input must be segmented to fixed-size blocks prior LTI filtering. Successive blocks are processed one at a time and the output blocks are fitted together…

n n ny h x

Assuming that hn is an M-sequence, we form an N-sequence (L - block length):

(4.31.1)

,

1

0n

m n

x mL n mL Lx

otherwise

We can do it by FFT: IFFT{FFT{x}FFT{h}}…

N >> M; L >> M; N = L + M - 1 and is a power of 2

Problem: DFT implies circular convolution – aliasing!

(4.31.2)


32

Linear filtering via DFT (cont)

Next, we compute N-point DFTs of xm,n and hn, and form

, , 0,1,... 1m k k m k k NY H X

, ,m k m nIDFT

Y y

(4.32.1)

- no aliasing!

Since each data block was terminated with M -1 zeros, the last M -1 samples from each block must be overlapped and added to first M – 1 samples of the succeeded block.

An Overlap-Add method.


33

Linear filtering via DFT (cont 2)

Alternatively:

Each input data block contains M -1 samples from the previous block followed by L new data samples; multiply the N-DFT of the filter’s impulse response and the N-DFT of the input block, take IDFT.

Keep only the last L data samples from each output block.

An Overlap-Save method.

The first block is padded by M-1 zeros.


34

DFT properties: General

from Mitra’s book

Btw, g[n] = gn


35

DFT properties: Symmetry

from Mitra’s book

xn is a real sequence


36

DFT properties: Symmetry (cont)

from Mitra’s book

xn is a complex sequence


37

N-point DFTs of 2 real sequences via a single N-point DFT

Let gn and hn are two length N real sequences.

*

*

1

21

2

N

N

k k k

k k k

G X X

H X Xj

Form xn = gn + jhn Xk

* *

N Nk N kNote that X X

(4.37.1)

(4.37.2)

(4.37.3)

(4.37.4)


38

Summary

Algorithm Time Frequency

CTFT Continuous Continuous

DTFT Discrete Continuous

DFT Discrete Discrete

elen 5346/4304 dsp and filter design fall 2008 1 lecture 4: frequency domain representation, dtft,...

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