digital signal processing, fall 2006 -...

1

Digital Signal Processing, II, Zheng-Hua Tan, 20061

Digital Signal Processing, Fall 2006- E-Study -

Zheng-Hua Tan

Department of Electronic SystemsAalborg University, Denmark

[email protected]

Lecture 2: Fourier transforms and frequency response


Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

Systemstructures

Filter structures Filter design

Filter

z-transform

MM1

MM2

MM9,MM10MM3

MM6

MM4

MM7 MM8

Sampling andreconstruction MM5

Systemanalysis

System

2


Part I: System impulse response

System impulse responseLinear constant-coefficient difference equations Fourier transforms and frequency response


FIR systems – reflected in the h[n]Ideal delay

Forward difference

Backward difference

Finite-duration impulse response (FIR) systemThe impulse response has only a finite number of nonzero samples.

integer. positive a ],[][ ],[][

dd

d

nnnnhnnnxny

−=∞<<∞−−=

δ

][]1[][][]1[][

nnnhnxnxny

δδ −+=−+=

]1[][][]1[][][

−−=−−=

nnnhnxnxny

δδ

nd

0

-10

3


IIR systems – reflected in the h[n]

Accumulator

Infinite-duration impulse response (IIR) systemThe impulse response is infinitive in duration.

StabilityFIR systems always are stable, if each of h[n] values is finite in magnitude.IIR systems can be stable, e.g.

][][][

][][

nuknh

kxny

n

k

n

k

==

=

∑

∑

−∞=

−∞=

δ 0

…

∞<−==

<=

∑∞ |)|1(1||

1|| with ][][

0aaS

anuanhn

n

∞<= ∑∞

−∞=

?|][|n nhS


Cascading systems

Causality Ideal delay

Any noncausal FIR system can be made cause by cascading it with a sufficiently long delay!

0 ,0][?

<= nnh

][][ dnnnh −= δ

][]1[][difference Forward

nnnh δδ −+=

]1[][][difference Backward

−−= nnnh δδ

]1[][delay sameple-One

−= nnh δ][nx

][nx ][ny

][ny

4


Cascading systems

Accumulator + Backward difference system

Inverse system:

][][systemr Accumulato

nunh =

][][ nnh δ=

]1[][][difference Backward

−−= nnnh δδ][nx

][nx ][nx

][nx

][][*][][*][ nnhnhnhnh ii δ==


Part II: LCCD equations


5


LCCD equations

An important class of LTI systems: input and output satisfy an Nth-order LCCD equations

Difference equation representation of the accumulator

][]1[][

]1[][][][][

][]1[

][][

1

1

nxnyny

nynxkxnxny

kxny

kxny

n

k

n

k

n

k

=−−

−+=+=

=−

=

∑

∑

∑

−

−∞=

−

−∞=

−∞=

∑∑==

−=−M

mm

N

kk mnxbknya

00][][

One-sampledelay

x[n]

y[n-1]

y[n]

Recursive representation


Part III: Fourier transforms

System impulse responseLinear constant-coefficient difference equationsFourier transforms and frequency response

Frequency-domain representation of discrete-time signals and systemsSymmetry properties of the Fourier transformFourier transform theorems

6


Signal representations

A sum of scaled, delayed impulse

Sinusoidal and complex exponential sequencesSinusoidal input sinusoidal response with the same frequency and with amplitude and phase determined by the systemComplex exponential sequences are eigenfunctions of LTI systems.signal representation based on sinusoids or complex exponentials

)cos(][ 0 φω += nAnx

∑∞

−∞=

−=k

knkxnx ][][][ δ

njenx ω=][

∑∞

−∞=

−=k

knhkxny ][][][


Eigenfunctions

Complex exponentials as input to system h[n]

Define Then

nj

k

kj

k

knjnj

nj

eekh

ekheTny

nenx

ωω

ωω

ω

)][(

][}{][

,][

)(

∑

∑∞

−∞=

−

∞

−∞=

−

=

==

∞<<∞−=

njjk

kjj

eeHny

ekheH

ωω

ωω

)(][

][)(

=

= ∑∞

−∞=

−

Eigenvalue Eigenfunction

A – linear operator

7


Eigenvalue – called frequency response

Frequency response is generally complex

Frequency response of the ideal delay system

Method 1: using the eigenfunction

Method 2: using the impulse response

.)(,1|)(|

).sin()(

),cos()(

dj

j

dj

I

dj

R

neHeH

neH

neH

ω

ω

ω

ω

ω

ω

ω

−=∠

=

−=

=

dnj

n

njd

j eenneH ωωω δ −∞

−∞=

−∑ =−= ][)(

d

dd

njj

njnjnnjnj

eeH

eeeeTnyωω

ωωωω

−

−−

=

===

)(

}{][ )(

)(|)(|

)()()(ωω

ωωω

jeHjj

jI

jR

j

eeH

ejHeHeH∠=

+=

describes changes in magnitude and phase

][][ ][][ dd nnnhnnxny −=→−= δ


Frequency response

The frequency response of discrete-time LTI systems is always a periodic function of the frequency variable w with period .

Only specify over the intervalThe ‘low frequencies’ are close to 0The ‘high frequencies’ are close to

)(][][)( 2)2()2( ωπωπωπω j

n

njnj

n

njj eHeenhenheH === ∑∑∞

−∞=

−−∞

−∞=

+−+

πωπ ≤<−

π±

π2

8


Ideal frequency-selective filters

For which the frequency response is unity over a certain range of frequencies, and is zero at the remaining frequencies.

Ideal low-pass filter: passes only low and rejects high


Ideal frequency-selective filters

9


Sinusoidal response of LTI systems

Sinusoidal input sinusoidal response with the same frequency and with amplitude and phase determined by the system

)cos(|)(| ][ then

)(2

)(2

][

22

)cos(][

0

0

0

0000

00

θφω

φω

ω

ωωφωωφ

ωφωφ

++=

+=

+=

+=

−−−

−−

neHAny

eeHeAeeHeAny

eeAeeAnAnx

j

njjjnjjj

njjnjj

θωωωω ω −∠−− === eeHeeHeHeH

nhjeHjjj j

|)(||)(|)()(

thenreal, is ][ if0

0000 )(*


Signal representation

More than sinusoids, a broad class of signals can be represented as a linear combination of complex exponentials:

If x[n] can be represented as a superposition of complex exponentials, output y[n] can be computed by using the frequency response, which is similar to the function of impulse response.

∑=k

njk

kenx ωα][

∑=∴k

njjk

kk eeHny ωωα )(][

10


Frequency-domain representation of x[n]

By Fourier transforms Fourier representation:

In general, Fourier transform is complex

)(|)(|

)()()(ωω

ωωω

jeXjj

jI

jR

j

eeX

ejXeXeX∠=

+=

ωπ

ωω deeX

x[n]

njj )(21

sinusoidscomplex smallmally infinitesi ofion superposit a asrepresent

njj

njj

enxeX

deeXnx

ωω

π

πωω ω

π−

∞

∞−

−

∑

∫

=

=

][)( where

)(21][ Inverse Fourier transform

w is continuous-time variable

Fourier transformn is discrete-time variable

Fourier spectrumSpectrumMagnitude spectrumAmplitude spectrumPhase spectrum


Frequency and impulse responses

Are a Fourier transform pair

Fourier transform is periodic with period

njj enxeX ωω −∞

∞−∑= ][)(

ωπ

π

πωω

ωω

deeHnh

enheH

njj

n

njj

∫

∑

−

∞

−∞=

−

=∴

=

)(21][

][)( Recall

π2

11


Sufficient condition for Fourier transform

Condition for the convergence of the infinite sum

X[n] is absolutely summable, then its Fourier transform exists (sufficient condition).

∞<≤

≤

=

∑

∑

∑

∞

∞−

−∞

∞−

−∞

∞−

|][|

|||][|

|][| |)(|

nx

enx

enxeX

nj

njj

ω

ωω


Example: ideal lowpass filter

Frequency response

Noncausal.Not absolutely summable.

⎩⎨⎧

≤<<

=πωω

ωωω

|| ,0|| ,1

)(c

cjlp eH

∞<<∞−== ∫− nn

ndenh cnj

lpc

c,

sin21][

πω

ωπ

ω

ωω

∑∞

−∞=

−=n

njcjlp e

nn

eH ωω

πωsin

)(

∑−=

−=M

Mn

njcjM e

nn

eH ωω

πωsin

)(

12


Example: Fourier transform of a constant

Constant sequence

Its Fourier transform is defined as the periodic impulse train

1][ =nx

)2(2)( reXr

j πωπδω ∑∞

−∞=

+=

njj

njj

enxeX

deeXnx

ωω

π

πωω ω

π−

∞

∞−

−

∑

∫

=

=

][)( where

)(21][





13


Symmetry properties of Fourier transform

Conjugate-symmetric sequenceConjugate-antisymmetric sequenceAny , ][nx

][][ * nxnx ee −=

][][ * nxnx oo −−=

][][][

][])[][(21][

][])[][(21][ define

])[][(21])[][(

21][

**

**

**

nxnxnx

nxnxnxnx

nxnxnxnx

nxnxnxnxnx

oe

oo

ee

+=∴

−−=−−=

−=−+=

−−+−+=

)())()((21)(

)())()((21)(

where)()()(

**

**

ωωωω

ωωωω

ωωω

jo

jjjo

je

jjje

jo

je

j

eXeXeXeX

eXeXeXeX

eXeXeX

−−

−−

−=−=

=+=

+=


Symmetry properties of Fourier transform

Table 2.1

14






Linearity of the Fourier Transform

)()(][][

)(][

)(][

2121

22

11

ωω

ω

ω

jjF

jF

jF

ebXeaXnbxnax

eXnx

eXnx

+↔+

↔

↔

15


Time shifting and frequency shifting

)(][

)(][

)(][

)( 00 ωωω

ωω

ω

−

−

↔

↔−

↔

jF

nj

jnjF

d

jF

eXnxe

eXennx

eXnx

d


Time reversal

)(][

real. is ][ if)(][

)(][

* ω

ω

ω

jF

jF

jF

eXnx

nxeXnx

eXnx

↔−

↔−

↔

−

16


Differentiation in frequency

ω

ω

ω

dedXjnnx

eXnxjF

jF

)(][

)(][

↔

↔


Parseval’s theorem

ωπ

π

π

ω

ω

∫∑ −

∞

−∞=

==

↔

deXnxE

eXnx

j

n

jF

22 |)(|21|][|

)(][

17


The convolution theorem

)()()(

][*][][][][

)(][

)(][

ωωω

ω

ω

jjjk

jF

jF

eXeHeY

nhnxknhkxny

eHnh

eXnx

=

=−=

↔

↔

∑∞

−∞=


The modulation or Windowing theorem

∫−−=

=↔

↔

π

π

θωθω

ω

ω

θπ

deWeXeY

nwnxnyeWnw

eXnx

jjj

jF

jF

)()(21)(

][][][)(][

)(][

)(

18


Fourier transform theorems

Table 2.2


Fourier transform pairs

Table 2.3

19


Example

Determining a Fourier transform using Tables 2.2 and 2.3

ω

ωω

ω

ωωωω

ωω

j

jj

n

j

jjjj

n

jj

Fn

aeeaeX

nuanxanxae

eeXeeX

nuanxnxae

eXnuanx

−

−

−

−−

−

−

−=

−==−

==

−=−=−

=↔=

1)(

])5[(i.e. ][][1

)()(

])5[(i.e. ]5[][1

1)(][][

552

5

5

15

2

512

11

]5[][ −= nuanx n


Summary



20


Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

Systemstructures

Filter structures Filter design

Filter

z-transform

MM1

MM2

MM9,MM10MM3

MM6

MM4

MM7 MM8

Sampling andreconstruction MM5

Systemanalysis

System

digital signal processing, fall 2006 -...

Documents