Digital Signal Processing, Fall 2006

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Digital Signal Processing, II, Zheng-Hua Tan, 2006 1

Digital Signal Processing, Fall 2006

- E-Study -

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Lecture 2: Fourier transforms and frequency response

Course at a glance

Discrete-time signals and systems

Fourier-domain representation

System structures

Filter MM1

MM2

MM6

MM4 Sampling and

reconstruction MM5

System analysis System

(2)

Part I: System impulse response

System impulse response

Linear 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) system

The impulse response has only a finite number of nonzero samples.

integer.

positive a

], [

] [

], [

] [

d d d

n n n n h

n n

n x n y

−

=

∞

<

∞

−

= δ

] [ ] 1 [ ] [

n n

n h

n x n

x n y

δ δ + −

=

− +

=

] 1 [ ] [ ] [

−

=

−

=

n n n h

n x n x n y

δ δ

n_d

0 -1

0

(3)

IIR systems – reflected in the h[n]

Accumulator

Infinite-duration impulse response (IIR) system

The impulse response is infinitive in duration.

Stability

FIR systems always are stable, if each of h[n] values is finite in magnitude.

IIR systems can be stable, e.g.

] [ ] [ ] [

] [ ] [

n u k n

h

k x n y

n

k n

k

=

∑

−∞

=

−∞

=

δ 0

…

∞

<

−

=

<

=

∑

^∞^| ^| ¹⁽¹ ^| ^|)

1

|

| with ] [ ] [

0 a a

S

a n

u a n h

n n

∞

<

=∑^∞=−∞

?

| ] [

n |hn

S

Cascading systems

Causality

Ideal delay

0 , 0 ] [

? <

= n

n h

] [ ]

[n n n_d

h =δ −

] [ ] 1 [ ] [

difference Forward

n n

n

h =δ + −δ

] 1 [ ] [ ] [

difference Backward

−

= n n

n

h δ δ

] 1 [ ] [

delay sameple -

One

−

= n n

h δ

] [n x

] [n

x y[n]

] [n y

(4)

Cascading systems

Accumulator + Backward difference system

Inverse system:

] [ ] [

system r Accumulato

n u n h =

] [ ]

[n n

h =δ

] 1 [ ] [ ] [

difference Backward

−

= n n

n

h δ δ

] [n x

] [n

x x[n]

] [n x

] [ ] [

* ] [ ] [

* ]

[n h n h n hn n

h _i = _i =δ

Part II: LCCD equations

Linear constant-coefficient difference 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

n x n y n y

n y n x k x n x n y

k x n

y

k x n y

n

k n

k

=

−

− +

= +

=

−

=

∑

−

−∞

=

−

−∞

=

−∞

=

∑

= =

−

=

− ^M

m m N

k

ky n k b xn m

a

0 0

] [ ]

[

One-sample delay x[n]

y[n-1]

y[n]

Recursive representation

Part III: Fourier transforms

Fourier transforms and frequency response

Frequency-domain representation of discrete-time signals and systems

Symmetry properties of the Fourier transform

Fourier transform theorems

(6)

Signal representations

A sum of scaled, delayed impulse

Sinusoidal and complex exponential sequences

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

Complex exponential sequences are eigenfunctions of LTI systems.

Æsignal representation based on sinusoids or complex exponentials

) cos(

]

[n =A ω₀n+φ x

∑

^∞

−∞

=

−

=

k

k n k x n

x[ ] [ ]δ[ ]

n

ej

n x[ ]= ^ω

∑

^∞

−∞

=

−

=

k

k n h k x n

y[ ] [ ] [ ]

Eigenfunctions

Complex exponentials as input to system h[n]

Define Then

n j k

k j k

k n j n

j n j

e e k h

e k h e

T n y

n e

n x

ω ω

ω

) ] [ (

] [ }

{ ] [

, ] [

) (

∑

∞

−∞

=

−

∞

−∞

=

−

=

∞

<

∞

−

=

n j j k

k j j

e e H n y

e k h e

H

ω ω

) ( ] [

] [ )

(

=

∑

^∞

−∞

=

−

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(

) (

d j

j

d j

I

d j

R

n e

H e H

n e

H

n e

H

ω ω ω

ω ω ω ω

−

=

∠

=

−

=

nd j n

n j d

j n n e e

e

H ^ω ^∞ δ ^ω ⁻^ω

−∞

=

∑

⁻ − ⁼

= [ ]

) (

d

d d

n j j

n j n j n n j n j

e e H

e e e

e T n y

ω ω

ω ω ω

ω

−

=

= ) (

} { ]

[ ⁽ ⁾

)

| (

) (

|

) ( ) ( ) (

ω ω

ω

ej H j j

j I j R j

e e H

e jH e H e H

= ∠

+

=

describes changes in magnitude and phase

] [ ] [ ] [ ]

[n xn n_d hn n n_d

y = − → =δ −

Frequency response

The frequency response of discrete-timeLTI systems is always a periodic function of the frequency variable wwith period .

Only specify over the interval

The ‘low frequencies’ are close to 0

The ‘high frequencies’ are close to

) ( ]

[ ]

[ )

( ⁽^ω ²^π⁾ ⁽^ω ²^π⁾ ^ω ²^π ^j^ω

n

n j n j n

n j

j hn e h ne e H e

e

H =

∑

=

∑

^∞ =

−∞

=

−

∞ −

−∞

=

+

− +

π ω π < ≤

− π

± π 2

(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 Æ sinusoidal response with the same frequency and with amplitude and phase determined by the system

) cos(

| ) (

| ] [ then

) 2 (

] [

2 2

) cos(

] [

0 0

0

0 0 0

0

0 0

θ φ ω φ

ω

ω ω φ ω

ω φ

ω φ ω

φ

+ +

=

+

=

+

=

+

=

−

− −

n e

H A n y

e e H Ae e

e H Ae n y

e Ae e

Ae n A

n x

j

n j j j n

j j j

n j j n

j j

ω θ ω

ω

ω −∠ ^ω −

− =H e = H e e = H e e

e H

n h

e j j H

j

j ^j

| ) (

|

| ) (

| ) ( ) (

then real, is ] [ if

0 0 0

0

0 * ( )

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

∑

=

k

n j ke ^k n

x[ ] α ^ω

∑

=

∴

k

n j j k

k

k e

e H n

y[ ] α ( ^ω ) ^ω

(10)

Frequency-domain representation of x[n]

By Fourier transforms ÆFourier representation:

In general, Fourier transform is complex

)

| (

) (

|

) ( ) ( ) (

ω ω

ω

ej X j j

j I j R j

e e X

e jX e X e X

= ∠

+

=

π ω

ω ω e d e

X

x[n]

n j j ) 2 (

1

sinusoids complex

small mally infinitesi of

ion superposit a

as represent

n j j

e n x e

X

d e e X n

x

ω ω

π π

ω

ω ω

π

∞ −

∞

−

∑

∫

=

] [ ) ( where

) 2 (

] 1 [

Inverse Fourier transform

wis continuous-time variable Fourier transform

nis discrete-time variable

Fourier spectrum Spectrum

Magnitude spectrum Amplitude spectrum Phase spectrum

Frequency and impulse responses

Are a Fourier transform pair

Fourier transform is periodic with period

n j

j x ne

e

X ^ω ⁻ ^ω

∞

∑

−

= [ ] )

(

π ω

π π

ω ω

d e e H n

h

e n h e

H

n j j n

n j j

∫

∑

−

∞

−∞

=

−

=

∴

=

) 2 (

] 1 [

] [ ) ( Recall

π 2

(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).

∞

<

≤

=

∑

∞

−

∞ −

∞

−

∞ −

∞

−

| ] [

|

||

] [

|

| ] [

| | ) (

|

n x

e n x

e n x e

X

n j n j j

ω ω ω

Example: ideal lowpass filter

Frequency response

Noncausal.

Not absolutely summable.

⎩⎨

⎧

≤

<

= <

π ω ω

ω

ω ω

|

| , 0

|

| , ) 1 (

c j c

lp e H

∞

<

∞

−

=

= ∫− n

n d n

e n

h_lp ^c ^j ⁿ ^c

c

sin , 2

] 1

[ π

ω ω π

ω ω

ω

∑^∞

−∞

=

= − n

n c j j

lp e

n e n

H ^ω ^ω

π ω ) sin

(

(12)

Example: Fourier transform of a constant

Constant sequence

Its Fourier transform is defined as the periodic impulse train

1 ] [n = x

) 2 ( 2 )

(e r

X

r

j^ω

∑

^∞ πδ ω π

−∞

=

+

=

n j j

e n x e

X

d e e X n

x

ω ω

π π

ω

ω ω

π

∞ −

∞

−

∑

∫

=

] [ ) ( where

) 2 (

] 1 [

System impulse response

(13)

Symmetry properties of Fourier transform

Conjugate-symmetric sequence

Conjugate-antisymmetric sequence

Any , x[n]

] [ ]

[n x ^* n x_e = _e −

] [ ]

[n x ^* n

x_o =− _o −

] [ ] [ ] [

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

] [ ]) [ ] [ 2( ] 1 [ define

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

*

n x n x n x

n x n x n x n x

n x n x n x n x n x

o e

o o

e e

+

=

∴

−

=

−

=

−

=

− +

=

−

− +

=

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

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

) ( ) ( ) (

*

ω ω

ω

j o j

j j

o

j e j j

j e

j o j e j

e X e

X e X e

X

e X e

X e X e

X

e X e X e X

−

=

−

=

= +

=

+

=

Symmetry properties of Fourier transform

Table 2.1

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Fourier transforms and frequency response

Linearity of the Fourier Transform

) ( )

( ]

[ ] [

) ( ]

[

) ( ] [

2 1

2 2

1 1

ω ω

j F j

F j F j

e bX e

aX n bx n ax

e X n x

+

↔ +

↔

(15)

Time shifting and frequency shifting

) (

] [

) ( ]

[

) ( ] [

)

( ₀

0 ω ω

ω

ω ω ω

−

↔

−

↔

F j n j

j n F j d F j

e X n x e

e X e n n x

e X n x

d

Time reversal

) ( ] [

real.

is ] [ if

) ( ] [

* ω

ω ω

F j F j F j

e X n x

n x

e X n x

↔

−

↔

−

↔

−

(16)

Differentiation in frequency

ω

ω ω

d e j dX n

nx

e X n

x

F j F j

) ] (

[

) ( ]

[

↔

Parseval’s theorem

π ω

π π

ω ω

∑ ∫

−

∞

−∞

=

↔

d e X n

x E

e X n x

j n

F j

2

2 | ( )|

2

| 1 ] [

| ) ( ] [

(17)

The convolution theorem

) ( ) ( ) (

] [

* ] [ ] [ ] [ ]

[

) ( ] [

ω ω ω

ω ω

j j j

k F j F j

e X e H e

Y

n h n x k n h k x n

y

e H n h

e X n x

=

−

=

↔

∑

^∞

−∞

=

The modulation or Windowing theorem

∫

−

= −

=

↔

π π

θ ω θ ω

ω ω

π ^X ^e ^W ^e ^dθ e

Y

n w n x n y

e W n w

e X n x

j j j

F j F j

) ( ) 2 (

) 1 (

] [ ] [ ] [

) ( ] [

) (

(18)

Table 2.2

Fourier transform pairs

Table 2.3

(19)

Example

Determining a Fourier transform using Tables 2.2 and 2.3

ω ω ω

ω ω

j j j

n j j j

j j

n j F j

n

ae e e a

X

n u a n

x a n x

ae e e

X e e X

n u a n

x n x

e ae X n u a n x

−

− −

−

= −

−

=

= −

=

−

=

−

=

= −

↔

=

) 1 (

]) 5 [ (i.e.

] [ ] [

) 1 ( )

(

]) 5 [ (i.e.

] 5 [ ] [

1 ) 1 ( ] [ ] [

5 5 2 5

5 1

5 2

5 1

2

1 1

] 5 [ ]

[n =a u n−

x ⁿ

Summary

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Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

System structures

Filter structures Filter design Filter

z-transform MM1

MM2

MM9,MM10 MM3

MM6

MM4

MM7 MM8

Sampling and

reconstruction MM5

System analysis System