Digital Signal Processing, Fall 2006 Part I: Introduction

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

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Lecture 1: Introduction,

Discrete-time signals and systems

Part I: Introduction

Introduction (Course overview)

Discrete-time signals

Discrete-time systems

Linear time-invariant systems

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General information

Course website

http://kom.aau.dk/~zt/cources/DSP/

Textbook:

Oppenheim, A.V., Schafer, R.W, "Discrete-Time Signal Processing", 2nd Edition, Prentice-Hall, 1999.

Readings:

Steven W. Smith, “The Scientist and Engineer's Guide to Digital Signal Processing”, California Technical Publishing, 1997. http://www.dspguide.com/pdfbook.htm (You can download the entire book!)

Kermit Sigmon, "Matlab Primer", Third Edition, Department of Mathematics, University of Florida.

V.K. Ingle and J.G. Proakis, "Digital Signal Processing using MATLAB", Bookware Companion Series, 2000.

General information

Duration

2 ECTS (10 Lectures)

Prerequisites:

Background in advanced calculus including complex variables, Laplace- and Fourier transforms.

Course type:

Study programme course (SE-course) , meaning a written exam at the end of the semester!

Lecturer:

Associate Professor, PhD, Zheng-Hua Tan Niels Jernes Vej 12, A6-319

[email protected], +45 9635-8686

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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

Course objectives (Part I)

To give the students a comprehension of the concepts of discrete-time signals and

systems

To give the students a comprehension of the Z- and the Fourier transform and their inverse

To give the students a comprehension of the relation between digital filters, difference equations and system functions

To give the students knowledge about the

most important issues in sampling and

reconstruction

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Course objectives (Part II)

To make the students able to apply digital filters according to known filter specifications

To provide the knowledge about the principles behind the discrete Fourier transform (DFT) and its fast computation

To make the students able to apply Fourier analysis of stochastic signals using the DFT

To be able to apply the MATLAB program to digital processing problems and

presentations

What is a signal ?

A flow of information.

(mathematically represented as) a function of independent variables such as time (e.g.

speech signal), position (e.g. image), etc.

A common convention is to refer to the

independent variable as time, although may

in fact not.

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Example signals

Speech: 1-Dimension signal as a function of time s(t);.

Grey-scale image: 2-Dimension signal as a function of space i(x,y)

Video: 3 x 3-Dimension signal as a function of space and time {r(x,y,t), g(x,y,t), b(x,y,t)} .

Types of signals

The independent variable may be either continuous or discrete

Continuous-time signals

Discrete-time signals are defined at discrete times and represented as sequences of numbers

The signal amplitude may be either continuous or discrete

Analog signals: both time and amplitude are continuous

Digital signals: both are discrete

Computers and other digital devices are restricted to discrete time

Signal processing systems classification follows the same lines

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Types of signals

From http://www.ece.rochester.edu/courses/ECE446

Digital signal processing

Modifying and analyzing information with computers – so being measured as

sequences of numbers.

Representation, transformation and

manipulation of signals and information they

contain

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Typical DSP system components

Input lowpass filter to avoid aliasing

Analog to digital converter (ADC)

Computer or DSP processor

Digital to analog converter (DAC)

Output lowpass filter to avoid imaging

ADC and DAC

Physical signals Analog signals Digital signals Transducers

e.g. microphones

Analog-to-digital converters

Digital-to-Analog converters Output devices

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Pros and cons of DSP

Pros

Easy to duplicate

Stable and robust: not varying with temperature, storage without deterioration

Flexibility and upgrade: use a general computer or microprocessor

Cons

Limitations of ADC and DAC

High power consumption and complexity of a DSP implementation: unsuitable for simple, low-power applications

Limited to signals with relatively low bandwidths

Applications of DSP

Speech processing

Enhancement – noise filtering

Coding

Text-to-speech (synthesis)

Next generation TTS @ AT&T

Recognition

Image processing

Enhancement, coding, pattern recognition (e.g. OCR)

Multimedia processing

Media transmission, digital TV, video conferencing

Communications

Biomedical engineering

Navigation, radar, GPS

Control, robotics, machine vision

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History of DSP

Prior to 1950’s: analog signal processing using electronic circuits or mechanical devices

1950’s: computer simulation before analog implementation, thus cheap to try out

1965: Fast Fourier Transforms (FFTs) by Cooley and Tukey – make real time DSP possible

1980’s: IC technology boosting DSP

Part II: Discrete-time signals

Introduction

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Sequences of numbers

Periodic sampling of an analog signal

integer an

is where

]}, [ {

n

n n

x

x= −∞< <∞

period.

sampling the

called is where

), ( ] [

T

n nT

x n

x = _a −∞< <∞

x[1]

x[2]

x[n]

x[-1]

x[0]

Sequence operations

The product and sum of two sequences x[n] and y[n]: sample-by-sample production and sum, respectively.

Multiplication of a sequence x[n] by a number : multiplication of each sample value by .

Delay or shift of a sequence x[n]

α α

integer an

is where

] [

]

[ ₀

n n n x n

y = −

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Basic sequences

Unit sample sequence(discrete-time impulse, impulse)

Any sequence can be represented as a sum of scaled, delayed impulses

More generally

] 5 [ ...

] 3 [ ]

3 [ ]

[n =a₋₃ n+ +a₋₂ n+ + +a₅ n−

x δ δ δ

∑

^∞

−∞

=

−

=

k

k n k x n

x[ ] [ ]δ[ ]

⎩⎨

⎧

=

= ≠

, 0 ,

1

, 0 ,

] 0

[ n

n n δ

Defined as

Related to the impulse by

Conversely,

Unit step sequence

⎩⎨

⎧

<

= ≥

, 0 ,

0

, 0 ,

] 1

[ n

n n u

∑

^∞

=

∞

−∞

=

−

=

−

=

+

− +

=

0

] [ ] [ ] [ ] [ or

...

] 2 [ ] 1 [ ] [ ] [

k k

k n k

n k u n u

n n

n n u

δ δ

δ

] 1 [ ] [ ]

[n =u n −u n− δ

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Exponential sequences

Extremely important in representing and analyzing LTI systems.

Defined as

If A and are real numbers, the sequence is real.

If and A is positive, the sequence values are positive and decrease with increasing n.

If , the sequence values alternate in sign, but again decrease in magnitude with increasing n.

If , the sequence values increase with increasing n.

1 0<α <

1

|

| α

>

α

A

n

x [ ] = α

0 1< <

− α

n n n

n x

2 2 ] [

) 5 . 0 ( 2 ] [

⋅

=

−

⋅

=

⋅

=

Combining basic sequences

An exponential sequence that is zero for n<0

⎩⎨

⎧

<

= ≥

0

, 0

, 0 ] ,

[

n

n n A

x

α

n

] [ ]

[

n A u n

x =

α

ⁿ

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Sinusoidal sequences

with A and real constants.

The with complex has real and imaginary parts that are exponentially weighted sinusoids.

φ

Aαn

) sin(

|

||

| ) cos(

|

||

|

||

|

| ]

[

then ,

|

| and

|

| If

0 0

) ( ₀

0 0

φ ω α

α α α

α α

φ ω φ ω

φ ω

+ +

+

=

+

n A

j n

A

e A

e e

A A

n x

e A A e

n n

n j n

n n j j

n

j j

α

n n

A n

x

[ ]

=

cos( ω

₀ +

φ ), for all

Complex exponential sequence

By analogy with the continuous-time case, the quantity is called the frequencyof the complex sinusoid or complex exponential and is call the phase.

nis always an integer Ædifferences between discrete-time and continuous-time

φ

ω

0

) sin(

|

| ) cos(

|

| ] [

, 1

|

| When

0 0

)

( ₀ ω φ ω φ

α

φ

ω = + + +

=

+ A n j A n

e A n

x ^j ⁿ

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An important difference – frequency range

Consider a frequency

More generally being an integer,

Same for sinusoidal sequences

So, only consider frequencies in an interval of such as

n j n

j n j n

j

Ae e Ae

Ae n

x [ ] =

⁽^ω⁰⁺²^π⁾

=

^ω⁰ ²^π

=

^ω⁰

)

2 ( ω

₀

+ π

π ω

π <

₀

≤ or 0 ≤

₀

< 2

−

r r ) , 2 ( ω

₀

+ π

) cos(

] )

2 cos[(

]

[ n = A ω

₀

+ π r n + φ = A ω

₀

n + φ x

π 2

n rn j

n j j n

r

j Ae e Ae

Ae n

x

[ ]

= ⁽^ω⁰⁺²^π ⁾ = ^ω⁰ ²^π = ^ω⁰

Another important difference – periodicity

In the continuous-time case, a sinusoidal signal and a complex exponential signal are both periodic.

In the discrete-time case, a periodic sequence is defined as

where the period N is necessarily an integer.

For sinusoid,

integer.

an is where

/ 2 or

2 that

requires which

) cos(

0 0

0

k

k N

N n

A n

A

ω π π

ω

φ ω

ω φ

ω

=

+ +

= +

n N

n x n

x [ ] = [ + ] , for all

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Another important difference – periodicity

Same for complex exponential sequence

So, complex exponential and sinusoidal sequences

are not necessarily periodic in nwith period

and, depending on the value of , may not be periodic at all.

Consider

k N

e e^j ⁿ ^N ^j ⁿ

π ω

ω ω

2 for

only true is which

,

0 )

( ₀

0

=

+ =

) / 2 ( π ω₀ ω0

16 of

period a

th wi ),

8 / 3 cos(

] [

8 of period a

th wi ),

4 / cos(

] [

2 1

=

N n

n x

N n

n x

π π

Increasing frequency Æincreasing period!

Another important difference – frequency

For a continuous-time sinusoidal signal

For the discrete-time sinusoidal signal

rapidly more

and more oscillates )

( increases, as

), cos(

) (

0

t x t A

t x

Ω

+ Ω

= φ

slower.

become ns

oscillatio the

, 2 towards from

increases as

rapidly more

and more oscillates ]

[ , towards 0

from increases as

), cos(

] [

0 0

0

π π

ω

π ω

φ ω

n x n

A n

x = +

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Frequency

Part II: Discrete-time systems

Introduction

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A transformation or operator that maps input into output

Examples:

The ideal delay system

A memoryless system

]}

[ { ]

[ n T x n y =

x[n] T{.} y[n]

∞

<

∞

−

= x n n n

n

y[ ] [ _d],

∞

<

∞

−

= x n n

n

y[ ] ( [ ])²,

Linear systems

A system is linear if and only if

Combined into superposition

Example 2.6, 2.7 pp. 19 constant arbitrary

an is where

] [ ]}

[ { ]}

[ { and

] [ ] [ ]}

[ { ]}

[ ] [

{ ₁ ₂ ₁ ₂ ₁ ₂

a

n ay n

x aT n

ax T

n y n y n x T n x T n x n x T

=

+

= +

additivity property

scaling property

] [ ] [ ]}

[ { ]}

[ ] [

{ax₁ n bx₂ n aT x₁ n aT x₂ n ay₁ n ay₂ n

T + = + = +

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Time-invariant systems

For which a time shift or delay of the input sequence causes a corresponding shift in the output sequence.

Example 2.8 pp. 20

] [ ] [ ] [ ]

[ ₀ ₁ ₀

1 n x n n y n y n n

x = − ⇒ = −

Causality

The output sequence value at the indexn=n₀ depends only on the input sequence values for n<=n₀.

Example

Causal for n_d>=0

Noncausal for n_d<0

∞

<

∞

−

=x n n n

n

y[ ] [ _d],

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Stability

A system is stable in the BIBO sense if and only if every bounded input sequence produces a

bounded output sequence.

Example stable

∞

<

∞

−

= x n n

n

y

[ ] ( [ ])

²

,

Part III: Linear time-invariant systems

Course overview

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Important due to convenient representations and significant applications

A linearsystem is completely characterised by its impulse response

Time invariance

] [

* ] [

] [ ] [ ]

[

n h n x

k n h k x n

y

k

=

−

=

∑

^∞

−∞

=

∑

∞

−∞

=

∞

−∞

=

∞

−∞

=

−

=

−

=

k

k k

k

n h k x k

n T k x

k n k x T n x T n y

] [ ] [ }

] [ { ] [

} ] [ ] [ { ]}

[ { ] [

δ

] [ ]

[n h n k h_k = −

Convolution sum

Forming the sequence h[n-k]

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Obtain the sequence h[n-k]

Reflecting h[k] about the origin to get h[-k]

Shifting the origin of the reflected sequence to k=n

Multiply x[k] and h[n-k] for

Sum the products to compute the output sample y[n]

Computation of the convolution sum

∑

^∞

−∞

=

−

=

k

k n h k x n

h n x n

y[ ] [ ]* [ ] [ ] [ ]

∞

<

−∞ k

Computing a discrete convolution

Example 2.13 pp.26 Impulse response

input

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

<

− −

−

≤

− ≤

− <

=

+

− +

. 1 1 ),

(

, 1 0

1 ,

1

, 0 , 0 ]

[

1 1

n a N

a

N a n

a

n n

y

N N

n n

] [ ]

[n a u n x = ⁿ

⎩⎨

⎧ ≤ ≤ −

=

−

=

otherwise.

, 0

, 1 0

, 1

] [ ] [ ] [

N n

N n u n u n h

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Properties of LTI systems

Defined by discrete-time convolution

Commutative

Linear

Cascade connection (Fig. 2.11 pp.29)

Parallel connection (Fig. 2.12 pp.30) ] [

* ] [ ] [

* ]

[n hn h n x n

x =

] [ ] [ ]

[n h₁ n h₂ n

h = +

] [

* ] [ ]

[n h₁ n h₂ n

h =

] [

* ] [ ] [

* ] [ ]) [ ] [ (

* ]

[n h₁ n h₂ n x n h₁ n x n h₂ n

x + = +

Properties of LTI systems

Defined by the impulse response

Stable

Causality

∑

^∞

−∞

=

∞

<

=

k

k h S | [ ]|

0 , 0 ]

[n = n<

h

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MATLAB

An interactive, matrix-based system for numeric computation and visualization

Kermit Sigmon, "Matlab Primer", Third Edition, Department of Mathematics, University of Florida.

Matlab Help (>> doc)

Summary

Course overview

Matlab functions for the exercises in this lecture are available at

http://kom.aau.dk/~zt/cources/DSP_E/MM1/

Thanks Borge Lindberg for providing the

Digital Signal Processing, Fall 2006 Part I: Introduction