DIGITAL SIGNAL PROCESSING - BASIC MATERIALS

8/9/2019 DIGITAL SIGNAL PROCESSING - BASIC MATERIALS

1/67

FUNDAMENTALS OF DIGITAL SIGNAL

PROCESSING(FDSP)


2/67

WHAT IS DSP(Digital SignalProcessing)?

Converting a continuously changing waveform(analog) into a series of discrete levels (digital)


3/67

WHAT IS DSP?

The analog waveform is sliced into equalsegments and the waveform amplitude ismeasured in the middle of each segment

The collection of measurements make up thedigital representation of the waveform


4/67

WHAT IS DSP?

0

0 . 2

2 0 . 4

4 0 . 6 4 0

. 8 2

0 . 9

8 1

. 1 1

1 . 2

1 . 2

4

1 . 2

7

1 . 2

4

1 . 2

1 . 1

1

0 . 9

8

0 . 8

2

0 . 6 4

0 . 4

4

0 . 2

2

0

- 0

. 2 2

- 0

. 4 4

- 0

. 6 4

- 0

. 8 2

- 0

. 9 8

- 1

. 1 1

- 1

. 2

- 1

. 2 6

- 1

. 2 8

- 1

. 2 6

- 1

. 2

- 1

. 1 1

- 0

. 9 8

- 0

. 8 2

- 0

. 6 4

- 0

. 4 4

- 0

. 2 2

0

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

1 3 5 7 9 1 1

1 3

1 5

1 7

1 9

2 1

2 3

2 5

2 7

2 9

3 1

3 3

3 5

3 7


5/67

DSP IS EVERYWHERESound applications

Compression, enhancement , special effects, synthesis, recognition,echo cancellation,Cell Phones, MP3 Players, Movies, Dictation, Text-to-speech,

CommunicationModulation, coding, detection, equalization, echo cancellation,Cell Phones, dial-up modem, DSL modem, Satellite Receiver,

AutomotiveABS, GPS, Active Noise Cancellation, Cruise Control, Parking,

MedicalMagnetic Resonance, Tomography, Electrocardiogram,

Military

Radar, Sonar, Space photographs, remote sensing,Image and Video ApplicationsDVD, JPEG, Movie special effects, video conferencing,

MechanicalMotor control, process control, oil and mineral prospecting,
http://www.eas.asu.edu/~dsp/grad/anand/java/Audio/Audio.htmlhttp://www.eas.asu.edu/~dsp/grad/anand/java/Audio/Audio.html


6/67

SIGNAL PROCESSINGHumans are the most advanced signal processors

speech and pattern recognition, speech synthesis,We encounter many types of signals in variousapplications

Electrical signals: voltage, current, magnetic and electricfields,Mechanical signals: velocity, force, displacement,Acoustic signals: sound, vibration,Other signals: pressure, temperature,

Most real-world signals are analog They are continuous in time and amplitudeConvert to voltage or currents using sensors and transducers

Analog circuits process these signals usingResistors, Capacitors, Inductors, Amplifiers,Analog signal processing examples

Audio processing in FM radiosVideo processing in traditional TV sets


7/67

LIMITATIONS OF ANALOG SIGNALPROCESSING

Accuracy limitations due toComponent tolerancesUndesired nonlinearities

Limited repeatability due to TolerancesChanges in environmental conditions

TemperatureVibration

Sensitivity to electrical noiseLimited dynamic range for voltage and currents

Inflexibility to changesDifficulty of implementing certain operations

Nonlinear operations Time-varying operations

Difficulty of storing information


8/67

8

DIGITAL SIGNAL PROCESSINGRepresent signals by a sequence of numbers

Sampling or analog-to-digital conversions

Perform processing on these numbers with a digital processorDigital signal processingReconstruct analog signal from processed numbers

Reconstruction or digital-to-analog conversion

A/D DSP D/Aanalogsignal

analogsignal

digitalsignal

digitalsignal

Analog input analog output Digital recording of music

Analog input digital output Touch tone phone dialing

Digital input analog output Text to speech

Digital input digital output Compression of a file on computer


9/67

PROS AND CONS OF DIGITALSIGNAL PROCESSING

ProsAccuracy can be controlled by choosing word lengthRepeatableSensitivity to electrical noise is minimalDynamic range can be controlled using floating pointnumbersFlexibility can be achieved with software implementationsNon-linear and time-varying operations are easier toimplementDigital storage is cheapDigital information can be encrypted for securityPrice/performance and reduced time-to-market

ConsSampling causes loss of informationA/D and D/A requires mixed-signal hardwareLimited speed of processorsQuantization and round-off errors


10/67

DSP APPLICATIONS

Image Processing Robotic vision, FAX,satellite weatherInstrumentation Spectrum analysis, noisereductionSpeech & Audio Speech recognition,equilizationMilitary Radar processing, missile guidance

Telecommunications Echo cancellation,video conferencing, VoIPBiomedical ECG analysis, patientmonitoringConsumer Electronics Cell phones, set topbox, video cameras


11/67

DSP APPLICATIONS


12/67

ANOTHER LOOK AT DSP APPLICATIONS

High-endWireless Base Station - TMS320C6000Cable modemgateways

Mid-endCellular phone - TMS320C540Fax/ voice server

Low endStorage products - TMS320C27Digital camera - TMS320C5000

Portable phonesWireless headsetsConsumer audioAutomobiles, toasters, thermostats, ...

I n c r e a

s i n g

C o s t

I n c r e a s i n g

v o l u m e


13/67

ADVANTAGES OF DSP

Guaranteed Accuracy Accuracy onlylimited by bit lengthPerfect Reproducibility Nocomponent tolerances, no componentdrift due to temperature or ageGreater Flexibility Functions andalgorithms can be changed throughsoftwareSuperior Performance Adaptivefiltering, linear phase responseSome Data Naturally Digital Images,computer files


14/67

DISADVANTAGES OF DSP

Speed and Cost ADC/DAC, uProcDesign Time Can be trickyFinite Word Length Issues


15/67

KEY DSP OPERATIONS

ConvolutionCorrelationFiltering

TransformationsModulation


16/67

CONVOLUTION

Many uses but a common use is determininga systems output if system input and systemimpulse response is known. For continuoussystem:

( ) ( ) ( ) ( )

=== d ht xd t h xt ht xt y )()()(

( )t x ( )t h ( ) ( ) ( )t ht xt y =


17/67

DISCRETE CONVOLUTION

We may however have a computer samplinga signal so that we have discrete data.So instead of continuous integration processwe have discrete summation.

( ) ( ) ( ) ( ) ( ) ,2,1,0 ===

=nk n xk hnhn xn y

k

Practically speaking though we would havefinite sequences x(n) and h(n) of lengths N 1 and N

2respectively, so this is then:

( ) ( ) ( ) ( ) ( ) ( )

1

1,,1,0

21

1

0

+=

===

=

N N M with

M nk n xk hnhn xn y M

k


18/67

CORRELATION

Correlation is essentially the same asconvolution (from a computationalstandpoint). You just dont flip anything.Instead of describing system output,correlation tells us information about thesignals.Cross-correlation function

Tells you a measure of similarities between twosignals.Application: Identifying radar return signals


19/67

Signals

What is a signal? A signal is a function of independent

variables such as time, distance,

position, temperature, pressure, etc. Most signals are generated naturally but

a signal can also be generatedartificially using a computer

Can be in any number of dimensions(1D, 2D or 3D)


20/67

Lecture 1 CE804 Autumn 2007/8 (copyright R.Palaniappan, 2008)

CLASSIFICATION OF SIGNALS

Signals can be classified into various types byNature of the independent variablesValue of the function defining the signals

Examples:Discrete/continuous functionDiscrete/continuous independent variableReal/complex valued functionScalar (single channel)/Vector (multi-channels)Single/Multi-trial (repeated recordings)Dimensionality based on the number of independent variables(1D/2D/3D)Deterministic/randomPeriodic/aperiodicEven/oddMany more.


21/67


CLASSIFICATION - DISCRETE/CONTINUOUSSIGNALS

Normally, the independent variable is time

Continuous time signal Time is continuousDefined at every instant of time

Discrete time signal Time is discreteDefined at discrete instants of time - it is a sequence of numbers

Four classifications based on time/amplitude -continuous/discrete:

Analogue, digital, sampled, quantised boxcar

CONTINUOUS AND DISCRETE


22/67

CONTINUOUS AND DISCRETESIGNALS

Continous signal

xa(t)

Discrete signal(sequence)

x[n]

x[n] = xa(nT )

T : sampling period

f s = 1/ T : sampling rate


23/67


CLASSIFICATION - DISCRETE/CONTINUOUSSIGNALS (CONT)

Amplitude- continuous

Time-continuous

Amplitude- continuous

Time-discrete

Amplitude- discrete

Time-discrete

Amplitude- discrete

Time-continuous


24/67

24

RANDOM V S DETERMINISTIC SIGNAL

Deterministic signalA signal that can be predicted using some methods like amathematical expression or look-up tableEasier to analyse

Random (stochastic)A signal that is generated randomly and cannot bepredicted ahead of timeMost biological signals fall in this categoryMore difficult to analyse


25/67


CLASSIFICATION PERIOD/APERIODIC

Periodic

Continuous time-signal isperiodic if it exhibitsperiodicity, i.e. x(t+T)=x(t),-


26/67


SINGULAR FUNCTIONSSingular functions

Important non-periodic signalsDelta/unit-impulse function is the most basic and all other singularfunctions can be derived from it

Unit impulse functions

Unit step functions

Unit ramp functions

Unit pulse function

== 1)(;0,0)( dt t t t 00

,0,1

{)(==

nn

n

00

,1,0

{)( >


27/67

CLASSIFICATION EVEN/ODDEven signal

Signal exhibit symmetry inthe time domainx(t)=x(-t) or x(n)=x(-n)

Odd signalSignal exhibit anti-symmetryin the time domainx(t)=-x(-t) or x(n)=-x(-n)

A signal can be expressed as a sum of its even andodd components

x(t)=x even (t)+x odd (t)where x even (t)=1/2[x(t)+x(-t)], x odd (t)=1/2[x(t)-x(-t)]


28/67


SIGNAL DESCRIPTION EXAMPLE

1 D Signal is a function of asingle independentvariable

Speech

2 - DSignal is a function of 2independent variables

Image

M - D Signal has more than 2independent variables

Video signal

DIMENSIONALITY


29/67


Continuous-time signalsThe signal is defined forevery instant of time in adefined range

Discrete-time signalThe independent

variable (time) isdiscrete. The signal isdefined at discreteinstants of time


30/67


Analog signalA continuous-timeand a continuous

amplitude

x(t )

t

x q(t) t

A Quantized Signal discrete in

amplitude butcontinuous in

time


31/67

CLASSIFICATION OF SIGNALSSampled data signalhas a continuousamplitude. Amplitudecan take any valuewithin a specified

range.

Digital signal is adiscrete-time signalwith discrete-valuedamplitudes


32/67

CLASSIFICATION OF SIGNALSA deterministic Signal

is one that isuniquely determinedby a well definedprocess such as amathematicalexpression or a look-up table

A random signal isone that isgenerated in arandom fashion andcannot be predicted

or reproduced


33/67

CLASSIFICATION OF SIGNALSDimension Type Symbol Independent

variable

1 - D Continuous-time v(t) t

1 - D Discrete - time {v(n)} n

2 - D Continuous-spatial v(x,y) x,y

2 - D Discrete - spatial {v(m,n)} m,n

3 - D Continuous-time and

spatial

v(x,y,t) x,y,t

3 - D Continuous-time andspatial

x,y,t

=),,(

),,(

),,(

),,(

t y xb

t y x g

t y xr

t y xu


34/67

DISCRETE-TIME SIGNALS:DISCRETE-TIME SIGNALS:TIME-DOMAIN REPRESENTATIONTIME-DOMAIN REPRESENTATION

Graphical representation of a discrete-timesignal with real-valued samples is as shownbelow:


35/67

DISCRETE-TIME SIGNALS:DISCRETE-TIME SIGNALS:

TIME-DOMAIN REPRESENTATIONTIME-DOMAIN REPRESENTATION

)(t xa

In some applications, a discrete-timesequence { x [n ]} may be generated by

sampling a continuous-time signal atuniform intervals of time


36/67


Here, n -th sample is given by

The spacing T between two consecutive samples iscalled the sampling interval or sampling period

Reciprocal of sampling interval T , denoted as ,

is called the sampling frequency :

),()(][ nT xt xn x anT t a == = ,1,0,1,2, =n

T F

T F

T

1=


37/67


Unit of sampling frequency is cycles persecond , or hertz (Hz), if T is in seconds

Whether or not the sequence { x [n ]} has beenobtained by sampling, the quantity x [n ] iscalled the n -th sample of the sequence

{ x [n ]} is a real sequence , if the n -th sample x [n ] is real for all values of n

Otherwise, { x [n ]} is a complex sequence


38/67


A complex sequence { x [n ]} can be written as

where

andare the real and imaginary parts of x [n ]

The complex conjugate sequence of { x [n ]} is given by

Often the braces are ignored to denote a sequence if there is no ambiguity

][n xre][n x

im

]}[{]}[{]}[{ n x jn xn x imre +=

]}[{]}[{]}[*{ n x jn xn x imre =


39/67


Two types of discrete-time signals: - Sampled-data signals in which samples arecontinuous-valued- Digital signals in which samples arediscrete-valuedSignals in a practical digital signal processingsystem are digital signals obtained byquantizing the sample values either by rounding or truncation


40/67


A discrete-time signal may be a finite-length or an infinite-length sequence

Finite-length (also called finite-duration orfinite-extent ) sequence is defined only for afinite time interval:where and with

Length or duration of the above finite-length sequence is

21 N n N 1 N


41/67


A length- N sequence is often referred to as anN-point sequence

The length of a finite-length sequence can be

increased by zero-padding , i.e., by appending itwith zeros

A right-sided sequence x [n ] has zero-valued

samples for

If a right-sided sequence is called acausal sequence

1 N n ,02 N

n N 1

2 N n


43/67

OPERATIONS ON SEQUENCES:OPERATIONS ON SEQUENCES:BASIC OPERATIONS BASIC OPERATIONS

Product ( modulation ) operation:

Modulator

An application is in forming a finite-lengthsequence from an infinite-length sequence bymultiplying the latter with a finite-lengthsequence called a window sequence Process called windowing

x [n ] y [n ]

w [n ] ][][][ nwn xn y =


44/67

BASIC OPERATIONSBASIC OPERATIONS

Addition operation:Adder

Multiplication operationMultiplier

][][][ nwn xn y +=

A

x [n ] y [n ]

][][ n x An y =

x [n ] y [n ]

w [n ]

+


45/67

BASIC OPERATIONS BASIC OPERATIONS

Time-shifting operation:where N is an integer

(i) If N > 0 , it is delaying operation

Unit delay

(ii) If N < 0 , it is an advance operation

Unit advance

][][ N n xn y =

y [n ] x [n ] z

1 z y [n ] x [n ] ][][ 1= n xn y

][][ 1+= n xn y


46/67

BASIC OPERATIONS BASIC OPERATIONS

Time-reversal ( folding ) operation:

Branching operation:Used to provide multiple copies of asequence

][][ n xn y =

x [n] x [n]

x [n]

ALIASING


47/67

ALIASINGAliasing :

If you sample too slow, the high frequency components will become irregularnoise at the sampling frequency

They are noises that are in the same frequency range of your signal!!!

Look at the samples

alone Can you tell which of the two frequenciesthe sampled seriesrepresents?

Either of the twosignals could producethe samples, i.e., thesignals are aliasesof each other


48/67

CLASSIFICATION OF SEQUENCES:CLASSIFICATION OF SEQUENCES:ENERGY AND POWER SIGNALS ENERGY AND POWER SIGNALS

Power Signal Power Si gnal An infinite energy signal with finite average power iscalled a power signal

Example - A periodic sequence which has a finite

average power but infinite energy


49/67

Energy SignalsEner gy Signals

A finite energy signal with zero averagepower is called an energy signal

Example - a finite-length sequence which hasfinite energy but zero average power

3( 1) ,[ ]

0, 0, 10

n others x n

n n

=<


50/67

OTHER TYPES OF CLASSIFICATIONS OTHER TYPES OF CLASSIFICATIONS

A sequence x [n ] is said to be bounded if

A sequence x [n ] is said to be absolutely summable if

A sequence x [n ] is said to be square-summable if


51/67

BASIC SEQUENCES

Unit impulse Unit step

Exponential

Periodic

Sinusoidal

Random


52/67

BASIC SEQUENCESBASIC SEQUENCES

Unit sample sequence -

Unit step sequence -

=

= 0,0 0,1][ nnn 1

4 3 2 1 0 1 2 3 4 5 6n

Time-Invariant

=> Time-variant

=> Time-variant

=> Time-variant


58/67

AliasingUnable to distinguish two continuous signals withdifferent frequencies based on samplesFrequencies higher than Nyquist frequency

Anti-aliasingLow-pass filter the frequencies above Nyquistfrequency


59/67

DISCRETE-TIME SYSTEM

Discrete-time system has discrete-timeinput and output signals


60/67

DIGITAL SYSTEM

A discrete-time system is digital if itoperates on discrete-time signals whoseamplitudes are quantizedQuantization maps each continuousamplitude level into a number

The digital system employs digitalhardware

1.explicitly in the form of logic circuits2. implicitly when the operations on the signalsare executed by writing a computerprogram


61/67

Discrete-time (DT) system is `sampled data system:Input signal u[k] is a sequence of samples (=numbers)

..,u[-2],u[-1],u[0],u[1],u[2],System then produces a sequence of output samples y[k]

..,y[-2],y[-1],y[0],y[1],y[2],

Will consider linear time-invariant (LTI) DT systems:Linear :

input u1[k] -> output y1[k]

input u2[k] -> output y2[k]hence a.u1[k]+b.u2[k]-> a.y1[k]+b.y2[k]

Time-invariant (shift-invariant)input u[k] -> output y[k], hence input u[k-T] -> output y[k-T]

u[k] y[k]


62/67

Causal systems:iff for all input signals with u[k]=0,k output y[k]=0,k output ,0,0,h[0],h[1],h[2],h[3],...

General input u[0],u[1],u[2],u[3]: (cfr. linearity & shift-invariance!)

=]3[]2[]1[]0[

.

]2[000]1[]2[00]0[]1[]2[0

0]0[]1[]2[00]0[]1[000]0[

]5[]4[]3[]2[]1[]0[

uuuu

hhhhhh

hhhhh

h

y y y y y y

`Toeplitz matrix


63/67

Convolution:

=

]3[

]2[

]1[]0[

.

]2[000

]1[]2[00

]0[]1[]2[0

0]0[]1[]2[00]0[]1[

000]0[

]5[

]4[

]3[

]2[]1[

]0[

u

u

uu

h

hh

hhh

hhhhh

h

y

y

y

y y

y

u[0],u[1],u[2],u[3] y[0],y[1],...

h[0],h[1],h[2],0,0,...

][*][][.][][ k uk hiuik hk yi

== = `convolution sum


64/67

Z-Transform:

=

i

i z ih z H ].[)(

[ ] [ ]=

]3[]2[]1[ ]0[

.

]2[000]1[]2[00]0[]1[]2[0

0]0[]1[]2[ 00]0[]1[

000]0[

.1

]5[]4[]3[]2[ ]1[

]0[

.1

3211).()(

5432154321

uuuu

hhhhhh

hhh hh

h

z z z z z

y y y y y

y

z z z z z

z z z z H z Y

=

i

i z i y z Y ].[)( =

i

i z iu z U ].[)(

)().()( z U z H z Y = H(z) is `transfer function


65/67

Z-Transform :input-output relationmay be viewed as `shorthand notation

(for convolution operation/Toeplitz-vector product)stability

bounded input u[k] -> bounded output y[k]--iff

--iff poles of H(z) inside the unit circle(for causal,rational systems)

)().()( z U z H z Y =

k

k h ][


66/67

Example-1 : `Delay operator

Impulse response is ,0,0,0, 1,0,0,0, Transfer function is

Example-2 : Delay + feedback Impulse response is ,0,0,0, 1,a,a^2,a^3 Transfer function is

1)( = z z H

u[k]

y[k]=u[k-1]

x

+

a

u[k]

y[k]

1

1

11

433221

.1)(

)(.)(

......)(

=

=

++++=

z a z

z H

z z H z a z H

z a z a z a z z H

LINEAR TIME INVARIANT


67/67

LINEAR TIME-INVARIANT (LTI) SYSTEM

Discrete-time system is LTI if itsinput-output relationship can be described by thelinear constant coefficients difference equation

The output sample y( ) might depend on all input samplesthat can be represented as

))(()( k x y =

DIGITAL SIGNAL PROCESSING - BASIC MATERIALS

Documents

Transcript of DIGITAL SIGNAL PROCESSING - BASIC MATERIALS