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Digital Signal Processing, I, Zheng-Hua Tan, 20061
Digital Signal Processing, Fall 2006
Zheng-Hua Tan
Department of Electronic Systems Aalborg University, Denmark
zt@kom.aau.dk
Lecture 1: Introduction,
Discrete-time signals and systems
Digital Signal Processing, I, Zheng-Hua Tan, 20062
Part I: Introduction
Introduction (Course overview)Discrete-time signals Discrete-time systemsLinear time-invariant systems
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Digital Signal Processing, I, Zheng-Hua Tan, 20063
General information
Course websitehttp://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.
Digital Signal Processing, I, Zheng-Hua Tan, 20064
General information
Duration2 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 TanNiels Jernes Vej 12, A6-319zt@kom.aau.dk, +45 9635-8686
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Digital Signal Processing, I, Zheng-Hua Tan, 20065
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, I, Zheng-Hua Tan, 20066
Course objectives (Part I)
To give the students a comprehension of the concepts of discrete-time signals and systemsTo give the students a comprehension of the Z- and the Fourier transform and their inverseTo give the students a comprehension of the relation between digital filters, difference equations and system functionsTo give the students knowledge about the most important issues in sampling and reconstruction
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Digital Signal Processing, I, Zheng-Hua Tan, 20067
Course objectives (Part II)
To make the students able to apply digital filters according to known filter specificationsTo provide the knowledge about the principles behind the discrete Fourier transform (DFT) and its fast computationTo make the students able to apply Fourier analysis of stochastic signals using the DFTTo be able to apply the MATLAB program to digital processing problems and presentations
Digital Signal Processing, I, Zheng-Hua Tan, 20068
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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Digital Signal Processing, I, Zheng-Hua Tan, 20069
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)} .
Digital Signal Processing, I, Zheng-Hua Tan, 200610
Types of signals
The independent variable may be either continuous or discrete
Continuous-time signalsDiscrete-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 continuousDigital 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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Digital Signal Processing, I, Zheng-Hua Tan, 200611
Types of signals
From http://www.ece.rochester.edu/courses/ECE446
Digital Signal Processing, I, Zheng-Hua Tan, 200612
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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Digital Signal Processing, I, Zheng-Hua Tan, 200613
Typical DSP system components
Input lowpass filter to avoid aliasingAnalog to digital converter (ADC)Computer or DSP processorDigital to analog converter (DAC)Output lowpass filter to avoid imaging
Digital Signal Processing, I, Zheng-Hua Tan, 200614
ADC and DAC
Physical signals Analog signals Digital signals
Transducerse.g. microphones
Analog-to-digitalconverters
Digital-to-Analogconverters
Output devices
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Digital Signal Processing, I, Zheng-Hua Tan, 200615
Pros and cons of DSP
ProsEasy to duplicateStable and robust: not varying with temperature, storage without deteriorationFlexibility and upgrade: use a general computer or microprocessor
ConsLimitations of ADC and DACHigh power consumption and complexity of a DSP implementation: unsuitable for simple, low-power applicationsLimited to signals with relatively low bandwidths
Digital Signal Processing, I, Zheng-Hua Tan, 200616
Applications of DSP
Speech processingEnhancement – noise filteringCoding Text-to-speech (synthesis)
Next generation TTS @ AT&TRecognition
Image processingEnhancement, coding, pattern recognition (e.g. OCR)
Multimedia processingMedia transmission, digital TV, video conferencing
CommunicationsBiomedical engineeringNavigation, radar, GPS Control, robotics, machine vision
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Digital Signal Processing, I, Zheng-Hua Tan, 200617
History of DSP
Prior to 1950’s: analog signal processing using electronic circuits or mechanical devices1950’s: computer simulation before analog implementation, thus cheap to try out1965: Fast Fourier Transforms (FFTs) by Cooley and Tukey – make real time DSP possible1980’s: IC technology boosting DSP
Digital Signal Processing, I, Zheng-Hua Tan, 200618
Part II: Discrete-time signals
IntroductionDiscrete-time signalsDiscrete-time systemsLinear time-invariant systems
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Digital Signal Processing, I, Zheng-Hua Tan, 200619
Discrete-time signals
Sequences of numbers
Periodic sampling of an analog signal
integeran is where]},[{
nnnxx ∞<<∞−=
period. sampling thecalled is where),(][
TnnTxnx a ∞<<∞−=
x[1]
x[2]
x[n]x[-1]
x[0]
Digital Signal Processing, I, Zheng-Hua Tan, 200620
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]
αα
integeran is where][][ 0
nnnxny −=
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Digital Signal Processing, I, Zheng-Hua Tan, 200621
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[][ 523 −+++++= −− nanananx δδδ
∑∞
−∞=
−=k
knkxnx ][][][ δ
⎩⎨⎧
=≠
=,0,1,0,0
][nn
nδ
Digital Signal Processing, I, Zheng-Hua Tan, 200622
Defined as
Related to the impulse by
Conversely,
Unit step sequence
⎩⎨⎧
<≥
=,0,0,0,1
][nn
nu
∑∑∞
=
∞
−∞=
−=−=
+−+−+=
0
][][][][
or...]2[]1[][][
kk
knknkunu
nnnnu
δδ
δδδ
]1[][][ −−= nununδ
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Digital Signal Processing, I, Zheng-Hua Tan, 200623
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.
10 << α
1|| >α
α
nAnx α=][
01 <<− α
n
n
n
nxnxnx
22][)5.0(2][
)5.0(2][
⋅=
−⋅=
⋅=
Digital Signal Processing, I, Zheng-Hua Tan, 200624
Combining basic sequences
An exponential sequence that is zero for n<0
⎩⎨⎧
<≥
=0 ,0,0,
][nnA
nxnα
][][ nuAnx nα=
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Digital Signal Processing, I, Zheng-Hua Tan, 200625
Sinusoidal sequences
with A and real constants.The with complex has real and imaginary parts that are exponentially weighted sinusoids.
φnAα
)sin(||||)cos(||||
||||
||||][
then,|| and || If
00
)( 0
0
0
φωαφωα
α
αα
αα
φω
ωφ
φω
+++=
=
==
==
+
nAjnA
eA
eeAAnx
eAAe
nn
njn
njnjn
jj
α
nnAnx allfor ),cos(][ 0 φω +=
Digital Signal Processing, I, Zheng-Hua Tan, 200626
Complex exponential sequence
By analogy with the continuous-time case, the quantity is called the frequency of the complex sinusoid or complex exponential and is call the phase.n is always an integer differences between discrete-time and continuous-time
φ0ω
)sin(||)cos(||||][
,1||When
00)( 0 φωφω
αφω +++==
=+ nAjnAeAnx nj
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Digital Signal Processing, I, Zheng-Hua Tan, 200627
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
njnjnjnj AeeAeAenx 000 2)2(][ ωπωπω === +
)2( 0 πω +
πωπωπ 20or 00 <≤≤<−
rr ,)2( 0 πω +
)cos(])2cos[(][ 00 φωφπω +=++= nAnrAnxπ2
njrnjnjnrj AeeAeAenx 000 2)2(][ ωπωπω === +
Digital Signal Processing, I, Zheng-Hua Tan, 200628
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/2or 2 that requireswhich
)cos()cos(
00
000
kkNkN
NnAnAωππω
φωωφω==
++=+
nNnxnx allfor ,][][ +=
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Digital Signal Processing, I, Zheng-Hua Tan, 200629
Another important difference – periodicity
Same for complex exponential sequence
So, complex exponential and sinusoidal sequences are not necessarily periodic in n with period and, depending on the value of , may not be periodic at all.
Consider
kNee njNnj
πω
ωω
2for only trueiswhich ,
0
)( 00
==+
)/2( 0ωπ0ω
16 of period ath wi),8/3cos(][ 8 of period ath wi),4/cos(][
2
1
====
NnnxNnnx
ππ
Increasing frequency increasing period!
Digital Signal Processing, I, Zheng-Hua Tan, 200630
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
0
txtAtx
Ω+Ω= φ
slower. become nsoscillatio the,2 towards from increases asrapidly more and more oscillates ][ , towards0 from increases as
),cos(][
0
0
0
ππωπω
φωnx
nAnx +=
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Digital Signal Processing, I, Zheng-Hua Tan, 200631
Frequency
Digital Signal Processing, I, Zheng-Hua Tan, 200632
Part II: Discrete-time systems
Introduction Discrete-time signals Discrete-time systemsLinear time-invariant systems
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Digital Signal Processing, I, Zheng-Hua Tan, 200633
Discrete-time systems
A transformation or operator that maps input into output
Examples:The ideal delay system
A memoryless system
]}[{][ nxTny =
T{.}x[n] y[n]
∞<<∞−−= nnnxny d ],[][
∞<<∞−= nnxny ,])[(][ 2
Digital Signal Processing, I, Zheng-Hua Tan, 200634
Linear systems
A system is linear if and only if
Combined into superposition
Example 2.6, 2.7 pp. 19
constantarbitrary an is where][]}[{]}[{
and][][]}[{]}[{]}[][{ 212121
anaynxaTnaxT
nynynxTnxTnxnxT
==
+=+=+additivity property
scaling property
][][]}[{]}[{]}[][{ 212121 naynaynxaTnxaTnbxnaxT +=+=+
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Digital Signal Processing, I, Zheng-Hua Tan, 200635
Time-invariant systems
For which a time shift or delay of the input sequencecauses a corresponding shift in the output sequence.
Example 2.8 pp. 20
][][][][ 0101 nnynynnxnx −=⇒−=
Digital Signal Processing, I, Zheng-Hua Tan, 200636
Causality
The output sequence value at the index n=n0depends only on the input sequence values for n<=n0.
Example
Causal for nd>=0Noncausal for nd<0
∞<<∞−−= nnnxny d ],[][
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Digital Signal Processing, I, Zheng-Hua Tan, 200637
Stability
A system is stable in the BIBO sense if and only ifevery bounded input sequence produces a bounded output sequence.
Example
stable∞<<∞−= nnxny ,])[(][ 2
Digital Signal Processing, I, Zheng-Hua Tan, 200638
Part III: Linear time-invariant systems
Course overviewDiscrete-time signals Discrete-time systemsLinear time-invariant systems
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Digital Signal Processing, I, Zheng-Hua Tan, 200639
Linear time-invariant systems
Important due to convenient representations and significant applicationsA linear system is completely characterised by itsimpulse response
Time invariance
][*][
][][][
nhnx
knhkxnyk
=
−= ∑∞
−∞=
∑∑
∑∞
−∞=
∞
−∞=
∞
−∞=
=−=
−==
kk
k
k
nhkxknTkx
knkxTnxTny
][][}][{][
}][][{]}[{][
δ
δ
][][ knhnhk −=
Convolution sum
Digital Signal Processing, I, Zheng-Hua Tan, 200640
Forming the sequence h[n-k]
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Digital Signal Processing, I, Zheng-Hua Tan, 200641
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
knhkxnhnxny ][][][*][][
∞<<−∞ k
Digital Signal Processing, I, Zheng-Hua Tan, 200642
Computing a discrete convolution
Example 2.13 pp.26Impulse response
input
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
<−−−
−≤≤−−
<
=
+−
+
.1 ),1
1(
,10 ,1
1,0 ,0
][
1
1
nNa
aa
Nna
an
nyN
Nn
n
][][ nuanx n=
⎩⎨⎧ −≤≤
=
−−=
otherwise. ,0,10 ,1][][][
NnNnununh
22
Digital Signal Processing, I, Zheng-Hua Tan, 200643
Properties of LTI systems
Defined by discrete-time convolutionCommutative
Linear
Cascade connection (Fig. 2.11 pp.29)
Parallel connection (Fig. 2.12 pp.30)
][*][][*][ nxnhnhnx =
][][][ 21 nhnhnh +=
][*][][ 21 nhnhnh =
][*][][*][])[][(*][ 2121 nhnxnhnxnhnhnx +=+
Digital Signal Processing, I, Zheng-Hua Tan, 200644
Properties of LTI systems
Defined by the impulse responseStable
Causality
∑∞
−∞=
∞<=k
khS |][|
0 ,0][ <= nnh
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Digital Signal Processing, I, Zheng-Hua Tan, 200645
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)
Digital Signal Processing, I, Zheng-Hua Tan, 200646
Summary
Course overviewDiscrete-time signals Discrete-time systemsLinear time-invariant systems
Matlab functions for the exercises in this lecture are available athttp://kom.aau.dk/~zt/cources/DSP_E/MM1/Thanks Borge Lindberg for providing the functions.
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