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Introduction to Digital Signal Processingpds/Lect01.pdf · 2012-10-23 · P.A. Lynn and W. Fuerst...
Transcript of Introduction to Digital Signal Processingpds/Lect01.pdf · 2012-10-23 · P.A. Lynn and W. Fuerst...
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Introduction to Digital Signal Processing
John Chiverton
Digital Signal Processing (1502432)School of Information Technology
Mae Fah Luang University
June 9th 2009
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Lecture Contents
Course FormatCourse Outline
Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers
A Typical Digial Signal Processing System
SummaryLecture summary
![Page 3: Introduction to Digital Signal Processingpds/Lect01.pdf · 2012-10-23 · P.A. Lynn and W. Fuerst Introductory Digital Signal Processing John Wiley J.G. Proakis and D. G. Manolakis](https://reader030.fdocuments.in/reader030/viewer/2022040608/5ec51999775d7944a7394952/html5/thumbnails/3.jpg)
Outline
Course FormatCourse Outline
Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers
A Typical Digial Signal Processing System
SummaryLecture summary
![Page 4: Introduction to Digital Signal Processingpds/Lect01.pdf · 2012-10-23 · P.A. Lynn and W. Fuerst Introductory Digital Signal Processing John Wiley J.G. Proakis and D. G. Manolakis](https://reader030.fdocuments.in/reader030/viewer/2022040608/5ec51999775d7944a7394952/html5/thumbnails/4.jpg)
Objectives
After finishing the course students should be able to
I Demonstrate knowledge of digital signal processingtechniques;
I Solve and analyze digital signal processing problems;
I Design systems using knowledge obtained from the course;
I Apply knowledge to other related topics.
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Course Description
I Time-varying signals
I Z-transformation
I Discrete Fourier Transformation
I Fourier Analysis for Time-varying signals
I Digital filter design
I Random signals
I Power spectrum estimation
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Assessment
1. Reports 30%2. Project/ presentation 10%3. Midterm examination 30%4. Final examination 30%
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Course Content
Period Topic8-14 June Introduction15-21 June Discrete time domain analysis22-28 June Discrete frequency domain analysis (Fourier)29-5 July Discrete frequency domain analysis (Z-Transform)6-12 July Design of non-recursive digital filters13-19 July Design of recursive digital filters20-26 July Implementation of discrete-time systems27-2 Aug. Review material & Applications3-9 Aug. Mid-Term Examinations10-16 Aug. Discrete and Fast Fourier Transforms (DFT, FFT)17-23 Aug. Sampling and reconstruction of signals24-30 Aug. Random signals review31-6 Sept. Linear prediction and optimum linear filters7-13 Sept. Adaptive Filters14-20. Sept. Applications21-27 Sept. Revision and review28> Sept. Examinations
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Text Books
Primary Texts:
P.A. Lynn and W. Fuerst Introductory Digital Signal Processing John Wiley
J.G. Proakis and D. G.Manolakis
Digital Signal Processing Principles,Algorithms and Applications
Pearson Ed-ucation
Other:
E.C. Ifeacher and B.W.Jervis
Digital Signal Processing A PracticalApproach
Addison-Wesley
J. Van de Vegte Fundamentals of Digital Signal Pro-cessing
Prentice Hall
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Outline
Course FormatCourse Outline
Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers
A Typical Digial Signal Processing System
SummaryLecture summary
![Page 10: Introduction to Digital Signal Processingpds/Lect01.pdf · 2012-10-23 · P.A. Lynn and W. Fuerst Introductory Digital Signal Processing John Wiley J.G. Proakis and D. G. Manolakis](https://reader030.fdocuments.in/reader030/viewer/2022040608/5ec51999775d7944a7394952/html5/thumbnails/10.jpg)
What is Digital Signal Processing?
Techniques include (e.g. )
I Filtering
I Frequency domaintechniques (i.e. Fourier)
I Time domain techniques
I Random signals
I Predication andEstimation (e.g. timeseries estimation)
Example Applications
I Audio processing
I Communication systems
I Image processing
I Video processing
I Data compression
I Vehicle control
I Financial engineering
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What is a Signal?
A simple example.
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y (a
rbitr
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ampl
itude
uni
ts)
x (seconds)
A simple signal: sinusoidal wave (sine wave)
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What is a Signal?
Can contain information for
I Communication
I Storage
I Calculation
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0.000 0.002 0.004 0.006 0.008 0.010
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
A simple signal: sinusoidal wave (sine wave)
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Example of What is a Signal?
Information is carried inthe
I amplitude, “a”;
I period, “T”;
I frequency, “f = 1/T”;
I and phase, “φ”.
Equation for a sine wave:
y(x) = a sin(2πfx+ φ)
where “x” is time in secondsfor this example. Amplitude“a = 1” controls the heightof the wave.
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0.000 0.002 0.004 0.006 0.008 0.010
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
A simple signal: sinusoidal wave (sine wave)
period
ampl
itude
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Frequency and Period
Equation for a sine wave:
y(x) = a sin(2πfx+ φ)
I f is the frequency
I Measured in Hertz orHz
I Here period,T = 0.002s
I f = 1/T Hz, thereforef = 1/0.002 = 500Hz. -1.50
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0.000 0.002 0.004 0.006 0.008 0.010
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
A simple signal: sinusoidal wave (sine wave)
period
ampl
itude
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Phase
Equation for a sine wave:
y(x) = a sin(2πfx+ φ)
I φ is the phase
I Here φ = 0
Therefore here,
y(x) = y(x, φ = 0) = a sin(2πfx).
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0.000 0.002 0.004 0.006 0.008 0.010
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
A simple signal: sinusoidal wave (sine wave)
period
ampl
itude
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Phase examples
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y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=-0.5× 2π
phase shift
complete cycle
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-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=-0.4× 2π
phase shift
complete cycle
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-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=-0.3× 2π
phase shift
complete cycle
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-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=-0.2× 2π
phase shift
complete cycle
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Phase examples cont’d
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-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=-0.1× 2π
phase shift
complete cycle
-1.50
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0.00
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1.00
1.50
-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=0.1× 2π
phase shift
complete cycle
-1.50
-1.00
-0.50
0.00
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1.00
1.50
-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=0.2× 2π
phase shift
complete cycle
-1.50
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0.00
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1.00
1.50
-0.004 -0.003 -0.002 -0.001 0.000 0.001 0.002 0.003
y (a
rbitr
ary
ampl
itude
uni
ts)
x (seconds)
phase angle=0.3× 2π
phase shift
complete cycle
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Cosine Vs Sine
Cosine and Sine functions areequivalent except for a phaseshift (1/4×period).
I cos(2πfx) = sin(2πfx+ φ) where φ = π/2.
I sin(2πfx) = cos(2πfx+ φ) where φ = −π/2.
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Angular Frequency
I Frequency, f = 1/T
I Angular frequency,ω = 2πf
I 1 period or cycle = 2πradians
y(x) = sin(2πfx+ φ)
= sin(ωx+ φ)
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0Pi 1Pi 2Pi 3Pi 4Pi 5Pi 6Pi 7Pi 8Pi 9Pi 10Pi
y(x)
ω x (seconds)
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Angular Frequency
I Frequency, f = 1/T
I Angular frequency,ω = 2πf
I 1 period or cycle = 2πradians
y(x) = sin(2πfx+ φ)
= sin(ωx+ φ)
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0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010
y(x)
x (seconds)
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Phasor Representation
A cosine (or sine) wave:
y(x) = a cos(ωx+ φ)
can be represented as a phasor.A phasor is a complex number:
z = x+jy = a(cos(φ)+j sin(φ))
where x is known as the real partor Re(z) = x and y is known asthe imaginery part or Im(z) = y.
x and y can be calculated withx = a cos(φ) and y = a sin(φ).
Also remember j =√−1.
Argand or Phasor Diagram:
-1.5j
-1.0j
-0.5j
0.0j
0.5j
1.0j
1.5j
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Real axis
Imag
iner
y ax
is
a
x
y
ω rad/s rotation
φ
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Complex Numbers
The square root of minus oneis not defined so a symbol, j isused (sometimes i):
j =√−1.
Powers:
I j2 = −1I j3 = −jI j−1 = 1/j = −j
If z = x+ jy (rectangularform) then alternativerepresentations are:
I Polar form: z = a∠φ
I Exponential form:z = a exp(jφ)
where a =√x2 + y2 and
φ = tan−1(y/x).
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Properties of Complex Numbers
If z = x+ jy, z1 = x1 + jy1 and z2 = x2 + jy2 then
I Addition:z1 + z2 = x1 + x2 + j(y1 + y2)
I Subtraction:z1 − z2 = x1 − x2 + j(y1 − y2)
I Multiplication:z1z2 = a1a2∠(φ1 + φ2)
I Division:z1/z2 = a1/a2∠(φ1 − φ2)
I Reciprocal: 1/z = 1/a∠(−φ)
I Square root:√z =√a∠(φ/2)
I Complex conjugate:z∗ = x− jy = a∠− φ
The polar form simplifies some operations such as multiplication and division of
complex numbers.
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Phasor Representation
Euler’s identity:exp(jφ) = cos(φ) + j sin(φ)
Therefore
I cos(φ) = Re(exp(jφ)) −→ or the real part, x
I sin(φ) = Im(exp(jφ)) −→ or the imaginary part, y
Recall the cosine wave:y(x) = cos(ωx+ φ)
which can be written as:
y(x) = Re(a exp(j(ωx+ φ))) = Re(a exp(jωx) exp(jφ))
= Re(A exp(jωx))
where A is the phasor representation of y(x) given by
A = a exp(jφ) = a∠(φ).
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Complex Exponentials, Sines and Cosines
Given
I y1(x) = b exp(jωx) = b cos(ωx) + jb sin(ωx)
I y2(x) = b exp(−jωx) = b cos(ωx) + jb sin(−ωx)as
I cos(−ωx) = cos(ωx) (even function)I sin(−ωx) = − sin(ωx) (odd function)
Theny1(x) + y2(x) = 2b cos(ωx).
So thatb cos(ωx) =
a
2exp(jωx) +
a
2exp(−jωx).
A similar approach can be used to derive a sine function.
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Outline
Course FormatCourse Outline
Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers
A Typical Digial Signal Processing System
SummaryLecture summary
![Page 27: Introduction to Digital Signal Processingpds/Lect01.pdf · 2012-10-23 · P.A. Lynn and W. Fuerst Introductory Digital Signal Processing John Wiley J.G. Proakis and D. G. Manolakis](https://reader030.fdocuments.in/reader030/viewer/2022040608/5ec51999775d7944a7394952/html5/thumbnails/27.jpg)
A Typical Digital Signal Processing System
Digital SignalProcessor
Analog to DigitalConverter
ConverterDigital to Analog
Analog Input
Analog OutputAnalog Filter
Analog Filter
Output
InputI Input Analog Filter
(antialiasing):Limits frequency range
I Analog to Digital ConverterConverts signal to digital
samples
I Digital Signal ProcessorStorage, Communication and
or Calculations
I Digital to Analog ConverterConvert to continuous signal
I Output Analog FilterRemoves sharp transitions
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Analog to Digital Converter (ADC)
I Real world is typically analog (continuous)
I Digital signal approximates analog signal with discretequantised samples
I ADC converts an analog signal to a digital signalI Signal is digitised in two ways:
I Signal is sampled at a sampling rate or frequency: Informationis collected about the signal at regular intervals.
I The continuous or analog signal is then quantised: i.e. put intodigital form, where only a finite set of numbers are represented.
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Quantisation using Truncation
I Signal can be quantised using e.g. truncation where numbersfollowing specified position are removed.
I Examples:I 5.7 truncated to integer is 5I 5.11 truncated to 1 decimal place is 5.1
I Negative numbers are truncated in the same way (notedifferent to the common floor function in matlab), e.g.
I -5.78 truncated to integer is -5I -5.135 truncated to 2 decimal places is -5.13
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Truncation Quantisation examples
I Errors can be seen between the sampled and the sampled andquantized signals.
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Quantisation using Rounding
I Rounding can be a quantization method associated withsmaller errors, e.g.
I 5.7 rounded to nearest integer is 6I 5.11 rounded to 1 decimal place is 5.1I -5.78 rounded to nearest integer is -6I -5.135 rounded to 2 decimal places is -5.14
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Rounding Quantisation examples
I Errors can be seen between the sampled and the sampled andquantized signals.
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Sampling
I Sampling also affects the quality of the digitised signal.
I Higher sampling rate reduces error and enables betterrepresentation of the original analog signal in digital form.
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Sampling examples
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Sampling examples cont’d
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Input Analog Filter: Antialiasing Filter
I Analog to Digital Converter (ADC) requires signal below aparticular frequency (Nyquist Frequency)
I ∴ Limit frequency range to below Nyquist frequency (fs/2)before Analog to Digital Conversion.
mag
nitu
de
high0 low
100%
sf /2 sf
frequencysampling
frequencycut−off
71%
frequencies
I Otherwise next stageproduces frequency errors(i.e. aliasing)
I Sampling produces copiesof signal at multiples ofsampling frequency
I Aliasing occurs whencopies of signal overlapeach other
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Digital Signal Processor
I After digitisation (with the ADC) digital signal processing maythen be performed on the digitised signal.
I Simple exampleI Averaging filter:
y[n] =x[n] + x[n− 1] + ...+ x[n− k + 1]
k
for window width k = 3
y[n] =x[n] + x[n− 1] + x[n− 2]
3
where x[n] is an input value at sample time n and y[n] is anoutput at sample time n
[x n] [y n]
FilterAveraging
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Averaging Filter Examples
Window width k controls the response of the filter. If k is too low,there is little benefit on output signal.
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Averaging Filter Examples cont’d
Window width k controls the response of the filter. If k is toohigh, the filter removes all of the output signal.
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Outline
Course FormatCourse Outline
Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers
A Typical Digial Signal Processing System
SummaryLecture summary
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What have we covered today?
I Course content
I Definition of digital signal processing
I Description of phase
I Cosine and Sine functions
I Complex numbers and alternative representations
I A typical digital signal processing system