introduction to digital signal processingpds/lect01.pdf · 2012-10-23 · p.a. lynn and w. fuerst...

Introduction to Digital Signal Processing

John Chiverton

Digital Signal Processing (1502432)School of Information Technology

Mae Fah Luang University

June 9th 2009

Lecture Contents

Course FormatCourse Outline

Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers

A Typical Digial Signal Processing System

SummaryLecture summary

Outline

Course FormatCourse Outline

Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers

A Typical Digial Signal Processing System

SummaryLecture summary

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.

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

Assessment

1. Reports 30%2. Project/ presentation 10%3. Midterm examination 30%4. Final examination 30%

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

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

Outline

Course FormatCourse Outline

Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers

A Typical Digial Signal Processing System

SummaryLecture summary

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

What is a Signal?

A simple example.

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

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)

What is a Signal?

Can contain information for

I Communication

I Storage

I Calculation

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

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)

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.

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

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

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

-1.00

-0.50

0.00

0.50

1.00

1.50

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

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).

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

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

Phase examples

-1.50

-1.00

-0.50

0.00

0.50

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=-0.5× 2π

phase shift

complete cycle

-1.50

-1.00

-0.50

0.00

0.50

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=-0.4× 2π

phase shift

complete cycle

-1.50

-1.00

-0.50

0.00

0.50

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

-1.50

-1.00

-0.50

0.00

0.50

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

Phase examples cont’d

-1.50

-1.00

-0.50

0.00

0.50

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

0.50

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

0.50

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

-1.00

-0.50

0.00

0.50

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

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.

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)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0Pi 1Pi 2Pi 3Pi 4Pi 5Pi 6Pi 7Pi 8Pi 9Pi 10Pi

y(x)

ω x (seconds)

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)

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

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)

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

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).

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.

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).

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.

Outline

Course FormatCourse Outline

Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers

A Typical Digial Signal Processing System

SummaryLecture summary

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

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.

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

Truncation Quantisation examples

I Errors can be seen between the sampled and the sampled andquantized signals.

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

Rounding Quantisation examples

I Errors can be seen between the sampled and the sampled andquantized signals.

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.

Sampling examples

Sampling examples cont’d

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

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

Averaging Filter Examples

Window width k controls the response of the filter. If k is too low,there is little benefit on output signal.

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.

Outline

Course FormatCourse Outline

Digital Signal ProcessingWhat is Digital Signal Processing?PhasePhasors and Complex Numbers

A Typical Digial Signal Processing System

SummaryLecture summary

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