Lecturer: Dr. Peter Tsang Room: G6505 Phone: Website: Files: SIG01.ppt, SIG02.ppt, SIG03.ppt, SIG04.ppt Restrict access to students taking this course.

Suggested reference books 1. M.L. Meade and C.R. Dillon, “Signals and Systems”, Van Nostrand Reinhold (UK). 2. N.Levan, “Systems and Signals”, Optimization Software, Inc. 3. F.R. Connor, “Signals”, Edward Arnold. 4 *. A. Oppenheim, “Digital Signal Processing”, Prentice Hall. Note: Students are encouraged to select reference books in the library. * Supporting reference

Course outline Week 2-4: Lecture Week 6: Test Week 7-10: Lecture Week 11: Test Scores Tests: 30% (15% for each test) Exam: 70%

Tutorials Group 01: Friday Weeks: 2,3,4,7,8,9 Group 02: Monday Weeks: 3,4,5,7,8,9 Group 03: Thursday Weeks: 2,3,4,7,8,9

Course outline 1. Time Signal Representation. 2. Continuous signals. 3. Fourier, Laplace and z Transform. 4. Interaction of signals and systems. 5. Sampling Theorem. 6. Digital Signals. 7. Fundamentals of Digital System. 8. Interaction of digital signals and systems.

Coursework Tests on week 6 and 11: 30% of total score. Notes in Powerpoint Presented during lectures and very useful for studying the course. Study Guide A set of questions to build up concepts. Discussions Strengthen concepts in tutorial sessions. Reference books Supplementary materials to aid study.

Expectation from students Attend all lectures and tutorials. Study all the notes. Participate in discussions during tutorials. Work out all the questions in the study guide at least once. Attend the test and take it seriously. Work out the questions in the test for at least one more time afterwards.

SIGNALS Information expressed in different forms Stock Price Transmit Waveform $1.00, $1.20, $1.30, $1.30, … Data File x(t)x(t) Primary interest of Electronic Engineers

SIGNALS PROCESSING AND ANALYSIS Processing: Methods and system that modify signals System y(t)y(t)x(t)x(t) Analysis: What information is contained in the input signal x(t)? What changes do the System imposed on the input? What is the output signal y(t)? Input/StimulusOutput/Response

SIGNALS DESCRIPTION To analyze signals, we must know how to describe or represent them in the first place. tx(t) Detail but not informative

TIME SIGNALS DESCRIPTION 1. Mathematical expression: x(t)=Asin(  t  2. Continuous (Analogue) 3. Discrete (Digital) x[n]x[n] n

TIME SIGNALS DESCRIPTION 4. Periodic x(t)= x(t+T o ) ToTo Period = T o 5. Aperiodic

TIME SIGNALS DESCRIPTION 6. Even signal Exercise: Calculate the integral 7. Odd signal

TIME SIGNALS DESCRIPTION 8. Causality Analogue signals: x(t) = 0 for t < 0 Digital signals: x[n] = 0 for n < 0

TIME SIGNALS DESCRIPTION 9. Average/Mean/DC value Exercise: Calculate the AC & DC values of x(t)=Asin(  t  with TMTM 10. AC value DC: Direct Component AC: Alternating Component

TIME SIGNALS DESCRIPTION 11. Energy Exercise: Calculate the average power of x(t)=Acos(  t  12. Instantaneous Power 13. Average Power Note: For periodic signal, T M is generally taken as T o

TIME SIGNALS DESCRIPTION 14. Power Ratio In Electronic Engineering and Telecommunication power is usually resulted from applying voltage V to a resistive load R, as The unit is decibel (db) Alternative expression for power ratio (same resistive load):

TIME SIGNALS DESCRIPTION 15. Orthogonality Exercise: Prove that sin(  t  and cos(  t  are orthogonal for Two signals are orthogonal over the interval if

TIME SIGNALS DESCRIPTION 15. Orthogonality: Graphical illustration x1(t)x1(t) x2(t)x2(t) x 1 (t) and x 2 (t) are correlated. When one is large, so is the other and vice versa x1(t)x1(t) x2(t)x2(t) x 1 (t) and x 2 (t) are orthogonal. Their values are totally unrelated

TIME SIGNALS DESCRIPTION 16. Convolution between two signals Convolution is the resultant corresponding to the interaction between two signals.

1. Dirac delta function (Impulse or Unit Response)  (t) 0 t where Definition: A function that is zero in width and infinite in amplitude with an overall area of unity. SOME INTERESTING SIGNALS

2. Step function u(t) 0 t SOME INTERESTING SIGNALS 1 A more vigorous mathematical treatment on signals

Deterministic Signals A continuous time signal x(t) with finite energy Can be represented in the frequency domain Satisfied Parseval’s theorem

Deterministic Signals A discrete time signal x(n) with finite energy Can be represented in the frequency domain Satisfied Parseval’s theorem Note:is periodic with period =

Deterministic Signals Energy Density Spectrum (EDS) Equivalent expression for the (EDS) where * Denotes complex conjugate

Two Elementary Deterministic Signals Impulse function: zero width and infinite amplitude Discrete Impulse function Given x(t) and x(n), we have and

Two Elementary Deterministic Signals Step function: A step response Discrete Step function

Random Signals Infinite duration and infinite energy signals e.g. temperature variations in different places, each have its own waveforms. Ensemble of time functions (random process): The set of all possible waveforms Ensemble of all possible sample waveforms of a random process: X(t,S), or simply X(t). t denotes time index and S denotes the set of all possible sample functions A single waveform in the ensemble: x(t,s), or simply x(t).

Random Signals x(t,s0)x(t,s0) x(t,s1)x(t,s1) x(t,s2)x(t,s2)

Deterministic Signals Energy Density Spectrum (EDS) Equivalent expression for the (EDS) where * Denotes complex conjugate

Random Signals Each ensemble sample may be different from other. Not possible to describe properties (e.g. amplitude) at a given time instance. Only joint probability density function (pdf) can be defined. Given a sequence of time instants the samplesIs represented by: A random process is known as stationary in the strict sense if

Properties of Random Signals is a sample at t=t i The lth moment of X(t i ) is given by the expected value The lth moment is independent of time for a stationary process. Measures the statistical properties (e.g. mean) of a single sample. In signal processing, often need to measure relation between two or more samples.

Properties of Random Signals are samples at t=t 1 and t=t 2 The statistical correlation between the two samples are given by the joint moment This is known as autocorrelation function of the random process, usually denoted by the symbol For stationary process, the sampling instance t 1 does not affect the correlation, hence

Properties of Random Signals Average power of a random process Wide-sense stationary: mean value m(t 1 ) of the process is constant Autocovariance function: For a wide-sense stationary process, we have

Properties of Random Signals Variance of a random process Cross correlation between two random processes: When the processes are jointly and individually stationary,

Properties of Random Signals Cross covariance between two random processes: When the processes are jointly and individually stationary, Two processes are uncorrelated if

Properties of Random Signals Power Spectral Density: Wiener-Khinchin theorem An inverse relation is also available, Average power of a random process

Properties of Random Signals Cross Power Spectral Density: Average power of a random process For complex random process,

is a sample at instance n. The lth moment of X(n) is given by the expected value Properties of Discrete Random Signals Autocorrelation Autocovariance For stationary process, let

The variance of X(n) is given by Properties of Discrete Random Signals Power Density Spectrum of a discrete random process Inverse relation: Average power:

Mathematical description of signal Signal Modelling are the model parameters. Harmonic Process model Linear Random signal model

Rational or Pole-Zero model Signal Modelling Autoregressive (AR) model Moving Average (MA) model

SYSTEM DESCRIPTION 1. Linearity System y1(t)y1(t)x1(t)x1(t) y2(t)y2(t)x2(t)x2(t) IF System y 1 (t) + y 2 (t)x 2 (t) + x 2 (t) THEN

SYSTEM DESCRIPTION 2. Homogeneity System y1(t)y1(t)x1(t)x1(t) ay 1 (t)ax 1 (t) IF THEN Where a is a constant

SYSTEM DESCRIPTION 3. Time-invariance: System does not change with time System y1(t)y1(t)x1(t)x1(t) y 1 (t  x 1 (t  IF THEN t x1(t)x1(t) t y1(t)y1(t) t x 1 (t  t y 1 (t  

SYSTEM DESCRIPTION 3. Time-invariance: Discrete signals System y 1 [n]x1[n]x1[n] System y 1 [n - m  x 1 [n - m  IF THEN tt tt x1[n]x1[n] x 1 [n - m  y 1 [n] y 1 [n - m  m m

SYSTEM DESCRIPTION 4. Stability The output of a stable system settles back to the quiescent state (e.g., zero) when the input is removed The output of an unstable system continues, often with exponential growth, for an indefinite period when the input is removed 5. Causality Response (output) cannot occur before input is applied, ie., y(t) = 0 for t <0

THREE MAJOR PARTS Signal Representation and Analysis System Representation and Implementation Output Response

Signal Representation and Analysis An analogy: How to describe people? (A) Cell by cell description – Detail but not useful and impossible to make comparison (B) Identify common features of different people and compare them. For example shape and dimension of eyes, nose, ears, face, etc.. Signals can be described by similar concepts: “Decompose into common set of components”

Periodic Signal Representation – Fourier Series Ground Rule: All periodic signals are formed by sum of sinusoidal waveforms (1) (2) (3)

Fourier Series – Parseval’s Identity Energy is preserved after Fourier Transform (4)

Fourier Series – Parseval’s Identity

Periodic Signal Representation – Fourier Series t x(t)x(t) -t 1 T/4-T/4 tx(t)x(t) -T/2 to –T/4 -T/4 to +T/4+1 +T/4 to +T/2 -T/2T/2

Periodic Signal Representation – Fourier Series t x(t)x(t) -t 1 T/4-T/4 tx(t)x(t) -T/2 to –T/4 -T/4 to +T/4+1 +T/4 to +T/2

Periodic Signal Representation – Fourier Series t x(t)x(t) -t 1 T/4-T/4 tx(t)x(t) -T/2 to –T/4 -T/4 to +T/4+1 +T/4 to +T/2 zero for all n We have,

Periodic Signal Representation – Fourier Series t x(t)x(t) -t 1 T/4-T/4 tx(t)x(t) -T/2 to –T/4 -T/4 to +T/4+1 +T/4 to +T/2 It can be easily shown that b n = 0 for all values of n. Hence, Only odd harmonics are present and the DC value is zero The transformed space (domain) is discrete, i.e., frequency components are present only at regular spaced slots.

Periodic Signal Representation – Fourier Series t x(t)x(t) -t A tx(t)x(t) -  /2 to –  /2 A -T/2 to -  /2 0 +  /2 to +T/2 0 -  /2  /2 -T/2T/2

Periodic Signal Representation – Fourier Series t x(t)x(t) -t A tx(t)x(t) -  /2 to –  /2 A -T/2 to -  /2 0 +  /2 to +T/2 0 -  /2  /2 -T/2T/2 It can be easily shown that b n = 0 for all values of n. Hence, we have

Periodic Signal Representation – Fourier Series Note: Hence: