SIGNALS PROCESSING AND ANALYSIS Processing: Methods and system that modify signals System x(t) y(t) Input/Stimulus Output/Response 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)?

SIGNALS DESCRIPTION To analyze signals, we must know how to describe or represent them in the first place. t x(t) 1 5 2 8 3 10 4 Detail but not informative

TIME SIGNALS DESCRIPTION 1. Mathematical expression: x(t)=Asin(wt+f) 2. Continuous (Analogue) x[n] n 3. Discrete (Digital)

TIME SIGNALS DESCRIPTION 4. Periodic x(t)= x(t+To) Period = To To 5. Aperiodic

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

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

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

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

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

TIME SIGNALS DESCRIPTION 15. Orthogonality Two signals are orthogonal over the interval if Exercise: Prove that sin(wt) and cos(wt) are orthogonal for

TIME SIGNALS DESCRIPTION 15. Orthogonality: Graphical illustration x2(t) x2(t) x1(t) x1(t) x1(t) and x2(t) are correlated. When one is large, so is the other and vice versa x1(t) and x2(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.

SOME INTERESTING SIGNALS 1. Dirac delta function (Impulse or Unit Response) d(t) 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) 1 t 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 Note: is periodic with period = Satisfied Parseval’s theorem

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,s1) 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 samples Is represented by: A random process is known as stationary in the strict sense if

Properties of Random Signals is a sample at t=ti The lth moment of X(ti) 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=t1 and t=t2 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 t1 does not affect the correlation, hence

Properties of Random Signals Average power of a random process Wide-sense stationary: mean value m(t1) 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 Average power of a random process For complex random process, Cross Power Spectral Density: For complex random process,

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

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

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

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

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

SYSTEM DESCRIPTION 2. Homogeneity System IF x1(t) y1(t) System ax1(t) ay1(t) THEN Where a is a constant

SYSTEM DESCRIPTION 3. Time-invariance: System does not change with time System IF x1(t) y1(t) System x1(t-t) y1(t-t) THEN x1(t) y1(t) t t x1(t-t) y1(t-t) t t t t

SYSTEM DESCRIPTION 3. Time-invariance: Discrete signals System IF x1[n] y1 [n] System x1[n - m] y1[n - m] THEN x1[n] y1 [n] t t y1[n - m] x1[n - m] t t 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

Signal Representation and Analysis 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 x(t) -T/2 T/2 1 t x(t) -T/2 to –T/4 -1 -T/4 to +T/4 +1 +T/4 to +T/2 -t t -1 -T/4 T/4

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

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

Periodic Signal Representation – Fourier Series x(t) 1 t x(t) -T/2 to –T/4 -1 -T/4 to +T/4 +1 +T/4 to +T/2 -t t -1 -T/4 T/4 It can be easily shown that bn = 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 x(t) -T/2 T/2 A t x(t) -t/2 to –t/2 A -T/2 to - t /2 + t /2 to +T/2 -t t -t/2 t/2

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

Periodic Signal Representation – Fourier Series Note: Hence: