1. 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)?

2. 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

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

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

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

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

7. TIME SIGNALS DESCRIPTIONTM
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

8. 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)

9. 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):

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

11. 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

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

13. 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.

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

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

16. 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

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

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

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

20. 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).

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

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

23. 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

24. 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.

25. 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

26. 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

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

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

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

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

31. 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

32. 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:

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

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

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

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

37. 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

38. 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

39. 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

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

41. 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”

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

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

44. Fourier Series – Parseval’s Identity

45. 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

46. 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

47. 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,

48. 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.

49. 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

50. 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

51. Periodic Signal Representation – Fourier Series
Note:
Hence: