1. Principle of Digital Communication (PDC) – EC 2004
Anupam Samui
KIIT University

2. Lesson Plan

4. Brief Overview of Communication System
What is communication: -
Communication is everywhere.
Alarm bell wakes us up in the morning informing and communicating its time for days work to start.
Throughout the day, with all our activities we try to show how hard we are working….!!
So it’s a method of conveying some information to a different entity by one entity.

5. Requirement:
Entity which is sending information is – sender, source or transmitter.
Entity receiving the information is – receiver, destination.
In between the source and receiver there is channel.
So the basic structure of a communication system is source, receiver and channel.

6. Basic Structure:-
Source
Receiver
channel

7. Signal
When a source transmits the information through the channel to the receiver - it will exploit some properties of the channel for conveying this information.
This gives the concept of signal.
It could be defined as a physical quantity that varies with time or any other independent variable and contains some information.
It can also be defined as a wave (electro-magnetic) used to convey information (data) from a transmitter to a receiver.

8. Signal
How it can be represented..!!

9. Signals can be classified as Analog & Digital.
Analog signals are continuous and can have infinite no of values in a given range (i.e., the signal is smooth).
Digital signals are discrete & can have only a limited number of values.

10. Analog & Digital Signal:

11. Signal
Many types of signals are available and they could be classified in different ways. One of them is according to periodicity –
Periodic
Non-periodic / A-periodic
Periodic Wave has some definite time period (and thus some definite frequency), To after which it repeats itself.
Non Periodic / A-periodic wave don’t have any definite time period to produce its exact replica in future (or its time period is infinite).

12. Signal
Periodic Signal:
A-periodic Signal:

13. Composite Signal
Periodic analog signals can be classified as simple or composite.
A simple periodic analog signal, a sine wave, cannot be decomposed into simpler signals.
A composite periodic analog signal is decomposed into many simple sine waves of discrete frequencies.
If the composite signal is non-periodic, the decomposition gives a combination of sine waves with continuous frequencies.
If a signal does not change at all its frequency is zero. But if it changes continuously its frequency is infinite.

14. Analog to digital conversion
Composite periodic Signal:

15. Some Signals Unit Step Function:- u(t) u(t) = 1 for t ≥ 0 1
= 0 for t < t
u(t-a) = 1 for t > a
= 0 for t < a a t

16. Unit ramp : - r(t) r (t) = t for t ≥ 0 = 0 for t < 0 t
Or r(t) = t.u(t)

17. Impulse function:-
and for
x(t) t

18. Some properties of unit impulse function:1. 2.

19. sinc function:- sinc (t) = sin (t)/t
It oscillates with period 2π and decays with increasing t and value is 0 at nπ, n – integers.

20. Real exponential signal:-
A A A
t t t

21. Complex exponential signal:-

22. Signal Presentation Signals could be presented in two domains:
Time
Frequency
A complete sine wave in time domain can be represented by a single spike in the frequency domain.
The frequency domain is more compact and useful when we are dealing with more than one sine wave.

23. Time domain & frequency domain plot of a sine wave:

24. Time & frequency domain presentation of three sine waves:

25. Signals in communication:
A single-frequency sine wave is not useful in data communications.
We need to send a composite signal, a signal made of many simple sine waves.
According to Fourier analysis, any composite signal is a combination of simple sine waves with different frequencies, amplitudes, and phases.

26. Decomposition of a periodic composite signal in time & Frequency domain:

27. Decomposition of a non periodic signal in time & frequency domain:

28. Bandwidth:
The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal.

29. Bandwidth of periodic & Non periodic composite signals:

30. Fourier Analysis
It’s a special method by which we convert time domain signal to a frequency domain signal & vice versa.
Every composite periodic signal can be represented with a series of sine and cosine functions.
The functions are integral harmonics of the fundamental frequency “f” of the composite signal.
Using the series we can decompose any periodic signal into its harmonics.

31. Fourier Analysis Any signal x(t) be one of the following;
Energy Signal
Power Signal
Neither if P = ∞

32. Fourier Analysis Example:
So this is neither an energy nor power signal.
So this is an energy signal.
This is a power signal.

33. Fourier Analysis
A periodic waveform x(t) (infinite energy/finite power) has a Fourier Series representation – power carried at discrete frequencies.
A non-periodic waveform x(t) (finite energy/zero power) has a Fourier Transform representation – energy carried at all (a continuum of) frequencies.
A ‘random’ waveform x(t), or sample sequence from a random process (infinite energy/finite power), has a Power Spectral Density representation, Sx(f) – power carried at all (a continuum of) frequencies.

34. Condition for Fourier Series
It may not be possible to represent a periodic signal as a Fourier series, if:
The signal is not integrable over any period
Over a finite interval of time, the signal has infinite number of variations
Over a finite interval of time, the signal has infinite number of discontinuities.
- these conditions are called Dirichlet’s Condition.

35. Dirichlet Condition:

36. Fourier Series
Three forms of Fourier series exists:
Where wo = 2πfo ; fo being the fundamental frequency and also wot could be represented with x.

37. Type I with Period: 2π

38. Type I with Period: 2L

39. Fourier Series If we re-arrange the series, we will have -
+ (a1 cos t + b1 sin t) + (a2 cos 2t + b2 sin 2t) + (a3 cos 3t + b3 sin 3t) + ...
Where,
The term (a1 cos t + b1 sin t) is known as the fundamental.
The term (a2 cos 2t + b2 sin 2t) is called the second harmonic.
The term (a3 cos 3t + b3 sin 3t) is called the third harmonic, etc.

40. Product of even and odd is odd
Fourier Series
Even function:
Odd function:
Key facts
Even
Odd
Product of even and odd is odd

41. Fourier Series Fourier cosine series f(x) is an even function T o=2L
Fourier sine series
f(x) is a odd function
T o=2L

42. Other forms: Type II: Cn = √((An * An )+(Bn * Bn)) tan Φn = (Bn / An )
Co = Ao
Cn = √((An * An )+(Bn * Bn))
tan Φn = (Bn / An )
Type III:
αn =

43. Fourier Half Range Expansion:-
Sometimes if one only needs Fourier series of a function to be defined in the range of (0, L) , it may be preferable to use a sine or cosine series instead of a regular Fourier series.
This can be accomplished by extending the definition of the given function to the following intervals
For Even (to have a Cosine Series):
(-L, 0) ,
(O,L) and
(L,2L)
For Odd (to have a Sine Series):
(0,L)
Such Fourier series are called Half Range expansion.

44. Properties of Fourier Series:
Given two periodic signals with same period T and fundamental frequency 0=2/T:
Linearity:
Time-Shifting:
Time-Reversal (Flip):

45. Linearity:

46. Time Shifting:

47. Time Reversal:

48. Fourier Transform
A transform takes one function (or signal) and turns it into another function (or signal).
Continuous Fourier Transform
Discrete Fourier Transform

49. Summery of Fourier Transform Properties

50. Summery of Fourier Transform Properties

51. Fourier transform for a periodic pulse train of duration T(-T/2 to T/2) and period –T0/2 to T0/2:

52. A single rectangular pulse of amplitude A and duration –T/2 to T/2.
A rectangular pulse is defined as-

53. If v(t)=cos(w0t) , find V(f).
As v(t) is periodic, therefore its having both Fourier series and Fourier transform.
Fourier Series representation:

54. Fourier Transform
Fourier transform of a periodic signal consists of impulses located at each harmonic frequency of the signal, i.e., at fn=n/T0.

55. A signal m(t) is multiplied by sinusoidal waveform of frequency fc
A signal m(t) is multiplied by sinusoidal waveform of frequency fc. The product signal is v(t)=m(t)cos(2πfct). If the Fourier transform of m(t) is M(f), i.e., Find V(f).
Solution:

56. Convolution
It’s a mathematical way of combining two signals to form a third signal.
It states that under suitable conditions the Fourier transform of a convolution is the pointwise product of Fourier transforms.
convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g. frequency domain).
Let ,
f and g be two functions with convolution f*g.
F denotes the fourier transform, so F(f) & F(g) denotes the fourier transform of f & g.
Then, denotes point wise multiplication.
Convolution for these two functions are defined as

57. Proof of Convolution Theorem…

58. Example of Convolution
If we apply a signal to a network having frequency response then, to find the output of the network we apply convolution principle.

59. Spectral Density
It describes how the energy or the power of a signal is distributed with frequency.
Its classified in two types
Energy Spectral Density
Power Spectral Density

60. Energy Spectral Density
It describes how the energy of a signal is distributed with frequency. If f(t) is a finite-energy signal, the spectral density Φ(ω) of the signal is the square of the magnitude of the continuous Fourier transform of the signal (here energy is taken as the integral of the square of a signal, which is the same as physical energy of the signal).
- where ω is the angular frequency (2π times the cycle frequency) and F(ω) is the continuous Fourier transform of f(t), and F * (ω) is its complex conjugate.

61. Energy Spectral Density
If the signal is discrete with values fn, over an infinite number of elements, we still have an energy spectral density:
- where F(ω) is the discrete-time Fourier transform of fn.

62. Power Spectral Density
It describes how the power of a signal is distributed with frequency. Here power can be the actual physical power can be defined as the squared value of the signal, that is, as the actual power if the signal was a voltage applied to a 1-ohm load.
The power of the signal in a given frequency band can be calculated by integrating over positive and negative frequencies,
- where, S(f) is the sum of normalized power contributed by each power spectral line up to frequency f.

63. Power Spectral Density
Application:
Sales estimation – which period of time is having better selling.
In Spectrum Analyzer - which frequency components are having required amplitude etc.

64. Parseval’s Theorem
States that power computed in either domain equals the power in the other. And its given by -