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Principle of Digital Communication (PDC ) – EC 2004. Anupam Samui KIIT University. Lesson Plan. Brief Overview of Communication System. What is communication: - Communication is everywhere.
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Principle of Digital Communication (PDC) – EC 2004 Anupam Samui KIIT University
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.
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.
Basic Structure:- Source Receiver channel
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.
Signal How it can be represented..!!
Signal • Signals can be classified as Analog & Digital. • Analog signals are continuousand 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.
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).
Signal • Periodic Signal: • A-periodic Signal:
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.
Analog to digital conversion • Composite periodic Signal:
Some Signals • Unit Step Function:- u(t) • u(t) = 1 for t ≥ 0 1 = 0 for t < 0. t u(t-a) = 1 for t > a 1 = 0 for t < a a t
Unit ramp : - r(t) • r (t) = t for t ≥ 0 = 0 for t < 0 t Or r(t) = t.u(t)
Impulse function:- and for x(t) t
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.
Real exponential signal:- A A A 0 t 0 t 0 t
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.
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.
Decomposition of a periodic composite signal in time & Frequency domain:
Decomposition of a non periodic signal in time & frequency domain:
Bandwidth: The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal.
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.
Fourier Analysis • Any signal x(t) be one of the following; • Energy Signal • Power Signal • Neither if P = ∞
Fourier Analysis Example: So this is neither an energy nor power signal. So this is an energy signal. This is a power signal.
Fourier Analysis • A periodic waveformx(t) (infinite energy/finite power) has a Fourier Series representation – power carried at discrete frequencies. • A non-periodic waveformx(t) (finite energy/zero power) has a Fourier Transform representation – energy carried at all (a continuum of) frequencies. • A ‘random’ waveformx(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.
Condition for Fourier Series • It may not be possible to represent a periodic signal as a Fourier series, if: • The signal is not integrableover 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.
Fourier Series • Three forms of Fourier series exists: • Where wo = 2πfo; fo being the fundamental frequency and also wotcould be represented with x.
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.
Fourier Series • Even function: • Odd function: Key facts Even Odd Product of even and odd is odd
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
Other forms: • Type II: • Co = Ao • Cn = √((An * An )+(Bn * Bn)) • tan Φn = (Bn / An ) • Type III: • αn =
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.
Properties of Fourier Series: • Given two periodic signals with same period T and fundamental frequency 0=2/T: • Linearity: • Time-Shifting: • Time-Reversal (Flip):
Fourier Transform • A transform takes one function (or signal) and turns it into another function (or signal). • Continuous Fourier Transform • Discrete Fourier Transform
