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**Communication Theory (EC 2252)**

Prof.J.B.Bhattacharjee K.Senthil Kumar ECE Department Rajalakshmi Engineering College

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**Review of Spectral characteristics**

Periodic and Non-periodic Signals: A signal is said to be periodic, if it exhibits periodicity. i.e., x(t +T)=x(t) , for all values of t. Periodic signal has the property that it is unchanged by a time shift of T. A signal that does not satisfy the above periodicity property is called a non-periodic signal. Periodic signals can be represented using the Fourier Series. Non-periodic signals can be represented using the Fourier Transform. Both Fourier series and Fourier Transform deal with the representation of the signals as a combination of sine and cosine waves.

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Fourier Series Fourier series: a complicated waveform analyzed into a number of harmonically related sine and cosine functions A continuous periodic signal x(t) with a period T may be represented by: x(t)=Σ∞k=1 (Ak cos kω t + Bk sin kω t)+ A0 Dirichlet conditions must be placed on x(t) for the series to be valid: the integral of the magnitude of x(t) over a complete period must be finite, and the signal can only have a finite number of discontinuities in any finite interval

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**Fourier Series Equations**

The Fourier series represents a periodic signal Tp in terms of frequency components: We get the Fourier series coefficients as follows: The complex exponential Fourier coefficients are a sequence of complex numbers representing the frequency component ω0k.

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**Periodic signals represented by Fourier Series have Discrete spectra.**

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The Fourier Transform Fourier transform is used for the non-periodic signals. A Fourier transform converts the signal from the time domain to the spectral domain. Continuous Fourier Transform:

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**Non-periodic signals represented by Fourier transform have Continuous spectra.**

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**Fourier Transform Pairs Note: Π stands for rectangular function**

Fourier Transform Pairs Note: Π stands for rectangular function. Λ stands for triangular function.

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**Introduction to Communication Systems**

Communication – Basic process of exchanging information from one location (source) to destination (receiving end). Refers – process of sending, receiving and processing of information/signal/input from one point to another point. Source Flow of information Destination Figure 1 : A simple communication system

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Electronic Communication System – defined as the whole mechanism of sending and receiving as well as processing of information electronically from source to destination. Example – Radiotelephony, broadcasting, point-to-point, mobile communications, computer communications, radar and satellite systems.

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Objectives Communication System – to produce an accurate replica of the transmitted information that is to transfer information between two or more points (destinations) through a communication channel, with minimum error.

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**NEED FOR COMMUNICATION**

Interaction purposes – enables people to interact in a timely fashion on a global level in social, political, economic and scientific areas, through telephones, electronic-mail and video conference. Transfer Information – Tx in the form of audio, video, texts, computer data and picture through facsimile, telegraph or telex and internet. Broadcasting – Broadcast information to masses, through radio, television or teletext.

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**Terms Related To Communications**

Message – physical manifestation produced by the information source and then converted to electrical signal before transmission by the transducer in the transmitter. Transducer – Device that converts one form of energy into another form. Input Transducer – placed at the transmitter which convert an input message into an electrical signal. Example – Microphone which converts sound energy to electrical energy. Input Transducer Message Electrical Signal

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Output Transducer – placed at the receiver which converts the electrical signal into the original message. Example – Loudspeaker which converts electrical energy into sound energy. Signal – electrical voltage or current which varies with time and is used to carry message or information from one point to another. Electrical Signal Output Transducer Message

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**Elements of a Communication System**

The basic elements are : Source, Transmitter, Channel, Receiver and Destination. Information Source Transmitter Channel Transmission Medium Receiver Destination Noise Figure : Basic Block Diagram of a Communication System

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**Function of each Element.**

Information Source – the communication system exists to send messages. Messages come from voice, data, video and other types of information. Transmitter – Transmit the input message into electrical signals such as voltage or current into electromagnetic waves such as radio waves, microwaves that is suitable for transmission and compatible with the channel. Besides, the transmitter also do the modulation and encoding (for digital signal).

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**Block Diagram of a Transmitter**

Transmitting Antenna 5 minutes exercise; Describe the sequence of events that happen at the radio waves station during news broadcast? Modulating Signal Audio Amplifier Modulator RF Amplifier Carrier Signal

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Channel/Medium – is the link or path over which information flows from the source to destination. Many links combined will establish a communication networks. There are 5 criteria of a transmission system; Capacity, Performance, Distance, Security and Cost which includes the installation, operation and maintenance. 2 main categories of channel that commonly used are; line (guided media) and free space (unguided media)

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Receiver – Receives the electrical signals or electromagnetic waves that are sent by the transmitter through the channel. It is also separate the information from the received signal and sent the information to the destination. Basically, a receiver consists of several stages of amplification, frequency conversion and filtering.

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**Block Diagram of a Receiver**

Receiving Antenna Destination – is where the user receives the information, such as loud speaker, visual display, computer monitor, plotter and printer. RF Amplifier Intermediate Frequency Amplifier Demodulator Destination Mixer Audio Amplifier Local Oscillator

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**Analog Modulation Baseband Transmission**

Baseband signal is the information either in a digital or analogue form. Transmission of original information whether analogue or digital, directly into transmission medium is called baseband transmission. Example: intercom (figure below) Voice Voice Microphone Audio Amplifier Speaker Audio Amplifier Wire

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**Baseband signal is not suitable for long distance communication….**

Hardware limitations Requires very long antenna Baseband signal is an audio signal of low frequency. For example voice, range of frequency is 0.3 kHz to 3.4 kHz. The length of the antenna required to transmit any signal at least 1/10 of its wavelength (λ). Therefore, L = 100km (impossible!) Interference with other waves Simultaneous transmission of audio signals will cause interference with each other. This is due to audio signals having the same frequency range and receiver stations cannot distinguish the signals.

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Modulation Modulation – defined as the process of modifying a carrier wave (radio wave) systematically by the modulating signal. This process makes the signal suitable for transmission and compatible with the channel. Resultant signal – modulated signal 2 types of modulation; Analog Modulation and Digital Modulation. Analogue Modulation – to transfer an analogue low pass signal over an analogue bandpass channel. Digital Modulation – to transfer a digital bit stream the carrier is a periodic train and one of the pulse parameter (amplitude, width or position) changes according to the audio signal.

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**Purpose of Modulation Process in Communication Systems**

To generate modulated signal that is suitable for transmission and compatible with the channel. To allow efficient transmission – increase transmission speed and distance, eg; By using high frequency carrier signal, the information (voice) can travel and propagate through the air at greater distances and shorter transmission time Also, high frequency signal is less prone to noise and interference. Certain types of modulation have the useful property of suppressing both noise and interference For example, FM use limiter to reduce noise and keep the signal’s amplitude constant. PCM systems use repeaters to generate the signal along the transmission path.

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**Amplitude Modulation (AM)**

Objectives:- Recognize AM signal in the time domain, frequency domain and trigonometric equation form Calculate the percentage of modulation index Calculate the upper sidebands, lower sidebands and bandwidth of an AM signal by given the carrier and modulating signal frequencies Calculate the power related in AM signal Define the terms of DSBSC, SSB and VSB Understand the modulator and demodulator operations

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**Introduction Modulation Schemes Modulation Carrier Signal**

The alteration of the amplitude, phase or frequency of an oscillator in accordance with another signal. Input signal is encoded in a format suitable for transmission A low frequency information signal is encoded over a higher frequency signal Carrier Signal Sinusoidal wave, Modulating Signal/Base band Information signal, Modulated Wave Higher frequency signal which is being modulated Modulation Schemes To counter the effects of multi path fading and time-delay spread

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**Modulation Schemes Carrier Signal, Modulating Signal, Vm**

Vc Modulating Signal, Vm Modulated Signal VAM VPM VFM FM means frequency modulation, or changing the frequency of a carrier signal based on the amplitude of an information (baseband) signal. Frequency modulation is a special case of angle modulation (phase modulation is another special case) in which the angle of the carrier is varied proportionately to the amplitude of the baseband signal. In angle modulation, the amplitude of the carrier signal is kept constant; hence, angle modulation methods are constant envelope transmissions. The diagram above illustrates amplitude, phase, and frequency modulation by a sine wave. The remainder of this module will focus on FM for mobile radio.

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Amplitude Modulation Time Domain Frequency Domain

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AM Modulator Modulator Information Signal Output Carrier Signal

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Amplitude Modulation Vc - Vc Vm - Vm Vam - Vam

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**Modulation Index Modulation Index, m**

Indicates the amount that the carrier signal is modulated. It is an expression of the amount of power in the sidebands. Modulation level ranges = 0-1 where 0 = no modulation 1 = full modulation >1 = distortion

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Modulation Index

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Modulation Index Vmax Vmin Vmax (p-p) Vmin (p-p)

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Modulation Index m = 0 m = 0.5 m = 1

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Bandwidth VC Bandwidth for AM signal, fc-fm fc fc+fm

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**Power Distributions Total transmitted power, PT If R= 1, fc-fm fc**

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**Double Side Band Suppressed Carrier (DSBSC)**

It is a technique where it is transmitting both the sidebands without the carrier (carrier is being suppressed/cut) Characteristics: Power content less Same bandwidth Disadvantages - receiver is complex and expensive.

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Single Side Band (SSB) Improved DSBSC and standard AM, which waste power and occupy large bandwidth SSB is a process of transmitting one of the sidebands of the standard AM by suppressing the carrier and one of the sidebands Advantages: Saving power Reduce BW by 50% Increase efficiency, increase SNR Disadvantages Complex circuits for frequency stability

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**Vestigial Side Band (VSB)**

VSB is mainly used in TV broadcasting for their video transmissions. TV signal consists of Audio signal – transmitted by FM Video signal – transmitted by VSB A video signal consists a range of frequency and fmax = 4.5 MHz. If it transmitted using conventional AM, the required BW is 9 MHz (BW=2fm). But according to the standard, TV signal is limited to 7 MHz only So, to reduce the BW, a part of the LSB of picture signal is not fully transmitted.

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**Vestigial Side Band (VSB)**

The frequency spectrum for the TV signal / VSB: Video Carrier Audio Carrier Total TV signal bandwidth = 7 MHz 4.5 MHz Lower Video Bands Upper Video Bands Lower Audio Bands Upper Audio Bands f (MHz) 1.25 5.75 6.25 6.75 7.0

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Modulator Circuits Modulating Signal Output Carrier A B C D E

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**Modulator Circuits A. Modulating Signal B. Carrier**

C. Sum of carrier and modulating signal D. Diode current E. AM output across tuned circuit

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Demodulator AM Signal A B C

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**Demodulator A. AM signal B. Current pulses through diode**

C. Demodulating signal D. Modulating signal

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**Frequency Modulation (FM)**

Objectives:- Recognize FM signal in the time domain, frequency domain and trigonometric equation form Calculate the percentage of modulation index Calculate the upper sidebands, lower sidebands and bandwidth of an FM signal by Carsons’s Rule and Bessel Function Table Calculate the power related in FM signal Understand the modulator and demodulator of FM

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Introduction FM is the process of varying the frequency of a carrier wave in proportion to a modulating signal. The amplitude of the carrier is kept constant while its frequency is varied by the amplitude of the modulating signal. In all types of modulation, the carrier wave is varied by the AMPLITUDE of the modulating signal. FM signal does not have an envelope, therefore the FM receiver does not have to respond to amplitude variations it can ignore noise to some extent.

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Frequency Modulation FM means frequency modulation, or changing the frequency of a carrier signal based on the amplitude of an information (baseband) signal. Frequency modulation is a special case of angle modulation (phase modulation is another special case) in which the angle of the carrier is varied proportionately to the amplitude of the baseband signal. In angle modulation, the amplitude of the carrier signal is kept constant; hence, angle modulation methods are constant envelope transmissions. The diagram above illustrates amplitude, phase, and frequency modulation by a sine wave. The remainder of this module will focus on FM for mobile radio.

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**Frequency Modulation The importance features about FM waveforms are:**

The frequency varies The rate of change of carrier frequency changes is the same as the frequency of the information signal The amount of carrier frequency changes is proportional to the amplitude of the information signal The amplitude is constant

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**Frequency Modulation Carrier Signal Modulating Signal/Base band**

Sinusoidal wave Modulating Signal/Base band Information signal Modulated Wave Higher frequency signal which is being modulated Where

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Frequency Modulation Time Domain Frequency Domain

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FM Modulator

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FM Modulator Modulator Information Signal Output Carrier Signal

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**Frequency Minimum Frequency Carrier Swing Carrier Frequency**

As in FM system, carrier frequency in FM systems must be higher than the information signal frequency. Maximum Frequency Minimum Frequency Carrier Swing

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**Modulation Index Modulation Index, m @ β**

Indicates the amount that the carrier signal is modulated. It is an expression of the amount of power in the sidebands. Modulation level ranges = 0 – Where Δf = fd = frequency deviation fm = modulating frequency Vm = amplitude of modulating signal

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Modulation Index β = 1 β = 5

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Modulation Index β = 25

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Modulation Index

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**Bandwidth Using Bessel Function, the bandwidth for FM signal,**

n = number of pairs of the significant sidebands fm = the frequency the modulating signal

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Bandwidth Using Carson’s Rule, to estimate the bandwidth for an FM signal transmission. Δf = peak frequency deviation fm(max) = highest modulating signal frequency

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Power Distributions FM transmitted power, PFM where

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**Narrowband FM and Wideband FM**

Narrowband FM has only a single pair of significant sidebands. The value of modulation index β <1. Wideband FM has a large number (theoretically infinite) number of sidebands. The value of modulation index β >=1.

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**Generation of Narrowband FM (NBFM)**

_ INTEGRATOR PRODUCT MODULATOR Σ NBFM WAVE + The modulator splits the carrier into two paths. One path is direct. The other path contains a -90 degree phase shift unit and a product modulator. The difference between the signals in the two paths produces the NBFM signal. -90 PHASE SHIFTER CARRIER WAVE MODULATING WAVE

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Frequency Modulators A frequency modulator is a circuit that varies carrier frequency in accordance with the modulating signal. There are two types of frequency modulator circuits. (1) Direct FM: Carrier frequency is directly varied by the message through voltage-controlled oscillator. Eg: Varactor diode modulator. (2) Indirect FM: Generate NBFM first, then NBFM is frequency multiplied for targeted Δf. Eg: Armstrong modulator

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FM Varactor Modulator

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**The Operation of the Varactor Modulator**

The info signal is applied to the base of the input transistor and appears amplified and inverted at the collector. This low freq signal passes through the RF choke (L1) and is applied across the varactor diode. Varactor diode behaves as voltage controlled capacitor. When low reverse biased voltage is applied, more capacitance is generated and thus decrease the frequency.

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When high reverse biased voltage is applied, less capacitance is generated and thus increase the frequency. The varactor diode changes its capacitance in sympathy with the info signal and therefore changes the total value of the capacitance in the tuned circuit. The changing value of capacitance causes the oscillator freq to increase and decrease under the control of the information signal. The output is therefore an FM signal.

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**Armstrong of indrect FM generation**

In this method the message signal is first subjected to NBFM modulator using a crystal-controlled oscillator for generating carrier. Crystal control provides frequency stability. The NBFM wave is next multiplied in frequency by using a frequency multiplier so as to produce the desired wideband FM.

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**Frequency Demodulator**

The FM demodulating circuits used to recover the original modulating signal. Any circuit that will convert a frequency variation in the carrier back into a proportional voltage variation can be used to demodulate or detect FM signals. A popular method used for FM demodulation is the Frequency discriminator.

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**Frequency discriminator**

Output of the Frequency discriminator

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**The Frequency discriminator circuit consists of the slope ciruit followed by the envelope detector.**

The slope circuit converts the instantaneous frequency variations of the FM input signal to instantaneous amplitude variations. These amplitude variations are rectified by the envelope detector to provide a DC output voltage which varies in amplitude and polarity with the input signal frequency.

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**FM vs AM: Advantages Disadvantages Better noise immunity**

Rejection of interfering signals because of capture effect Better transmitter efficiency Excessive use of spectrum More complex and costly circuits

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Review of Probability Sample Space：the space of all possible outcomes (δ) Event：a collection of outcomes：subset of δ Probability：a “measure” assigned to the events of a sample space with the following properties： for all event A in S If A and B are mutually exclusive, Theorem: The Conditional probability of an event A given the occurrence of event B is

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**Two events A and B are independent if**

Random Variables A rule which assigns a numerical value to each possible outcomes of a chance experiment. If the experiment is flipping a coin. Then a random variable X can be defined as : S1 H X(S1)=1 S2 T X(S2)=-1

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**Cumulative Distribution Function (CDF)**

≜ Properties of CDF： 1. 2. 3. Probability Density Function (PDF) Properties of PDF： , ,

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**The totality of all sample**

Random Processes: A random process is a mapping from the sample space to an ensemble of time functions. X1(t) X2(t) XN(t) Sample function t The totality of all sample functions is called an ensemble For a specific time X(tk) is a random variable

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Gaussian process A random process X(t) is a Gaussian process if for all n and for all (t1 t2 ... tn), the sequence of random variables { X(t1), X(t2)... X(tn) } has a jointly Gaussian density function. Central limit theorem The sum of a large number of independent and identically distributed(i.i.d) random variables getting closer to Gaussian distribution. Thermal noise can be closely modeled by Gaussian process.

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Property 1 For Gaussian process, knowledge of the mean(m) and covariance(C) provides a complete statistical description of process. Property 2 If a Gaussian process X(t) is passed through a LTI system, the output of the system is also a Gaussian process. The effect of the system on X(t) is simply reflected by the change in mean(m) and covariance(C) of X(t).

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Noise Theory Shot noise: It results from the shot effect in the amplifying devices and active device. It is caused by random variation in the arrival of electrons (or holes) at the output of the devices. For diode, the rms shot noise current is given by:

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Thermal noise is the electrical noise arising from the random motion of electrons in a conductor. The noise power generated by a resistor is given by:

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White noise: It is the idealized form of noise, whose spectrum is independent of the operating frequency. The power spectral density of white noise w(t) is Sw(f)=N0 /2. The autocorrelation Rw(t) of white noise is an impulse as shown below. Sw(f) Rw() f

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**w(t) n(t) Narrow band noise (Ideal case)**

filtered noise is narrow-band noise n(t) = nI(t)cos(2fCt) - nQ(t)sin(2fCt) where nI(t) is inphase, nQ(t) is quadrature component filtered signal x(t) x(t) = s(t) + n(t) - Average Noise Power = N0BT BPF

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Noise Figure Consider a signal source. The signal to noise ratio (SNR) available from the source is given by: Consider that the source is connected to an amplifier with gain G. Since all amplifiers contribute noise, the available output SNR will be less than the SNR of the source.

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**The noise power at the output of the amplifier will be**

The noise factor F is defined as : When noise factor is expressed in decibels, it is called noise figure. Noise figure = (F) dB = 10logF

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**The noise power expressed in terms of a temperature is callled Noise Temperature.**

If the amplifier noise is Pna , then the equivalent noise temperature Te of the amplifier is given by the equation

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**AM SUPERHETERODYNE RECEIVER**

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RF section: It generally consists of a pre-selector and an amplifier stage. The pre-selector is a broad tuned band-pass filter with adjustable center frequency that is tuned to the desired carrier frequency. The other functions of the RF section are detecting, band limiting and amplifying the received RF signals. Mixer/converter section: It is the stage of down-converts the received RF frequencies to intermediate frequencies (IF) which are simply frequencies that fall somewhere between the RF and information frequencies, hence the name intermediate. This section also includes a local oscillator (LO).

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IF Section: IF or intermediate frequency section is the stage where its primary functions are amplification and selectivity. AM detector Section: AM detector section is the stage that demodulates the AM wave and converts it to the original information signal. Audio section: Audio section is the stage that amplifies the recovered information.

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**Performance of CW Modulation Systems**

Introduction - Receiver Noise (Channel Noise) : additive, White, and Gaussian Receiver Model 1. RX Model Sw(f) Rw() f N0 = KTe where K = Boltzmann’s constant Te = equivalent noise Temp. Average noise power per unit bandwidth

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**SNR The signal x(t) available for demodulation is defined by**

The output signal-to-noise ratio (SNR)O is defined as the ratio of the average power of the demodulated message signal to the average power of the noise, both measured at the receiver output. The channel signal-to-noise ratio, (SNR)C is defined as the ratio of the average power of the modulated signal to the average power of the channel noise in the message bandwidth, both measure at the receiver input. For the purpose of comparing different CW modulation systems, we normalize the receiver performance by dividing (SNR)O by (SNR)C. This ratio is called figure of merit for the receiver and is defined as

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**Noise in DSB-SC Receivers**

Let’s consider the case of DSB-SC. The expression for the modulated signal is given as The carrier wave is statistically independent of the message signal. The average power of DSB-SC modulated component of s(t) is

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With a noise PSD of N0/2 the average noise power in the message bandwidth W equals WN0 (baseband scenario). Pm is the power of the message. Hence we have Finding an expression for (SNR)O, we have

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Output of the LPF is The power of the signal component at the receiver output is The average power of the filtered noise is 2WN0. The average noise power at the receiver output is Hence we have,

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**Noise in AM receiver using envelope detection**

The expression for AM signal is given as where it is assumed that The average power of the carrier in the AM signal s(t) is The average power of the information bearing component is Average power of the full AM signal s(t) is

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**Hence, the channel signal to noise ratio for AM is**

Finding an expression for (SNR)O, we have

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Threshold Effect When carrier-to-noise ratio is small as compared to unity the noise term dominates the performance of the envelope detector and is completely different. Representing the narrowband noise n(t) in terms of its envelope and phase, we have The phasor diagram for x(t) = s(t) + n(t) becomes

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**The noise envelope is used as a reference here due to its dominance**

The noise envelope is used as a reference here due to its dominance. Here it is assumed that Ac is small as compared to r(t). If we neglect the quadrature component of the signal with respect to the noise we have Hence, when carrier-to-noise ratio is small the detector has no component that is strictly proportional to the message signal m(t). Recalling that is uniformly distributed over radians. Hence, it follows that we have a complete loss of information at the detector output (as expected value will be zero). This loss of information m(t) at the output of the envelope detector is called the threshold effect.

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**Pre-emphasis and De-emphasis**

FM results is an unacceptably low SNR at the high frequency end of the message spectrum. To offset this undesirable occurrence, pre-emphasis and de-emphasis technique is used. Pre-emphasis consists in artificially boosting the spectral components in the higher part of the message spectrum. This is accomplished by passing message signal m(t) , through the pre-emphasis filter, denoted Hpe(f) . The pre-emphasized signal is used to frequency modulate the carrier at the transmitting end. In the receiver, the inverse operation, de-emphasis, is performed. This is accomplished by passing the discriminator output through a filter, called the de-emphasis filter, denoted Hde(f ) .

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**Pre-emphasis and de-emphasis in FM**

P.S.D. of noise at FM Rx output P.S.D. of typical message signal Commercial FM radio에서 사용

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**Information theory What is information theory ?**

Information theory is needed to enable the communication system to carry information (signals) from sender to receiver over a communication channel it deals with mathematical modelling and analysis of a communication system its major task is to answer to the questions of signal compression and data transfer rate. Those answers can be found and solved by entropy and channel capacity

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**Information is a measure of uncertainty**

Information is a measure of uncertainty. The less is the probability of occurrence of a certain message, the higher is the information. Since the information is closely associated with the uncertainty of the occurrence of a particular symbol, When the symbol occurs the information associated with its occurrence is defined as:

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Entropy Entropy is defined in terms of probabilistic behaviour of a source of information In information theory the source output are discrete random variables that have a certain fixed finite alphabet with certain probabilities Entropy is an average information content for the given source symbol. (bits/message)

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**‘H’ is the average number of bits of information per message. Hence**

Rate of information: If a source generates at a rate of ‘r’ messages per second, the rate of information ‘R’ is defined as the average number of bits of information per second. ‘H’ is the average number of bits of information per message. Hence R = rH bits/sec

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Source Coding Source coding (a.k.a lossless data compression) means that we will remove redundant information from the signal prior the transmission. Basically this is achieved by assigning short descriptions to the most frequent outcomes of the source output and vice versa. The common source-coding schemes are prefix coding, huffman coding, lempel-ziv coding.

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Source Coding Theorem Source coding theorem states that the output of any information source having entropy H units per symbol can be encoded into an alphabet having N symbols in such a way that the source symbols are represented by code words having a weighted average length not less than H/logN. Hence source coding theorem says that encoding of messages from a source with entropy H can be done, bounded by the fundamental information theoretic limitation that the Minimum average number of symbols/message is H/logN.

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Source coding example Prefix coding has an important feature that it is always uniquely decodable and it also satisfies Kraft-McMillan (see formula p. 624) inequality term Prefix codes can also be referred to as instantaneous codes, meaning that the decoding process is achieved immediately

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Shannon-Fano Coding: In Shannon–Fano coding, the symbols are arranged in order from most probable to least probable, and then divided into two sets whose total probabilities are as close as possible to being equal. All symbols then have the first digits of their codes assigned; symbols in the first set receive "0" and symbols in the second set receive "1". As long as any sets with more than one member remain, the same process is repeated on those sets, to determine successive digits of their codes. When a set has been reduced to one symbol, of course, this means the symbol's code is complete and will not form the prefix of any other symbol's code.

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Huffman Coding: Create a list for the symbols, in decreasing order of probability. The symbols with the lowest probability are assigned a ‘0’ and a ‘1’. These two symbols are combined into a new symbol with the probability equal to the sum of their individual probabilities. The new symbol is placed in the list as per its probability value. The procedure is repeated until we are left with 2 symbols only for which 0 and 1 are assigned. Huffman code is the bit sequence obtained by working backwards and tracking sequence of 0’s and 1’s assigned to that symbol and its successors.

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Lempel-Ziv Coding: A drawback of Huffman code is that knowledge of probability model of source is needed. Lempel-Ziv coding is used to overcome this drawback. while Huffman’s algorithm encodes blocks of fixed size into binary sequences of variable length, Lempel-Ziv encodes blocks of varying length into blocks of fixed size. Lempel-Ziv coding is performed by parsing the source data into segments that are the shortest subsequences not encountered before.

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Mutual Information Source X Channel Receiver Y Consider a communication system with a source of entropy H(X). The entropy on the receiver side be H(Y). H(X|Y) and H(Y|X) are the conditional entropies, and H(X,Y) is the joint entropy of X and Y. Then the Mutual information between the source X and the receiver Y can be expressed as: I(X,Y) = H(X) - H(X|Y) H(X) is the uncertainty of source X and H(X/Y) is the uncertainty of X given Y. Hence the quantity H(X) - H(X|Y) represents the reduction in uncertainty of X given the knowledge of Y. Hence I(X,Y) is termed mutual information.

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Channel Capacity Capacity in the channel is defined as a intrinsic ability of a channel to convey information. Using mutual information the channel capacity of a discrete memoryless channel is the maximum average mutual information in any single use of channel over all possible probability distributions. Thus Channel capacity C=max( I(X,Y) ).

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**Shannon’s Channel Coding theorem**

The Shannon theorem states that given a noisy channel with channel capacity C and information transmitted at a rate R, then if R < C there exist codes that allow the probability of error at the receiver to be made arbitrarily small. This means that theoretically, it is possible to transmit information nearly without error at any rate below a limiting rate, C. The converse is also important. If R > C, an arbitrarily small probability of error is not achievable. All codes will have a probability of error greater than a certain positive minimal level, and this level increases as the rate increases. So, information cannot be guaranteed to be transmitted reliably across a channel at rates beyond the channel capacity.

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**Shannon-Hartley theorem or Information Capacity Theorem**

An application of the channel capacity concept to an additive white Gaussian noise channel with B Hz bandwidth and signal-to-noise ratio S/N is the Information Capacity Theorem. It states that for a band-limited Gaussian channel operating in the presence of additive Gaussian noise, the channel capacity is given by C = B log2(1 + S/N) where C is the capacity in bits per second, B is the bandwidth of the channel in Hertz, and S/N is the signal-to-noise ratio.

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**Band width and SNR tradeoff**

As the bandwidth of the channel increases, it is possible to make faster changes in the information signal, thereby increasing the information rate. However, as B , the channel capacity does not become infinite since, with an increase in bandwidth, the noise power also increases. As S/N increases, one can increase the information rate while still preventing errors due to noise. For no noise, S/N and an infinite information rate is possible irrespective of bandwidth.

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**Implications of the Information Capacity Theorem**

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**Rate distortion theory**

Rate distortion theory is the branch of information theory addressing the problem of determining the minimal amount of entropy or information that should be communicated over a channel such that the source can be reconstructed at the receiver with a given distortion. Rate distortion theory can be used for the given below situations: 1. Source coding in which the coding alphabet cannot exactly represent the source information. 2. when the information is to be transmitted at a rate greater than channel capacity.

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**Lower the bit rate R by allowing some acceptable distortion D of the signal**

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**Rate Distortion Function:**

The functions that relate the rate and distortion are found as the solution of the following minimization problem. In the above equation, I(X,Y) is the Mutual information.

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**Rate distortion function for Gaussian memory-less source**

If Px(X) is Gaussian, variance is 2 and if we assume that successive samples of the signal x are stochastically independent, we find the following analytical expression for the rate distortion function.

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**A Plot of the Rate distortion function for Gaussian source**

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Lossy Source Coding Lossy source coding is the representation of the source in digital form with as few bits as possible while maintaining an acceptable loss of information. In lossy source coding, the source output is encoded at a rate less than the source entropy. Hence there is reduction in the information content of the source. Eg: It is not possible to digitally encode an analog signal with a finite number of bits without producing some distortion.

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Frequency modulation and circuits

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