MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI

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1 MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI QUESTION BANK DEPARTMENT: ECE SEMESTER: V SUBJECT CODE / Name: EC2301 DIGITAL COMMUNICATION UNIT-I DIGITAL COMMUNICATION SYSTEMS PART -A (2 Marks) 1. Draw the typical digital communication system[auc NOV/DEC2011] [AUC NOV/DEC2012] 2. How can BER of an system be improved [AUC NOV/DEC2012] Increasing the transmitted signal power Employing modulation and demodulation technique Employing suitable coding and decoding methods Reducing noise interference with help of improved filtering 3. Define half power bandwidth [AUC NOV/DEC2011] half power bandwidth is the bandwidth whre PSD of the signal drops to half (3dB) of its maximum value.it is called 3dB bandwidth EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 1

2 4. Give an example for time limited and time unlimited signals [AUC APR/MAY 2011] time limited- rectangular pulse, triangular pulse time unlimited signals - sinusoidal signal,exponential signal and step signal 5. Give the advantages and disadvantages of digital communication Advantage Speech, video and other data can be transmitted simultaneously Wide dynamic range is possible since data is digital DisAdvantage digital communication required synchronization data rate are very high [AUC APR/MAY 2011] 6. Which parameter is called figure of merit of a digital communication system and why? The ration E b /No or bit energy to noise power spectral density is called figure of merit of a digital communication system [AUC NOV/DEC 2010] 6. What is meant by distortion less transmission? [AUC NOV/DEC 2010] For distortion less transmission, the transfer function of the system if given as, H(w)=Ke -jwto K- Constant magnitude response The transfer function impose two requirements on the system 1. The system response must have constant magnitude response 2. The system phase shift response must be linear with frequency 7. Define BER BER is defines as the number of bits that are wrongly transmitted.it is normally given as the probability of bit error. 8. What are the advantages of PAM? PAM can easily generated and detected PAM forms the basis for many other pulse modulation techniques such as PCM,DM,ADM 9. What is meant by basis set? The set of signal which are orthogonal to each other is called basis set 10. What is the condition for orthogonal? 11. Define noise equivalent bandwidth The response between ideal and practical filter.the area inside the filter is called noise equivalent bandwidth EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 2

3 12. State Dimensionality theorem The Dimensionality theorem states that a real waveform can be completely specified by N independent piece of information where N is given by N=2BT o where N dimension of the waveform B= Bandwidth of the signal T o =Time 13. What is GSOP? Gram Schmidt ortogonalization procedure is the tool to obtain the orthonormal basis fuction Φ i (t) 14. Write the expression for Linear filter channel. r(t)=x(t)*h(t)+n(t) PART-B (16 Marks) 1. Explain in detail about the GRAM Schmidt orthogonalisation procedure (16) [AUC NOV/DEC 2011, AUC NOV/DEC 2012] 2. Explain any three communication channel models(12) [AUC NOV/DEC2012] 3. State the advantage and disadvantage of digital communication system(4) [AUC NOV/DEC2012] 4. Discuss in detail about mathematical mode of communication channel (16) [AUC NOV/DEC 2011] 5. Explain how PWM and PPM signals are generated.(16 ) [AUC APR/MAY 2011] 6. Classify channels. Explain the mathematical model of any two communication channels (16) [AUC APR/MAY 2011] 7. Draw a neat block diagram of a typical digital communication system and explain the function of the key signal processing blocks.( 16) [AUC NOV/DEC 2010] 8. Distinguish between base band and band pass signalling. (6) [AUC NOV/DEC 2010] 9. Explain Binary symmetric channel and Gaussian channel with their mathematical models. (10) [AUC NOV/DEC 2010] 10. Derive Geometrical representation of signal.(8) 11. Explain the procedure for obtaining from the basis set.(8) 12. Explain the mathematical models of communication channel 13. Explain the concept of PWM and PAM 14. Obtain the orthonormal basis function for the set of waveforms using GSOP EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 3

4 QUESTIONS WITH ANSWERS 1. Explain in detail about the GRAM Schmidt orthogonalisation procedure (16) [AUC NOV/DEC 2011, AUC NOV/DEC 2012] Obtain the orthonormal basis function for the set of waveforms using GSOP Gram-Schmidt Orthogonalization The principle of Gram-Schmidt Orthogonalization (GSO) states that, any set of M energy signals, {si(t)}, 1 i M can be expressed as linear combinations of N orthonormal basis functions, where N M. If s1(t), s2(t),.., sm(t) are real valued energy signals, each of duration T sec, The ϕj(t)-s are the basis functions and sij -s are scalar coefficients. We will consider real-valued basis functions ϕj (t) - s which are orthonormal to each other, i.e., Note that each basis function has unit energy over the symbol duration T. Now, if the basis functions are known and the scalars are given, we can generate the energy signals G-S-O procedure Part I: We show that any given set of energy signals, {si (t)}, 1 i M over 0 t < T, can be completely described by a subset of energy signals whose elements are linearly independent. To start with, let us assume that all si(t) -s are not linearly independent. Then, there must exist a set of coefficients {ai}, 1 < i M, not all of which are zero, such that, a1s1 (t) + a2s2 (t) + + am sm (t) = 0, 0 t < T Verify that even if two coefficients are not zero, e.g. a1 0 and a3 0, then s1(t) and s3(t) are dependent signals. Let us arbitrarily set, am 0. Then, EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 4

5 Consider a reduced set with (M-1) signals {si(t)}, i = 1,2,.., (M 1). This set may be either linearly independent or not. If not, there exists a set of {bi}, i = 1,2, (M 1), not all equal to zero such that, Arbitrarily assuming that bm-1 0, we may express sm-1(t) as Now, following the above procedure for testing linear independence of the remaining signals, eventually we will end up with a subset of linearly independent signals. Let {si(t)}, i = 1, 2,., N M denote this subset. Part II : We now show that it is possible to construct a set of N orthonormal basis functions ϕ1(t), ϕ2(t),.., ϕn(t) from {si(t)}, i = 1, 2,.., N. Let us choose the first basis function as, So, we verified that the function g2(t) is orthogonal to the first basis function. This gives us a clue to determine the second basis function. Now, energy of g2(t) EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 5

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7 Gram-Schmidt Orthogonalization procedure: If the signal set {sj(t)} is known for j = 1, 2,.., M, 0 t <T, Derive a subset of linearly independent energy signals, {si(t)},i = 1, 2,.., N M. Find the energy of s1(t) as this energy helps in determining the first basis function ϕ1(t), which is a normalized form of the first signal. Note that the choice of this first signal is arbitrary. Find the scalar s21, energy of the second signal (E 2), a special function g2(t) which is orthogonal to the first basis function and then finally the second orthonormal basis function ϕ2(t) Follow the same procedure as that of finding the second basis function to obtain the other basis functions. 2. Discuss in detail about mathematical mode of communication channel (16) Classify channels. Explain the mathematical model of any two communication channels (16) Noise Cha [AUC NOV/DEC 2011]e In the design of communication systems for transmitting information through physical channels, we find it convenient to construct mathematical models that reflect the most important characteristics of the transmission medium. Then, the mathematical model for the channel is used in the design of the channel encoder and modulator at the transmitter and the demodulator and channel decoder at the receiver. The Additive Noise Channel. The simplest mathematical model for a communication channel is the additive noise channel, In this model the transmitted signal s(t) is corrupted by an additive random noise process The additive noise channel s(t). Physically, the additive noise process may arise from electronic components and amplifiers at the receiver of the communication system, or from interference encountered in transmission, as in the case of radio signal transmission. If the noise is introduced primarily by electronic components and amplifiers at the receiver, it may be characterized as thermal noise. This type of noise is characterized statistically as a Gaussian noise process. Hence, the resulting mathematical model for the channel is usually called the additive Gaussian noise channel. Because this channel model applies to a broad class of physical communication channels and because of its mathematical tractability, this is the predominant channel model used in our communication system analysis and design. Channel EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 7

8 attenuation is easily incorporated into the model. When the signal undergoes attenuation in transmission through the channel, the received signal is where a represents the attenuation factor. Linear filter channel In some physical channels such as wire line telephone channels, filters are used to ensure that the transmitted signals do not exceed specified bandwidth limitations and, thus, do not interfere with one another. Such channels are generally characterized mathematically as linear filter channels with additive noise, as illustrated in Figure 1.9. Hence, if the channel input is the signal s(t), the channel output is the signal where h(t) is the impulse response of the linear filter and denotes convolution.inear Time-t Linear time invariant filter Channel Physical channels such as underwater acoustic channels and ionospheric radio channels which result in time-variant multipath propagation of the transmitted signal may be characterized Mathematically as time-variant linear filters. Such linear filters are characterized by time-variant channel impulse response h(t; t) where h(t; t) is the response of the channel at time t, due to an impulse applied at time t t. Thus, t represents the "age" (elapsed time) variable. The linear time-variant filter channel with additive noise is illustrated Figure For an input signal s(t), the channel output signal is A good model for multipath signal propagation through physical channels, such as the ionosphere (at frequencies below 30 MHz) and mobile cellular radio channels, is a special case of Equation EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 8

9 where the {ak(t)} represent the possibly time-variant attenuation factors for the L multipath propagation paths Hence, the received signal consists of L multipath components, where each component is attenuated by {ak} and delayed by {tk}. The three mathematical models described above adequately characterize a large majority of physical channels encountered in practice. These three channel models are used in this text for the analysis and design of communication systems. 3. Explain how PWM and PPM signals are generated. Explain the concept of PWM and PAM Pulse Width Modulation & Pulse Position Modulation Pulse Time Modulation (PTM) is a class of signaling technique that encodes the sample values of an analog signal onto the time axis of a digital signal. The two main types of pulse time modulation are: 1. Pulse Width Modulation (PWM) 2. Pulse Position Modulation (PPM) In PWM the sample values of the analog waveform are used to determine the width of the pulse signal. Either instantaneous or natural sampling can be used.in PPM the analog sample values determine the position of a narrow pulse relative to the clocking time. It is possible to obtain PPM from PWM by using a mono-stable multivibrator circuit. EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 9

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11 PWM generation using instantaneous sampling PWM signal generation using natural sampling EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 11

12 The PWM or PPM signals may be converted back to the corresponding analog signal by a receiving system EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 12

13 For PWM detection the PWM signal is used to start and stop the integration of the integrator. After reset integrator starts to integrate during the duration of the pulse and will continue to do so till the pulse goes low. If integrator has a DC voltage connected as input, the output will be a truncated ramp. After the PWM signal goes low, the amplitude of the truncated ramp will be equal to the corresponding PAM sample value. Then it goes to zero with reset of the integrator. 4. State the advantage and disadvantage of digital communication system(4) [AUC NOV/DEC2012] EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 13

14 Draw a neat block diagram of a typical digital communication system and explain the function of the key signal processing blocks.( 16) State the advantage and disadvantage of digital communication system MODEL OF A COMMUNICATION SYSTEM(ANALOG) The purpose of a Communication System is to transport an information bearing signal from a source to a user destination via a communication channel. The three basic elements of every communication systems are Transmitter, Receiver and Channel. The Overall purpose of this system is to transfer information from one point (called Source) to another point, the user destination. The message produced by a source, normally, is not electrical. Hence an input transducer is used for converting the message to a time varying electrical quantity called message signal. Similarly, at the destination point, another transducer converts the electrical waveform to the appropriate message. The transmitter is located at one point in space, the receiver is located at some other point separate from the transmitter, and the channel is the medium that provides the electrical connection between them. The purpose of the transmitter is to transform the message signal produced by the source of information into a form suitable for transmission over the channel. The received signal is normally corrupted version of the transmitted signal, which is due to channel imperfections, noise and interference from other sources.the receiver has the task of operating on the received signal so as to reconstruct a recognizable form of the original message signal and to deliver it to the user destination. Communication Systems are divided into 3 categories: 1. Analog Communication Systems are designed to transmit analog information using analog modulation methods. 2. Digital Communication Systems are designed for transmitting digital information using digital modulation schemes, and 3. Hybrid Systems that use digital modulation schemes for transmitting sampled and quantized values of an analog message signal. ELEMENTS OF DIGITAL COMMUNICATION SYSTEMS: The figure 1.2 shows the functional elements of a digital communication system. Source of Information: 1. Analog Information Sources. 2. Digital Information Sources. Analog Information Sources Microphone actuated by a speech, TV Camera scanning a scene, continuous EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 14

15 amplitude signals. Digital Information Sources These are teletype or the numerical output of computer which consists of a sequence of discrete symbols or letters. An Analog information is transformed into a discrete information through the process of sampling and quantizing. Digital Communication System SOURCE ENCODER / DECODER: The Source encoder ( or Source coder) converts the input i.e. symbol sequence into a binary sequence of 0 s and 1 s by assigning code words to the symbols in the input sequence. For eg. :-If a source set is having hundred symbols, then the number of bits used to represent each symbol will be 7 because 27=128 unique combinations are available. The important parameters of a source encoder are block size, code word lengths, average data rate and the efficiency of the coder (i.e. actual output data rate compared to the minimum achievable rate) At the receiver, the source decoder converts the binary output of the channel decoder into a symbol sequence. The decoder for a system using fixed length code words is quite simple, but the decoder for a system using variable length code words will be very complex. Aim of the source coding is to remove the redundancy in the transmitting information, so that bandwidth required for transmission is minimized. Based on the probability of the symbol code word is assigned. Higher the probability, shorter is the codeword. Ex: Huffman coding. CHANNEL ENCODER / DECODER: Error control is accomplished by the channel coding operation that consists of systematically adding extra bits to the output of the source coder. These extra bits do not convey any information but helps the receiver to detect and / or correct some of the errors in the information bearing bits. There are two methods of channel coding: 1. Block Coding: The encoder takes a block of k information bits from the source encoder and adds r error control bits, where r is dependent on k and error control capabilities desired. 2. Convolution Coding: The information bearing message stream is encoded in a continuous fashion by continuously interleaving information bits and error control bits. The Channel decoder recovers the information bearing bits from the coded binary stream. Error detection and possible correction is also performed by the channel decoder. The important parameters of coder / decoder are: Method of coding, efficiency, error control capabilities and complexity of the circuit. MODULATOR: The Modulator converts the input bit stream into an electrical waveform suitable for transmission over the communication channel. Modulator can be effectively used to minimize the effects of channel noise, to match the frequency spectrum of transmitted signal with channel characteristics, to provide the capability to multiplex many signals. DEMODULATOR: EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 15

16 The extraction of the message from the information bearing waveform produced by the modulation is accomplished by the demodulator. The output of the demodulator is bit stream. The important parameter is the method of demodulation. CHANNEL: The Channel provides the electrical connection between the source and destination. The different channels are: Pair of wires, Coaxial cable, Optical fibre, Radio channel, Satellite channel or combination of any of these. The communication channels have only finite Bandwidth, non-ideal frequency response, the signal often suffers amplitude and phase distortion as it travels over the channel. Also, the signal power decreases due to the attenuation of the channel. The signal is corrupted by unwanted, unpredictable electrical signals referred to as noise. The important parameters of the channel are Signal to Noise power Ratio (SNR), usable bandwidth, amplitude and phase response and the statistical properties of noise. Advantages of Digital Communication 1. The effect of distortion, noise and interference is less in a digital communication system. This is because the disturbance must be large enough to change the pulse from one state to the other. 2. Regenerative repeaters can be used at fixed distance along the link, to identify and regenerate a pulse before it is degraded to an ambiguous state. 3. Digital circuits are more reliable and cheaper compared to analog circuits. 4. The Hardware implementation is more flexible than analog hardware because of the use of microprocessors, VLSI chips etc. 5. Signal processing functions like encryption, compression can be employed to maintain the secrecy of the information. 6. Error detecting and Error correcting codes improve the system performance by reducing the probability of error. 7. Combining digital signals using TDM is simpler than combining analog signals using FDM. The different types of signals such as data, telephone, TV can be treated as identical signals in transmission and switching in a digital communication system. 8. We can avoid signal jamming using spread spectrum technique. Disadvantages of Digital Communication: 1. Large System Bandwidth:- Digital transmission requires a large system bandwidth to communicate the same information in a digital format as compared to analog format. 2. System Synchronization:- Digital detection requires system synchronization whereas the analog signals generally have no such requirement. Channels for Digital Communications The modulation and coding used in a digital communication system depend on the characteristics of the channel. The two main characteristics of the channel are BANDWIDTH and POWER. In addition the other characteristics are whether the channel is linear or nonlinear, and how free the channel is free from the external interference. Five channels are considered in the digital communication, namely: telephone channels, coaxial cables, optical fibers, microwave radio, and satellite channels. Telephone channel: It is designed to provide voice grade communication. Also good EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 16

17 for data communication over long distances. The channel has a band-pass characteristic occupying the frequency range 300Hz to 3400hz, a high SNR of about 30db, and approximately linear response. For the transmission of voice signals the channel provides flat amplitude response. But for the transmission of data and image transmissions, since the phase delay variations are important an equalizer is used to maintain the flat amplitude response and a linear phase response over the required frequency band. Transmission rates upto16.8 kilobits per second have been achieved over the telephone lines. Coaxial Cable: The coaxial cable consists of a single wire conductor centered inside an outer conductor, which is insulated from each other by a dielectric. The main advantages of the coaxial cable are wide bandwidth and low external interference. But closely spaced repeaters are required. With repeaters spaced at 1km intervals the data rates of 274 megabits per second have been achieved. Optical Fibers: An optical fiber consists of a very fine inner core made of silica glass, surrounded by a concentric layer called cladding that is also made of glass. The refractive index of the glass in the core is slightly higher than refractive index of the glass in the cladding. Hence if a ray of light is launched into an optical fiber at the right oblique acceptance angle, it is continually refracted into the core by the cladding. That means the difference between the refractive indices of the core and cladding helps guide the propagation of the ray of light inside the core of the fiber from one end to the other. Compared to coaxial cables, optical fibers are smaller in size and they offer higher transmission bandwidths and longer repeater separations. Microwave radio: A microwave radio, operating on the line-of-sight link, consists basically of a transmitter and a receiver that are equipped with antennas. The antennas are placed on towers at sufficient height to have the transmitter and receiver in line-of-sight of each other. The operating frequencies range from 1 to 30 GHz. Under normal atmospheric conditions, a microwave radio channel is very reliable and provides path for high-speed digital transmission. But during meteorological variations, a severe degradation occurs in the system performance. Satellite Channel: A Satellite channel consists of a satellite in geostationary orbit, an uplink from ground station, and a down link to another ground station. Both link operate at microwave frequencies, with uplink the uplink frequency higher than the down link frequency. In general, Satellite can be viewed as repeater in the sky. It permits communication over long distances at higher bandwidths and relatively low cost. Bandwidth: Bandwidth is simply a measure of frequency range. The range of frequencies contained in a composite signal is its bandwidth. The bandwidth is normally a difference between two numbers. For example, if a composite signal contains frequencies between 1000 and 5000, its bandwidth is , or If a range of 2.40 GHz to 2.48 GHz is used by a device, then the bandwidth would be 0.08 GHz (or more commonly stated as 80MHz).It is easy to see that the bandwidth we define here is closely related to the amount of data you can transmit within it - the more room in frequency space, the more data you can fit in at a given moment. The term bandwidth is often used for something we should rather call a data rate, as in my Internet connection has 1 Mbps of bandwidth, meaning it can transmit data at 1 megabit per second EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 17

18 5. Derive Geometrical representation of signal. Basis Vectors The set of basis vectors {e1, e2,, en} of a space are chosen such that: Should be complete or span the vector space: any vector a can be expressed as a linear combination of these vectors. Each basis vector should be orthogonal to all others Each basis vector should be normalized: A set of basis vectors satisfying these properties is also said to be a complete orthonormal basis Signal Space In an n-dim space, we can have at most n basis vectors EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 18

19 Basic Idea: If a signal can be represented by n-tuple, then it can be treated in much the same way as a n-dim vector. Let φ1(t), φ2(t),., φn(t) be n signals Consider a signal x(t) and suppose that If every signal can be written as above ~ ~ basis functions and we have a n-dim signal space Orthonormal Basis Signal set {φk(t)}n is an orthogonal set if If cj 1 j {φk(t)} is an orthonormal set. Basis Functions for a Signal Set Consider a set of M signals (M-ary symbol) {si(t), i=1,2,,m } with finite energy. That is Then, we can express each of these waveforms as weighted linear combination of orthonormal signals where N M is the dimension of the signal space and are called the orthonormal basis functions Let, for a convenient set of {ϕj (t)}, j = 1,2,,N and 0 t <T, Now, we can represent a signal si(t) as a column vector whose elements are the scalar coefficients sij, j = 1, 2,.., N : EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 19

20 These M energy signals or vectors can be viewed as a set of M points in an N dimensional Euclidean space, known as the Signal Space. Signal Constellation is the collection of M signals points (or messages) on the signal space. EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 20

21 6. Distinguish between base band and band pass signalling Communication systems can be classified into two groups depending on the range of frequencies they use to transmit information. These communication systems are classified into BASEBAND or PASSBAND system. Baseband transmission sends the information signal as it is without modulation (without frequency shifting) while passband transmission shifts the signal to be transmitted in frequency to a higher frequency and then transmits it, where at the receiver the signal is shifted back to its original frequency. Almost all sources of information generate baseband signals. Baseband signals are those that have frequencies relatively close to zero such as the human voice (20 Hz 5 khz) and the video signal from a TV camera (0 Hz 5.5 MHz). A plot of an audio signal and its frequency spectrum are shown below, where it is seen that the most of the power of the audio signal is concentrated in the frequency range from (0 4 khz). The telephone system used for homes and offices, for example, may transmit the baseband audio signal as it is when the call is local (from your home to your neighbor s home). However, when the telephone call is a long distance call that is transmitted via microwave or satellite links, the baseband audio signal becomes unsuitable for transmission and the communication system becomes a passband system. Similarly, transmitting the video signal from your camera to your TV using a wire represents a baseband communication while transmitting that video signal via satellites passband transmission. Therefore, baseband transmission, which is easier than passband transmission, is usually used when communicating over wires, while over the air transmission requires passband transmission 7.Explain Binary symmetric channel and Gaussian channel with their mathematical models Binary Symmetric Channel (BSC) Let us consider a channel input alphabet X = {a1, a2} and a channel output alphabet Y = { b1, b2}. Further, let P(b1 a1) = P(b2 a2) = 1-ε and P(b1 a2) = P(b2 a1) = ε. This is an example of a EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 21

22 binary channel as both the input and output alphabets have two elements each. Further, the channel is symmetric and unbiased in its behavior to the two possible input letters a1and a2. Representation of a binary symmetric channel; ε indicates the probability of an error during transmission. Usually, if a1 is transmitted through the channel, b1 will be received at the output provided the channel has not caused any error. So, ε in our description represents the probability of the channel causing an error on an average to the transmitted letters. Let us assume that the probabilities of occurrence of a1 and a2 are the same, i.e. PX(a1) = PX(a2) = 0.5. A source presenting finite varieties of letters with equal probability is known as a discrete memory less source (DMS). Such a source is unbiased to any letter or symbol. GAUSSIAN CHANNEL EC2301 DIGITAL COMMUNICATION V Sem ECE R.Vanitha Asst.Prof./ECE Page 22