Digital Communication Pulse code modulation.ppt

1. DIGITAL COMMUNICATION COURSE OBJECTIVES: 1. Familiarize the students with elements of digital communication system and waveform coding techniques like PCM, DPCM, DM and ADM. 2. Introduce the concepts of information theory and source coding 3. Familiarize the students with channel coding techniques such as LBC, BCC and convolution codes 4. Introduce the concepts of baseband digital data transmission and analyze the error performance of different digital carrier modulation schemes like ASK, FSK, PSK etc. 5. Familiarize the students with the concepts of spread spectrum communication with emphasis on DSSS and FHSS. COURSE OUTCOMES: CO1: Classify the different types of digital modulation techniques PCM, DPCM, DM and ADM and compare their performance by SNR. CO2: Illustrate the classification of channels and Source coding methods. CO3:Distinguish different types of Error control codes along with their encoding/decoding algorithms. CO4: Examine the Performance of different Digital Carrier Modulation schemes of Coherent and Non-coherent type based on Probability of error. CO5:Generation of PN sequence using Spread Spectrum and characterize the Acquisition Schemes for Receivers to track the signals. MATRUSRI ENGINEERING COLLEGE

2. UNIT I- ELEMENTS OF DIGITAL COMMUNICATION SYSTEM: Check with autonomous comparison of Digital and Analog communication systems, analog to digital conversion, quantization and encoding techniques, PCM. Companding in PCM systems - u law and a law, applications of PCM: introduction to linear prediction theory. Modulation and demodulation of DPCM, DM and ADM. Comparison of PCM, DPCM, DM and ADM. SNRQ of PCM and DM. UNIT-I OUTCOMES: Familiarize the students with elements of digital communication system and waveform coding techniques like PCM, DPCM, DM and ADM. MATRUSRI ENGINEERING COLLEGE

3. TEXT BOOKS /REFERENCES TEXT BOOKS: 1. Simon Haykin, “Communication systems” 4/e, Wiley India 2011 2. Sam Shanmugam K, “Digital and Analog Communication systems”, Wiley 1979. 3. B.P.Lathi, “Modern digital and analog communication systems” 3/e, OxfordUniversityPress. 1998. 4. Leon W.Couch II., Digital and Analog Communication Systems, 6th Edn, Pearson Education inc., New Delhi, 2001. 5. R.E.Zimer&R.L.Peterson : Introduction to Digital Communication, PHI, 2001. REFERENCES: 1. P. Ramakrishna Rao, “Digital Communication”, TMH, 2011. 2. Dr. Sanjay Sharma, “Digital and Analog Communication”, Mc Graw Hill Publication, 2009. 3. Bernard Sklar “Digital Communications – Fundamentals and Applications” / 2nd Edition, Prentice Hall. 4. John G. Proakis” Digital Communications” Fourth Edition (textbook) McGraw Hill. MATRUSRI ENGINEERING COLLEGE

4. LESSON PLAN: UNIT I- : Elements of Digital Communication System MATRUSRI ENGINEERING COLLEGE S. No. Topic(S) No. of Hrs Relevant COs Text Book/ Reference Book 1. Check with autonomous Comparison of Digital and Analog Communication Systems 01 CO1 T1,T2,T5, R1,R2,R4 2. Analog to Digital Conversion 01 CO1 T1,T2,T5, R1,R2,R4 3. Quantization, PCM 02 CO1 T1,T2,T5, R1,R2,R4 4. Encoding techniques, SNRQ of PCM 02 CO1 T1,T2,T5, R1,R2,R4 5. Companding in PCM systems - u law and a law, Applications of PCM 01 CO1 T1,T2,T5, R1,R2,R4 6. Introduction to Linear Prediction Theory 01 CO1 T1,T2,T5, R1,R2,R4 7. Modulation and demodulation of DPCM 01 CO1 T1,T2,T5, R1,R2,R4 8. Modulation and demodulation of DM,SNRQ 02 CO1 T1,T2,T5, R1,R2,R4 9 Modulation and demodulation of ADM, Comparison of PCM, DPCM, DM and ADM 01 CO1 T1,T2,T5, R1,R2,R4 Total 12

5. CONTENTS: -ELEMENTS OF DIGITAL COMMUNICATION - COMPARISON OF DIGITAL WITH ANALOG SYSTEMS. OUTCOMES: Distinguish between analog and digital systems. MODULE-I MATRUSRI ENGINEERING COLLEGE

6. THE WORD DIGITAL COMES FROM THE LATIN WORD DIGIT AND DIGITUS (THE LATIN WORD FOR FINGER), AS FINGERS. WHAT IS DIGITAL COMMUNICATION? DIGITAL COMMUNICATION IS A MODE OF COMMUNICATION WHERE THE INFORMATION OR THE THOUGHT IS ENCODED DIGITALLY AS DISCRETE SIGNALS AND ELECTRONICALLY TRANSFERRED TO THE RECIPIENTS. Digital communication covers a broad area of communications techniques including: • Digital transmission is the transmission of digital pulses between two or more points in a communication system. • Digital radio is the transmitted of digital modulated analog carriers between two or more points in a communication system. Introduction MATRUSRI ENGINEERING COLLEGE

7. FIRST COMMUNICATION –DIGITAL IN NATURE BY SAMUEL MORSE IN 1837. MATRUSRI ENGINEERING COLLEGE

8. Example of Digital Communication: Manager wanted to meet all his team members at the Conference room to discuss their key responsibility areas and areas of expertise. He didn’t have the time to go to their workstations and invite them individually. Instead he opted an easier and cheaper mode to communicate his idea. He sent an email marking a cc to all the participants, inviting them for the meeting. This is an example of Digital communication where the information was sent electronically. In digital communication Information flows in a digital form and the source is generally the keyboard of the computer. A single individual is capable of digital communication. It also saves wastage of manpower and is one of the cheapest modes of communication. Digital communication is also a really quick way to communicate. The information can reach the recipient within a fraction of a second. An individual no longer has to wait to personally meet the other individual and share his information. MATRUSRI ENGINEERING COLLEGE

9. Some examples of digital communication are • E-mailing- Computers • Texting- Cell Phones • Fax • Teleconferencing • Video conferencing Basic Digital Communication Nomenclature: Textual Message: information comprised of a sequence of characters. Binary Digit (Bit): the fundamental information unit for all digital systems. Symbol (mi where i=1,2,…M): for transmission of the bit stream; groups of k bits are combined to form new symbol from a finite set of M such symbols; M=2k. Digital Waveform: voltage or current waveform representing a digital symbol. Data Rate: Symbol transmission is associated with a symbol duration T. Data rate R=k/T [bps]. Baud Rate: number of symbols transmitted per second [baud] MATRUSRI ENGINEERING COLLEGE

10. MATRUSRI ENGINEERING COLLEGE

12. Building blocks of Digital Communication System MATRUSRI ENGINEERING COLLEGE

14. Low Pass Filter Sampler Quantizer Channel Encoder Line Encoder Pulse Shaping Filters Source Encoder Modulator Multiplexer Input Signal Analog/ Digital To Channel Detector Receiver Filter De- Modulator From Channel Channel Decoder Digital-to-Analog Converter De- Multiplexer Signal at the user end Carrier Analog Digit al Carrier Recovery Symbol timing Recovery Digit al Analog Twisted Pair Co-axial Cable Optical Fiber Wireless Space Building blocks of Digital Communication System MATRUSRI ENGINEERING COLLEGE

20. ADVANTAGES: 1. COMPLEXITY 2. COST 3. ROBUSTNESS 4.STORAGE & RETRIEVAL 5. FLEXIBILITY 6. EFFECT OF NOISE AND INTERFERENCE 7. LONG-HAUL COMMUNICATION USING A NUMBER OF REPEATERS 8. SECRECY OF COMMUNICATION DISADVANTAGES: 1. MORE BANDWIDTH THAN ANALOG SYSTEMS. 2. DIGITAL COMPONENTS GENERALLY CONSUME MORE POWER AS COMPARED TO ANALOG COMPONENTS. MATRUSRI ENGINEERING COLLEGE

21. 1. What are the major design parameters of an digital communication systems? 2. Distinguish between baseband and bandpass digital modulation? 3. What are the advantages of digital communication over analog systems? ANSWER! MATRUSRI ENGINEERING COLLEGE

22. CONTENTS: A/D Conversion OUTCOMES: Illustrate the mechanism of how an analog signal is converted to digital signal. MODULE-II MATRUSRI ENGINEERING COLLEGE

23. Digital transmission refers to transmission of digital signals between two or more points in a communication system. if the original signal is in analog form then it needs to be converted to digital pulses prior to transmission and converted back to analog signals in the receiver. The conversion of analog signal to digital pulses is known as WAVEFORM CODING. The digitized signals may be in the form of binary or any other form DIGITAL transmission of discrete level digital pulses . A/D Conversion MATRUSRI ENGINEERING COLLEGE

24. The process of converting continuous time signals into equivalent discrete time signals, can be termed as sampling. A/D Conversion MATRUSRI ENGINEERING COLLEGE

25. Sampling theorem: A bandlimited signal with no spectral components beyond , can be uniquely determined by values sampled at uniform intervals of The sampling rate, is called Nyquist rate. Sampling MATRUSRI ENGINEERING COLLEGE

26. Sampling MATRUSRI ENGINEERING COLLEGE

27. Def: It is the process of assigning to each one of the sample values of the message signal, a discrete value from a prescribed set of finite number of such discrete values called ‘quantum value’. The total dynamic range of the analog signal is divided into equal number of finite number of levels or segments. We round off a sample value falling within a particular segment to the value represented by the prescribed level passing through the middle of the level. Quantization MATRUSRI ENGINEERING COLLEGE

28. STEP SIZE Q is called quantum levels , n –number of bits. Quantization MATRUSRI ENGINEERING COLLEGE max min ( ) 2 p p p V V V V V Q Q Q        2n Q  2 2 p n V  

29. When the quantization levels are uniformly distributed over the full amplitude range of the input signal, the quantizer is called an uniform or linear quantizer. UNIFORM QUANTIZATION MATRUSRI ENGINEERING COLLEGE a) Mid tread Type b) Midrise Type

30. When the quantization levels are not uniformly distributed over the full amplitude range of the input signal, the quantizer is called an non-uniform quantization. A non-uniform quantization practically gives a SNR ratio that remains essentially constant for wide r range of input voltage levels. A non-uniform quantizer is called Robust Quantizer COMPANDING The non-uniform quantization is practically achieved through a process called Companding. “The process of compressing message signals like speech at the transmitter and expanding them at receiver is called Companding” NON-UNIFORM QUANTIZATION MATRUSRI ENGINEERING COLLEGE

31. Compression of the input samples is accomplished according to a specific law governing the relationship between amplitudes of the input and output samples. μ-law Companding: in North America and japan , μ-law Companding is used. When μ = 0, it corresponds to uniform quantization. 8-bit PCM code and μ=255, the output SNR MATRUSRI ENGINEERING COLLEGE   max max ln 1 sgn ln 1 x x y y x                      2 2 3 ln(1 ) o Q SNR    1 for x 0 sgn 1 for x<0 x       

32. A-law Companding : European countries and India prefer A-law companding to approximate true logarithmic companding. A = 1 corresponds to uniform quantization The standard value of A used in digital telephony is A=87.6 MATRUSRI ENGINEERING COLLEGE max max max max max max 1 sgn 0 1 lnA 1 ln 1 sgn 1 1 lnA x A x x y x x A y x A x x y x x A                                         

33. ENCODING MATRUSRI ENGINEERING COLLEGE

34. ENCODING MATRUSRI ENGINEERING COLLEGE

35. ENCODING-UNIPOLAR MATRUSRI ENGINEERING COLLEGE

36. ENCODING-POLAR MATRUSRI ENGINEERING COLLEGE

37. ENCODING-POLAR MATRUSRI ENGINEERING COLLEGE

40. 1. What is the need of non-uniform quantization? 2. What happens when the analog signal is passed through compander? 3. Why there is always a defined upper limit to the analog information signal frequency that can be transmitted in digital communication system? 4. Differentiate between mid-tread and mid-rise types of uniform quantizers. Which one is preferred and why? 5. Represent the binary data 1 0 1 1 0 0 0 1 using the following line coding techniques: (i) Unipolar NRZ (ii) Bipolar RZ (iii) BP-RZ-AMI Answer! MATRUSRI ENGINEERING COLLEGE

41. CONTENTS: PCM APPLICATION OF PCM OUTCOMES: Understand the process of PCM and its application. MODULE-III MATRUSRI ENGINEERING COLLEGE

42. PULSE CODE MODULATION (PCM) IS A SPECIAL FORM OF A/D CONVERSION. IT CONSISTS OF SAMPLING, QUANTIZING, AND ENCODING STEPS. IT IS WIDELY POPULAR BECAUSE: - USED FOR LONG TIME IN TELEPHONE SYSTEMS - INEXPENSIVE ELECTRONICS EXISTS - ERRORS CAN BE CORRECTED DURING LONG HAUL TRANSMISSION - CAN USE TIME DIVISION MULTIPLEXING NOTE: PCM IS NOT A MODULATION TECHNIQUIE , IT IS THE NAME GIVEN TO CLASS OF BASEBAND SIGNALS OBTAINED FROM THE QUANTIZED PAM SIGNALS BY ENCODING EACH QUANTIZED SAMPLE INTO A DIGITAL WORD. PULSE CODE MODULATION MATRUSRI ENGINEERING COLLEGE

43. PCM TRANSMISTTER & RECEIVER PULSE CODE MODULATION MATRUSRI ENGINEERING COLLEGE

44. PCM TRANSMISTTER & RECEIVER PULSE CODE MODULATION MATRUSRI ENGINEERING COLLEGE

45. Process : MATRUSRI ENGINEERING COLLEGE

46. MATRUSRI ENGINEERING COLLEGE Figure shows an analog signal x(t) limited in its excursions to the range -4 to +4V. The step size between quantization levels has been set at 1V. Thus, eight quantization levels are employed. These are located at -3.5, -2.5,……+3.5V. Assign the code number 0 to the level at -3.5V, code number 1 to the level at -2.5V, and so on, until the level at 3.5V, which is assigned the code number 7.

47. Regenerative Repeater circuit: MATRUSRI ENGINEERING COLLEGE

48. Regenerative Repeater circuit: MATRUSRI ENGINEERING COLLEGE

49. Regenerative Repeater Process : MATRUSRI ENGINEERING COLLEGE

50. Bandwidth of PCM signal: MATRUSRI ENGINEERING COLLEGE

51. When a large number of PCM signals are to be transmitted over a common channel, multiplexing of these PCM signals are required PCM-TDM SYSTEM: T1 CARRIER SYSTEM (APPLICATION) MATRUSRI ENGINEERING COLLEGE

52. The frame structure of T-1 carrier system is shown . Instead of using a separate channel for signaling purposes, the LSB slots normally used for voice information, are themselves used once in six frames, for the purpose of signaling referred to as ‘channel associated signaling’. Frame synchronization: For proper synchronization one frame bit is included at the beginning of every frame. The pattern formed by 12 such frame bits occurring in 12 successive frames gives a 12-bit code called the frame sync word. PCM-TDM SYSTEM: T1 CARRIER SYSTEM (APPLICATION) MATRUSRI ENGINEERING COLLEGE

53. PCM-TDM SYSTEM: T1 CARRIER SYSTEM (APPLICATION) MATRUSRI ENGINEERING COLLEGE   1 l n 1 2 b m sampling period T tota umber of bits nN f      1 2 m nN f  2 m BW nNf 

54. 1. What is the objective of including equalizer circuit in PCM signal regenerators? 2. An audio signal is required to transmit with sampling rate of 40kHz and 14bits/sample using linear PCM. Calculate minimum transmission data rate? 3. What are the merits and de-merits of PCM systems? 4. Twenty four voice signals are sampled uniformly and TDM-ed. The highest frequency component is 3.4KHz. (i)What is the minimum channel bandwidth required , if signals are pulse amplitude modulated using Nyquist rate. (ii)If the signals are Pulse code modulated with 8-bit encoder, what would be the sampling rate? The bit rate is given as 1.5 Mbits/sec. ANSWER! MATRUSRI ENGINEERING COLLEGE

55. CONTENTS: -LINEAR PREDICTIVE CODING -DPCM OUTCOMES: Use the knowledge of prediction theory to understand the design process of DPCM. MODULE-IV MATRUSRI ENGINEERING COLLEGE

56. When adjacent samples of a message have good correlation, as in the case of audio and video message samples encoded using PCM, it is possible to predict the value of a future sample by making use of the present and some previous samples. -P PREVIOUS SAMPLES. - WEIGHTS A LINEAR COMBINATION OF THE PREVIOUS SAMPLE VALUES IS USED FOR OBTAINING THE PREDICTED VALUE, THE PREDICTION PROCESS IS CALLED ‘LINEAR PREDICTION’. LINEAR PREDECTIVE THEORY MATRUSRI ENGINEERING COLLEGE 1 2 ( ) ( 1 ) ( 2 ) ( ) s s s p s x nT h x n T h x n T h x n pT        ( 1 ), ( 2 ), ( ) s s s x n T x n T x n pT    1 2 , , p h h h

57. (A) P-th order filter transmitter THE PREDICTOR WEIGHTS OR COEFFICIENTS MUST BE CHOSEN THAT THE ‘PREDICTION ERROR’ IS MINIMIZED (B) P-th order filter Receiver LINEAR PREDECTIVE THEORY MATRUSRI ENGINEERING COLLEGE

58. Differential PCM MATRUSRI ENGINEERING COLLEGE

59. DPCM MATRUSRI ENGINEERING COLLEGE

61. DPCM TRANSMITTER DPCM RECEIVER DPCM MATRUSRI ENGINEERING COLLEGE

63.  The principle of DPCM is “over sampling “ and “Prediction” DPCM MATRUSRI ENGINEERING COLLEGE

64. irrespective pf properties of prediction filter, the quantized signal at the prediction filter i/p differs by original i/p by quantizing error. DPCM MATRUSRI ENGINEERING COLLEGE

65. PROCESSING GAIN: with the prediction order of 5, DPCM gives 11dB improvement in SNR compared to PCM and with sampling rate of 8KHZ, DPCM gives saving bitrate of 1 or 2 bits/sample i.e 8 to 16 kbps compared to PCM DPCM MATRUSRI ENGINEERING COLLEGE

66. 1. What are the benefits of reducing the amount of redundant information between adjacent sample? 2. What are the advantages of using a predictor in DPCM? 3. What is the conceptual difference between the conceptual PCM and DPCM? 4. What is meant by oversampling? ANSWER! MATRUSRI ENGINEERING COLLEGE

67. CONTENTS: -DELTA MODULATION -ADAPTIVE DM -COMPARISION OF WAVEFORM CODING’S OUTCOMES: Understand necesscesity of DM and overcome its drawbacks and also Compare various waveform coding techniques MODULE-V MATRUSRI ENGINEERING COLLEGE

68. The type of modulation, where the sampling rate is much higher and in which the step size after quantization is of a smaller value Δ, such a modulation is termed as delta modulation. If this sampling interval in differential PCM is reduced considerably, the sample to-sample amplitude difference is very small, as if the difference is 1-bit quantization, then the step-size will be very small i.E., Δ delta. Delta modulation is a simplified form of DPCM technique, also viewed as 1-bit DPCM scheme. As the sampling interval is reduced, the signal correlation will be higher. DELTA MODULATION MATRUSRI ENGINEERING COLLEGE

71. MATRUSRI ENGINEERING COLLEGE 

84. (2) GRANULAR NOISE. The granular noise occurs when the step size is too large relative to the local slope characteristics of the input wave form x(t), thereby causing the staircase approximation u(t) to hunt around a relatively flat segment of the input waveform; the granular noise is analogous to quantization noise in a PCM system. Advantages of DM Over DPCM 1-bit quantizer Very easy design of the modulator and the demodulator However, there exists some noise in DM. Slope Over load distortion (when Δ is small) Granular noise (when Δ is large) MATRUSRI ENGINEERING COLLEGE

85. ADAPTIVE DELTA MODULATION MATRUSRI ENGINEERING COLLEGE

92. NUMBER OF SUCCESSIVE 1’S OR 0’S STEP SIZE 1  2  3 2 4 4 . . MATRUSRI ENGINEERING COLLEGE

93. COMPARISION OF PCM,DPCM,DM,ADM MATRUSRI ENGINEERING COLLEGE

94. 1. How is DM different from pcm and DPCM? 2. What is the common algorithm followed in ADM? 3. Why is it necessary to use higher sampling rate for DM than that for pcm? 4. Indicate the possible situations in which the use of DM scheme is recommended? 5. What are the distortions encountered in DM and how to overcome? ANSWER! MATRUSRI ENGINEERING COLLEGE

95. CONTENTS: - QUANTIZATION ERROR - SNR OF PCM -BANDWIDTH-POWER TRADEOFF -SNR OF DM OUTCOMES: Derive the SNR expressions for PCM and DM and understand tradeoff mechanisms. MODULE-VI MATRUSRI ENGINEERING COLLEGE

96. The difference between the actual sampled values to the approximated quantized value is called quantization error. Maximum quantization error should not exceed Mean square value of the error = avg.Power in quantization noise QUANTIZATION NOISE MATRUSRI ENGINEERING COLLEGE ( ) ( ) e s q s Q x nT x nT   2   2 2 e Q       2 2 2 2 ( )d e e e e Q Q f Q Q       2 12 q N   

97. Let the sampled value of the signal x(t) is assumed to lie in range (-vp ,vp ), then the step size of uniform quantizer is assume that input x(t) is normalized , i.e Vp=1 Also if the destination signal power ‘P’ is normalized i.e P≤1 watt FOR NORMALIZED INPUT AND POWER, MATRUSRI ENGINEERING COLLEGE 2 2 p n V   2 2 1 2 3 n q p N V   2 2 3 [ ] 2 n o p P SNR V            4.77 6.02 q dB S n dB N         

98. STEP-1 : The quantization noise power Case (i): consider n-bit binary PCM, a message signal which is uniformly distributed between mean-squared value of a RVX S= SNR of PCM MATRUSRI ENGINEERING COLLEGE 2 12 q N    max max x and x  max max 2 2 2 max max 1 2 3 x x x X x dx x           2 2 max max 2 2 4 2 12 2 3 q n n x x N                  max max 2 2 , 2n x x where Q    6 q dB S n dB N        

99. CASE (II): Let us assume that a message signal is of sinusoidal nature, signal power, s = CASE (III): Suppose that the baseband signal x(t) is modelled as the sample function of a Gaussian random Process of zero mean and that the amplitude range of x(t) at the quantizer input extends from -4Arms to 4Arms. Avg.signal power, S= It is observed that a 1-bit increase per sample increases SNR by 6db. In general as number of bits per sample increases from n to n+k, the SNR increases by “6ndb/6kdb”. Hence it is called “6ndb rule”. MATRUSRI ENGINEERING COLLEGE ( ) cos2 m m x t A f t   max min m 2 V V A Q Q     2 m 2 3 2 q n A N    2 m 2 A   1.76 6.02 q dB S n dB N         2 rms A 8 2 rms n A     6 7.2 q dB S n dB N         

100. As n increases, SNR increases rapidly (exponentially). At the same time, the required transmission bandwidth BT also increases. Without increasing the transmitter power we can just increase n and get an improved destination SNR, there is a power- bandwidth trade-off possible in PCM. For radio broadcasting applications, SNR required is 60dB for a PCM systems but with b=6, the same SNR is achieved by FM but for PCM b needs to be increased beyond 8 to achieve 60dB SNR. Hence, in broadcasting applications FM is preferred over PCM. Bandwidth-power tradeoff PCM MATRUSRI ENGINEERING COLLEGE   2 2 l SNR c    2 2 2 ( ) 3 ( ) , 3 ( ) ln 1 p m t uncompanded m where c companded                   2 2 2 2 T B b B SNR c c   , exp T B b called bandwidth ansion factor B 

101. Assume that no slope overload distortion Consider a sinusoidal message signal , avg.signal power with in a filter bandwidth W SNR of DM MATRUSRI ENGINEERING COLLEGE max ( ) s d m t T dt   ( ) sin2 m m t A f t   2 s m f A f    2 2 A S  2 2 2 2 2 2 2 8 s m s m f f f S f              3 3 q N    3 3 2 3 3 8 80 s s q m m DM f f S N f f                        3 2 2 3 8 s q m DM f S N f W           

102. 1. Why PCM is not used for radio broad casting applications? 2. Consider an analog input signal to PCM whose bandlimited to 4KHz and varies from -3.8 V to +3.8 V, with average power of 30mW. The required SNR is given as 20dB.Assumimg uniform quantization, determine number of bits required per sample. 3. Compute SNR for PCM and DM systems for 8-bits . Comment on result. 4. An audio signal comprising of single term s(t)=3cos(2π1000t) is quantized using DM. Determine SNR of DM with sampling frequency 8-times the Nyquist rate. ANSWER! MATRUSRI ENGINEERING COLLEGE

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