2Course BooksText: Digital Communications: Fundamentals and Applications,By “Bernard Sklar”, Prentice Hall, 2nd ed, 2001.Taub’s Principles of Communication Systems3rd editionHerbert Taub , donald L Schilling , Goutam SahaDigital CommunicationR.N. Mutagi ( Oxford Press )Probability and Random Signals for Electrical Engineers, Neon GarciaReferences:Digital Communications, Fourth Edition, J.G. Proakis, McGraw Hill, 2000.
3Course Outline Review of Probability Signal and Spectra (Chapter 1) Formatting and Base band Modulation (Chapter 2)Base band Demodulation/Detection (Chapter 3)Channel Coding (Chapter 6, 7 and 8)Band pass Modulation and Demod./Detect.(Chapter 4)Spread Spectrum Techniques (Chapter 12)Synchronization (Chapter 10)Source Coding (Chapter 13)Fading Channels (Chapter 15)
4Today’s GoalReview of Basic ProbabilityDigital Communication Basic
5Question Bank :Q-1 Explain basic communication system
6Communication Source Transmitter Receiver Recipient Channel Main purpose of communication is to transfer information from a source to a recipient via a channel or medium.Basic block diagram of a communication system:SourceTransmitterChannelReceiverRecipient
7Brief Description Source: analog or digital Transmitter: transducer, amplifier, modulator, oscillator, power amp., antennaChannel: e.g. cable, optical fibre, free spaceReceiver: antenna, amplifier, demodulator, oscillator, power amplifier, transducerRecipient: e.g. person, (loud) speaker, computer
8Types of informationVoice, data, video, music, etc.Types of communication systemsPublic Switched Telephone Network (voice,fax,modem)Satellite systemsRadio,TV broadcastingCellular phonesComputer networks (LANs, WANs, WLANs)
9Information Representation Communication system converts information into electrical electromagnetic/optical signals appropriate for the transmission medium.Analog systems convert analog message into signals that can propagate through the channel.Digital systems convert bits(digits, symbols) into signalsComputers naturally generate information as characters/bitsMost information can be converted into bitsAnalog signals converted to bits by sampling and quantizing (A/D conversion)
11Why digital?Digital techniques need to distinguish between discrete symbols allowing regeneration versus amplificationGood processing techniques are available for digital signals, such as medium.Data compression (or source coding)Error Correction (or channel coding)(A/D conversion)EqualizationSecurityEasy to mix signals and data using digital techniques
16Basic Digital Communication Transformations Formatting/Source CodingTransforms source info into digital symbols (digitization)Selects compatible waveforms (matching function)Introduces redundancy which facilitates accurate decoding despite errorsIt is essential for reliable communicationModulation/DemodulationModulation is the process of modifying the info signal to facilitate transmissionDemodulation reverses the process of modulation. It involves the detection and retrieval of the info signalTypesCoherent: Requires a reference info for detectionNoncoherent: Does not require reference phase information
17Basic Digital Communication Transformations Coding/DecodingTranslating info bits to transmitter data symbolsTechniques used to enhance info signal so that they are less vulnerable to channel impairment (e.g. noise, fading, jamming, interference)Two CategoriesWaveform CodingProduces new waveforms with better performanceStructured SequencesInvolves the use of redundant bits to determine the occurrence of error (and sometimes correct it)Multiplexing/Multiple Access Is synonymous with resource sharing with other usersFrequency Division Multiplexing/Multiple Access (FDM/FDMA
19Q -4 Which are the basic performance metrics for digital communication system ? ( Advantages )
20Performance Metrics Metric is fidelity: want Analog Communication SystemsMetric is fidelity: wantSNR typically used as performance metricDigital Communication SystemsMetrics are data rate (R bps) and probability of bit errorSymbols already known at the receiverWithout noise/distortion/sync. problem, we will never make bit errors
21Main PointsTransmitters modulate analog messages or bits in case of a DCS for transmission over a channel.Receivers recreate signals or bits from received signal (mitigate channel effects)Performance metric for analog systems is fidelity, for digital it is the bit rate and error probability.
22Q –5 Why we prefer Digital communication system ?
23Why Digital Communications? Easy to regenerate the distorted signalRegenerative repeaters along the transmission path can detect a digital signal and retransmit a new, clean (noise free) signalThese repeaters prevent accumulation of noise along the pathThis is not possible with analog communication systemsTwo-state signal representationThe input to a digital system is in the form of a sequence of bits (binary or M_ary)Immunity to distortion and interferenceDigital communication is rugged in the sense that it is more immune to channel noise and distortion
24Why Digital Communications? Hardware is more flexibleDigital hardware implementation is flexible and permits the use of microprocessors, mini-processors, digital switching and VLSIShorter design and production cycleLow costThe use of LSI and VLSI in the design of components and systems have resulted in lower costEasier and more efficient to multiplex several digital signalsDigital multiplexing techniques – Time & Code Division Multiple Access - are easier to implement than analog techniques such as Frequency Division Multiple Access
25Why Digital Communications? Can combine different signal types – data, voice, text, etc.Data communication in computers is digital in nature whereas voice communication between people is analog in natureThe two types of communication are difficult to combine over the same medium in the analog domain.Using digital techniques, it is possible to combine both format for transmission through a common mediumEncryption and privacy techniques are easier to implementBetter overall performanceDigital communication is inherently more efficient than analog in realizing the exchange of SNR for bandwidthDigital signals can be coded to yield extremely low rates and high fidelity as well as privacy
26Q-6 Explain disadvantages of Digital communication system
27Why Digital Communications? DisadvantagesRequires reliable “synchronization”Requires A/D conversions at high rateRequires larger bandwidthNongraceful degradationPerformance CriteriaProbability of error or Bit Error Rate
28Q-7 Why designing digital communication system what care we must take ?
29Goals in Communication System Design To maximize transmission rate, RTo maximize system utilization, UTo minimize bit error rate, PeTo minimize required systems bandwidth, WTo minimize system complexity, CxTo minimize required power, Eb/No
30Comparative Analysis of Analog and Digital Communication Q-8 Compare Digital and analog communication systems.Q- 9 What is the advantage of digital over analog communication system ?Comparative Analysis of Analog andDigital Communication
31Digital Signal Nomenclature Information SourceDiscrete output values e.g. KeyboardAnalog signal source e.g. output of a microphoneCharacterMember of an alphanumeric/symbol (A to Z, 0 to 9)Characters can be mapped into a sequence of binary digits using one of the standardized codes such asASCII: American Standard Code for Information InterchangeEBCDIC: Extended Binary Coded Decimal Interchange Code
32Digital Signal Nomenclature Digital MessageMessages constructed from a finite number of symbols; e.g., printed language consists of 26 letters, 10 numbers, “space” and several punctuation marks. Hence a text is a digital message constructed from about 50 symbolsMorse-coded telegraph message is a digital message constructed from two symbols “Mark” and “Space”M - aryA digital message constructed with M symbolsDigital WaveformCurrent or voltage waveform that represents a digital symbolBit RateActual rate at which information is transmitted per second
33Digital Signal Nomenclature Baud RateRefers to the rate at which the signaling elements are transmitted, i.e. number of signaling elements per second.Bit Error RateThe probability that one of the bits is in error or simply the probability of error
351.2 Classification Of Signals 1. Deterministic and Random Signals A signal is deterministic means that there is no uncertainty with respect to its value at any time.Deterministic waveforms are modeled by explicit mathematical expressions, example:A signal is random means that there is some degree of uncertainty before the signal actually occurs.Random waveforms/ Random processes when examined over a long period may exhibit certain regularities that can be described in terms of probabilities and statistical averages.
362. Periodic and Non-periodic Signals A signal x(t) is called periodic in time if there exists a constantT0 > 0 such that(1.2)t denotes timeT0 is the period of x(t).
373. Analog and Discrete Signals An analog signal x(t) is a continuous function of time; that is, x(t) is uniquely defined for all tA discrete signal x(kT) is one that exists only at discrete times; it is characterized by a sequence of numbers defined for each time, kT, wherek is an integerT is a fixed time interval.
384. Energy and Power Signals The performance of a communication system depends on the received signal energy; higher energy signals are detected more reliably (with fewer errors) than are lower energy signalsx(t) is classified as an energy signal if, and only if, it has nonzero but finite energy (0 < Ex < ∞) for all time, where:(1.7)An energy signal has finite energy but zero average power.Signals that are both deterministic and non-periodic are classified as energy signals
394. Energy and Power Signals Power is the rate at which energy is delivered.A signal is defined as a power signal if, and only if, it has finite but nonzero power (0 < Px < ∞) for all time, where(1.8)Power signal has finite average power but infinite energy.As a general rule, periodic signals and random signals are classified as power signals
405. The Unit Impulse Function Dirac delta function δ(t) or impulse function is an abstraction—an infinitely large amplitude pulse, with zero pulse width, and unity weight (area under the pulse), concentrated at the point where its argument is zero.(1.9)(1.10)(1.11)Sifting or Sampling Property(1.12)
421.3 Spectral DensityThe spectral density of a signal characterizes the distribution of the signal’s energy or power in the frequency domain.This concept is particularly important when considering filtering in communication systems while evaluating the signal and noise at the filter output.The energy spectral density (ESD) or the power spectral density (PSD) is used in the evaluation.
431. Energy Spectral Density (ESD) Energy spectral density describes the signal energy per unit bandwidth measured in joules/hertz.Represented as ψx(f), the squared magnitude spectrum(1.14)According to Parseval’s theorem, the energy of x(t):(1.13)Therefore:(1.15)The Energy spectral density is symmetrical in frequency about origin and total energy of the signal x(t) can be expressed as:(1.16)
442. Power Spectral Density (PSD) The power spectral density (PSD) function Gx(f ) of the periodic signal x(t) is a real, even, and nonnegative function of frequency that gives the distribution of the power of x(t) in the frequency domain.PSD is represented as:(1.18)Whereas the average power of a periodic signal x(t) is represented as:(1.17)Using PSD, the average normalized power of a real-valued signal is represented as:(1.19)
461.4 Autocorrelation 1. Autocorrelation of an Energy Signal Correlation is a matching process; autocorrelation refers to the matching of a signal with a delayed version of itself.Autocorrelation function of a real-valued energy signal x(t) is defined as:(1.21)The autocorrelation function Rx(τ) provides a measure of how closely the signal matches a copy of itself as the copy is shiftedτ units in time.Rx(τ) is not a function of time; it is only a function of the time difference τ between the waveform and its shifted copy.
471. Autocorrelation of an Energy Signal The autocorrelation function of a real-valued energy signal has the following properties:symmetrical in about zeromaximum value occurs at the originautocorrelation and ESD form a Fourier transform pair, as designated by the double-headed arrowsvalue at the origin is equal to the energy of the signal
482. Autocorrelation of a Power Signal Autocorrelation function of a real-valued power signal x(t) is defined as:(1.22)When the power signal x(t) is periodic with period T0, the autocorrelation function can be expressed as(1.23)
492. Autocorrelation of a Power Signal The autocorrelation function of a real-valued periodic signal has the following properties similar to those of an energy signal:symmetrical in about zeromaximum value occurs at the originautocorrelation and PSD form a Fourier transform pairvalue at the origin is equal to the average power of the signal
50Digital Communication Systems Lecture-2 PULSE MODULATION AND DIGITAL TRANSMISSION OF ANALOG SIGNAL
53Example 1:In ASCII alphabets, numbers, and symbols are encoded using a 7-bit codeA total of 27 = 128 different characters can be represented usinga 7-bit unique ASCII code (see ASCII Table, Fig. 2.3)
54Formatting Transmit and Receive Formatting Transition from information source digital symbols information sink
55Character Coding (Textual Information) A textual information is a sequence of alphanumeric charactersAlphanumeric and symbolic information are encoded into digital bits using one of several standard formats, e.g, ASCII, EBCDIC
56Q-13 Explain process of sampling with neat sketches.
57Transmission of Analog Signals Structure of Digital Communication TransmitterAnalog to Digital Conversion
58SamplingSampling is the processes of converting continuous-time analog signal, xa(t), into a discrete-time signal by taking the “samples” at discrete-time intervalsSampling analog signals makes them discrete in time but still continuous valuedIf done properly (Nyquist theorem is satisfied), sampling does not introduce distortionSampled values:The value of the function at the sampling pointsSampling interval:The time that separates sampling points (interval b/w samples), TsIf the signal is slowly varying, then fewer samples per second will be required than if the waveform is rapidly varyingSo, the optimum sampling rate depends on the maximum frequency component present in the signal
59Analog-to-digital conversion is (basically) a 2 step process: SamplingConvert from continuous-time analog signal xa(t) to discrete-time continuous value signal x(n)Is obtained by taking the “samples” of xa(t) at discrete-time intervals, TsQuantizationConvert from discrete-time continuous valued signal to discrete time discrete valued signal
60Sampling Sampling Rate (or sampling frequency fs): The rate at which the signal is sampled, expressed as the number of samples per second (reciprocal of the sampling interval), 1/Ts = fsNyquist Sampling Theorem (or Nyquist Criterion):If the sampling is performed at a proper rate, no info is lost about the original signal and it can be properly reconstructed later onStatement:“If a signal is sampled at a rate at least, but not exactly equal to twice the max frequency component of the waveform, then the waveform can be exactly reconstructed from the samples without any distortion”
61Sampling If Rs < 2B, aliasing (overlapping of the spectra) results If signal is not strictly bandlimited, then it must be passed through Low Pass Filter (LPF) before samplingFundamental Rule of Sampling (Nyquist Criterion)The value of the sampling frequency fs must be greater than twice the highest signal frequency fmax of the signalTypes of samplingIdeal SamplingNatural SamplingFlat-Top Sampling
62Q-14 Describe different sampling techniques with neat sketches.
63Ideal Sampling ( or Impulse Sampling) Is accomplished by the multiplication of the signal x(t) by the uniform train of impulses (comb function)Consider the instantaneous sampling of the analog signal x(t)Train of impulse functions select sample values at regular intervalsFourier Series representation:Ref: ex 2.12 BPL Ch .2
64Ideal Sampling ( or Impulse Sampling) Therefore, we have:Take Fourier Transform (frequency convolution)Ref : next slide
69Ideal Sampling ( or Impulse Sampling) This shows that the Fourier Transform of the sampled signal is the Fourier Transform of the original signal at rate of 1/Ts
70Ideal Sampling ( or Impulse Sampling) This means that the output is simply the replication of the original signal at discrete intervals, e.g
71Ideal Sampling ( or Impulse Sampling) As long as fs> 2fm,no overlap of repeated replicas X(f - n/Ts) will occur in Xs(f)Minimum Sampling Condition:Sampling Theorem: A finite energy function x(t) can be completely reconstructed from its sampled value x(nTs) withprovided that =>
72Ts is called the Nyquist interval: It is the longest time interval that can be used for sampling a bandlimited signal and still allow reconstruction of the signal at the receiver without distortion
73It is practically difficult to create a train of impulses Practical SamplingIn practice we cannot perform ideal samplingIt is practically difficult to create a train of impulsesThus a non-ideal approach to sampling must be usedWe can approximate a train of impulses using a train of very thin rectangular pulses:Note:Fourier Transform of impulse train is another impulse trainConvolution with an impulse train is a shifting operation
74Natural SamplingIf we multiply x(t) by a train of rectangular pulses xp(t), we obtain a gated waveform that approximates the ideal sampled waveform, known as natural sampling or gating (see Figure 2.8)
75Each pulse in xp(t) has width Ts and amplitude 1/Ts The top of each pulse follows the variation of the signal being sampledXs (f) is the replication of X(f) periodically every fs HzXs (f) is weighted by Cn Fourier Series CoeffiecientThe problem with a natural sampled waveform is that the tops of the sample pulses are not flatIt is not compatible with a digital system since the amplitude of each sample has infinite number of possible valuesAnother technique known as flat top sampling is used to alleviate this problem
76Flat-Top SamplingHere, the pulse is held to a constant height for the whole sample periodFlat top sampling is obtained by the convolution of the signal obtained after ideal sampling with a unity amplitude rectangular pulse, p(t)This technique is used to realize Sample-and-Hold (S/H) operationIn S/H, input signal is continuously sampled and then the value is held for as long as it takes to for the A/D to acquire its valueEffect of the hold operation is the significant attenuation of the higher frequency spectral replicates.
78Taking the Fourier Transform will result to where P(f) is a sinc function
79Flat top sampling (Frequency Domain) Flat top sampling becomes identical to ideal sampling as the width of the pulses become shorter
80Q –15 Explain the method of reconstruction of signal from sampled signal. Q- 16 What is aliasing ? How we can avoid aliasing ?
81Recovering the Analog Signal One way of recovering the original signal from sampled signal Xs(f) is to pass it through a Low Pass Filter (LPF) as shown belowIf fs > 2B then we recover x(t) exactlyElse we run into some problems and signal is not fully recovered
82Undersampling and Aliasing If the waveform is undersampled (i.e. fs < 2B) then there will be spectral overlap in the sampled signalThe signal at the output of the filter will bedifferent from the original signal spectrumThis is the outcome of aliasing!This implies that whenever the sampling condition is not met, an irreversible overlap of the spectral replicas is produced
83x(t) containing higher frequency than were expected This could be due to:x(t) containing higher frequency than were expectedAn error in calculating the sampling rateUnder normal conditions, undersampling of signals causing aliasing is not recommended
84Solution 1: Anti-Aliasing Analog Filter All physically realizable signals are not completely bandlimitedIf there is a significant amount of energy in frequencies above half the sampling frequency (fs/2), aliasing will occurAliasing can be prevented by first passing the analog signal through an anti-aliasing filter (also called a prefilter) before sampling is performedThe anti-aliasing filter is simply a LPF with cutoff frequency equal to half the sample rate
85Case 2Aliasing is prevented by forcing the bandwidth of the sampled signal to satisfy the requirement of the Sampling TheoremCase 1
86Solution 2: Over Sampling and Filtering in the Digital Domain The signal is passed through a low performance (less costly) analog low-pass filter to limit the bandwidth.Sample the resulting signal at a high sampling frequency.The digital samples are then processed by a high performance digital filter and down sample the resulting signal.The alias frequency is given byfa = fs / 2 – ( f – fs / 2 ) = fs - f
87Example : A 5. 5 kHz tone is sampled at 8 KHz Example : A 5.5 kHz tone is sampled at 8 KHz . Find the alias frequency generated.Solution :Here,f = 5.5 kHz and fs = 8 kHzFromfa = fs / 2 – ( f – fs / 2 ) = fs – ffa = 8000 / 2 – ( 5500– 8000 / 2 ) = = 2500 = 2.5 kHz
88Q-17 Explain the method of sampling band pass signals.
89Sampling of Bandpass Signals X(f)(a)fX(f)(b)fs2fsffX(f)(c)X(f)f(d)band-pass signal (b) signal sampled at fs>2f2(c) Signal sampled at fs>2(f2-f1) (d) Band-pass filter required for signal recovery
90A band-pass signal occupies a frequency band from f1 to f2. It has one sided spectrum as shown in figure (a).Sampling a band-pass signal at Nyquist rate spectrum is shown in fig (b).Clearly there are gaps in this spectrum.To avoid the spectrum overlap we can reduce the sampling frequency.Sampling frequency so arranged avoid overlapping of the spectrum fig(c).Minimum sampling frequency rate for a band-pass signal from the f1 to f2 , with bandwidth B=f2-f1 is given byMin fs = 2f2 / N where N is integer part of the ratio f2/BFDM signals and sub-band signals used in speech coding are examples of band-pass signals.
91Example : a FM signal at 10.7 MHz IF needs to be digitized for demodulation in a digital domain. If the bandwidth of the signal is 200 kHz, find the minimum usable sampling frequencySolution :The Bandwidth B= f2-f1= 200 kHz.The center frequency , fIF = (f1+f2)/2 = 10,700 kHz.Therefore the highest frequency f2 = 10, = 10,800 kHzN = f2/B = 10,800/200 = 54Minimum sampling frequency fs = 2*f2/N = 2 * (10,800 / 54 ) = 400 kHz
92Example : A triangular waveform with 10 ms period is to be digitized Example : A triangular waveform with 10 ms period is to be digitized. If the waveform fidelity is to be maintained up to its 10th harmonic, what should be the sampling frequency ?Solution :The fundamental frequency of the triangular waveform isf0 = 1/T = 1 / ( 10 x 10-3 ) = 100 HzThe 10th harmonic is , thereforefm = 10 * f0 = 10 x 100 = 1000 HzHence the sampling frequency isfs = 2 x 1000 = 2000 Hz
93Summary Of Sampling Ideal Sampling (or Impulse Sampling) Natural Sampling(or Gating)Flat-Top SamplingFor all sampling techniquesIf fs > 2B then we can recover x(t) exactlyIf fs < 2B) spectral overlapping known as aliasing will occur
94Example 1: Example 2: Consider the analog signal x(t) given by What is the Nyquist rate for this signal?Example 2:Consider the analog signal xa(t) given byWhat is the Nyquist rate for this signal?What is the discrete time signal obtained after sampling, if fs=5000 samples/s.What is the analog signal x(t) that can be reconstructed from the sampled values?
95Practical Sampling Rates Speech- Telephone quality speech has a bandwidth of 4 kHz (actually 300 to 3300Hz)- Most digital telephone systems are sampled at samples/secAudio:- The highest frequency the human ear can hear is approximately 15kHz- CD quality audio are sampled at rate of 44,000 samples/secVideo- The human eye requires samples at a rate of at least 20 frames/sec to achieve smooth motion
96Q-18 Explain PCM communication system with necessary blocks.
97Pulse Code Modulation (PCM) Pulse Code Modulation refers to a digital baseband signal that is generated directly from the quantizer outputSometimes the term PCM is used interchangeably with quantization
101Each quantized sample is represented by a word consisting of three bits in the example. Space between words (i.e. samples) allow multiplexing.
102Q-20 Explain PCM system and mention its advantage
103Pulse Code Modulation Figure The basic elements of a PCM system. EE 541/451 Fall 2006
104Advantages of PCM:Relatively inexpensiveEasily multiplexed: PCM waveforms from different sources can be transmitted over a common digital channel (TDM)Easily regenerated: useful for long-distance communication, e.g. telephoneBetter noise performance than analog systemSignals may be stored and time-scaled efficiently (e.g., satellite communication)Efficient codes are readily availableDisadvantage:Requires wider bandwidth than analog signals
105Q- 21 Mention different source of corruption in sampling and quantization process.
1062.5 Sources of Corruption in the sampled, quantized and transmitted pulses Sampling and Quantization EffectsQuantization (Granularity) Noise: Results when quantization levels are not finely spaced apart enough to accurately approximate input signal resulting in truncation or rounding error.Quantizer Saturation or Overload Noise: Results when input signal is larger in magnitude than highest quantization level resulting in clipping of the signal.Timing Jitter: Error caused by a shift in the sampler position. Can be isolated with stable clock reference.Channel EffectsChannel NoiseIntersymbol Interference (ISI)
107Channel Noise : Thermal noise , interference from other users and interference from circuit switching transients can cause errors in detecting the pulses carrying the digitized samples.Channel induced error degrades the reconstructed signal quality.Rapid degradation of output signal quality with channel induced error is called threshold effect.Intersymbol Interference : The channel is always band limited .A band limited channel disperses or spreads a pulse waveform passing through it.Channel BW > Pulse BW then spreading will slightWhen channel BW close to signal BW the spreading will exceed a symbol duration and cause signal pulses to overlap.This is called Inter symbol interference , ISI which degrades the system performanceRising the signal power can not overcome the error performance
108Signal to Quantization Noise Ratio The level of quantization noise is dependent on how close any particular sample is to one of the L levels in the converterFor a speech input, this quantization error resembles a noise-like disturbance at the output of a DAC converter
109Q- 22 Explain uniform quantization Q- 23 Explain non uniform quantizationQ-24 Compare Uniform and non uniform quantization
110i.e. when all values within the range are equally likely Uniform QuantizationA Quantizer with equal quantization level is a Uniform QuantizerEach sample is approximated within a quantile intervalUniform Quantizer are optimal when the input distribution is uniformi.e. when all values within the range are equally likelyMost ADC’s are implemented using uniform QuantizerError of a uniform quantizer is bounded by
111Q –25 What is quantization noise ? Q-26 What is quantization error ?Q-27 Derive the equation for Quantization error
112Figure illustrates L- level quantizer for a signal having peak to peak voltage range Vpp = Vp – (- Vp) = 2Vp volts.The step size between quantization levels, called the QUANTILE interval , is denoted by q volts.Each sample value is approximated with a quantized pulse ; the approximation will result in error no larger than q/2 or –q/2 in positive and negative direction respectively.
113Signal to Quantization Noise Ratio A useful figure of merit for the uniform quantizer is the quantizer variance.The mean-squared value (noise variance) of the quantization error is given by:Where p(e) =1/q is the uniform probability density function of the quantization error.
114Q-28 Derive the equation for signal to quantization noise ratio or SQNR
115The variance σ2 , corresponds to the average quantization noise power. The peak power of the analog signal (normalized to 1 Ω )can be expressed as:Therefore the Signal to Quantization Noise Ratio is given by:L is no of quantization levels.L x q = 2 Vp= Vpp Vp = L x q / 2L=2nFor perfect understanding refer next few slides
116Average signal power of the applied input signal is given by (1)The quantization noise is(2)The number of quantization levels is L then L x q = 2VpSo thatVp = L x q / 2(3)
117Q-29 Derive the equation for signal to quantization noise ratio or SQNR Ref : Taub and Shilling ( Principles of communication systems)
119Since L = 2n, SNR = 22n or in decibels where L = 2n is the number of quantization levels for the converter. (n is the number of bits).Since L = 2n, SNR = 22n or in decibelsIf q is the step size, then the maximum quantization error that can occur in the sampled output of an A/D converter is qVpp= 2V= qL
120Q-30 Explain PWM and PPM modulation and demodulation with necessary sketches. Q-31 Explain different pulse modulation techniques with necessary sketches.
121Pulse Modulation Two Types: Pulse Amplitude Modulation (PAM) Recall that analog signals can be represented by a sequence of discrete samples (output of sampler)Pulse Modulation results when some characteristic of the pulse (amplitude, width or position) is varied in correspondence with the data signalTwo Types:Pulse Amplitude Modulation (PAM)The amplitude of the periodic pulse train is varied in proportion to the sample values of the analog signalPulse Time ModulationEncodes the sample values into the time axis of the digital signalPulse Width Modulation (PWM)Constant amplitude, width varied in proportion to the signalPulse Duration Modulation (PDM)sample values of the analog waveform are used in determining the width of the pulse signal
123Pulse Width Modulation (PWM) and Pulse Position Modulation (PPM) - 1 In PWM, message modulates the width of the pulse and in PPM, message modulates position of the fixed width pulse. These are not suitable for TDM.
124Pulse Width Modulation (PWM) and Pulse Position Modulation (PPM) - 2
125Pulse Width Modulation (PWM) and Pulse Position Modulation (PPM) PWM the message modulates the width of the pulse .PPM the position of the arrival of a fixed width pulse in each sample period is modulated by the message signal.Disadvantage : the randomness of the width in PWM and randomness in position in PPM , not suitable for time division multiplexing scheme.As a consequence their contribution to communication is limited.PWM finds its application in motor control, in delivery of power which is precisely regulated by regulating the width of the pulse. ( Power electronics , Chopper )Together PWM and PPM are known as Pulse Time Modulation or PTM.
126Pulse Width Modulation (PWM) We have a comparator , one input of which is fed by input message signal and the other by a sawtooth signal which operated at a carrier frequency.The maximum of the input signal ( both +ve and –ve side) should be less than that of sawtooth signal.
127Output of comparator will be PWM wave PWM pulses occur at regular interval, its rising edge coinciding with the falling edge sawtooth signal.
128When sawtooth signal as its minimum, which is always less than the minimum of input signal the +Ve input of the comparator is at higher potential and the comparator output is positive.When the sawtooth signal rises with a fixed slope and crosses input signal value the –Ve input of comparator is at higher potential and the comparator output will be –Ve.The duration for which the comparator stays at high is thus dependent on input signal magnitude and this decides the width of the pulse generated.Message information gets reflected in the time during comparator output is at HIGH ( +Ve) or the width of the pulse generated at its output which is directly proportional to amplitude of the message signal at that instant.
129Pulse Position Modulation (PPM) PPM generation is usually a post processing of PWM signal and is shown in figure (b)PWM signal generated as above is sent to an inverter which reveres polarity of the pulses.it is followed by a differentiator , we will have +Ve spike where in original PWM signal pulse was going from HIGH to LOW and –Ve spikes where LOW to HIGH.
130these spikes are then fed to a +Ve edge triggered fixed width pulse generator which generates pulses of fixed width when a +Ve spike appeared , coinciding with the falling edge of original PWM signal.
131the occurrence of these falling edges were dependent ( proportional to amplitude of message) on input message and hence the delay in occurrence of these fixed width pulses are proportional to the amplitude of the message at that instant.IC 710 Comparator, The inverter – differentiator block of fig(b) designed using op-amp and RC components.The fixed width pulse generator can be a monoshot device like IC , IC 555 etc.Demodulation of PWMFor PWM demodulation , start a ramp at the positive edge and stop it when the negative edge comes.Since the widths are different these ramps will reach different heights in each cycle which is directly proportional to pulse width and in turn the amplitude of the modulating signal.
132This when passed through a low pass filter will follow the envelope i This when passed through a low pass filter will follow the envelope i.e. message signal and the demodulation is done.PPM DemodulationSimilar scheme is employed , now the ramp starts at one positive edge of the pulse and stops at the positive edge of the next pulse.Thus the delay between the pulses decides the height of the ramp generated and in turn closely follows the modulating message amplitude.Low pass filter after that filters out the envelope information as demodulated signal.Transistor and RC combination can be used both for ramp generation and filtering to implement a demodulator circuit.If synchronous clock is available then PPM can be converted to PWM ( fig) and then PWM demodulator can be used to get the message signal back.Between PWM and PPM , the latter gives better performance in a noisy system
133SR edge triggered flip-flop is set by +Ve edge of the clock. It remains set so that output Q is High, till a +ve edge from PPM resets it.The more the delay in arrival , the longer the duration Q remains high.It is again set in the next clock period by the rising edge of clock pulse.Thus the output of the flip-flop is a train of pulses , the width of which is decided by how late PPM pulses arrive in a particular clock period in which again the message information is contained.
134Pulse Width Modulation (PWM) and Pulse Position Modulation (PPM) - 3
135Q-32 Explain quantization and quantization error Q-32 Explain quantization and quantization error. Derive the necessary question for quantization error. ( Ref : Taub and Schilling )
136Quantization for Digital Representation-1 Understanding of quantization from Taub and Schilling
137Quantization error : the quantized signal and the original signal from which it was derived differ from one another in a random manner.This difference or error may be viewed as a noise due to the quantization process and is called QUANTIZATION ERROR.Calculation of Quantization error : where e is the difference between original and quantized signal voltage.Let us divide total peak-to-peak range of the message signal m(t) into M equal voltage intervals, each of magnitude S volts.At the center of the each voltage interval we locate a quantization level m1,m2,m3,…., mM as shown in figure (a) ( next slide).Dashed level instantaneous value of the message signal m(t) at a time t.-- m(t) happens to be closest o the level mk , the quantizer output will be mk, the voltage corresponding to that level.The error is e = m(t) – mk
138Quantization for Digital Representation-2 Right : Range of voltage over which signal m(t) makes excursion is divided into M quantization range each of size S.Left : The error voltage e(t) as a function of the instantaneous value of he signal m(t).
139Quantization for Digital Representation-3 Quantization error :f(m) is pdf and considered constant for each quantization levelSubstituting x=(m-mk)f(x) S is the probability that the signal voltage m(t) will be in the all quantization range so the sum of terms in the parenthesis has a total value of unity
140Q-33 Explain COMPANDING ? What is the advantage ? Q- 34 Explain different companding techniques .
141COMPANDINGThe dynamic range can be improved by companding i.e. by first compressing and then expanding. A small amplitude signal will range through more quantization region.
142Nonuniform Quantization Nonuniform quantizers have unequally spaced levelsThe spacing can be chosen to optimize the Signal-to-Noise Ratio for a particular type of signalIt is characterized by:Variable step sizeQuantizer size depend on signal size
143Many signals such as speech have a nonuniform distribution See Figure on next page (Fig. 2.17)Basic principle is to use more levels at regions with large probability density function (pdf)Concentrate quantization levels in areas of largest pdfOr use fine quantization (small step size) for weak signals and coarse quantization (large step size) for strong signals
144Statistics of speech Signal Amplitudes Figure 2.17: Statistical distribution of single talker speech signal magnitudes (Page 81)
146Nonuniform quantization using companding Companding is a method of reducing the number of bits required in ADC while achieving an equivalent dynamic range or SQNRIn order to improve the resolution of weak signals within a converter, and hence enhance the SQNR, the weak signals need to be enlarged, or the quantization step size decreased, but only for the weak signalsBut strong signals can potentially be reduced without significantly degrading the SQNR or alternatively increasing quantization step sizeThe compression process at the transmitter must be matched with an equivalent expansion process at the receiver
147The signal below shows the effect of compression, where the amplitude of one of the signals is compressedAfter compression, input to the quantizer will have a more uniform distribution after samplingAt the receiver, the signal is expanded by an inverse operationThe process of COMpressing and exPANDING the signal is called compandingCompanding is a technique used to reduce the number of bits required in ADC or DAC while achieving comparable SQNR
148Basically, companding introduces a nonlinearity into the signal This maps a nonuniform distribution into something that more closely resembles a uniform distributionA standard ADC with uniform spacing between levels can be used after the compandor (or compander)The companding operation is inverted at the receiverThere are in fact two standard logarithm based companding techniquesUS standard called µ-law compandingEuropean standard called A-law companding
149Input/Output Relationship of Compander Logarithmic expression Y = log X is the most commonly used companderThis reduces the dynamic range of Y
150Types of Companding -Law Companding Standard (North & South America, and Japan) wherex and y represent the input and output voltages is a constant number determined by experimentIn the U.S., telephone lines uses companding with = 255Samples 4 kHz speech waveform at 8,000 sample/secEncodes each sample with 8 bits, L = 256 quantizer levelsHence data rate R = 64 kbit/sec ( 8,000 x 8 = 64,000) = 0 corresponds to uniform quantization
151A-Law Companding Standard (Europe, China, Russia, Asia, Africa) wherex and y represent the input and output voltagesA = 87.6A is a constant number determined by experiment
156PCM Waveform TypesThe output of the A/D converter is a set of binary bitsBut binary bits are just abstract entities that have no physical definitionWe use pulses to convey a bit of information, e.g.,In order to transmit the bits over a physical channel they must be transformed into a physical waveformA line coder or baseband binary transmitter transforms a stream of bits into a physical waveform suitable for transmission over a channelLine coders use the terminology mark for “1” and space to mean “0”In baseband systems, binary data can be transmitted using many kinds of pulses
157There are many types of waveforms. Why? performance criteria! Each line code type have merits and demeritsThe choice of waveform depends on operating characteristics of a system such as:Modulation-demodulation requirementsBandwidth requirementSynchronization requirementReceiver complexity, etc.,
158Goals of Line Coding (qualities to look for) A line code is designed to meet one or more of the following goals:Self-synchronizationThe ability to recover timing from the signal itselfThat is, self-clocking (self-synchronization) - ease of clock lock or signal recovery for symbol synchronizationLong series of ones and zeros could cause a problemLow probability of bit errorReceiver needs to be able to distinguish the waveform associated with a mark from the waveform associated with a spaceBER performancerelative immunity to noiseError detection capabilityenhances low probability of error
159Spectrum Suitable for the channel Spectrum matching of the channele.g. presence or absence of DC levelIn some cases DC components should be avoidedThe transmission bandwidth should be minimizedPower Spectral DensityParticularly its value at zeroPSD of code should be negligible at the frequency near zeroTransmission BandwidthShould be as small as possibleTransparencyThe property that any arbitrary symbol or bit pattern can be transmitted and received, i.e., all possible data sequence should be faithfully reproducible
160Line CoderThe input to the line encoder is the output of the A/D converter or a sequence of values an that is a function of the data bitThe output of the line encoder is a waveform:where f(t) is the pulse shape and Tb is the bit period (Tb=Ts/n for n bit quantizer)This means that each line code is described by a symbol mapping function an and pulse shape f(t)Details of this operation are set by the type of line code that is being used
161Summary of Major Line Codes Categories of Line CodesPolar - Send pulse or negative of pulseUnipolar - Send pulse or a 0Bipolar (a.k.a. alternate mark inversion, pseudoternary)Represent 1 by alternating signed pulsesGeneralized Pulse ShapesNRZ -Pulse lasts entire bit periodPolar NRZBipolar NRZRZ - Return to Zero - pulse lasts just half of bit periodPolar RZBipolar RZManchester Line CodeSend a 2- pulse for either 1 (high low) or 0 (low high)Includes rising and falling edge in each pulseNo DC component
162Turn the pulse ON for a ‘1’, leave the pulse OFF for a ‘0’ When the category and the generalized shapes are combined, we have the following:Polar NRZ:Wireless, radio, and satellite applications primarily use Polar NRZ because bandwidth is preciousUnipolar NRZTurn the pulse ON for a ‘1’, leave the pulse OFF for a ‘0’Useful for noncoherent communication where receiver can’t decide the sign of a pulsefiber optic communication often use this signaling formatUnipolar RZRZ signaling has both a rising and falling edge of the pulseThis can be useful for timing and synchronization purposes
163Alternating like this eliminates the DC component Bipolar RZA unipolar line code, except now we alternate between positive and negative pulses to send a ‘1’Alternating like this eliminates the DC componentThis is desirable for many channels that cannot transmit the DC componentsGeneralized GroupingNon-Return-to-Zero: NRZ-L, NRZ-M NRZ-SReturn-to-Zero: Unipolar, Bipolar, AMIPhase-Coded: bi-f-L, bi-f-M, bi-f-S, Miller, Delay ModulationMultilevel Binary: dicode, doubinaryNote:There are many other variations of line codes (see Fig. 2.22, page 80 for more)
164Commonly Used Line Codes Polar line codes use the antipodal mappingPolar NRZ uses NRZ pulse shapePolar RZ uses RZ pulse shape
165Where Xn is the nth data bit Unipolar NRZ Line CodeUnipolar non-return-to-zero (NRZ) line code is defined by unipolar mappingIn addition, the pulse shape for unipolar NRZ is:where Tb is the bit periodWhere Xn is the nth data bit
166Bipolar Line CodesWith bipolar line codes a space is mapped to zero and a mark is alternately mapped to -A and +AIt is also called pseudoternary signaling or alternate mark inversion (AMI)Either RZ or NRZ pulse shape can be used
167Manchester Line CodesManchester line codes use the antipodal mapping and the following split-phase pulse shape:
170Comparison of Line Codes Self-synchronizationManchester codes have built in timing information because they always have a zero crossing in the center of the pulsePolar RZ codes tend to be good because the signal level always goes to zero for the second half of the pulseNRZ signals are not good for self-synchronizationError probabilityPolar codes perform better (are more energy efficient) than Unipolar or Bipolar codesChannel characteristicsWe need to find the power spectral density (PSD) of the line codes to compare the line codes in terms of the channel characteristics
171Comparisons of Line Codes Different pulse shapes are usedto control the spectrum of the transmitted signal (no DC value, bandwidth, etc.)guarantee transitions every symbol interval to assist in symbol timing recovery1. Power Spectral Density of Line Codes (see Fig. 2.23, Page 90)After line coding, the pulses may be filtered or shaped to further improve there properties such asSpectral efficiencyImmunity to Intersymbol InterferenceDistinction between Line Coding and Pulse Shaping is not easy2. DC Component and BandwidthDC ComponentsUnipolar NRZ, polar NRZ, and unipolar RZ all have DC componentsBipolar RZ and Manchester NRZ do not have DC components
172First Null BandwidthUnipolar NRZ, polar NRZ, and bipolar all have 1st null bandwidths of Rb = 1/TbUnipolar RZ has 1st null BW of 2RbManchester NRZ also has 1st null BW of 2Rb, although the spectrum becomes very low at 1.6Rb
173Generation of Line Codes The FIR filter realizes the different pulse shapesBaseband modulation with arbitrary pulse shapes can be detected bycorrelation detectormatched filter detector (this is the most common detector)
174Bits per PCM word and M-ary Modulation Section 2.8.4: Bits per PCM Word and Bits per SymbolL=2lSection 2.8.5: M-ary Pulse Modulation WaveformsM = 2kProblem 2.14: The information in an analog waveform, whose maximum frequency fm=4000Hz, is to be transmitted using a 16-level PAM system. The quantization must not exceed ±1% of the peak-to-peak analog signal.(a) What is the minimum number of bits per sample or bits per PCM word that should be used in this system?(b) What is the minimum required sampling rate, and what is the resulting bit rate?(c) What is the 16-ary PAM symbol Transmission rate?