Digital Communication and System Concepts

2Digital Communication System Concepts Vijay K. G a r g Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois, USA Yih-Chen Wang Lucent Technologies, Naperville, Illinois, USA 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Digital Communication System .............................................................. Messages, Characters, and Symbols ......................................................... Sampling Process ................................................................................. Miasing .............................................................................................. Quantization ....................................................................................... Pulse Amplitude Modulation ................................................................. Sources of Corruption .......................................................................... Voice Communication .......................................................................... Encoding ............................................................................................ 2.9.1 EncodingSchemes 957 957 957 959 960 960 961 963 964 2.1 Digital Communication System Figure 2.1 shows a block diagram of a typical digital c o m m u nication system. We focus primarily on formatting and transmission of baseband signal. Data already in a digital format would bypass the formatting procedure. Textual information is transformed into binary digits by use of a coder. Analog information is formatted using three separate processes: • Sampling • Quantization • Encoding In all cases, the formatting steps result in a sequence of 2.2 Messages, Characters, and Symbols When digitally transmitted, the characters are first encoded into a sequence of bits, called a bit stream or baseband signal. Groups of n bits can be combined to form a finite symbol set or w o r d o f M = 2 ~ for such symbols. A system using a symbol set size of M is called an M-ary system. The value of n or M represents an important initial choice in the design of any digital communication system. For n = 1, the system is referred to as binary, the size of symbol set is M = 2, and the modulator uses two different waveforms to represent the binary 1 and the binary O. In this case, the symbol rate and the bit rate are the same. For n = 2, the system is called quaternary or 4-ary (M = 4). At each symbol time, the modulator uses one of the four different waveforms to represent the symbol (see Figure 2.2). binary digits. These digits are transmitted through a baseband channel, such as a pair of wires or a coaxial cable. However, before we transmit the digits, we must transform the digits into waveforms that are compatible with the channel. For baseband channels, Compatible waveforms are pulses. The conversion from binary digits to pulse waveform takes place in a wave encoder also called a baseband modulator. The output of the waveform encoder is typically a sequence of pulses with characteristics that correspond to the binary digits being sent. After transmission through the channel, the received waveforms are detected to produce an estimate of the transmitted digits, and then the final step is (reverse) formatting to recover an estimate of the source information. Copyright © 2005 by Academic Press. All rights of reproduction in any form reserved. 2.3 Sampling Process Analog information must be transformed into a digital format. The process starts with sampling the waveform to produce a discrete pulse-amplitude-modulated waveform (see Figure 2.3). The sampling process is usually described in a time domain. This is an operation that is basic to digital signal processing and digital communication. Using the sampling 957 958 Vijay K. Garg and Yih-Chen Wang Digital information I Textual I informationl - [ Sampler I I I I_ ~ II Quantizer Coder I I J WavefOrmencoder ~[Transmitter Binarydigits Channel Pulsewaveform Analoginformation q Textual 9 Low-passI. Filter I Decoder [ [ Waveform~ decoder Receiver Digitalinformation FIGURE 2.1 BlockDiagram of a Typical Digital Communication System 1 where g~(t) is the ideal sampled signal and where 8(t - nT~) is the delta function positioned at time t = nTs. A delta function is closely approximated by a rectangular pulse of duration At and amplitude g ( n Ts) / A t; the smaller we make At, the better will be the approximation: oo -1 FIGURE 2.2 Binary and Quaternary Systems m = gs(t) = f~ ~_, G(f - mr,), ~x~ (2.2) process, we convert the analog signal in a corresponding sequence of samples that are usually spaced uniformly in time. The sampling process can be implemented in several ways, the most popular being the sample-and-hold operation. In this operation, a switch and storage mechanism (such as a transistor and a capacitor, or shutter and a film strip) form a sequence of samples of the continuous input waveform. The output of the sampling process is called pulse amplitude modulation (PAM) because the successive output intervals can be described as a sequence of pulses with amplitudes derived from the input waveform samples. The analog waveform can be approximately retrieved from a PAM waveform by simple low-pass filtering, provided we choose the sampling rate properly. The ideal form of sampling is called instantaneous sampling. We sample the signal g(t) instantaneously at a uniform rate off~ once every T~ sec. Thus, we can write: where G(f) is the Fourier transform of the original signal g(t) and f~ is sampling rate. Equation 2.2 states that the process of uniformly sampling a continuous-time signal of finite energy results in a periodic spectrum with a period equal to the sampling rate. Taking the Fourier transform of both side, of Equation 2.1 and noting that the Fourier transform of the delta function 8(t - nT,) is equal to e j2"rrnfFs: G~(f) = ~ n~--OC g(nTs)e j2~nfr,. (2.3) ga(t) = ~ g(nTs)8(t- nTs), 11~--0C (2.1) Equation 2.3 is called the discrete-time Fourier transform. It is the complex Fourier series representation of the periodic frequency function C~(t), with the sequence of samples g(nT,) defining the coefficients of the expansion. We consider any continuous-time signal g(t) of finite energy and infinite duration. The signal is strictly band-limited with no frequency component higher than W Hz. This implies that the Fourier transform G(f) of the signal g(t) has the property that G(f) is zero for ]fl-> W. If we choose the 2 Digital Communication System Concepts g(t) Ts(t) 959 T FIGURE 2.3 Sampling Process sampling period T, = 1/2W, then the corresponding spectrum is given as: width W Hz is called the Nyquist rate and 1/2 W sec is called the Nyquist interval. We discuss the sampling theorem by assuming that signal g(t) is strictly band-limited. In practice, however, an inforC~(f) = g (~w)e j~- = fsG(f) + f~ G(f - mf~) mation-bearing signal is not strictly band-limited, with the m--~, me0 result that some degree of under sampling is encountered. (2.4) Consequently, some aliasing is produced by the sampling process. Aliasing refers to the phenomenon of a high-frequency component in the spectrum of the signal seemingly taking on Consider the following two conditions: the identity of a lower frequency in the spectrum of its sampled (1) G(f) = 0 for If[ > W. version. (2) fs = 2W. We find from equation 2.4 by applying these conditions, 2.4 Aliasing (2.5) Figure 2.4 shows the part of the spectrum that is aliased due to under sampling. The aliased spectral components represent ambiguous data that can be retrieved only under special conditions. In general, the ambiguity is not resolved and ambiguous data appear in the frequency band between (fs -fro) and G(f)=2-~G~(f) .'. G(f) = ~ n~--CXD - W < f < W. g e-(L~) - W < f < W. Thus, if the sample value g(n/2W) of a signal g(t) is specified for all n, then the Fourier transform G(f) of the signal is uniquely determined by using the discrete-time Fourier transform of equation 2.5. Because g(t) is related to G(f) by the inverse Fourier transform, it follows that the signal g(t) is itself uniquely determined by the sample values g(n/2 W) for -cxD < n < c~. In other words, the sequence {g(n/2W)} has all the information contained in g(t). We state the sampling theorem for band-limited signals of finite energy in two parts that apply to the transmitter and receiver of a pulse modulation system, respectively. (1) A band-limited signal of finite energy with no frequency components higher than W Hz is completely described by specifying the values of signals at instants of time separated by 1/2 W sec. (2) A band-limited signal of finite energy with no frequency components higher than W Hz may be completely recovered from a knowledge of its samples taken at the rate of 2 W samples/sec. This is also known as the uniform sampling theorem. The sampling rate of 2 W samples per second for a signal band- fro. In Figure 2.5, we show a higher sampling rate Jj to eliminate the aliasing by separating the spectral replicas. Figures 2.6 and 2.7 show two ways to eliminate aliasing using antialiasing filters. The analog signal is preflltered so that the new maximum frequency fm is less than or equal to f J 2 . Thus, there are no aliasing components seen in Figure 2.6 because f~ > 2f' Eliminating aliasing terms prior to sampling m. is a good engineering practice. When the signal structure is lx~(ol FIGURE 2.4 SampledSignal Spectrum 960 Vijay K. Garg and Yih-Chen Wang trade-off is needed between the cost of a small transition bandwidth and costs of the higher sampling rate, which are those of more storage and higher transition rates. In many systems, the answer has been to make the transition bandwidth 10 and 20% of the signal bandwidth. If we account for the 20% transition bandwidth of the antialiasing filter, we have an engineering version of Nyquist sampling rate: fm fs'-fm FIGURE 2.5 Higher Sampling Rate to Eliminate Aliasing ~f f,>_ZZfm. Example 3 We want to produce a high-quality digitalization of a 20-kHz bandwidth music signal. The sampling rate of greater than or equal to 22 ksps should be used. The sampling rate for compact disc digital audio player is 44.1 ksps, and the standard sampling rate for studio-quality audio player is 48 ksps. f'm fs-f'm t~ ,f 2.5 Quantization In Figure 2.8, each pulse is expressed as a level from a finite number of predetermined levels; each such level can be represented by a symbol from a finite alphabet. The pulses in Figure 2.8 are called quantized samples. When the sample values are quantized to a finite set, this format can interface with a digital system. After quantization, the analog waveform can still be recovered but not precisely; improved reconstruction fidelity of the analog waveform can be achieved by increasing the number of quantization levels (requiring increased system bandwidth). FIGURE 2.6 Prefiltering to Eliminate Aliasing IX~(f)l i 1 I, fn,' td2 FIGURE 2.7 Postfiltering to Eliminate Aliasing Portion of the Spectrum well known, the aliased terms can be eliminated after sampling with a linear pass filter (LPF) operating on the sampled data. In this case, the aliased components are removed by postfiltering after sampling. The filter cutoff frequency f'm removes the aliased components; i'm needs to be less than (f~-fm). It should be noted that filtering techniques for eliminating the aliased portion of the spectrum will result in a loss of some signal information. For this reason, the sample rate, cutoff bandwidth, and filter type selected for a particular signal bandwidth are all interrelated. Realizable filters require a nonzero bandwidth for the transition between the passband and the required out-of-band attenuation. This is called the transition bandwidth. To minimize the system sample rate, we desire that the antialiasing filter has a small transition bandwidth. Filter complexity and cost rise sharply with narrower transition bandwidth, so a 2.6 Pulse Amplitude Modulation There are two operations involved in the generation of the pulse amplitude modulation (PAM) signal: (1) Instantaneous sampling of the message signal m(t) every Ts sec, where fs = 1/Ts is selected according to the sampling theorem (2) Lengthening the duration of each sample obtained to some constant value T g(t) -*IT FIGURE 2.8 Flattop Quantization t 2 Digital Communication System Concepts h(t) 961 1.0 f)l ctrummagnitude 0 -3/T -1/T 0 1/T 3/T Pulse 3/T~~_ - arg[H(f)] _~I_/T., j Spectrum phase 3/T FIGURE 2.9 Rectangular Pulse and Its Spectrum (1) Assuming a sampling rate of 8 kHz, calculate the spacing between successive pulses of the multiplexed signal. (2) Repeat your calculations using Nyquist rate sampling. 106 T~ -- - -- 125 Ixs. 8000 For 25 channels (24 voice channels +1 sync), time allocated for each channel is 125/25 = 5 b~s. Since the pulse duration is 1 izs, the time between pulses is (5 - 1) = 4 b~s. The Nyquist rate is 7.48 Hz (2.2 x 3.4). In addition: 106 7480 These two operations are jointly referred to as sample and hold. One important reason for intentionally lengthening the duration of each sample is to avoid the use of an excessive channel bandwidth because bandwidth is inversely proportional to pulse duration. The Fourier transform of the rectangular pulse h(t) is given as (see Figure 2.9): H(f) = Tsinc(fT)e -j2"~fr. (2.6) We observe that by using flattop samples to generate a PAM signal, we introduce amplitude distortion as well as a delay of T/2. This effect is similar to the variation in transmission frequency that is caused by the finite size of the scanning aperture in television. The distortion caused by the use of PAM to transmit an analog signal is called the aperture affect. This distortion may be corrected by using an equalizer (see Figure 2.10). The equalizer has the effect of decreasing the in-band loss of the filter as the frequency increases in such a manner to compensate for the aperture effect. For T/Ts < O.1, the amplitude distortion is less than 0.5%, in which case the need of equalization may be omitted altogether. Ts -Tc -- -- 134 p~s. 134 -- 5.36 Izs. 25 The time between pulses is 4.36 Vs. Example 4 Sampled uniformly and then time-division multiplexed are 24 voice signals. The sampling operation involved flattop samples with 1 Ixs duration. The multiplexing operation includes provision for synchronization by adding an extra pulse of sufficient amplitude and also 1 Ixs duration. The highest frequency component of each voice signal is 3.4 kHz. 2.7 Sources of Corruption The sources of corruption include sampling and quantization effects as well as channel effects, as described in the following bulleted list. • Quantization noise: The distortion inherent in quantization is a roundoff or truncation error. The process of PAMsignal s(t) IlL Filter(LPF)I~ FIGURE 2.10 [ EqualizerJ ] . Message m(t) signal An Equalizer Application 962 encoding the PAlVl waveform into a quantized waveform involves discarding some of the original analog information. This distortion is called quantization noise; the amount of such noise is inversely proportional to the number of levels used in the quantization process. Quantizer saturation: The quantizer allocates L levels to the task of approximating the continuous range of inputs with a finite set of outputs (see Figure 2.11). The range of inputs for which the difference between the input and output is small is called the operating range of the converter. If the input exceeds this range, the difference between the input and output becomes large, and we say that the converter is operating in saturation. Saturation errors are more objectionable than quantizing noise. Generally, saturation is avoided by use of automatic gain control (AGC), which effectively extends the operating range of the converter. Timing jitter: If there is a slight jitter in the position of the sample, the sampling is no longer uniform. The effect of the jitter is equivalent to frequency modulation (FM) of the baseband signal. If the jitter is random, a low-level wideband spectral contribution is induced whose properties are very close to those of the quantizing noise. Timing jitter can be controlled with very good power supply isolation and stable clock reference. Channel 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 errors can degrade the reconstructed signal quality quite quickly. The rapid degradation of the output signal quality with channelinduced errors is called a threshold effect. Intersymbol interference: The channel is always bandlimited. A band-limited channel spreads a pulse waveform passing through it. When the channel bandwidth is much greater than pulse bandwidth, the spreading of the pulse will be slight. When the channel bandwidth is close to the signal bandwidth, the spreading will exceed a Vijay iv(. Garg and Yih-Chen Wang symbol duration and cause signal pulses to overlap. This overlapping is called inter-symbol interference ISI), ISI causes system degradation (higher error rates); it is a particularly insidious form of interference because raising the signal power to overcome interference will not improve the error performance: • q/2 0-2= Ie2p(e)de= q/2 Ie21de=q2 -q/2 -q/2 q 12 = average quantization noise power. Vd z (V_~)2 (~)2_ L2q2~ ~ q q2/12 -- 3L2. (2.7) (2.8) • In the limit as L ---, oc, the signal approaches the PAM format (with no quantization error) and signal-to-quantization noise ratio is infinite. In other words, with an infinite number of quantization levels, there is zero quantization error. 2vp Typically L = 2R, R = Log2L, and q = G- = (2Vp)/2 R. • 1"2Vp\ 2 ---=_1V2 2_2R Let P denote the average power of the message signal re(t), and then: (SNR)o -- ~ -\ P/ 22R. ° (2.9) The output SNR of the quantizer increases exponentially with increasing number of bits per sample, R. An increase in R requires a proportionate increase in the channel bandwidth. Example 5 We consider a full-load sinusodial modulating signal of amplitude A that uses all representation levels provided. The average signal power is (assuming a load of 1 fl): The equations are written and solved as follows: L levels 1 L A2 P ~--. 2 0-2 = fA22 -2R. 3 2 A (SNR) o m (I/3A22_2R) 2 _~ (22R) = 1.8 + 6R dB. FIGURE 2.11 Uniform Quantization 2 Digital Communication System Concepts L R [bits] SNR [decibels] 963 In equation 2.10, IX is constant, x and y are the input and output voltages, ~ = 0 represents uniform quantization, and ix = 255 is the standard value used in North America. • A-Law, used in Europe, is as follows: 32 64 128 256 5 6 7 8 31.8 37.8 43.8 49.8 y _ A(lx[/Xm~)sgnx, yma× 1 + In A 0 < Ixl < 1 Xma~- A sgnx, 1 Ixl < 1. A < Xma~ (2.11a) (2.11b) 2.8 V o i c e C o m m u n i c a t i o n = For most voice c o m m u n i c a t i o n , very low speech volumes predominate; about 50% of the time, the voltage characterizing detected speech energy is less than 1/4 of the root-meansquare (rms) value. Large amplitude values are relatively rare; only 15% of the time does the voltage exceed the rms value. The quantization noise depends on the step size. When the steps are uniform in size, the quantization is called the uniform q u a n t i z a t i o n . Such a system would be wasteful for speech signals; many of the quantizing steps would rarely be used. In a system that uses equally spaced quantization levels, the quantization noise is same for all signal magnitudes. Thus, with uniform quantization, the signal-to-noise ratio (SNR) is worse for low-level signals than for high-level signals. Nonuniform q u a n t i z a t i o n can provide fine quantization of the weak signals and coarse quantization of the strong signals. Thus, in the case of nonuniform quantization, quantization noise can be made proportional to signal size. Improving the overall SNR by reducing the noise for predominant weak signals, at the expense of an increase in noise, can be done for rarely occurring signals. The nonuniform quantization can be used to make the SNR a constant for all signals within the input range. For voice, the signal dynamic range is 40 dB. N o n u n i f o r m q u a n t i z a t i o n is achieved by first distorting the original signal with logarithmic compression characteristics and then using a uniform quantizer. For small magnitude signals, the compression characteristics have a much steeper slope than the slope for large magnitude signals. Thus, a given signal change at small magnitudes will carry the uniform quantizer through more steps than the same change at large magnitudes. The compression characteristic effectively changes the distribution of the input signal magnitude so there is no preponderance of low-magnitude signals at the output of the compressor. After compression, the distorted signal is used as an input to a uniform quantizer. At the receiver, an inverse compression characteristic, called expansion, is used so that the overall transmission is not distorted. The whole process (compression and expansion) is called c o m p a n d i n g . • The ~-Law, used in North America, is as follows: y ym~x In [1 + Ixl/xmax] in [1 + ix] sgnx. 1 + ln(lx[/Xmax) 1 + in A The A is the positive constant, and A = 87.6 is the standard value used in Europe. Example 6 The information in an analog waveform with maximum frequency fm= 3 kHz is transmitted over an M-ary PCM system, where the number of pulse levels is M = 32. The quantization distortion is specified not to exceed 4- 1% of the peak-to-peak analog signal. (1) What is minimum number of bits/sample or bits/ PCM word that should be used? (2) What is minimum sampling rate, and what is the resulting transmission rate? (3) What is the PCM pulse or symbol transmission rate? Solutions: lel _<pyre, where p is fraction of the peak-to-peak analog voltage• • le axl- Vpp 2L ' • \2t < pv,,). - • 2R:L>--. -- 2p 2R > I 1 - 2 × 0.01 -- 50, useR=6. .'.(R>5.64) • f~ = 2fro = 6000 samples/sec. fs = 6 x 6000 = 36 kbps. • M = 2/, = 32. (2•10) b = 5 bits/symbol. 36000 5 -- 7200 symbols/sec. sgnx = 1, x _ 0. sgnx = - 1 , x < 0. Rs-- 964 Vijay K. Garg and Yih-Chen Wang 2.9 Encoding are shown in Figure 2.12 in which the codeword is 4-bit representation of each quantized sample. In the bit duration p o r t i o n o f Figure 2.12, each binary 1 is represented by a pulse, and each binary 0 is represented by the absence of a pulse. If we increase the pulse width to the m a x i m u m possible (equal to bit duration, t), we have the waveform shown in the + V and - V b o t t o m p o r t i o n of Figure 2.12. Rather than describe this waveform as a sequence of present or absent pulses, we can describe it as a sequence of transitions between two levels. W h e n the waveform occupies the upper voltage level, it represents a b i n a r y 1; when it occupies the lower voltage, it represents a binary 0. We need an e n c o d i n g process to translate the discrete sets of sample value to a more appropriate form of signal. Any plan to represent each of the discrete sets of value as a particular arrangement of discrete events is called a code. One of the discrete events in a code is called a code s y m b o l or symbol. A particular arrangement of symbols used in a code to represent a single value of the discrete set is called a c o d e w o r d or character. Most c o m m o n l y used pulse code modulation (PCM) waveforms are classified into the following groups: • • • • Nonreturn to zero (NRZ) Return to zero (RZ) Phase-encoded Multilevel binary 2.9.1 Encoding Schemes The following encoding schemes are often used. (1) N o n r e t u r n to Zero-Level (NRZ-L) • 1 = high level • 0 = low level (2) N o n r e t u r n to Z e r o - M a r k (NRZ-M) • 1 = transition at the beginning of interval • 0 = no transition (3) N o n r e t u r n to Zero-Space (NRZ-S) • 1 = no transition • 0 = transition at the beginning of the interval (4) R e t u r n to Zero (RZ) • 1 = pulse in first half of bit interval • 0 = no pulse (5) Biphase-level (Manchester) • 1 = transition from high to low in middle of interval • 0 = transition from low to high in middle of interval (6) B i p h a s e - M a r k • Always a transition at the beginning of interval • 1 = transition in middle of interval • 0 = no transition in middle of interval (7) B i p h a s e - s p a c e • Always a transition at the beginning of interval • 1 = no transition in middle of interval • 0 = transition in middle of interval (8) Differential Manchester • Always a transition in middle of interval • 1 = no transition at the beginning of interval • 0 = transition at beginning of interval (9) Delay m o d u l a t i o n (Miller) • 1 = transition in middle of interval • 0 = no transition if followed by 1, or transition at end of interval if followed by 0 (10) Bipolar • 1 = pulse in first half of interval, alternating polarity from pulse to pulse • 0 = no pulse Codeword t i m e s l o t s The reason for the large selection relates to the differences in performance that characterize each waveform. In selecting a coding scheme for a particular application, some of the parameters worth examining are: • • • • • • The dc c o m p o n e n t Self-clocking Error detection Bandwidth compression Noise i m m u n i t y Biphase level (Manchester code) Code word time slot I. 1 Bit duration I 0 1 1 "1 I 1 0 0 N" +v I I H FIGURE 2.12 I I -V Bit Sequence and Waveform