NCM-and-energy-BW-trade-off - Non-coherent detection of...

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Unformatted text preview: Non-coherent detection of Carrier-Modulated Signals Until now, detection is performed when all parameters are known. These are: 0 amplitude a timing parameters 0 phase of the carrier coherent detection When carrier phase is not known, it is usually assumed as a random variable uniformly distibuted in [0,275). ----> non-coherent detection In carrier modulated systems oscillators of the transmitter and receiver must be locked. Otherwise, assume e.g. 1 Hz offset where fc = 100 MHz ----> phase error of 27: rad in 1 sec. One solution: Estimate carrier phase using a PLL , disadvantage: complexity Not knowing the phase will cause a penalty (some performance loss) This loss is made small with clever design, at least in the symbol-by-symbol detection mentioned below. Remark: non-coherent detection is only an issue in BP transmission Structure of Optimal Non'coherent Demodulator Consider an M-ary modulation system: air) = HIU}CDSIWJ + moi. T.- 5r 5 Tr. or in complex envelope notation: w) = Re {a,-(:)eiri“ieiw}, r, < r < '1}. where o ai(t) : real carrier amp. 0 yi(t) : phase modulation of ith signal Energy of ith signal: T; 2 i T: E, :1. s,- (f)d! = —[ a?(t)dr. r. 2 T. assuming carrier freq. is much greater than the signal bandwidth. MCH Q o H‘Cpflj Ob’l‘kogomxl SCGflaH'lhfi M‘A‘h Sequemceo $11M) 2 A M109 00—5 (“Ci (1 row $€quwC€ AVON c; \o‘mm‘rs LIKXM HQAMMA mcfirhf The Mcnkvad haicbm.) Com ‘0: ve.fr¢5m4ecl cu: P142] —.= g;(-£—t) + nae) = Sifi’c,‘c,87 + NH w‘nU-L 53431797: mam) 093(wa + wet-Z) *9), (P3: 0/1,--./3N" Hart 9: Lug—C (F; a {chime skrrf+ due 7L0 Je/CUJ C. If {OQWFed' byme'HmMj IFS known (2+ “Hm receJ'Ver :2) ‘er. effluf oi +ramami+ +fm; Adagf 21/ 0/1 Alfie modLLZQ+qu Comfp/peydj 3% be Pen/[Quecp' nggual‘" H”. carri‘ew“ phage, Q, :3 frermed wflmowq mum-“fin” _ m WU C m G eflada “k!— Gyrq ne3\ig§k\e COMPMQ‘Q +0 T5) :) 9 moo‘e\t.a\ Ob unifarm in [IO/1R). De r1 \rfirfgflrlmmgfi 091“me Dg‘l‘e c. 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Corm’aj‘JW '_‘ nTJ 0’“! ) T) ——> Xgn :j r(-&] 00>(Udc't') 0H +fer) wQ/uct‘ 11-31;”)0H’ ((1,1) T) (fl-0 TJ (fl'f)TJ HT} gm :j me) SAL (war) Ont Jrj m; .r—Lamcu (SM) A+ QM” (fi-IIT: @14ch (Si/m 1 h3p0}1’w$;1ed /\qC/‘e mwf a} L 1 fly“ ’1 - IV“ (A 1’" 120/3"qu q L ‘ 2 H) "hm : [QM + PC“ cm 3”“ —— r‘gn . M gym] 1 + [QM + Q“ 003 5m + {En W Sim] gkj kapr‘ Hc+a+l°4 , r ’7. '2. +2) Uzi/M] a-ijm :ifn + If»: e / . . . _ , I Bmgng phase: notation for the two consecutive measurements. we can see that 2” IS equivalent to ,i ~ . .. _....m.«..-—~.-1--' '1. n r ", i=0.1,....M—l, (3.5.8b) H H a + :1 7a E X12. where r“ = rt.” +yr'r5a represents the integrator output at Equation (3.5.8b) can be i ' 3‘" interpreted as flaring the previous phasor by the hypothesized phase advance and then the magnitude-squared expressidn in (3.5.8b) and-eliminating COmmon terms for each Q “a; statistic. we may equivalently a ___,__ ‘T . . H_ _ -Wun.wm ; summingwuh thecurrent phasor__an fornnngt emagnttude squared. Upon expanding . _... _ I _........._..- l a Wit & '} maximize Re lranIie—flmlw ii (3.59:1) \/ If we interpret rn as a vector in the plane and r:_IeJ3”M as another, the decision should maximize rn - (rwleh’zmlm). '1 {3.5.9b) i___ I I _4 l -..._I' which is just the vector inner or dot product between the current measurement and the rotated previous measurement vectors. Thus. optimal processing can be reinterpreted as follows: rotate the first phasor by the hypothesized phase advance; then compute the vector inner product with the next phasor and decide in favor of the largest. Still another formulation of the optimal differential detector __fo_llo_w_s from applying Euler’s relation to (3.57975): yielding i _ _ _ 2H! maximize |r,,||r,.-1lcos (ya — 12..-] — (3.5.10) 1 from which it becomes clear that the Wthe phasors are irrelevant, and only 4—-—_..__——..._._ 6,, = y” — y,,_l is important. Thus, the phase—differencing demodulator is optimal when the decrsion_1s_based on two consecutive symbol intervals. (for small “B. the DPSK receiver can be implemented in a manner that avoids much of the comp exity of Figure 3.5.1, in particular the inverse—tangent operation. In the binary DPSK case, (3.5.10) simplifies to testing whether the measured phase difference 5,, exceeds rr/2 in magnitude. [1:30, then the decision 1 is produced; else {I is decided. Wm vector inner product oi two consecutive (nonrotated) phasors is negative or positive. In terms of the data produced within the demodulator, we have the test 0 omw+amq:o gun I which defines the binary receiver of Figure 3.5.33. Similarly. in the M : 4 case. the processing may be interpreted as sign tests on the vector inner product and cross product of consecutive measurements (Exercise 3.5.3). '— An altemative binary DPSK receiver is shown in Figure 3.5.3b, involving a front- end filter matched to the signal over one interval. that is, a constant phase sinusoid of duration T; seconds, followed by a delay line and a sampled phase detector. .[It is imperative that the delay be nearly equivalent to a multiple of 2a radians at the Operating center frequency of the detector. certainly an implementation difficulty.) Sec. 3.5 Phase Comparison or Dilferentially Coherent Demodulation oi PSK 219 Delay TS (c) Figure 3.5.3 Two implementations of binary DPSK demodulator. [a3 base— band inner product form; (b) matched filterfdclay line form; (c) wrong imple- mentation of binary DPSK demodulator. In the detector of Figure 3.5.310. all essential noise filtering is performed by the matched filter. Prevalem in textbooks. but suboptimal. is the receiver shown in Fig- ure 3.5.3c, which reverses the order. Notice that the phase detection step (multiplication of two bandpass signals} indicated in Figure 3.5.3b is nonlinear with respect to the input. and we simply cannot commute the order of the operations. Although this receiver pro- duces correct decisions for sufficiently high SNR. its performance is substantially worse than the optimal DPSK receiver at SNRS of interest. 3.5.2 Performance Evaluation for M -DPSI( To evaluate the demodulator error probability. we consider first the binary case. One helpful way of visualizing binary DPSK is as a binarv orthogonal design lasting 2?", seconds. Each interval uses the previous bit as a phase referenccfsfiefiight write the 220 Modulation and Detection Chap. 3 two signal hypotheses as concatenations in time of consec tive PSK si inals: / 25$ “3 25E “3 r sot!) <——+ C05(0)c!+9), T eos(wct+i9}, 1 T5 3 253 “2 25. "2 51(r) <——+ T cosmic! +9). - T C05fwcf + 6). S .‘F (We recmphasize that a new bit is transmitted every T; seconds, and the previous signal becomes the new reference signal.) We have just argued that the DPSK receiver is in effect a noncoherent detector over two intervals. Since in the binary case the signal pairs in (3.5.12) are orthogonal, the bit error probability for binary DPSK can be evaluated using binary noncohereptoflhoggnal performance, exce t with the effective symbeo‘l-KEfier’gthIié-hHi;Th'usfiiising (3.4.26) _ _.P m arid t IS conversion, we find that 1 = gin-EMU, binary DPSK. AWGN (3.5.13) . r. When M = 2, DPSK has only slight loss in energy efficiency relative to coherent PSK. At Pb = 10—5, DPSK requires about 10.4 dB lib/ND. whereas coherent PSK requires about 9.6 dB. Differentially encoded, coherently detected PSK requires about 9.9 dB. Some propensity exists for paired or back-to-back symbol errors with DPSK, but it is not so strong as in differentially encoded coherent PSK. where an isolated error on one bit produces a paired error upon differential decoding. To see why the tendency is less. consider again the case of M : 2. A common error event is of the following form: the first phasor experiences a phase error of. say. final = —50". while the second phase error is 13,, = +45”. The phase difference 5,. thus exceeds 90", inducing a decision error. However, if the next phase error is less than 45'”, a subsequent symbol error is not made. This pattern is far more likely than “error near 0‘, error of _100". error near 0“ which would induce paired symbol errors. Thus. back-to—back decision errors are not as predominant as might be expected. Salz and Salzherg [28] and Oberst and Schilling [29] give an analysis of this double-error effect. In any case. the marginal Pb is correctly expressed in (3.5.13). Returning‘lm‘M-ary DPSK case, the calculation of__syrnbol__ qugprgbgp‘ility would Est-calculate the puff for the _modulo‘24fr _phase'I-difference _of twophas‘ors cor- rupted __ twofdimensional independent Gaussian pfd’ff. lisiifo‘riiiulatcd'in ' ' ' '“ _.-. ..s,._--m-n—-m'fin.--r Pawtlla el al. ['30]: i I 1 “3 E «253 R.) f(5) : tsinx) 1+ ——S(l -l- cosésinx) -exp (i w coséisinx) (1);. 2r: 0 N0 N0 (3.5.14) In Figure 3.5.4, we show the p.d.f. for the phase difference measurement when ES/N” : 10 dB. given that zero phase difference occurred at the transmitter. This p.d.f. can be integrated numerically over the region Nil 3 “if/£10 produce P5, and presentation of this analysis is found in Lindsey and Simon [12]. Figure “3—375 -—-—__L presents the results graphically. showing M —ary coherent detection for comparison. As Sec. 3.5 Phase Comparison or Differentially Coherent Demodulation of PSK 221 | :3.er0 = 10 dB \ ESIND = 5 dB —tt 0 rt [3 Figure 3.5.4 Phase difference p.d.f. expected. the demodulator‘s lack of aanhowledg-ze always costs inverter.qu efficiency; the difference is small for binary DPSK, but for itifgé’thrf‘s'aflELEhSK, the penalty is nearly 3 dB. We might have anticipated this, because in DP K detection two ' noise vectors influence the phase difference. At high signal-to-noise ratio, the phase error foreach symbol is nearly Gatflsianz and thus the pmflttt‘EEEfimE'fcihfih‘if'dfiiiié‘fian (Eat: Figure 3.5.4 tEFéiEfifile). but with tWice remittance due-ta'fitrepeaaeaeeer the measurements. (This is pursued funhcr in Exercise 3.5.2.) An alternative method to calculating error probability uses the p.d.f.‘s for the ran dom variables in (3.5.7), following the earlier analysis of the noncoherent detector. The signals are not orthogonal, however. over 2'1"F in the nonbinary case. and noncentral chi~Squared statistics are enc0untered [31]. Because the energy efficiency of M-DPSK is quite poor for M 3 es cially rel- ative to the coherent counte art, these designs are rarely found in modern practice when energie‘fhmency is a primary concern. Binary DPSK, however, represents an effective alternative to binary PSK. with or without additional coding. and 4-DPSK was selected as a modulation technique for one of the first high-speed modems. the Bell model 201 HOG—bps telephone channel modem, implemented in 1962. There. receiver simplicity was of paramount concern. as well as bandwidth economy. and channel SNR was nom- inally rather high. 4-ary DPSK, combined with channel coding, has been selected as the modulation method for next-generation digital cellular telephony in the United States. If bit error probability is to be minimized, the phase changes should be Gray- coded. since the most likely phase difference error is to an adjacent region, and such cases should produce minimal bit errors. Under such conditions, PJ— Pb % log2 M . (3.5.15) as for coherent PSK. Although the use of DPSK avoids needing to know absolute carrier phase, it is important that the receiver be well synchronized in frequency. If it is not. the measured 222 Modulation and Detection Chap. 3 0 3 6 9 12 15 18 21 Eb/Ng, dB Figure 3.5.5 Symbol error probability for M—ary DPSK. phase increments will be biased away from the middle of the decision zones, increasing the symbol error probability. A good rule of thumb for binary DPSK reception is to maintain Auin 5 0.1, where Aw is the radian frequency offset [32]. This ensures that the carrier phase 6 drifts by less than 0.1 radian during one bit interval. Generalizing this to the M—ary case, we might then require that AmTj 5 (ll/M, since the decision regions shrink inversely with M. The frequency offset Af must then be held to less than about RS/oUM, where R, is the symbol rate. Example 3.12 4-ary DPSK Suppose in a 4-ary DPSK receiver that the following sequence of phase measurements [in degrees} is observed over five consecutive symbol intervals: 39. l 10. 50. 239. 21. The phase difference sequence, modulo Err. is 7'1"). 300°, 189°, 142", and these are mapped to symbols 1.3.12 according to {3.5.1}. If Eb/No = 9 dB on this channel, then the probability of a symbol error from Figure 3.5.5 is 3- 10—3. Sec. 3.5 Phase Comparison or Difterentialty Coherent Demodulation of PSK 223 For R; = 24 ksps, approximately the rate for the 15—54 digital time-division Cellular standard in North America, then by the above rule the required frequency accuracy must be less than about Af 5 24,000/(60 -4} a 100 Hz. This constitutes the allowable frequency offset for oscillator instability and Doppler shift combined. It is possible to improve the performance of M-DPSK by forming decisions based on more than two consecutive symbols, which is called multisymbol detection of M-I)PSK. Specifically, we can employ a sliding (or blOckl window of length (N + HT, to decide N consecutive data symbols, or perhaps just the oldest symbol of a sliding block. The qualitative notion is that a longer observation window allows effectively the establishment of a higher-quality phase reference for detection than that obtained from just the previous symbol. In some sense such detectors are acting as short-memory phase estimating schemes, and in the limit of large observation interval, the performance ap- proaches that of coherent detection with differential detection. For M > 2, the potential gains in energy efficiency are significant. and it has been Shevvn that use of N = 3 provides at least half the available gain. However. the receiver processing. if optimal noncoherent detection is pursued, is considerably larger, for M” hypotheses need to be examined for a window of length (N + UT: seconds. Furthermore, the constant phase as- sumption about the channel becomes more questionable. The interested reader is referred to [33] and [34] for a discussion of these possibilities. its PERFORMANCE ON THE SLOW, NONsELECIIvERAYLEtGH FADING CHANNEL We now study the perfomtance of the previous modulation and detection strategies on the slow, fiat~fading Rayleigh channel and will observe a fundamentally different dependence on signal—to—noise ratio than seen thus far for the nonfading, Gttussian-fioise channel. Specifically, instead of a negative exponential dependence on Eb/No common to all cases in Sections 3.3, 3.4, and 3.5, we shall find that the infrequent, very deep amplitude fading events induce a much weaker (inverse) dependence of P, on average Eb/Ng. This will be true for all uncoded transmission strategies, and the potential performance penalties due to fading are enormous for high-reliability systems. However, various channel coding tech- niques studied in later chapters will be able to substantially mitigate the effect of fading. To recall the model assumptions made at the beginning of the chapter, we assume the channel gain Aft) is a Rayleigh random process, but essentially fixed over the duration of one sytnbol's decision interval. In actuality, the amplitude is a slowly varying random process, and our primary interest is in the average error probability computed over the fading distribution. Assuming ergodicin holds for the process, the ensemble average performance we will compute would correspond to the time-averaged performance on an actual link. We should be aware though, that for any given channel the “instantaneous” error probability will fluctuate. A practical difficulty associated with fading channels is that the demodulator must know the channel’s scale factor A for optimal detection in those cases where the signals are not equal energy, for example. with on—off keying or lfi-QAM. Because this is some- times difficult to establish and because performance is sensitive to errors in this estimate, 224 Modulation and Detection Chap. 3 Summary of Energy and Spectrum Efficiency of Modulation Techniques M-ary orthogonal: a many signal-space dimensions per symbol a most BW consumptive M-PSK or M-QAM : 0 less energy efficient 0 BW economic Now, compare these schemes against the bounds on this energy-BW trade-off provided by channel capacity For fair comparison the required Eb/ No of each technique required to attain Pe = 10’5 Remember Nyquist result that to transmit R5 symbols/ sec, a BW of RS/ 2 Hz is required. For BP signalling a BW of RS Hz is required. But this loss can be regained using quadrature modulation as in PSK or QAM. Now, consider PSK/QAM/PAM class: Let bit rate = Rb o symbol rate: R5: Rb/(logz M) a required BW: B = Rb/( logz M) o spectral efficiency: Rb / B = log; M bps/Hz (QAM/PAM/PSK) Examples: ( Rb/ B)QPSK = 2 bps/Hz ( Rb/ B)64-QAM = 6 bps/Hz (these limits are optimistic, achievable spectral efficiencies may be 25 % less) Consider orthogonal/biorthogonal formats: Let bit rate = Rb bps o # of orthogonal dim. = Rb M / logz (M) dim/sec (since M-ary orthogonal signalling occupies M dim.s, and symbol rate, R5: Rb / logz (M) sps) Baseband transmission: B =( Rb M) / (2 logz (M)) dim/sec BP transmission: B is the same as in Baseband when quadrature modulation is utilized --> Rb/ B = (2 logz (M) / M ) bps/Hz (orthogonal signals) Alternative derivation: For M-FSK: Af = R5/2 (for orthogonality) BW required: M 7“ Rs/Z = Rb * M/ (2 "“ logz (M)) Hz --> same result. biorthogonal signalling --> a factor of 2 gain in spectral efficiency Consider the figure: (From Digital Comm. by Proakis and Salehi 5‘h ed.) 9 dB gap to capacity bound for most mod. schemes (i.e. 9 dB more power is used to achieve a given spectral efficieny) --> aim of comm. eng.s to gain this power ( —-> error correcting coding) {TllunnL'l 1', capacity limzl a.“ lfl M 'E- Q-‘i‘fl .'l-i‘ — *1 With“. {Si-d: . M iii 0AM M - if Pi". H [H l-ill'i P5”: or m . Tips]: .w — 4 I435. .rr :- PAM 5553;; I'd! .Mymrllclle' SE11: pm la-i',I y. [Li “j. lid—3;" {if—64 Ehhlagnml ‘i'HTIil-J. E'ollcfim’: dL'LEL'LIoIr. I,'|,' I’IIZLlT RIC ain‘t—I Comparison of several modulation schemes at P, = 10'5 symbol enor probability. ...
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NCM-and-energy-BW-trade-off - Non-coherent detection of...

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