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Unformatted text preview: Noncoherent detection of CarrierModulated 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). > noncoherent 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 symbolbysymbol detection
mentioned below. Remark: noncoherent detection is only an issue in BP transmission Structure of Optimal Non'coherent Demodulator Consider an Mary 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.
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.. _....m.«..—~.1' '1. n r ", i=0.1,....M—l, (3.5.8b) H
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a
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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 magnitudesquared expressidn in (3.5.8b) andeliminating 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‘lKEﬁer’gthIiéhHi;Th'usfiiising (3.4.26) _ _.P m arid t IS conversion, we ﬁnd that 1
= ginEMU, 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 backtoback 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. backto—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 doubleerror effect. In any case. the marginal Pb is correctly
expressed in (3.5.13). Returning‘lm‘Mary DPSK case, the calculation of__syrnbol__ qugprgbgp‘ility
would Estcalculate the puff for the _modulo‘24fr _phase'Idifference _of twophas‘ors cor
rupted __ twofdimensional independent Gaussian pfd’ff. lisiifo‘riiiulatcd'in ' ' ' '“ _.. ..s,._mn—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 aanhowledgze 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 signaltonoise 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 dueta'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 MDPSK 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 4DPSK was selected as a modulation technique for one of the ﬁrst highspeed 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. 4ary DPSK, combined with channel coding, has been selected as the
modulation method for nextgeneration 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 4ary DPSK Suppose in a 4ary 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 timedivision 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 MDPSK by forming decisions based
on more than two consecutive symbols, which is called multisymbol detection of
MI)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 higherquality phase reference for detection than that obtained from
just the previous symbol. In some sense such detectors are acting as shortmemory 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, Gttussianﬁoise 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 highreliability 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 timeaveraged 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 lﬁQAM. 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
Mary orthogonal: a many signalspace dimensions per symbol
a most BW consumptive MPSK or MQAM : 0 less energy efficient
0 BW economic Now, compare these schemes against the bounds on this energyBW tradeoff 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)64QAM = 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 Mary 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 MFSK: 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
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Comparison of several modulation schemes at P, = 10'5 symbol enor probability. ...
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