COHERENT DIRECT-SEQUENCE SYSTEMS
1.7 SLOW FADING MULTIPATH CHANNELS
In many radio channels signals reflect off the surface of water, buildings, trees, etc., causing multiple signal terms at the receiver. The atmosphere also causes reflections where sometimes the reflected signals are used as the pri- mary means of sending signals from transmitters to receivers. Examples include shortwave ionospheric radio communication at HF (3 MHz to 30 MHz), tropospheric scatter radio communication at UHF (300 MHz to 300 MHz) and SHF (3000 MHz to 30,000 MHz), and ionospheric forward scat- ter radio communication at VHF (30 MHz to 300 MHz). These fading mul- tipath channels are usually modeled as having a randomly time-varying filter together with noise and interference [33].
Examination of the DS/BPSK system in a fading multipath channel begins by defining the DS/BPSK signal in (1.4) with an arbitrary phase term u, i.e., (1.158) The simplest multipath example is where thee is an unfaded direct path sig- nal and one reflected path signal. The received signal has the form
(1.159) Hereais the reflected signal amplitude term,tis its delay relative to the direct signal, and uis its phase relative to the direct signal. Here x(t)⫽ c(t)s(t) is the DS/BPSK signal with no fading and J(t) is the jamming signal.
y1t2⫽x1t; 02 ⫹ax1t⫺t;u2 ⫹J1t2. x1t;u2 ⫽c1t2d1t222S cos30t⫹u4.
Es>NJ
Es>NJ
Pb⫽12re1⫺B
r1Eb>NJ2 1⫹r1Eb>NJ2f.
The usual DS/BPSK receiver would multiply y(t) with the PN waveform c(t) and then compute the cosine component of the carrier,
(1.160) where
(1.161) and
(1.162) Evaluating (1.160) using (1.161) and (1.162) gives
(1.163) where
(1.164) is a Gaussian random variable with variance NJ/2 and
(1.165) whered0is the value of the data signal d(t) during the interval (0,Tb).
Suppose in (1.159) the multipath delay tsatisfies
(1.166) whereTcis the PN chip time. Then for each t,c(t) and c(t⫺ t) are inde- pendent and n⬘ is a sum of independent random variables which can be approximated as a Gaussian random variable with variance no greater than a2S/(2Wss). Since typically SVJis assumed, for all realistic values of a, this multipath noise term is negligible compared to the noise term n0which is due to jamming. Thus we have the approximation
(1.167) for multiplath delay tsatisfying (1.166).
When the multipth delay tis greater than the chip time Tc, there is neg- ligible degradation due to multipath. This also applies when the direct path experiences slowly varying frequency non-selective fading as discussed in the previous section. In general, however, it is possible to do better. Assuming the channel is slowly varying so that the multipath parameters a,t, and u
r0⬵d02Eb⫹n0
n0œ
tⱖTc
⫽ad022S冮tTbc1t2c1t⫺t2cos3v01t⫺t2⫹u4fc1t2dt
n0œ ⫽ 冮0Tbc1t2ax1t⫺t;u2fc1t2dt
n0⫽ 冮0Tbc1t2J1t2fc1t2dt
r0⫽d02Eb⫹n0œ ⫹n0
0ⱕtⱕTb. fc1t2⫽B
2 Tb
cos v0t
⫽s1t2⫹c1t2ax1t⫺t;u2⫹c1t2J1t2 r1t2⫽c1t2y1t2
r0⫽ 冮0Tbr1t2fc1t2dt
are known to the receiver, the receiver can multiply y(t) with c(t⫺t) and find the cosine component corresponding to the coordinate
(1.168) This results in the cosine component relative to the multipath signal of the form (ignoring th direct path noise term)
(1.169) where
(1.170) is a Gaussian random variable with variance NJ/2. Thus there are two out- puts of the channel given by r0andr1.
Except for some unrealistic waveforms for the jammer, the correlation betweenn0andn1is zero and thus these are independent Gaussian random variables. The optimum decision rule5based on r0andr1is to compare
(1.171) with zero as in (1.19). The bit error probability is thus
(1.172) This bit error probability is better than using a conventional DS/BPSK receiver which only uses r0in its decision.
Condition (1.166) for multipath delay results in a diversity system where two independent channel outputs are available. Thus DS/BPSK spread-spec- trum signals not only provide protection against jamming but also can resolve multipath and take advantage of the natural diversity available.
For a multipath channel with Lpaths and a resulting channel output (1.173) it is possible to compute at the receiver the Loutputs
(1.174) wherefc(t⫺t;u) is given in (1.168). This assumes the receiver has complete knowledge of the multipath statistics which include amplitudes {al}, delays
rl⫽ 冮tTb⫹tl
l
c1t⫺tl2y1t2fc1t⫺tl;ul2dt y1t2 ⫽ a
L
i⫽1alx1t⫺tl;ul2⫹J1t2 Pb⫽Qa
B
11⫹a222Eb
NJ
b.
⫽d011⫹a222Eb⫹n0⫹an1
r⫽r0⫹ar1
n1⫽ 冮tTb⫹tc1t⫺t2J1t2fc1t⫺t;u2dt
r1⬵d0a2Eb⫹n1
tⱕtⱕTb⫹t. fc1t⫺t;u2⫽
B 2 Tb
cos3v01t⫺t2⫹u4
5This can be obtained by using the maximum-likelihood rule of comparing p(r0,r10d⫽ ⫺1) with p(r0,r10d⫽1).
{tl}, and carrier phases {ul}. If now
(1.175) then we have the approximation
(1.176) where {nl} are independent Gaussian random variables with variance NJ/2.
Here the optimum decision rule is given by (1.19) with
(1.177) and the bit error probability is
(1.178) whereEbis the energy of any single multipath signal when the amplitude term is unity (a⫽1). The energy due to all multipath terms is
(1.179) and (1.178) can be rewritten as
(1.180) Next suppose that each multipath signal has a slowly varying independent fade. The Lreceiver outputs are then given by
(1.181) where {Al} are independent fade random variables. Conditioned on
(1.182) the bit error probability and its Chernoff bound are given by
(1.183)
⫽ 1 2 q
L l⫽1
e⫺Al2Eb>NJ.
ⱕ12e⫺1al⫽1L Al22Eb>NJ
Pb1A2⫽Q° R
a a
L l⫽1
Al2b2Eb
NJ
¢ A⫽ 1A1,A2,p,AL2
l⫽1, 2,p,L rl ⬵ d0Al2Eb⫹nl
Pb⫽Qa B
2ET
NJ
b. ET⫽ a a
L l⫽1al2bEb
Pb⫽Q°Ba a
L i⫽1
al 2b2Eb
NJ
¢ r⫽ a
L l⫽1
alrl
l⫽1, 2,p,L r1 ⬵ d0a2Eb⫹nl
冨ti⫺tj0 ⱖTc for all i⫽j
AssumingAlhas probability density function pl(⭈) for each l⫽1, 2, . . . ,L the average bit error bound is
(1.184) which for Rayleigh amplitudes with
(1.185) becomes
(1.186) where is the average signal energy in the l-th multipath signal given by (1.187) Note that when all the multipath energy terms are identical the exact expression for Pbis given by (1.146) with d⫽L. The general exact expres- sion will be shown in (1.201).
In many channels, such as the tropospheric scatter channel, it is more appropriate to view the received signal as consisting of a continuum of mul- tipath components. Such channels are usually characterized with channel output (see Proakis [33]),
(1.188) whereH(t;t) is a randomly time-varying filter.
Associated with the randomly time-varying filter H(t;t) are two basic parameters;Tmdenotes the total multipath delay spread of the channel and Bddenotes the Doppler spread of the channel. From these define
(1.189) as the “coherence bandwidth” and
(1.190) as the “coherence time” of the channel. Roughly, if two CW signals of fre- quency separation greater than ⌬fcwere transmitted through the channel, then the output signals at the two carrier frequencies would have indepen- dent channel disturbances (phase and envelope). Similarly, when a single CW signal is transmitted through the channel, its output sampled at time sepa- rations greater than ⌬tcwould have independent channel disturbances at the sample times.
¢tc⫽ 1 Bd
¢fc⫽ 1 Tm
y1t2⫽ 冮⫺qqH1t;t2x1t⫺t;1t4u2 1t⫺t2 2dt⫹J1t2
El⫽sl2
Eb; l⫽1, 2,p,L. El
Pbⱕ 1 2 q
L l⫽1
a 1
1⫹El>NJ
b sl2⫽ 冮0qa2pl1a2da; l⫽1, 2,p,L
Pbⱕ 1 2 q
L l⫽1
e冮0qe⫺a2EbNJp11a2daf
For the uncoded DS/BPSK signal one data bit is transmitted every Tbsec- onds. Assume
(1.191) so that intersymbol interference between data bits can be ignored. Also assume the channel is slowly varying so that
(1.192) Thus the channel disturbance is almost constant during a data bit time Tb. Finally, since our signal is a wideband signal of bandwidth Wssassume
(1.193) This model assumes many independent scatters are causing the continuum of multipath components. Thus the resulting channel output signal term is the sum of many independent scatters which justifies assuming it is a Gaussian random process. At any time it has a Rayleigh envelope probability distribution and an independent phase uniformly distributed over [0, 2p].
Skywave propagation where an HF signal is reflected off the ionosphere is an example whre this model applies. If, however, there also exists a strong unfaded signal component such as a groundwave at HF, which appears in shorter ranges between transmitter and receivers, the signal out of the chan- nel is assumed to have Rician fading statistics. In the following assume Rayleigh fading only.
Since the transmitted signal x(t) has bandwidth Wsscentered at carrier fre- quency v0radians, it can be represented in terms of samples of the inphase and quadrature components of the signal at sample times {n/Wss:n⫽. . . ,
⫺1, 0, 1, 2, . . . }. Thus the channel output can be modelled as
(1.194) where at any time tthe envelope terms are independent Rayleigh random variables and all phase terms are independent and uniformly distributed over [0, 2p]. Also, since the total multipath spread is Tm, for all practical purposes we can truncate the number of terms to
(1.195) The assumption regarding the slowly varying nature of the channel where (1.192) holds means that An(t) and un(t) are constant during the bit time Tb. Thus
(1.196) and the same situation as before with the finite number of distinct multipath components shown in (1.173) occurs here.
y1t2⫽ a
L
n⫽1Anxat⫺ n Wss
; unb ⫹J1t2 L⫽WssTm⫹1.
y1t2⫽ a
q
n⫽⫺qAn1t2xat⫺ n Wss
; un1t2 b ⫹J1t2 Wss W ¢fc.
Tb V ¢tc. Tb W Tm
For this case the receiver can compute outputs for individual multipath signals as follows,
(1.197) provided the phase terms {un} are known at the receiver. Also assuming envelopes {An} are known, the optimum receiver compares
(1.198) to zero to make the binary decision .
Next examine the optimum receiver structure implied by (1.197) and (1.198). Note that (1.197) can be rewritten as
(1.199) and so (1.198) becomes
(1.200) Figure 1.17 shows a block diagram for the optimum demodulator.
The ideal tapped delay line receiver of Figure 1.17 attempts to collect coherently the signal energy from all the received signal paths that fall within the span of the delay line and carry the same information. Because its actions act like a garden rake this has been coined the “Rake receiver” [23].
The bit error bound using the ideal Rake receiver is given by (1.186). An exact expression is given by (see Proakis [3], Chapter 7)
(1.201) where
(1.202) The ideal Rake receiver assumes complete knowledge of the phase and envelope terms which appear in the correlation functions {Anc(t)fc(t;un)}.
When the fading is slow this estimate is quite good.
A simpler form of the optimum receiver can be obtained using complex baseband representations for the radio signals on a carrier frequency of
l⫽1, 2,p,L. pl⫽ q
L k⫽1 k⫽l
a El
El⫺Ek
b Pb⫽ 1
2 a
L l⫽1
plc1⫺ B
El>NJ
1El⫺Ek
d r⫽ 冮0Tbcn⫽1aL yat⫹ Wnss
bAnc1t2fc1t;un2dt. n⫽1, 2,p,L
rn⫽ 冮0Tbyat⫹ Wnss
bc1t2fc1t;un2dt dˆ
r⫽ a
L n⫽1Anrn
n⫽1, 2,pL rn⫽ 冮n>WTb⫹n>Wss
ss
cat⫺ n Wss
by1t2fcat⫺ n
Wss
; unbdt;
v0radians. In general, the radio signal f(t) with a carrier frequency v0has representation
(1.203) whereg(t) and h(t) are real-valued functions and
(1.204) is the complex baseband signal representing the radio signal f(t). Using script letters to represent complex baseband signals
(1.205) where
(1.206) Assuming the carrier frequency is much greater than the signal bandwidth, the form for rin (1.200) is given by
(1.207) The receiver structure of Figure 1.17 can implement this complex form of the optimum receiver by replacing y(t) by (t) and Anc(t)⌽(t;un) by
for each n.
We now examine a way of estimating which is required in a practical Rake receiver. Note that
(1.208)
⫹ a
L n⫽1 n⫽l
Anc1t2cat⫺ n⫺l Wss
bdat⫺ n⫺l
Wss
b22Sejun.
⫽Alejul22Sd1t2⫹c1t2j1t2
⫹c1t2j1t2 yat⫹ l
Wss
bc1t2⫽c1t2a
L
n⫽1Ancat⫺ n⫺l Wss
bdat⫺ n⫺l
Wss
b22Sejun Ane⫺jun
Ane⫺junc1t2 f
⫽Ree冮0Tbcn⫽1aL B 2
Tbyat⫹ n Wss
bAne⫺junc1t2 ddtf. r⫽Ree冮0Tbcn⫽aL1
yat⫹ n Wss
bAnc1t2£*1t;un2 ddtf y1t2⫽ a
L n⫽1
Ancat⫺ n Wss
bdat⫺ n
Wss
b22Seju⫹j1t2.
£1t;u2⫽B 2 Tb
eju; x1t;u2⫽c1t2d1t222Seju,
y1t2⫽Re5y1t2ejv0t6 J1t2⫽Re5j1t2ejv0t6, f1t;u2⫽Re6£1t;u2ejv0t6, x1t;u2⫽Re5x1t;u2ejv0t6,
f1t2 ⫽g1t2ejh1t2
⫽Re5f1t2ejv0t6 f1t2⫽g1t2cos3v0t⫹h1t2 4
Figure 1.17.Ideal Rake receiver.
Recall that c(t) is independent of c(t⫺(n⫺l)/Wss) for each value of twhen n⫽land thus,
(1.209) where d0is the data bit in (0,Tb) and nlis a Gaussian random variable.
This suggests that the estimate for be given by the conjugate of (1.209). This estimate, however, includes the data bit d0. Assuming
remains unchanged over 2Tbseconds, an estimate can be based on the pre- viousTbsecond channel output signal. This estimate is shown in the com- plex form of the Rake receiver illustrated in Figure 1.18.
Ane⫺jun Ane⫺jun
冮0Tbyat⫹ W1ssbc1t2dt⫽Alejul22Sd0#Tb⫹nl
Figure 1.18. Rake with estimates.
Thus far the results in this section have applied only to uncoded DS/BPSK signals in slow fading multipath channels. With coding and ideal interleavers and deinterleavers the channel disturbance can be assumed to be indepen- dent for each transmitted coded bit. Assuming the soft decision metric of (1.120) where ris given by (1.207) and xH{⫺1, 1} the pairwise error bound for two sequences xand is given by (1.131) where
(1.210) From (1.181) and (1.198)
(1.211) where {nl} are i.i.d. zero-mean Gaussian random variables with variance NJ/2.
Thus
(1.212) where the amplitudes {Al} are again assumed Rayleigh distributed with vari- ance as in (1.185). Then, define parameter
(1.213)
where is given by (1.187) with Ebreplaced by Es.
Suppose is the total average energy and each multipath energy term is the same. That is,
(1.214) El⫽ ET
L l⫽1, 2,p,L. ET
El
⫽min
sⱖ0q
L
l⫽1a 1
1⫹21El>N012s⫺s22b D⫽min
lⱖ0q
L l⫽1
a 1
1⫹2sl212l2Es⫺l2NJ2b
⫽ q
L
l⫽1a 1
1⫹2sl212l2Es⫺l2NJ2b
⫽ q
L l⫽1
E5e⫺A2l12l2Es⫺l2NJ26
⫽ q
L
l⫽1E5e⫺2lAl22EsE5e⫺2lAlnl0Al66
⫽ q
L l⫽1
E5E5e⫺2l3Al22Es⫹Alnl40Al66 D1l2⫽ q
L
l⫽1E5e⫺2l3Al22Es⫹Alnl46
⫽ a
L
l⫽11d0A2l2Es⫹Alnl2 r⫽ a
L l⫽1
Alrl
D1l2 ⫽E5elr1xˆ⫺x20x6, xˆ ⫽x. xˆ
Then s⫽1 minimizes the bound and
(1.215) For the hard decision channel with the usual unweighted metric we have simply
(1.216) where now ⫽Pbgiven by (1.201) and (1.202).