and frequency-selective channel
EXTENDED MLSE DIVERSITY RECEIVER FOR THE TIME- AND
FREQUENCY-SELECTIVE CHANNEL
Brian Hart (hartbd@elec.canterbury.ac.nz), D.P.Taylor (taylor@elec.canterbury.ac.nz)
Department of Electrical and Electronic Engineering, University of Canterbury, New Zealand
Abstract This paper develops an MLSE diversity receiver for
the time- and frequency-selective channel corrupted by additive
Gaussian noise, when linear constellations are employed. The
paper extends Ungerboeck’s derivation of the Extended MLSE
receiver for the purely frequency-selective channel to the more
general channel with diversity. The new receiver structure and
metric assume a known channel. The major contributions of this
paper are as follows: (i) the derivation of a finite-complexity
diversity receiver that is Maximum Likelihood (ML) for all
linear channel models and sources of diversity; (ii) a benchmark,
in that the new receiver’s performance is a lower bound on the
performance of practical systems; (iii) insight into matched
filtering and ML diversity receiver processing for the time- and
frequency-selective channel; (iv) bounds on the new receiver’s
BER for ideal CSI, in a fast Rayleigh fading channel with
multiple independently faded paths.
I. INTRODUCTION
Shannon’s channel capacity theorem[1] proves that coding
schemes exist to transmit information with an arbitrary low
probability of error, assuming the information rate of the source
does not exceed the channel capacity. However, the coding
delay is not constrained and there is no guarantee that coding is
actually helpful in a system when the transmission delay is fixed.
For instance, even the simplest two-state interleaved trellis code
in the frequency-flat, Rayleigh fading channel requires an
interleaving delay of at least ∆s for optimal performance,
according to [2]
∆ > 153
. fD
(1)
where f D = vf c c is the one-sided Doppler spread; v is the
receiver speed; fc is the carrier frequency; and c is the speed of
light. For a receiver operating at 900MHz, travelling at walking
pace (4km/h), the Doppler spread is 3.3Hz and the interleaving
delay is 0.46s.
Furthermore, the code’s performance is
significantly impaired by reducing the interleaving. In channels
with significant frequency-selectivity, there is implicit delay
diversity, the error rate is less dominated by long, deep fades,
and so codes can enhance the system performance. The codes
still require the interleaving of (1) for optimal performance.
However, a robust communication system cannot rely on
Doppler or delay spread, so we argue that coding will be
insufficient for future systems that offer ATM-like services: i.e. a
common digital transmission system providing high bandwidth, a
low error rate, and a low delay.
Therefore, this paper analyses receiver structures for
frequency diversity and receiver space diversity. Maximal Ratio
Combining (MRC)[3] is optimal for slowly varying, frequencyselective channels. However, rapidly varying channels introduce
additional pulse distortion which must be explicitly removed to
avoid an ISI-induced error floor[4]. By following Ungerboeck’s
Extended MLSE receiver for the frequency-selective channel[5],
this paper derives an MLSE diversity receiver, assuming the
channel is known (ideal channel state information, CSI). The
new receiver extends the idea of MRC to rapidly varying
channels, and is known as the “Extended MLSE diversity
receiver for the time- and frequency-selective channel,” or the
EMLSE-tf diversity receiver.
II. RECEIVER DERIVATION
The transmitter maps a binary source to a sequence of M-ary
complex phasors, {ai}. The transmission interval is assumed to
extend sufficiently far into the past and future that the end points
can be assumed to be ±∞ without penalty. In complex baseband,
the transmitter sends
a(t ) = ∑ α i h(t − iT )
(2)
i
where h(t) is the transmitter pulse shape; and T is the symbol
[ ] = 1.
period. The phasors are normalised so that E α i
2
The diversity receiver receives D signals (threads). They
may arise from different antennae or different carrier
frequencies. The dth thread (d = 1 ... D) is the transmitted signal
distorted in time and frequency by the channel, modelled
generally by a complex, time-varying, filter, zd(t,x), and
complex Gaussian noise, nd(t), is added, resulting in the D
threads, y(t) = [y1(t) ... yD(t)]T, where each thread is described by
y d (t ) = a(ξ)∗ z d (t , ξ)
t =ξ
+ nd (t ) =
∞
d
d
∫ a(t − ξ)z (t , ξ)dξ + n (t )
(3)
−∞
For space diversity, the noise comes from different antennae and
front-end electronics.
For frequency diversity, the noise
occupies different bandwidths.
Therefore we assume
independent noise in distinct threads. The channel, zd(t,x), may
be correlated between threads, but we shall see that this has no
influence on the receiver structure for ideal CSI. Substituting (2)
into (3), we have
y d (t ) = ∑ α i c d (t − iT , iT ) + n d (t )
(4)
i
where cd(t-iT,iT) is the effective pulse shape, combining the
effects of the transmitter pulse shape and the time-varying
distortion of the dth channel, as expressed by
c d (t − iT , iT ) =
∞
d
∫ h(t − ξ − iT )z (t , ξ)dξ
−∞
(5)
Equation (4) is the familar notation for linear modulations, but
now there is an extra parameter, iT, since the effective pulse
shape is time varying.
The MLSE receiver searches all allowed symbol sequences in
the transmission interval and chooses the one with maximum
likelihood. Conditioned on the symbol sequence and the channel
process (i.e. ideal CSI), the log-likelihood of y(t) matches the
log-likelihood of the complex, zero-mean, Gaussian noise vector,
n(t) = [n1(t) ... nD(t)]T,
(
)
l y α ,z y α , z = ln (n) =
∑ ln (n d (t )) = ∑ l y α ,z ( y d (t ) α, z d ) (6)
D
D
d =1
d =1
which is just the sum of each thread’s noise log-likelihood,
(
)
∞ ∞
y (t1 ) − ∑ αi c (t1 − iT , iT )
i
−∞ −∞
ly α ,z y d (t ) α, z d ~ − ∫
×
(
−1
Rnn
t1
∫
(7)
− t2 ) y(t2 ) − ∑ α k c(t2 − kT , iT ) dt1dt2
k
since the noise is independent between threads. The overbar
denotes complex conjugation; the noise autocovariance is
defined by Rnn (τ) =
(
)
convolution. Following [5], the log-likelihood of each thread
can be transformed as
)
{
}
l y α ,z y d α, z d ~ ∑ 2ℜ α i mid − ∑ ∑ α l sld,k α k
where
d
l ,k
d
i
m and s
mid =
i
l
(8)
k
d
−1
d
∫ ∫ y (t 2 )Rnn (t1 − t 2 )c (t1 − iT , iT )dt1dt 2
−1
= y d (t1 )∗ Rnn
(t1 )∗ c d (−t1 , iT )
(9)
t1 =iT
∞ ∞
∫
d
d
−1
∫ c (t1 − lT , lT )Rnn (t1 − t 2 )c (t 2 − kT , kT )dt1dt 2
−∞ −∞
In slow fading, the presence of c d (.) in the mid equation is
equivalent to MRC, since it removes the channel’s phase
distortion, weights the signal according to the depth of the fade,
and then matched filters the signal. As with Ungerboeck’s
EMLSE-f receiver[5], there is no need for a noise-whitening
filter.
The sld,k term cancels or equalises the ISI introduced by the
channel’s time- and frequency-selectivity. In the time-invariant
channel, the ISI is fixed, so Ungerboeck’s EMLSE-f receiver is
able to precompute sl,k. This is not possible in time-varying
channels.
Substituting (8) into (6) leads to
l y α ,z (y α , z) ~
d =1
∑ 2ℜ{α i mi } −∑ ∑ α l sl ,k α k
i
l
where mi and sl,k are the sum of all the threads’
sld,k terms respectively,
(10)
k
mid and
D
∑ sld,k
(11)
d =1
sl ,k = sk ,l , can exploited by
properly grouping the double summation of (10) to obtain the
iterative metric[5],
∞
i −1
∞
) ∑ 2ℜαi mi − ∑ si ,k α k − αi 2 si,i = ∑ li
(
ly α , z y α , z ~
i =−∞
k =−∞
i =−∞
(12)
which can be calculated more efficiently. As we shall see, this
additive metric leads to a straightforward implementation using
the Viterbi algorithm.
Having assumed ideal CSI, the statistics of the channel are
irrelevant to the metric. The analysis thus applies to all noiselimited, linear channels, including dispersive, L-path[7],
Rayleigh, Ricean, log-normal and AWGN. When ideal CSI is
not available, the performance of this receiver still leads to a
lower bound on a practical receiver’s performance.
III. WHITE NOISE
For the important case of white noise with a two-sided
passband power spectral density of N0/2, we have in complex
baseband Rnn (τ) = N 0 δ(τ) . Since the metric of (12) is used for
comparisons only, the scale factor of N0 can be neglected. Then
δ(τ )
−1
using Rnn
∝ δ(τ) simplifies mi and si,k to
( τ) =
N0
mi =
∞ ∞
sl , k =
The conjugate symmetry of sl,k,
are defined as
−∞ −∞
sld,k =
D
∑ mid
−1
E n d (t )n d (t + τ) ; Rnn
(τ) is its inverse
1
2
−1
kernel[6], and satisfies Rnn (τ)∗ Rnn
(τ) = δ(τ) , where ∗ denotes
(
mi =
si ,k =
D ∞
∑ ∫ y d (t1 )c d (t1 − iT , iT )dt1
d =1 −∞
D ∞
(13)
∑ ∫ c (t1 − iT , iT )c (t1 − kT , kT )dt1
d
d
d =1 −∞
We note that mi can be considered as the output at t = iT (i.e.
symbol-rate sampled) of the sum of matched filters for the timeand frequency-selective channel (MF-tf). The signal is filtered
by a filter matched to the effective pulse shape in each thread.
Modern digital receivers operate in discrete time. The
receiver first applies a zonal anti-aliasing filter to the received
signal, with transfer function, rect ( fTr ) , where Tr = T/r. The
filtered signal is sampled at t = lTr, so the receiver takes r 1
samples per symbol period. Subscripts are used to denote time,
so that yl = y(lTr), cl-ir,ir = c(lTr-irTr,irTr), nl = n(lTr). The
integrals of (13) are replaced by summations, resulting in
min {ir , kr }
mi =
D ir + Lr −1
∑ ∑ yld cld−ir ,ir
d =1 l = ir
si ,k =
D
∑
+ Lr −1
∑
cld−ir ,ir cld− kr ,kr (14)
d =1 l = max {ir , kr }
where the factor of Tr is neglected, since the metrics are used for
comparison only. The absolute bandwidth of a time-limited
pulse is infinite, so strictly speaking a sampling rate of 1/Tr is
inadequate. However, we can normally use another bandwidth
definition (such as the -40dB bandwidth), add the maximum
Doppler spread, and sample accordingly. In this case, the
discrete and continuous time receivers are equivalent.
For the special case of effective pulses of less than one
symbol in duration (i.e. a short transmitted pulse that is shorter
than T even after delay-spread smearing), there can be no ISI, the
receiver makes hard, symbol-by-symbol decisions, and the
branch metric reduces to a Euclidean distance, li ~ mi − α i si ,i
2
. Exact BERs can be calculated for this case. Here, the
EMLSE-tf diversity receiver’s decisions have the same geometric
interpretation as the decisions of a receiver for the AWGN
channel, except that the signal space is warped in a time-varying
manner, according to si,i.
V. PERFORMANCE EVALUATION
We seek the discrete-time receiver’s BER in a fast Rayleigh
fading channel with multiple independently faded paths in white
noise. The analysis of the trellis-based receiver follows
Forney[8]. The BER is upper bounded by summing the pairwise
error probabilities between the transmitted sequence and some
erroneous sequence, weighted by the number of bit errors that
the error event introduces, and the probability that the
transmitted sequence is sent. For the Rayleigh fading channel,
the required random variables are Gaussian quadratic forms in
IV. RECEIVER STRUCTURE
cld−ir ,ir and nld , and their probability is evaluated by computing
The MLSE receiver employs exhaustive comparison to find
the maximum metric and thus the maximum likelihood sequence.
This task is undertaken efficiently by a Viterbi processor, which
has finite complexity provided the ISI extends over only a finite
number of symbols. This condition is satisfied if the pulse shape
has a finite duration, h(t) = 0, t < 0, HT ≤ t, and the delay spread
is finite, zd(t,x) = 0, x < 0, t ≤ x, for some pulse length, H,
and maximum delay, t. Define L = H + τ T . Then the ISI
residues. The problem is a common one in Rayleigh fading
channels, so we omit the details here, since the solution is well
covered in the literature[5,8,9]. Suffice it that the autocorrelation
term, si,k, in (13) is zero for k − i ≥ L , and the branch metric is
1
2
(
)
E clb−ir ,ir cmd− kr =
1
2
(
cld−ir ,ir , is given by
∞ ∞
∫ ∫ h((l − ir )Tr − ξ1)h ((m − kr )Tr − ξ2 ) ×
−∞ −∞
(16)
)
E z (lTr , ξ1 )z (mTr , ξ2 ) dξ1dξ2
b
d
A Wide Sense Stationary, Uncorrelated Scattering[10]
channel model is assumed, with the conventional J 0 2πλx
( )
given by
i −1
2
li ~ 2ℜα i mi − ∑ α i si ,k α k − α i si ,i
k =i − L +1
of the effective pulse samples,
(15)
This is a function of L symbols, (α i − L +1K α i ) , known as the
hypothesis vector. A partially-connected trellis is built up, with
ML-1 states, labelled by the first L-1 symbols of the vector,
autocorrelation function in space, x[11]. Diversity is achieved
through a receiver antenna array, comprising D antennae in a
row, equispaced at A wavelength intervals. For frequency-flat
fading, the expectation in (16) reduces to
1
2
State
64
748
$ i − L +1Kα$ i −1 , α$ i ), where the last symbol labels the current
(α
branch.
(
J0 2 π
(
)
E zb (lTr , ξ1 )z d (mTr , ξ2 ) =
× J0 2 π
During the ith symbol period, the receiver computes one
value of mi and L values of si,k according to (11) or (13). The ML
branch metrics are calculated according to (15), and then applied
to a standard Viterbi processor. The EMLSE-tf space diversity
structure is shown in Figure 1.
((d − b) Acosθ + (m − l) f DTr )2 + ((d − b) Asinθ)2
(17)
where θ is the angle between the antenna array and the receiver
velocity.
For time- and frequency-selective channels, we assume a
delay profile comprising N taps with with equal mean power.
Then
1
2
Figure 1: The EMLSE-tf space diversity receiver. The antenna icon
encompasses reception, amplification, and translation to baseband.
)
E zb (lTr , ξ1 )z d (mTr , ξ2 ) = δ(ξ1 )δ(ξ1 − ξ2 ) ×
1
N
N −1
τn
∑ δ ξ1 − N − 1δ(ξ1 − ξ2 )
n =0
((d − b) Acosθ + (m − l ) f DTr )2 + ((d − b) Asinθ)2
(18)
In practice, only the dominant short error events are
considered, where up to E symbols may be in error. If the fading
is fast and the SNR high, then the dominant error events are
short, so the truncated bound can validly neglect long error
events. However, with slow fading, most errors occur in the
deep fades. If the fading is very slow then the deep fade lasts
hundreds of symbols at reasonable SNRs, and so do the error
events. Thus the upper bound is tight and easily calculated only
for fast fading and high SNR. The union of E = 1 error events
due to the nearest neighbours of the errored symbol is
asymptotically correct at high SNRs.
The average bit energy to noise spectral density is given by
∞
Eb
=
N0
1
2
E
∫
2
∞
∫ h(t1 − ξ 1 )z(t1 , ξ1 )dξ 1 dt1
−∞ −∞
N 0 log 2 M
∞
=
∫
−∞
space diversity dwarves the implicit channel diversity, and it is
reliably available for all channels.
h(t1 ) dt1
2
N 0 log 2 M
(19)
VI. RESULTS
For figures 2 - 4, we use a pulse with a 50% excess
bandwidth square-root raised cosine spectrum, but truncated in
the time domain to H = 5 symbol periods. For figures 2 - 4, r = 2
samples per symbol are taken; for figures 5 and 6, r = 3 samples
per symbol are taken.
Figure 2 compares the performance of MRC and EMLSE-tf
diversity receivers. Higher diversity can significantly reduce the
required SNR to achieve a specified BER. The MRC receiver is
normally constructed with a channel estimator to track the timeselective channel, but the very limited ability of the LMS and
RLS (with an exponential forget factor) algorithms to predict the
channel makes them unsuitable for fast fading channels, so an
error floor results. Thus the conventional MRC diversity
receiver cannot easily tolerate fast fading without an error floor,
so instead it is plotted for slow fading. This slow fading curve is
a lower bound on the MRC’s performance in fast fading. We see
that the EMLSE-tf diversity receiver is able to exploit the
implicit Doppler diversity of the channel[12], but the gain that it
provides diminishes as the number of antennae increases.
Notwithstanding, there is no error floor for the EMLSE-tf
diversity receiver.
Three curves are plotted for the EMLSE-tf diversity receiver:
a lower bound that considers only one, length E = 1, error event;
a nearest neighbour curve that for QPSK considers two length E
= 1, error events and is asymptotically correct; and a union
bound that considers all error events of up to E = 3 symbols in
length. At low SNR, the longer events dominate since the bound
is very loose, and truncation is not valid. At high SNR, the
shortest error event dominates, and the bound is both valid and
effective.
Figure 3 repeats Figure 2, but now with both time- and
frequency-selectivity. Again the MRC receiver is plotted for
slow fading. Both receivers can exploit the channel’s implicit
delay diversity, but the EMLSE-tf diversity receiver outperforms
the MRC receiver by exploiting the implicit Doppler diversity.
Figure 4 examines the effect of antenna separation on the
BER. Most of the achievable diversity gain is achieved for
separations above 0.15λ. Maximum diversity is achieved at
0.383λ, at the first zero of the autocorrelation function. We see
that space diversity can be achieved without an undue antennae
volume for 900MHz and higher carrier frequencies.
Figures 5 and 6 show the BER behaviour over a range of
fading rates and delay spreads. Figure 5 covers frequency-flat,
fast fading channels.
Figure 6 considers time-invariant,
frequency-selective channels. Implicit diversity gains of 15dB
and 12dB at a BER of 10-4 are observed for the extreme Doppler
and delay spreads, but the BER improvement for low Doppler
and delay spreads is minimal. By comparison, the gain due to
VII. CONCLUSIONS
An MLSE diversity receiver for the time- and frequencyselective channel has been derived, assuming ideal CSI and
linear signal constellations. A method of bounding the receiver’s
BER has been derived, and the bounds are tight for the
frequency-selective, fast Rayleigh fading channel under
consideration.
The BER analysis shows that the implicit Doppler and delay
diversity of the channel mildly improve the receiver’s
performance, but that the explicit (space) diversity actually
provides most of the benefits. The performance enhancement
over conventional Maximal Ratio Combining is negligible at
normal fading rates, but its error floor is removed.
VIII. REFERENCES
[1] C.E.Shannon, “A Mathematical Theory of Communication,”
Bell Syst. Tech. J., vol 27, July 1948, pp 379-423
[2] R.van Nobelen and D.P.Taylor, “Analysis of the Pairwise
Error Probability of Non-Interleaved Codes on the Rayleigh
Fading Channel,” in Globecom Communications Theory MiniConference, San Francisco, 1994, pp181-185
[3] D.G.Brennan, “Linear Diversity Combining Techniques,”
vol. 47, Proc. IRE, pp1075-1102, June 1959
[4] J.K.Cavers, “On the Validity of the Slow and Moderate
Fading Models for Matched Filter Detection of Rayleigh Fading
Signals,” Can. J. of Elect. & Comp. Eng., vol. 17, no. 4, pp183189, 1992
[5] G.Ungerboeck, “Adaptive Maximum-Likelihood Receiver for
Carrier-Modulated Data-Transmission Systems,” IEEE Trans.
Commun., vol 22, no. 5, pp624-636, May 1974
[6] H.L.Van Trees, “Detection, Estimation, and Modulation
Theory, Part 1,” New York, John Wiley and Sons, 1969
[7] V.-P. Kaasila, A.Mammela, “Bit Error Probability of a
Matched Filter in a Rayleigh Fading Multipath Channel,” IEEE
Trans. Commun., vol. 42, no. 2-4, pp826-828, 1994
[8] G.D.Forney, “Maximum-Likelihood sequence Estimation of
Digitial Sequences in the Presence of Intersymbol Interference,”
IEEE Trans. Inf. Theory, vol. 18, no. 3, pp363-378, May 1972
[9] B.D.Hart, D.P.Taylor, “Extended MLSE Receiver for the
Frequency-Flat, Fast Fading Channel and Linear Modulations,”
in Globecom Communications Theory Mini-Conference,
Singapore, 1995, pp157-161
[10] P.A.Bello, “Characterization of Randomly Time-Variant
Linear Channels, IEEE Transactions of Communication
Systems,” IEEE Trans. Commun. Sys., vol. 11, pp360-393, Dec.
1962
[11] R.H.Clarke, “A Statistical Theory of Mobile-Radio
Reception,” Bell Syst. Tech. J., vol 47, July-Aug. 1968, pp9571000
[12] W.C.Dam, “An Adaptive Maximum Likelihood Receiver
for Rayleigh Fading Channels,” M.Eng. thesis, Hamilton Ont.
Canada: McMaster University, 1990
Figure 2: Upper (union) bound, nearest neighbour estimate and lower
bound on the BER for EMLSE-tf diversity receiver with fDT = 0.1, and
nearest neighbour BER estimate for MRC diversity receiver with fDT = 0
(An MRC receiver in fast fading has a BER floor). Error events up to E
= 3 symbols are considered, t/T = 0, A = 0.383, θ = 0, ideal CSI and
QPSK.
Figure 3: Upper (union) bound, nearest neighbour estimate and lower
bound on the BER for EMLSE-tf diversity receiver with fDT = 0.1, and
and nearest neighbour BER estimate for MRC diversity receiver with
fDT = 0 (An MRC receiver in fast fading has a BER floor). Error events
up to E = 3 symbols are considered, t/T = 0.1, N = 2, A = 0.383, θ = 0,
ideal CSI and QPSK.
Figure 4: Nearest neighbour BER estimates for EMLSE-tf diversity
receiver, with fDT = 0,t/T = 0, D = 2, ideal CSI and QPSK.
Figure 5: Nearest neighbour BER estimates for EMLSE-tf diversity
receiver, with fDT = 0, 0.01, 0.03, 0.1, 0.3; t/T = 0; D = 1, 2, 3; A =
0.383; θ = 0; H = 5; r = 3; ideal CSI and QPSK.
Figure 6: Nearest neighbour BER estimates for EMLSE-tf diversity
receiver, with t/T = 0, 0.01, 0.03, 0.1, 0.3; fDT = 0; D = 1, 2, 3; A =
0.383; θ = 0; N = 4; r = 3; H = 5; ideal CSI and QPSK.
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