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Content:
2002 Conference on
Information Sciences and Systems,
Princeton University, March 2022, 2002
MultipleSymbol Detection of Differential Unitary SpaceTime Modulation in FastFading Channels with Known Correlation
Bijoy Bhukania and Phil Schniter 1 Department of
Electrical engineering The
Ohio State University 2015 Neil Ave,
Columbus, OH 43210 Email: {bhukanib, schniter}@ee.eng.ohiostate.edu
Abstract  With a fading channel, standard ML detection of differential Unitary SpaceTime modulation (DUST) leads to an error floor in the BERversusSNR curve since it is derived under the assumption that the channel remains constant during every consecutive pair of matrixsymbols. Assuming knowledge of the channel fading correlations, we design multiplesymbol ML differential detectors which drastically reduce this error floor, especially for fastfading channels. Multiplesymbol differential detection has the additional benefit of reducing the 3dB loss in SNR (relative to coherent detection) that characterizes standard onesymbol differential detection. To derive these new detectors, we make the simplifying assumption that the channel changes once per matrixsymbol (i.e., block fading) rather than once per channel use (i.e., continuous fading). However, the block fading assumption is not required when diagonal codes are used. The
Simulation results show that, in spite of the blockfading assumption, the new receivers far outperform standard singlesymbol detection under continuous fading as well. I. Introduction The capacity of wireless
communication systems over fading channels is significantly enhanced by the use of multiple antennas at the transmitter and/or at the receiver. Spacetime coding is a bandwidth and power efficient method of communication over fading channels that realizes benefits of multiple transmit antennas [1]. Although the majority of spacetime codes proposed so far assume knowledge of channel state information (CSI) at the receiver [1], the class of unitary spacetime codes [2] and differential unitary spacetime codes [3, 4] have been proposed for Rayleigh flatfading channels when neither the transmitter nor the receiver knows the fading coefficients. Differential unitary spacetime modulation (DUST) can be seen as an extension of differential phaseshift keying (DPSK), commonly used in single antenna systems when the channel is unknown to the receiver as well as to the transmitter. The information is encoded in the phase difference between two consecutive symbols, so that the information can be decoded from phase difference between two consecutive observations. In DUST, the transmitted symbols can be considered M ЧM unitary matrices, where M is the number of transmit antennas. The previously transmitted matrixsymbol is premultiplied by the current information matrixsymbol to form the current transmitted matrixsymbol. The standard differential receiver detects one information matrixsymbol from each pair of consecutive received matrices. 1Corresponding author
Ideally, we desire that the channel experienced by every pair of consecutively transmitted matrices is the same. However, this would require that the channel be forever constant a requirement which is never met in practice. The performance of DUST with standard singlesymbol maximumlikelihood (ML) decoding in continuouslyfading channel has been evaluated in [5], where it has been shown that an error floor is achieved when the error due to channel variation dominates that due to additive noise. In this paper, we present DUST detectors for use in fast fading channelschannels for which standard singlesymbol detection is unsuccessful. In deriving the new detectors, we make the simplifying assumption that the channel changes once per matrixsymbol (i.e., M channel uses), and that the receiver knows the fading correlation. We will see, however, that the block fading assumption is not necessary when diagonal spacetime codes [3] are used. Under these assumptions, single and multiplesymbol ML differential detectors are derived. Simulation results show that, in spite of the blockfading assumption, the new detectors exhibit significantly improved performance compared to the standard singlesymbol detector in fast continuouslyfading channels. The notations used in this paper are as follows: Matrices will be denoted by capital letters (e.g., X and X) and vectors by lower case bold (e.g., x). IN will denote identity matrix of size N Ч N . The operator vec(·), e.g., xn = vec(Xn), denotes stacking of the columns of matrix Xn in column vector xn. (·) denotes conjugate transposition, denotes the Kronecker product, tr(·) denotes the trace operator, and det(·) the determinant.
II. Background
The system model in a continuouslyfading channel is
2 s0,n 0
0
Xn
=
r M
6 6 6 4
0 ...
s1,n . . . ... . . .
0 3 2 H0,n 3
0 7 6 H1,n 7
...
76 76 54
...
7 7
+
Wn
5
0
0 . . . sM1,n
HM 1,n (1)
where Xn is the M ЧN received matrix during the nth matrix
symbol interval, and where M and N are the number of trans
mit and receive antennas, respectively. Hk,n is the M Ч N MIMO channel response matrix at the kth time instant in the
nth matrixsymbol interval, i.e., at the (nM + k)th channel
use. Sn = [s0,n s1,n . . . sM1,n] is the nth M Ч M trans
mitted matrixsymbol, encoded as Sn = Vzn Sn1. zn L = {0, 1, . . . , 2RM  1} is the timen integer index into matrix al
phabet A of size 2RM , so that Vzn A. Thus R is the number of bits per channel use. Wn is a matrix of i.i.d. unit variance
Gaussian entries and is the average SNR.
Under the assumption that the channel remains constant for one matrixsymbol interval, we have Hn = H0,n = · · · = HM1,n, which changes (1) to
r
Xn =
M
SnHn
+
Wn
(2)
Even without this assumption, the use of diagonal space time
codes Sn implies that the continuous fading model (1) simpli
fies to
r
Xn =
M
SnHn(c)
+
Wn
(3)
where the kth row of the "equivalent continuous fading chan
nel
matrix"
H (c) n
is the kth
row of Hk,n,
for k
=
0, . . . , M 1.
If
the MIMO fading process Hk,n is independent between anten
nas,
then
the
equivalent
fading
process
H (c) n
is
also
indepen
dent
between
antennas;
the
process
H (c) n
,
however,
is
M fold
"faster" than Hk,n.
Keeping
this
in
mind,
H (c) n
shall be de
noted as Hn from here onwards, and (2) will be used as the
system model. Note that (2) is an approximate model when
nondiagonal codes are used in continuous fading channel.
Since
Xn1
=
p M
Sn1 Hn1
+
Wn1 ,
on
realizing
that
Sn = Vzn Sn1, and assuming Hn = Hn1, we can write
Xn = VnXn1 + Wn  Vzn Wn1
(4)
 {z }
W^ n
Since Vzn is unitary, W^ n contains i.i.d. Gaussian entries with variance twice of those in Wn. From (4) it is straightforward to show that the ML detection rule for zn is [3]
z^n = arg max [tr{XnVzn Xn1}]
(5)
znL
Due to increased noise variance in W^ n, this detector loses 3dB in performance compared to coherent detection. Under the assumption of Hn = Hn1, the Chernoff upper bound on Pe for the detector in (5) converges to zero as [3]. However if the channel changes from one matrixsymbol interval to the next, i.e., if Hn = Hn1 + H, then (4) becomes
r
Xn = Vzn Xn1 +
M
SnH
+Wn

Vzn
Wn1
 {z }
X~ n
where the term X~n creates additional "noise" that induces an error floor in BER curve. Note that the detector in (5) ignores channel variation and is therefore suboptimal in a fading environment. In following sections we derive detection rules that exploit knowledge of the autocorrelation of the timevarying channel coefficients.
III. OneSymbol ML Differential Detection
Here we derive the ML detector for zn given Xn and Xn1 that exploits knowledge of the correlation between Hn and Hn1 (rather than assuming Hn = Hn1 as in standard detector [3]). Denoting hn = vec(Hn), xn = vec(Xn), and wn = vec(Wn),
» xn xn1
=
r » IN Sn
M
0
0
» hn
IN Sn1
hn1
 {z }
xn
+
» wn wn1
(6)
Assuming hn and wn contain zeromean unitvariance i.i.d. Gaussian
random variables, we have E[hnhn] = IMN . Furthermore, we assume that E[hnhnk] = kIMN , k > 0. Con ditioned Sn and Sn1, xn is a zeromean Gaussian vector with autocorrelation matrix
R(1) =
M
»
IM N 1IN Vzn
1IN Vzn IM N
+ I2MN
(7)
where we have used the facts that Sn is unitary and Sn = Vzn Sn1. Note that the distribution of xn depends on Vzn rather than Sn and Sn1. Equation (7) can be rewritten
R(1)
=
"
M
(1

1
)
+
" 1
I2M
N
"
+
M
p 1
1 1

IN
Vzn
# h
p 1
1 1

IN
Vzn
i
From the identity det(I + AB) = det(I + BA) it follows that det(R(1)) is independent of Vzn . Note also that
R(1) 1
=
1» d1
`1M +1M INґ
IM N Vzn
`1M+1M INґIMVNzn (8)
d1
=
`1 +
M
ґ2

`
M
ґ2
12
>0
Thus the ML detector for zn given Xn and Xn1 (equivalently given xn) becomes
z^n = arg max p(xnVzn ) znL
= arg max exnR(1)1xn
znL
h
=
arg min tr zn L
1 M d1
Xn
Vzn
Xn1
+
1 M d1
Xn1
Vzn
Xn
i
= arg max {tr[1XnVzn Xn1]}
(9)
znL
While detector (9) is similar to detector (5), it exploits the
known fading correlation 1. When 1 is real and positive,
however, it does not affect the decision rule and can be re
moved, making (9) identical to (5). For reasons that will be
come
clear
later,
we
denote
a(1) 0,1
=
1.
In order to further
improve performance we consider joint differential detection
of multiple symbols in the next section.
IV. Multiplesymbol ML differential detection Multiplesymbol differential detection of M PSK has been proposed as an effective way to reduce the 3dB SNR loss incurred by onesymbol differential detection [6], as well as to enhance performance in correlated Rayleigh fading channels [7]. We extend this idea to DUST and incorporate knowledge of fading correlation, assuming the blockfading channel described in the previous section. First we derive the 2 and 3symbol joint detectors and later generalize to an arbitrary number of symbols. A. TwoSymbol ML Detection We now collect Xn2, Xn1, and Xn, with the intention of
detecting zn1 and zn. Straightfor
Ward Extension of (6) yields pendent of {Vzn , Vzn1 , Vzn2 }. This leads to the ML detector
2 xn 3 2 wn 3
4 xn1 5 = 4 wn1 5 +
xn2
wn2
 {z }
x2n
p M
4
IN Sn 0
0
0 IN Sn1 0
0
3 2 hn 3
0
5 4 hn1 5
IN Sn2
hn2
{z^n, z^n1, z^n2} = arg max [tr{ zn ,zn1 ,zn2 L
(14)
a(3) 2,3
Xn2
Vzn2
Xn3
+
a(3) 1,2
Xn1
Vzn1
Xn2
+
a(3) 0,1
Xn
Vzn
Xn1
+
a(3) 1,3
Xn1
Vzn1
Vzn2
Xn3
+
a(3) 0,2
Xn
Vzn
Vzn1
Xn2
+
a(03,3) XnVzn Vzn1 Vzn2 Xn3}]
where
Under the channel and noise assumptions of Section III, the
received vector xn conditioned on Sn, Sn1 and Sn2 is zeromean Gaussian with an autocorrelation matrix R(2) that de
pends
only
on Vzn1 , Vzn , 1, 2
and
M
.
As before,
det(R(2) )
can be shown to be independent of Vzn and Vzn1. In this
case,
a(3) 2,3
=
a(3) 0,1
=
"`1
+
M
ґ2

` M
ґ2
" 12 b1
a(3) 1,2
=
b5
"` M
ґ2
12

M
`1
+
M
ґ 1"
+
b1
" 
` M
ґ2
13
+
M
`1
+
M
ґ" 2
a(3) 1,3
=
a(3) 0,2
=
b5
"` M
ґ2
12

M
`1
+
M
ґ" 2
+
R(2)1 =
2 1 4 d2
a(0a2,2)(02,1)aIN(0I2,0)NIMVNzVnzn1V1zn
a(02,1) IN Vzn1 a(1a2,2)(12,1)IINMN Vzn
a(02,2) IN Vzn Vzn1 3
a(12,2) IN Vzn
5 (10)
a(2) 2,2
IM
N
b1
" M
`1
+
M
ґ 3

` M
ґ2
" 12
a(3) 0,3
=
b1
" M
`1
+
M
ґ 2

` M
ґ2
" 12

b6
" M
`1
+
M
ґ 3

` M
ґ2
" 12
and
b1
=
`
M
ґ2
(12 3
+
1 2 2

112)

where
` M
ґ2
`1
+
M
ґ (12
+
23)
+
d2 a(2) 0,0 a(2) 1,1 a(2) 0,1
= = = =
` M
+ 1ґ3

`
M
ґ2
`
M
+ 1ґ (212
+ 22)+
` M
ґ3
(12 2
+
12 2 )
a(2) 2,2
=
`1
+
M
ґ2

` M
ґ2
12
`1
+
M
ґ2

` M
ґ2
2 2
a(2) 1,2
=
M
` M
+ 1ґ 1

`
M
ґ2
12
(11)
M
`1
+
ґ2 M
1
b5
=
`1
+
M
ґ3
+
2
`
M
ґ3
(123) 
` M
ґ2
`1
+
M
ґ `12
+
22
+
32ґ
b6
=
`1
+
M
ґ3
+
2
`
M
ґ3
`122ґ
Note that (14) is a straightforward extension of (13), except that the coefficients {a(k3,)l} are different from {a(k2,)l}. As be
a(2) 0,2
=
M
` M
+
1ґ
2

`
M
ґ2
12
(12)
fore,
if
1
=
2
=
3
=
0,
then
a(3) k,l
=
0
k
=
l,
imply
ing that differentiallyencoded symbols cannot be detected.
Hence the detection rule is given by
When k = 1 k, i.e., the channel is fixed, the detection rules (13) and (14) do not depend on SNR, since in that case
{z^n, z^n1} = arg max xnR(2)1xn zn1 ,zn L = arg max ^tr{a(02,1) XnVzn Xn1 zn1 ,zn L
+
a(2) 1,2
Xn1
Vzn1
Xn2
+
a(2) 0,2
Xn
Vzn
Vzn1
Xn2 }~
(13)
The main difference between detector (13) and onesymbol
detector (9) is that (13) makes use of additional channel pa
rameters. Thus we hope for improved performance in fast
fading channels. It is worth noticing that when the channels
in subsequent blocks are independent (i.e., 1 = 2 = 0), then
a(2) 0,1
=
a(2) 0,2
=
a(2) 1,2
=
0,
implying
that
differential
encoding
cannot be usedan intuitively satisfying observation.
B. Threesymbol ML detection
a(2) 0,1
=
a(2) 0,2
>
0
and
a(3) 0,1
=
a(3) 1,2
=
a(3) 0,2
=
a(3) 0,3
>
0.
Though we have not derived ML detection rules for non
diagonal DUST codes in continuouslyfading channels, the
multiplesymbol detection rules (13) and (14), derived for
the blockfading channel, should still show improvement over
the standard detector in a continuousfading environment.
The coefficients {a(im ,j)} used in this case would be recomputed with k defined such that E[h0,nh0,nk] = kIMN for h0,n = vec(H0,n). The simulation results in Section VI con
firm that detectors (13) and (14) significantly outperform the
standard singlesymbol detector (5) under fast continuous fad
ing.
C. mSymbol ML Detection
The ML joint detection rule for m symbols can be shown to be
We now give the structure of the 3symbol differential de
tector as a prelude to the general msymbol case and to
demonstrate that closedform expressions for the parame
ters
a(m) k,l
become
quite
lengthy
when
m
3.
As in pre
vious sections, we compute the inverse of the autocorrela
tion matrix of xn = [xn xn1 xn2 xn3], conditioned on
{Sn, Sn1, Sn2, Sn3}, and find that its determinant is inde
{z^n, z^n1, . . . , z^nm+1} =
arg max
zn ,zn1 ,...,znm+1 L
(15)
" tr
(mX1
mX k1
a(m) i,i+k+1
Xni
iY +k
!
)#
Vznj Xnik1
k=0 i=0
j=i
where
Qm1 j=0
(Aj
)
=
A0A1 · · · Am1.
From the previous two
subsections, we see that deriving closedform expressions for
the coefficients {a(km,l)} is quite difficult when m > 3. These coefficients can, however, be computed numerically by first
constructing an (m + 1)M N Ч (m + 1)M N autocorrelation
matrix as an extension of (7), but with the additional sim
plification that Vznj = IM j, then computing its inverse.
In
the
case
of
a
fixed
channel
it
can
be
shown
that
a(m) i,j
=
a(m) k,l
>0
i, j, k, l
{0, . . . , m} &
i = j, k
= l.
As
a
result,
the
detector
(15)
becomes
independent
of
coefficients
a(m) i,j
.
It is important to note that there does not appear to exist
a Viterbilike algorithm for msymbol ML detection that has
complexity linear in m. Observe that it is not possible to write
the detection metric as a sum of terms that contain strict sub
sets of the term with
tshyme fboorlmsettr{VXznn`,Q. .jm.=,V01zVnznm+j1ґ}X; nthemreЇ.isTahlwuas,yms oanxe
imization of the quantity in (15) can only accomplished using
a brute force search over all symbol combinations, yielding a
complexity exponential in m.
V. Suboptimal Sequence Detection
Practical applications require the detection of N 3 symbols. Yet, as we have seen, the joint ML differential detector for N symbols has a complexity that is exponential in N . Thus, we are motivated to consider suboptimal N symbol detection using some combination of msymbol ML detectors for, say, m 3. It is instructive to note the difference between the N symbol ML detector and any suboptimal N symbol detector constructed from msymbol ML detectors (m < N ). From (15), we see that the suboptimal sequence detector uses detection metrics that linearly combine terms based on subsets of {Vzn , . . . , Vznm+1} for n {m  1, . . . , N  1}. The combining coefficients are taken from the set {a(km,l)}. In contrast, the optimal N symbol detection metric linearly combines these terms with additional terms based on subsets of {VzN1, . . . , Vz0 } that are ignored by the suboptimal detector. In addition, the N symbol combining coefficients are {a(kN,l)}, which are, in general, different from {a(km,l)}. Thus, the task of constructing a "good" N symbol detector that employs (at most) msymbol optimal detections (m < N ) can be considered equivalent to the approximation of {a(kN,l)} by a sparse coefficient set. For the simulations in Section VI, we divide the (N + 1)matrix observation sequence into consecutive subsequences of length m + 1, each of which overlaps its neighbor by one matrixobservation. Then, msymbol detection is performed on each subsequence. This is equivalent to replacing the coefficients {a(kN,l)} with {a(km,l)} where defined, and replacing the rest with zeros. Schemes in which neighboring observation subsequences overlap by more than one matrixobservation lead to different approximations of the coefficient set {a(kN,l)} and are currently under investigation.
VI. Simulations We evaluate the performance of the detectors in two types of channel: the "block fading channel" (2) and the "continuous fading channel" (3). The correlation between fading coefficients k symbols apart is given by J0(2fDTsk) [8] in continuous fading channels, where fDTs is the normalized Doppler frequency. In block fading channel, correlation between channel coefficients m matrixsymbols apart is given by J0(2fDTsM m).
BER
100
101
102
103
(5) (9)
(13)  fixed channel
(13)
(14)  fixed channel
(14)
104 0
5
10
15
20
25
30
SNR [dB]
Fig. 1: Diagonal codes in continuous fading channel with fDTs = 0.1.
As shown in Section II, the use of diagonal codes in continuousfading channels yield the same model as general codes in blockfading channels, and hence detector performance with diagonal codes is identical for these two channel types. Therefore, we first present the performance of the detectors (5) (9) (13) and (14) in continuousfading channels using diagonal codes. The simulations assume two transmit and two receive antennas with R = 1 and the constellation specified in [3]. As described in Section V, detection is accomplished by detecting nonoverlapping portions of the transmitted symbol sequence using 1, 2, and 3symbol ML differential detectors (9), (13), and (14). Results will be presented for detectors with exact knowledge of fading correlations, as well as for detectors which assume that the channel is fixed (marked by "fixed" in the figures). In the figures shown here, solid, dashed, and dotted lines correspond to 1, 2 and 3 symbol detection, respectively. Fig. 1, where fDTs = 0.1, clearly illustrates the advantage of detectors which jointly detect multiple symbols and incorporate channel fading parameters. Note that detector (5), which ignores the fading correlation, succumbs to a very high error floor. Detector (9), designed to incorporate fading correlation into singlesymbol detection, performs same as (5) does, since Rayleigh flatfading model used in our simulations leads to 1 > 0. Thus, both forms of singlesymbol detection perform very suboptimally in the fading environment. The 2symbol detector (13) which incorporates fading correlation exhibits considerably improved performance, although still succumbing to an error floor. Meanwhile, the 2symbol detector which ignores fading correlation performs even worse than the onesymbol detector. Performance increases dramatically with the 3symbol detector that incorporates fading, and decreases with the 3symbol detector that ignores fading. As hinted by the plot, even the good 3symbol detector will succumb to an error floor at highenough SNR. The important point, however, is that the error floor has been pushed outside of the expected operating range. Figures 2 and 3, corresponding to normalized Doppler fre
100
100
BER BER
101
101
102
102
103
104
(5)
(9)
(13)  fixed channel
105
(13)
(14)  fixed channel
(14)
103
(5)
(9)
104
(13)  fixed channel
(13)
(14)  fixed channel
(14)
106 0
105
5
10
15
20
25
0
SNR [dB]
5
10
15
20
25
SNR [dB]
Fig. 2: Diagonal codes in continuous fading channel with Fig. 3: Diagonal codes in continuous fading channel with
fDTs = 0.05.
fDTs = 0.025.
quencies of fDTs = 0.05 and 0.025, respectively, mimic the results of Fig. 1 but in a less pronounced fashion. Again we see the improvement associated with multiplesymbol ML differential detectors that incorporate fading correlation. Next, the performance of the detectors has been evaluated in continuouslyfading channels with nondiagonal codes to illustrate the performance loss due to approximation in the system model (2). The nondiagonal codes are generated by right multiplying the diagonal codes by a fixed nondiagonal unitary matrix. Because such an operation does not change the product distance of the constellation [3], the comparison of diagonal to nondiagonal codes is fair. Figures 4 and 5 illustrate the performance of the detectors with nondiagonal codes in continuouslyfading channel with fDTs = 0.05 and 0.025, respectively, and compares them with performance of 3symbol detector with diagonal codes. Although the performance loss due to neglecting the channel variation within the matrixsymbol interval is significant when fDTs = 0.05 and negligible when fDTs = 0.025, in both cases 3 and 2symbol detectors that incorporate knowledge of fading correlations perform better than the single symbol detector in terms of reducing the error floor. The above observations lead us to important conclusions about the detectors described in this paper. First, multiplesymbol detection is essential to combat fading channels since the detectors (5) and (9) are often equivalent. Increasing the number of symbols being jointly detected improves the performance in terms of SNR loss and increases the robustness against fading channels when knowledge of fading correlations is properly incorporated into the detection rule. Generally, the faster the fading, the more symbols are required in joint detection to push the error floor out of the operating range. VII. Conclusions In this paper, we have demonstrated the efficacy of multiplesymbol ML differential detection that incorporates channel fading parameters in combating the error floor exhibited by standard onesymbol ML detection of DUST. In fact, we have shown that multiplesymbol (versus singlesymbol) detection
is essential to performance enhancement. Our multiplesymbol detection rules, which assume the channels to be blockfading when nondiagonal codes are used, have been shown to improve performance in continuouslyfading channels as well. We are currently investigating the robustness of these detectors to imperfect knowledge of fading correlation and SNR as well as reduced complexity implementation of the multiplesymbol detectors for large m. In addition, we are working to derive theoretical error bounds for these detectors. References [1] A. Naguib, N. Seshadri, and A.R. Calderbank, "Increasing data rate over wireless channels", IEEE Signal Processing Magazine, vol. 17, pp. 7692, May 2000. [2] B.M. Hochwald and T.L. Marzetta, "Unitary spacetime modulation for multipleantenna communications in Rayleigh flat fading",
IEEE Trans. on
information theory, vol. 46, no. 2, pp. 543564, Mar. 2000. [3] B.M. Hochwald and W. Sweldens, "Differential unitary spacetime modulation", IEEE trans. on Communications, vol. 48, no. 12, pp. 20412052, Dec. 2000. [4] B.L. Hughes, "Differential spacetime modulation", IEEE Trans. on Information Theory, vol. 46, no. 7, pp. 25672578, Nov. 2000. [5] C. Peel and A. Swindlehurst, "Performance of unitary spacetime modulation in continuously changing channel", in IEEE Internat. Conf. on Acoustics, Speech, and Signal Processing, 2001. [6] D. Divsalar and M.K. Simon, "Multiplesymbol differential detection of MPSK", IEEE trans. on Communications, vol. 38, no. 3, pp. 300308, March 1990. [7] P. Ho and D. Fung, "Error performance of multiple symbol differential detection of PSK signals transmitted over correlated Rayleigh fading channels", IEEE Intern. Conf. on Communications, vol. 2, pp. 568574, 1991. [8] W.C. Jakes, Microwave Mobile Communications, IEEE press, Piscataway, NJ, 1993.
BER
100
101
102
103
(5)
(9)
(13)fixed
(13)
104
(14)fixed
(14)
(14)diagonal codes
105 0
5
10
15
20
25
30
SNR [dB]
Fig. 4: Nondiagonal codes in continuouslyfading channel with fDTs = 0.05.
BER
100
101
102
(5)
(9)
103
(13)fixed
(13)
(14)fixed
(14)
(14)diagonal codes
104
0
2
4
6
8
10
12
14
16
18
SNR [dB]
Fig. 5: Nondiagonal codes in continuouslyfading channel with fDTs = 0.025.
B Bhukania, P Schniter