Multicode Chip-Interleaved DS-CDMA to Effect
Synchronous Correlation of Spreading Codes in
Quasi-Synchronous Transmission over Multipath Channels
Yu-Nan Lin and David W. Lin, Senior Member, IEEE
Abstract— The performance of conventional DS-CDMA sys-tems is greatly affected by asynchronous or multipath propa-gation. We show that, with a certain way of multicode assign-ment, chip-interleaved DS-CDMA (CIDS-CDMA) can perform better in such a condition thanks to its ability to preserve the synchronous correlations among the spreading codes. Compared to the recently proposed chip-interleaved block-spread CDMA (CIBS-CDMA) that requires a single-user equalizer, the pre-sented scheme can attain better or comparable performance in channel-coded transmission with a rake receiver that has lower complexity.
Index Terms— Chip interleaving, code division multiaccess, multipath channels.
I. INTRODUCTION
T
HE capacity of a DS-CDMA system highly dependson the correlation properties of the spreading codes. Some well-known classes of spreading codes that have low inter-code correlations are the Gold, the Kasami, and the m-sequences. Their property of low inter-code correlations, how-ever, may be lost in asynchronous transmission or multipath propagation, resulting in a performance close to that of random spreading [1].
By chip-interleaved block spreading (abbreviated CIBS-CDMA) [2], [3], the orthogonality among spreading codes can be maintained in asynchronous or multipath propagation, and the multiaccess interference (MAI) is eliminated deterministi-cally. Although intersymbol interference (ISI) will show up in multipath propagation, it can be dealt with using a single-user equalizer. However, an issue, heretofore unaddressed, arises when channel coding is employed. Briefly, if the residual ISI after equalization comes mainly from a small number of paths (in the equalized channel), then the performance of channel coding may suffer, because channel codes are normally designed for the white Gaussian noise condition whereas the residual ISI in this case is not white Gaussian. One remedy is to lengthen the equalizer for further ISI suppression. Another is to use turbo equalization that iterates
Manuscript received May 1, 2004; revised March 21, 2005; accepted April 14, 2006. The associate editor coordinating the review of this letter and approving it for publication was K. B. Lee. This work was supported by the National Science Council of the Republic of China under Grant NSC 92-2219-E-009-018.
Y.-N. Lin was with the Department of Electronics Engineering, Na-tional Chiao Tung University, Hsinchu, Taiwan, ROC. He is now with Realtek Semiconductor Corp., Hsinchu, Taiwan 300, ROC (e-mail: ynlin.ee87g@nctu.edu.tw).
D. W. Lin is with the Department of Electronics Engineering and Center for Telecommunications Research, National Chiao Tung University, Hsinchu, Taiwan 30010, ROC (e-mail: dwlin@mail.nctu.edu.tw).
Digital Object Identifier 10.1109/TWC.2006.04285
between equalization and channel decoding [4], [5]. Both entail additional complexity, with the turbo equalizer also increasing the decoding delay.
Herein we present a multicode chip-interleaved DS-CDMA (CIDS-CDMA) scheme, which can be viewed as a modifi-cation of CIBS-CDMA, that not only maintains the spread-ing codes’ synchronous correlations (to be defined later) in quasi-synchronous multipath channels but also facilitates low-complexity rake receiving for channel-coded transmission. In what follows, we first describe the CIDS-CDMA signals in Section II. In Section III, we describe how the proposed scheme maintains the synchronous code correlation in quasi-synchronous multipath propagation. We also give a system example that uses Gold sequences as the spreading codes. Section IV gives a brief comparison of the proposed scheme and CIBS-CDMA. Section V presents numerical results that illustrate the proposed scheme’s performance. Finally, Section VI gives a conclusion.
II. CIDS-CDMA SIGNALS
A. Transmitted Signal
In CIDS-CDMA, the modulated signals are formed as as follows. First, data symbols are partioned into blocks. After spreading, the symbols in each block are transmitted with their chips interleaved. Whereas CIBS-CDMA employs one spreading code for each data block of a user, we allow use of multiple spreading codes. Its implications will become clear later.
Let each block of interleaved data contain Mdsymbols and,
for the time being, assume that each symbol in the block is
spread with a different code. Assume that M0 zero chips are
padded after each Md data chips as in [2], where M0may be
null. Then the signal for block g of user k can be expressed as
sk(g) = √
2PCkbk(g), (1)
where√2P is the normalized signal amplitude of each user,
bk(g) = [bk(gMd), . . . , bk((g + 1)Md− 1)]T denotes the gth
data block, and
Ck= [diag(ck[0]), 0Md×M0, . . . , diag(ck[N − 1]), 0Md×M0]T
(2) is the “spreading matrix" with N being the spreading factor,
ck[n] (n = 0, . . . , N − 1) the vector of the nth chips of the Mdcodes, and0Md×M0 the all-zero matrix of size Md× M0.
The dimension of Ck is N M × Md where M = Md+ M0.
The number of output chips for each block after spreading is thus equal to N M .
Let c(h)k [n] (n = 0, . . . , N − 1) denote the spreading code for the hth data symbol of user k. Then, for block g,
ck[n] = [c(gMk d)[n], . . . , c((g+1)Mk d−1)[n]]T, (3)
where the absence of index g in the left-hand side reflects the fact that we use the same set of spreading codes for all blocks. (Change of spreading code assignment from block to block is not considered in this work. It may result in some performance gain whose amount depends on the operating condition, but the amount tends to be insignificant when users are many.) The
synchronous correlation between the two codes for the hth symbol of user m and the hth symbol of user k is defined as
Λm,k(h, h) =N−1 n=0
c(hm)[n]c(h)k [n]. (4)
B. Received Signal
Let the maximum possible delay spread for any user be
L − 1 chips and the channel response for user k be given by hk[n] =
L−1 l=0
αk[l]δ[n − τk− l] (5)
where τk is the initial delay of user k’s signal. The model
accommodates both synchronous and quasi-synchronous
trans-mission. For convenience, define αk[l] = 0 for l < 0 and
l ≥ L and let τmax = maxk{τk}. Let M0 ≥ τmax+ L − 1.
After channel propagation, each block of user signal is spread
over a time interval of no more than N M + τmax+ L chips.
In particular, the received signal in the interval [gN M, (g +
1)NM + τmax+ L − 2] (in unit of chips) due to user k’s
transmission can be expressed in vector form as
xk(g) = Hk,0sk(g) + Hk,−1sk(g − 1) + Hk,1sk(g + 1), (6)
where xk(g) is an (NM + τmax+ L − 1)-vector and Hk,0,
Hk,−1, andHk,1are (N M +τmax+L−1)×NM Toeplitz
ma-trices. MatricesHk,0andHk,1are lower triangular with their
first columns given by [01×τk, αk[0], . . . , αk[L−1], 0, . . . , 0]T
and [0, . . . , 0, αk[0], . . . , αk[L+τmax−τk−2]]T, respectively,
whereasHk,−1 is upper triangular with its first row given by
[0, . . . , 0, αk[L − 1], . . . , αk[0], 01×(τk−1)].
In (6), the second and the third right-hand-side terms are the interference from the previous and the subsequent data blocks and have been named interblock-interference (IBI). Thanks to zero padding, these IBI terms can be made of no influence to data detection, as already noted in [2]. Hence we may ignore them in subsequent derivation. Then the total received signal in the above time interval is given by
x(g) = K−1
k=0
xk(g) + ξ(g), (7)
where ξ(g) = [ξ[gN M ], . . . , ξ[(g + 1)N M + τmax+ L − 2]]T
is a vector of additive noise samples, assumed white Gaussian.
III. EFFECTINGSYNCHRONOUSCODECORRELATION
AFTERMULTIPATHPROPAGATION
The rake receiver, which is of lower complexity than many other receiver structures, can be used to receive the above CIDS-CDMA signal. To fully collect the energy in the lth path of user k’s signal, the despreader input must contain the
(τk+ l)th to the (NM − M0− 1 + τk+ l)th elements of x(g).
Denote byH(l)k,0the matrix composed of the lth to the (N M +
l − 1)th rows of Hk,0 and by ξ(l)(g) the vector consisting of the lth to the (N M + l − 1)th elements of ξ(g). Without loss of generality, consider detection of the 0th user signal. Despreading of the received signal for the lth path results in
an Md-vector given by y(l)0 (g) =√2P K−1 k=0 CT 0H(τk,00+l)Ckbk(g) + CT0ξ(τ0+l)(g). (8)
Let J(h) denote the N M × N M matrix whose elements are
all zero except for the hth diagonal where the elements are all ones, where h = 0 refers to the main diagonal, h > 0 a
sub-diagonal, and h < 0 a super-diagonal. (J(h) is an h-unit
delay operator.) Then CT 0H(τk,00+l)Ck = CT0 · L−1 d=0 αk[d]J(τk+d−τ0−l)Ck = L−1 d=0 αk[d]CT0J(τk+d−τ0−l)Ck. (9) The matrix CT0J(τk+d−τ0−l)C
k is all zero except for the
(τk + d − τ0− l)th diagonal. Let (A)i,j denote the (i, j)th
element of matrix A. Then for j = i − (τk+ d − τ0− l),
CT 0J(τk+d−τ0−l)Ck i,j = N−1 n=0 c(gM0 d+i)[n]c(gMk d+j)[n] = Λ0,k(gMd+ i, gMd+ j). (10) Hence CT 0J(τk+d−τ0−l)Ck= J(τk+d−τ0−l)D(τk+d−τ0−l) (11)
whereD(l) is a diagonal matrix whose i diagonal element is
given by Λ0,k(gMd+ i, gMd+ i − l). In summary, y(l)0 (g) =√2P K−1 k=0 R(l)0,kbk(g) + ξ(l)0 (g) (12) where R(l)0,k = L−1 d=0 αk[d]J(τk+d−τ0−l)D(τk+d−τ0−l), ξ(l)0 (g) = C0ξ(τ0+l)(g). (13)
From the above, it is clear that the despreading result is entirely determined by the synchronous correlation of the spreading codes although the transmission is over a multipath channel. Thus, unlike conventional DS-CDMA whose perfor-mance depends on both the even and the odd code correlations [6], the multicode CIDS-CDMA has a more controllable performance.
k
b [h]
s [n]
kRow Input, Column Output Block Interleaver
N
M
c
k [n] (h%L)Fig. 1. A CIDS-CDMA transmitter that employs L spreading codes periodically.
The rake combiner output for b0(g) is given by
ˆb0(g) =L−1
l=0
α∗0[l]y(l)0 (g), (14)
which contains ISI, MAI, and noise, in addition to the desired signal b0(g).
Since the number of spreading codes having low synchro-nous correlation is limited, we should conserve code usage. From the above discussion, we can see that it suffices to have
L spreading codes for each user, to be used periodically, when
the multipath delay spread is not greater than L−1 chips. This is illustrated in Fig. 1.
A. An Example Using Gold Sequences
Any set of sequences with low (including zero) synchronous correlation can be used in the proposed system. However, all types of sequences are not equal considering system capacity. For example, only N/L users can be accommodated if Walsh-Hadamard sequences are used, but other sequences may permit more. A recent paper presents optimal binary sequence design for many combinations of processing gain and number of used spreading sequences [7]. However, to change the number of used sequences may require a redesign of all used sequences. In addition, Gold sequences are shown to be optimal in various conditions [7]. Hence we consider use of Gold sequences for reasons of flexibility and performance.
A set of Gold sequences of length N is constructed from a preferred pair of m-sequences of length N . The set contains
N + 2 sequences, including the preferred pair. Dropping one
of the two m-sequences from the set results in a set where
the correlation value between any two sequences is −1. This
property is used in our code assignment strategy.
Unlike conventional spreading code assignment, we assign the spreading codes gluttonously as follows. We assign L of the N +1 Gold sequences to the first user, then another L to the second user, and so on, until we exhaust the N + 1 sequences. Then we assign the N +1 sequences which are one cyclic shift of the original Gold sequences, and then those which are two cyclic shifts, and so on. This way, we can assure that in the despreading at each rake finger, there are always N among the KL − 1 interfering signals that have a code correlation
equal to−1. Numerical evaluation shows that the mean-square
synchronous correlation of cyclically shifted Gold sequences is close to N , similar to random codes. Hence, by Gaussian approximation, the resultant bit error rate (BER) in random static channels when the multipaths are all of equal strength
is given by PCI = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Q 1 1 SNR+2N2LN +KL−N−12NL , KL > N + 1, Q 1 1 SNR+KL−12N2L , KL ≤ N + 1. (15) In comparison, the performance of conventional DS-CDMA under random-code spreading is given by
Pconv = Q 1 1 SNR+KL−12NL . (16)
Comparing (15) with (16), it is clear that the performance of the proposed system is always better than the conventional one, especially when KL is not much larger than N + 1.
IV. COMPARISONWITHCIBS-CDMA
As discussed previously, CIBS-CDMA assigns one spread-ing code to each user, where the spreadspread-ing codes are orthogo-nal. This deterministically eliminates MAI in multipath prop-agation, but results in ISI which can be dealt with by single-user equalization. In comparison, the proposed scheme may employ orthogonal or nonorthogonal codes and it may assign one or more codes to a user, depending on the desired capacity and the channel condition. The proposed code assignment has several advantages.
First, for CIBS-CDMA, the residual ISI after single-user equalization may be far from being Gaussian when the equal-ized channel consists of only a small number of taps. Such ISI may highly degrade the performance of channel coding as can be seen in the numerical results of various studies [4], [5], [8], [9]. With multicode CIDS-CDMA, the total interference approaches Gaussian more closely due to the presence of a larger number of interferers (consisting of the different user signals, spread with multiple spreading codes and propagated over multipath channels). This is beneficial to the error correction capability of channel coding.
Second, the proposed multicode CIDS-CDMA provides a soft capacity as conventional DS-CDMA does, in the sense that the performance degrades gracefully as more users are accommodated. In contrast, the capacity of CIBS-CDMA is limited by the number of orthogonal sequences. (One could consider overloading the CIBS-CDMA system with additional users whose signals are spread with nonorthogonal spreading sequences. But then the added users may experience perfor-mance similar to that of conventional DS-CDMA due to MAI.) Third, the proposed scheme offers added flexibility in lever-aging between receiver complexity and transmission perfor-mance. For low complexity, a rake receiver may be employed. Table I shows the relative complexity of some core functions in the rake receiver and the CIBS-CDMA equalizer in detecting one user symbol. In multicode CIDS-CDMA, assume that a user is assigned L spreading codes and the rake receiver has (at most) L fingers. The rake receiver needs to despread up to L codes per user symbol, but the CIBS-CDMA equalizer only one. However, despreading is usually of lower complexity compared to the other functions in Table I. For CIBS-CDMA, the equalizer complexity depends on its type. Table I lists the complexity involved with serial and block linear MMSE
TABLE I
RELATIVECOMPLEXITY OFRAKERECEIVER FORMULTICODE CIDS-CDMAANDEQUALIZER FORCIBS-CDMA
Despreading Filtering Filter Computation*
CIDS-CDMA L L 0
CIBS-CDMA (serial) 1 Lg (L + Lg+ 1)2/Md CIBS-CDMA (block) 1 Md+ M0 (Md+ M0)2/Md *Excluding computation of needed signal correlations.
5 10 15 20 25 10−6 10−5 10−4 10−3 10−2 10−1 Number of Users BER DS−CDMA (Simu.) DS−CDMA (Ana.) CIDS−CDMA (Simu.) CIDS−CDMA (Ana.)
Fig. 2. BER performance of conventional DS-CDMA and CIDS-CDMA under Gold-sequence spreading, for different user numbers atL = 6, N = 63, and SNR = 13 dB.
equalization employing the low-complexity method to
com-pute the equalizer coefficients [2], where Lg is the number of
taps for serial equalizer. To have better transmission efficiency
(=Md/(Md + M0)), Md should be much larger than M0.
Hence, block equalization is not suitable in such a system due to the huge demand of equalizer taps. For serial equalization,
experience shows that we usually need Lg> L to have good
results. Hence the proposed scheme will normally have a lower receiver complexity. Now Table I does not count in the computations needed of the signal correlations for determining the rake finger weights or that for setting up the equation for calculating the CIBS-CDMA equalizer coefficients. However, their relative complexity should be on the order of L versus
Lg. On the other hand, if better performance than what rake
can provide is desired in a multicode CIDS-CDMA system, an equalizer can also be employed. This is especially the case when the number of spreading codes assigned to the concerned user is smaller than L. This equalizer can have a lower complexity than that in CIBS-CDMA [10]. The details are omitted here.
V. SIMULATIONRESULTS
A. Comparison With Conventional DS-CDMA
Firstly, we present results for random static channels.
We let each initial delay τk be uniformly distributed in
{0, 1, . . . , N − 1}. Limiting τmax to N − 1 is for easier
comparison between conventional DS-CDMA and multicode CIDS-CDMA; it is not a fundamental limit to the proposed scheme. The multipaths have the same amplitude, with their
4 6 8 10 12 14 16 10−6 10−5 10−4 10−3 10−2 10−1 Number of Users BER DS−CDMA (v=3 km/hr) DS−CDMA (v=30 km/hr) DS−CDMA (v=60 km/hr) CIDS−CDMA (v=3 km/hr) CIDS−CDMA (v=30 km/hr) CIDS−CDMA (v=60 km/hr)
Fig. 3. BER performance of conventional DS-CDMA and CIDS-CDMA under Gold-sequence spreading, for different user numbers atL = 4, N = 31, and SNR = 13 dB under fading channels.
phases uniformly distributed in [0, 2π). Gold sequences are used in both systems. Fig. 2 shows the results for N = 63 and L = 6 at SNR = 13 dB. As pointed out in [1], the results for conventional DS-CDMA under asynchronous transmission match closely the theoretical result for random-code spreading (given in (16)). With the proposed code assignment, CIDS-CDMA clearly outperforms conventional DS-CIDS-CDMA espe-cially when the user number is small. The larger discrepancy between the simulation results and the theory at when K is about (N + 1)/L can be explained by noting that, because the number of spreading codes with different cyclic shifts is relatively small in this situation, the MAI has a greater variance about its mean than when K is larger.
The foregoing analysis and simulation have assumed that the channel responses remain unchanged during the whole chip-interleaving block so as to effect perfect synchronous correlation among the spreading codes. Therefore, question arises as to how multicode CIDS-CDMA performs in fading channels. Hence we consider transmission over multipath
Rayleigh fading channels next. Let N = 31, Md = 152,
M0 = 12, carrier frequency = 2 GHz, and chip rate = 3.84
Mcps (as in 3GPP [11]). Assume that τmax= 8 chips, which
corresponds to a maximum 625 m difference in initial path lengths, and asssume that L = 4. For simplicity, let the total path energy be normalized per chip-block length, which is 31 × 164 = 5084 chips and about two slots in 3GPP. The results for different moving speeds are shown in Fig. 3. As can be expected, channel fading rate has minor impact on DS-CDMA since we have assumed quite accurate power control. Comparatively, faster fading does more harm to CIDS-CDMA because channel fading alters the correlation among users. Nevertheless, under the simulated parameters, CIDS-CDMA still outperforms DS-CDMA with slight capacity loss.
B. Comparison With CIBS-CDMA
Now we simulate channel-coded transmission where the channel code is the rate-1/2 binary convolutional code of constraint length 7 with generators g1= 1338and g2= 1718.
7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 10−5 10−4 10−3 10−2 10−1 SNR (dB) BER CIBS (Lg=7) CIBS (Lg=15) CIBS (Block) CIDS (K=25) CIDS (K=31)
Fig. 4. BER performance of convolutionally coded CIBS-CDMA and CIDS-CDMA as a function of SNR withN = 32 and 31, respectively. Solid lines: before channel decoding; dashed lines: after channel decoding.
And we employ soft-decision Viterbi decoding with traceback
length = 5× 7 = 35. For spreading, we use Gold sequences
of length 31 in multicode CIDS-CDMA and Walsh-Hadamard sequences of length 32 in CIBS-CDMA. In both systems,
Md = 144 and M0 = 4. A 24 × 24 block bit-interleaver is
inserted between the channel coder and the spectrum spreader to disperse the sometimes bursty errors for better decoding performance. We simulate synchronous transmission over
4-path channels with average 4-path energies 0, −1, −2, and
−3 dB. The path coefficients are Rayleigh distributed. Each
simulation run involves one randomly generated channel. In each run, we simulate perfect power control by normalizing the total path energy. Fig. 4 shows the average results over 1, 000 simulation runs. For CIBS-CDMA, the serial
equal-ization delay is set to Lg/2. Observe that although
CIBS-CDMA with serial equalization has better performance before channel decoding, its performance after decoding improves less dramatically than multicode CIDS-CDMA due to non-Gaussian distribution of the residual ISI. Even under the very heavy system loads simulated, the latter can still outperform the former at a lower receiver complexity. CIBS-CDMA with block equalizer is also shown there and demonstrates a superior performance especially in high SNR. However, the required taps for equalization is 148, which is too complex
to be implemented. The performance of CIDS-CDMA can be further improved by applying some advanced interference cancellation techniques. However, it is beyond of the scope of this letter.
VI. CONCLUSION
We proposed a multicode chip-interleaved DS-CDMA scheme for transmission over slow-fading multipath channels. Through analysis and simulation, we showed that the proposed scheme had superior performance to conventional DS-CDMA in this condition, thanks to its ability to preserve the synchro-nous correlation among the spreading codes. In addition, the scheme was able to attain comparable or better performance than CIBS-CDMA at a lower receiver complexity, even under a heavy system load.
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