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3178 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 12, DECEMBER 1994
MMSE Interference Suppression for
Direct-Sequence Spread-Spectrum CDMA
Upamanyu Madhow and Michael L. Honig, Senior Member, ZEEE
Abstract- We consider interference suppression for direct- sequence spread-spectrumcode-divisionmultiple-access(CDMA) systems using the minimum mean squared error (MMSE) per- formance criterion. The conventional matched filter receiver suffers from the near-far problem, and requires strict power control (typically involving feedback from receiver to transmit- ter) for acceptable performance. Multiuser detection schemes previously proposed mitigate the near-far problem, but are complex and require explicit knowledge or estimates of the interference parameters. In this paper, we present and analyze several new MMSE interferencesuppression schemes, which have the advantage of being near-far resistant (to varying degrees, de- pending on their complexity),and can be implemented adaptively when interferenceparameters are unknown and/or time-varying. Numerical results are provided that show that these schemes offer significant performance gains relative to the matched filter receiver. We conclude that MMSE detectors can alleviate the need for stringent power control in CDMA systems, and may be a practical alternative to the matched filter receiver.
I. INTRODUCTION
EMODULATION of direct-sequence spread-spectrum D(DS/SS) code-division multiple-access (CDMA) signals is conventionally achieved with a matched filter receiver. Because the crosscorrelations between the spreading, or signature, sequences for different transmissions are nonzero, a nearby interferer can disrupt reception of a highly attenuated desired signal. Interference suppression schemes previously proposed [ 5 ] , [8]-[9], [14]-[16] can mitigate this near-fur problem by exploiting the structure of the multiple-access interference. These schemes are significantly more complex than the matched filter receiver and require explicit knowledge or estimates of interference parameters. Consequently, recent proposals for CDMA systems (e.g., see 141) assume a matched filter receiver, and solve the near-far problem by using power control, which requires feedback from the receiver to the transmitter. In this paper we propose and analyze several interference suppression schemes based on the minimum mean squared error (MMSE) criterion. This work, originally presented in
[lo]-[ll], was motivated by recent work which has shown that MMSE equalization techniques can be used to suppress both intersymbol interference (ISI) and crosstalk interference in wire channels [l], [7], [13]. A major advantage of MMSE schemes, relative to other previously proposed interference suppression schemes, is that explicit knowledge of interference parameters is not required, since filter parameters can be adapted to achieve the MMSE solution. Also, the complexity of these schemes, measured in number of filter coefficients, can be adjusted to achieve a given level of performance. The MMSE linear detector for a pulse-amplitude modu- lated data signal in the presence of interfering data signals consists of a bank of filters matched to the pulse shapes of all active users followed by symbol-rate samplers and an Infinite-length Impulse Response (IIR) multi-input/single- output digital filter (see [7] and the references therein). The interference suppression schemes proposed here can be viewed as finite-complexity approximations of this detector. The first scheme proposed consists of sampling the channel output at the
chip rate, and using an N-tap adaptive FIR filter to minimize
the mean squared error (MSE) between the transmitted and detected symbol where N is the processing gain. This scheme is motivated by the fact that the MMSE linear detector just described can be implemented as an infinite-length fractionally spaced tapped-delay line. For the special case of symbol- and chip-synchronous CDMA transmissions, the N-tap detector is in fact equivalent to the MMSE linear detector. If, however, transmissions are symbol-asynchronous,then the MMSE linear detector requires an infinite number of taps. Furthermore, if transmissions are chip-asynchronous, then the MMSE linear detector requires that the tap spacing be smaller than the chip duration. However, our numerical results demonstrate that even in the chip- and symbol-asynchronous situation, the N-tap MMSE detector can offer a dramatic performance improvement relative to the matched filter detector. Finally, in the absence of multiple-access interference, the N-tap detector reduces to the conventional matched filter receiver. The remaining interference suppression schemes proposed~~ Paper approved by J. S. Lehnert, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received October 13, 1992; revised April 15, 1993. This paper was presented in part at the First Intemational Conference Universal Personal Communications, Dallas, TX, September 28-October 1, 1992, and at the E E E GLOBECOM, Orlando, FL, December 6-9, 1992. 07960 USA. He is now with the Coordinated Science Laboratorv. Universitv
are motivated by the situation in which the processing gain is large, but the number Of strong interferers to be suppressed is relatively Small. This may occur if power control alone cannot provide reliable protection
detector in t h ~situation may be degraded bv slow convergence
and the number Of active
U. Madhow was with Bell Communications Research, Morristown, NJ against^ strong interferers.^ The^ performance^ Of^ the N-tap .. Y - of Illinois, Urbana, IL 61801 USA.
Evanston, IL^ 60208 USA.
due to the large number of adaptive taps. We therefore consider simpler schemes for interference suppression that have fewer
M. L. Honig is with the Department of EECS, Northwestern University,
IEEE Log Number 9406144. adaptive taps. In the first, the channel output is connected 009@6778/94$04.00 0 1994 IEEE
MADHOW AND HONIG: MMSE INTERFERENCE SUPPRESSION 3179
to a bank of D filters, each of which is a cyclically shifted version of the matched filter. In the second scheme a single matched filter is used, but its output is sampled D^ times per symbol interval. In each case, the decision statistic is a linear combination of the D samples obtained in a symbol interval, where the weights are selected to minimize the MSE. Since
D can be much smaller than the processing gain N , these
schemes should be easier to adapt than the N-tap MMSE
detector when N is large. The first scheme typically performs
somewhat better than the second scheme. Furthermore, the first
scheme can be implemented as a bank of D filters of length
N/D, which are each sampled D times per symbol interval.
This makes the complexity of the two schemes comparable. Interference suppression techniques for CDMA systems using the MMSE criterion have also been considered in [2], [5], [16], and [17]. An adaptive correlator, which is similar to the N-tap detector is proposed in [17]. However, the em- phasis in [ 171 is on supressing narrowband interference, rather than other wideband CDMA signals.The detector proposed in [16] uses an MMSE criterion to estimate a finite block of transmitted symbols. The front end of this detector consists of a bank of filters matched to the pulse shapes of all active users followed by symbol-rate samplers. Explicit knowledge or estimates of the interference parameters are therefore assumed. Also, the estimated symbols are obtained by processing all matched filter outputs which correspond to the entire block of transmitted symbols. In contrast, the detectors presented here process samples from within a single symbol interval to estimate the desired symbol, and require significantly less computation. The MMSE decision feedback equalizer (DFE) in the con- text of CDMA is considered in [2]. Since only decisions from the desired user are fed back, the feedback filter suppresses intersymbol interference, but not multiple-access interference. Since we focus on the latter in the present paper, we restrict attention to linear MMSE detectors of varying complexity. A multiuser decision-feedback detector, in which decisions on transmitted symbols from all users are fed back, is considered in [5]. Although the feedback filter in this case does suppress multi-user interference, this structure is more complex than the detectors considered in this paper, and is more difficult to adapt in the presence of unknown interferers. As the level of background noise tends to zero, or as the energies of the interferers increases to infinity, the MMSE linear detector converges to the decorrelating detector intro- duced in [8]-191, which eliminates multiple access interference at the expense of noise enhancement. Because the schemes considered here are finite complexity approximations of the MMSE linear detector, their performance is, in general, not as good as the performance of the decorrelating detector (which has the same complexity as that of the MMSE linear
detector) [ll]. Nevertheless, the results in Sections I11 and V
demonstrate that the schemes considered here are generally near-far resistant, in the sense defined in [9]. The next section presents the CDMA system model consid- ered, and the performance of the MMSE detectors presented in this paper is analyzed in Section 111. Section IV presents the two simpler interference suppression schemes. Numerical
results are presented and discussed in Section V, and Section VI contains our conclusions.
11. SYSTEM MODEL
The received signal is the sum of K simultaneous CDMA
transmissions plus additive white Gaussian noise. The received signal due to the jth user is given by
r j ( t ) = Jq b ; , j s j ( t - i T - v j ) ~ ~ s ( ~ , t + B j ) ,
00
2=-= l < j l K (1)
where T is the bit interval, b ; , j E {11-1) is the ith bit of the jth user, Pj, vj, and O j are the power, delay, and carrier phase of the jth user, respectively, w,^ is the carrier frequency, and s j ( t ) is a spreading, or signature, waveform given by N- s j ( t ) = -x-aj[Ic]$(t - IcT,) k=O
where aj[k] E {-1, 1) is the kth element of the spreading
sequence for user j , $(t) is the chip waveform, N is the
processing gain, and T, = TIN is the chip duration. We
assume that $ ( t ) has unit energy and duration T,.
The received signal is then K T ( t ) = C T j ( t ) + n(t) (3)
where n(t) is white Gaussian noise with power spectral density N0/2. The problem considered is to demodulate the first transmission, which will be referred to as the desired transmission. It is assumed that the receiver is synchronized to this transmission, so that the lcth sample at the output of the chip matched filter is
T [ k ] = h/
j=
(k+l)Tc+vi T ( t ) $ ( t ) COS (4 + 61) d t. (^) (4)
For the detectors considered in this paper all bit decisions are based on the discrete-time signal ~ [ k ]. (The notation z[lc] will be used to denote samples of a continuous-time signal spaced at the chip interval.) We assume that the power and delay of the desired signal
are, respectively, Pl = 1 and v1 = 0. For convenience we
also consider a carrier-synchronous system in which the carrier phase B j = 0 for each j , although our analysis is easily
modified to take nonzero O j into account. For 2 5 j 5 K ,
the relative delay vj = (rj + Sj)Tc where rj is an integer
between 0 and N - 1, and Sj = vj/T, - rj lies in the interval
[0, 1). An asynchronous system is assumed in which both rj and Sj may be nonzero. Each detection scheme presented here estimates a given transmitted symbol from received samples within a single symbol period. The estimate of bo, 1 therefore depends only
on the received signal for t E [0, TI, or equivalently, the
vector of received samples rT = (r[O],... ,T[N- 11). During this symbol interval the jth interfering signature sequence is
kTc +vi
MADHOW AND HONIG: MMSE INTERFERENCE SUPPRESSION 3181
0: PI
I----?
'.( 1y-y Interference Subspace S f
Fig. 1. Geometric representation of signal and interference vectors.
If A is ill-conditioned, then there are different solutions for c which result in nearly the same MSE. This can result in tap wandering during adaptation, which has been observed in single-user applications by Gitlin et al. [6]. One solution to this problem, which slightly increases the MSE, is the tap leakage algorithm proposed in [6].
A. Alternative Derivation Using Orthogonal Decompositions
In what follows it will be convenient to assume that the noise is white, i.e., that r = a 2 1 M. This assumption does not entail any loss of generality since the noise n can always be whitened by an orthonormal linear transformation V where VrVT = a 2 1 ~ , assuming I? is positive definite.
Define the integerence subspace SI as the subspace of RM
spanned by the interference vectors p z ,... , p L , and let Sy
denote the subspace of RM orthogonal to S I. The desired
vector p1 can then be expressed as
P 1 = P 1^ I^ + 01, I (14)
where p: denotes the projection of p1 onto SI and o: denotes
the projection of p1 onto Sy. Fig. 1 illustrates these geometric relationships. The MMSE solution c must lie in the subspace S spanned by the signal vectors p1 , p z ,... , p L since from (6), any vector w orthogonal to this space satisfies w T r = vTn, and therefore only contributes noise to the output. Furthermore, the
space S is the direct sum of the orthogonal subspaces SI and
the subspace spanned by 0:. so that the MMSE solution can be written as
c = CI + dlo: (^) (15)
where we have used the fact that cTpj = (c')'pj for 2 5 j 5
L. Solving (16)-(17) gives
where c' is given by
B. Asymptotic Behavior of the MMSE Solution The quantities in (18)-( 19) will be specified in terms of the crosscorrelations of the signal vectors p1 ,... , p L ; however, before doing this we consider the limiting behavior of the MMSE solution as 1 ) the noise level goes to zero, and 2) the interference vectors increase in energy. First consider the zero- noise situation in which a' = 0. If 0: # 0, then clearly the interference can be eliminated by choosing c to be a multiple of 0:. This choice of c is called the zero-forcing solution. If a' = 0, then the zero-forcing solution gives zero MSE, as indicated by (18). In this case, the MMSE solution is therefore
where the scaling is determined by the requirement lcTrl =
Now consider the near-far situation, in which the energy of one or more of the interference vectors can vary^ arbitrarily.
Define wj = (Ipj112 as the energy of the jth signal vector
where 1^5 j^5 L.^ For a given^ w1, we are interested in the MMSE solution c when the interference energies wz,... , W L assume values that maximize the MMSE. It is easily verified that the MMSE increases monotonically as a function of the interference energies so that we consider the asymptotic MMSE solution as wj -+ 03 for j E J,, for some subset J , ( 2 ,... , L }. In addition, we assume that wj, j $! Jm, are constant. In contrast to the worst case MMSE performance considered here, we note that for the maximum-likelihood de- tector, the interference energies that maximize error probability are not easily determined [141. It is shown in Appendix B that the MMSE solution satisfies
0: I 10 : II
c = -
lcTplI = lbll = 1.
w,lim -io3^ cTpj^ =^ 0,^ j^ E^ J,.
That is, c is asymptotically orthogonal to the space spanned by the set of interference vectors { p j } , j E J,. In the worst-case situation in which these vectors span the entire interference subspace S I , we have that c' = 0 asymptotically, and we
where c' is the projection of c onto S I , and dl is a scalar.
(14) and (15) gives
Projecting each side of (10) onto SI and Sy, and using
3182 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 12, DECEMBER 1994
The MMSE solution is “near-far resistant” in the sense that the worst case MSIR in (22) is greater than zero provided that 11 oi 11 > 0. We now determine the near-far resistance of the MMSE solution in the sense defined in [81-[91. Let
P,(n) be the average error probability for a given detector as
a function of the noise variance 2. The asymptotic efJiciency of the detector is defined in [8]-[9], [15] as
y = sup{r;: u-0 lim P e ( a ) / & ( m / a ) 2 0}, (23)
and is a limiting measure, as the noise level tends to zero, of how well the detector performs in the presence of multiple- access interference relative to its performance in the absence of multiple-access interference. The near-far resistance of the detector is defined in [8]-[9] as
77 = W 2 ,... , W L inf^ y.
That is, the near-far resistance is the asymptotic efficiency evaluated for worst case interference energies, and is a measure
of the robustness of the detector with respect to variations in
the received interference energies. As 0 -+ 0, the error probability for the MMSE detector considered here satisfies
The asymptotic efficiency of the MMSE detector is therefore 110:112/w1. Since this quantity is independent of the energies of the interference vectors, we have that
The near-far resistance of the MMSE detectors considered is therefore the (appropriately normalized) norm of the compo- nent of the desired signal vector which is orthogonal to the space spanned by the interference vectors. The near-far resistance is nonzero if and only if the desired vector p1 is not contained in the interference subspace S I. A necessary condition for this is to be true is that the dimension
of SI be strictly less than the dimension of the signal vectors
M. Since the dimension of SI is upper bounded by L - 1,
the number of interference vectors, it is reasonable to expect nonzero near-far resistance when L - 1 5 M - 1. We emphasize that this is neither a necessary nor a sufficient condition for nonzero near-far resistance, but is merely an approximate rule. Applying this rule to the N-tap MMSE
detector, the signal vector dimension M = N , so that this
detector has enough degrees of freedom to suppress ( N - 1)/2 asynchronous interferers. (Recall that each asynchronous interferer can generate at most two interference vectors, so that
L - 1 5 2(K - 1)). In contrast, for synchronous CDMA, the
N-tap detector has enough degrees of freedom to suppress at
most N - 1 interferers, since each interferer generates only
one interference vector (L - 1 = K - 1).
C. MMSE Pet$onnance in Terms of Signal Energies and Cross-Correlations Rewriting (18)-(19) in terms of signal energies and cross- correlations gives somewhat different (but equivalent) ex- pressions for the MMSE solution c, the MMSE, and MSIR than (11)-(13). Define the unit-energy signal vector p j by p j = wJ1’2pj, and the normalized crosscorrelation between the jth and kth signals as R j k = pTpk where 1 5 j , k < L.
Let RI = ( R j k ) z < j , k < L denote the nonnegative definite
(L - 1) x (L - 1) interference cross-correlation matrix, and let pT = (R12,... , R I L ) denote the vector of cross-correlations between the desired vector and the interference vectors. and 0:. The projection of the desired
signal onto the interference space SI can be written as p: =
~ i u i ~ F = ~ z j j i ~. TO compute ZT = ( z z ,... , zL) we note that
We first compute
pro: = j$(pl - p i ) = 0, 2^5 k^^5 L,^ (27) which gives
RIZ = p. (28)
The vector z need not be unique unless the interference vectors
are linearly independent, in which case z = Rllp. It follows
that
I (29) From (26), the near-far resistance of the MMSE detector can be written as
(30)
where z is any vector that satisfies (28). It is easily shown that 77 is unique even if (28) does not have a unique solution. To specify the MMSE solution we must compute &. By definition,
\ -
77 = 1 - pTz
L
cI = C d j p i ,
j = 2
where ti? = ( d 2 ,... , d L ) is determined from (19). It is shown in Appendix C that
dI = (MMSE)J.lleI (32) where e: = (e2,... , e L ) satisfies
(RIWIRI + ~ ‘ R I ) ~ I= P , (33)
and that
MMSE = [I + w l ( i - pTz)/o2 + wlpTeIj-l, (^) (34)
MSIR = (w1/a2)[(1 - pTz) + a 2 p T e ~ ] (35)
where the (L-1) x (L-1) matrix W I = diag[w2,... , w ~ ]. An important difference between (34)-(35) and the equivalent ex- pressions (12)-( 13) is that the vectors that appear in (34)-(35) have L - 1 components, whereas the vectors in (12)-(13) have
3184 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 42, NO. 12, DECEMBER 1994
.
(b) Fig. 2. (a) Detector with cyclically shifted filter bank. f o , ... , f D P 1 are cyclic shifts of the matched filter a l. (b) Reduced complexity CSFB for D = 2.
from the formulas in Section I11 by making the appropriate substitutions. We now show that the vector y can be generated with D filters each of length N / D , assuming that D divides N, so
that the total number of taps in the filter bank is N , instead
of ND. It is shown in Appendix D that the same reduction
in complexity can be achieved even if D does not divide
N (which is the case for the example used to generate our
numerical results).
The output of the ith filter f; after the zeroth symbol interval
is N - 1 y; = a1 [(j+ i A ) mod N]r[j] j = O D-la- = (^) E a l [ k +jA]r[(k + ( j - i)A)modN]. (42) k=O j=O
We now divide each filter f; into D disjoint contiguous
subji2ters of length N/D. The set of subfilters, which we
denote as {ek}, k = l ,... , N / D , is the same for each of
the cyclically shifted filters fi, and is specified by
[ejll~= a l [ j + k A ] , 0 5 j^5 A -^ 1,^0^5 k^5 D -^ 1. (43)
Let &[m] be the output of the kth subfilter e k at time m. Then (42) and (43) imply that
D - 1
yi = C2/k[((k-i+l)A-l)mod N], 0 5 i 5 D-1, (44)
that is, y; is the sum of the outputs of the subfilters ek, k =
l ,. -. , D , sampledat (chip) times [(k-i+l)A-11modN. To
generate all D components of y the output of each {ek} must
be sampled D times at chip times i N / D , i = 0,. , D - 1. This is illustrated in Fig. 2(b) for the case D = 2. Of course, to detect the mth bit bm,^^1 all chip samples used to generate the corresponding vector of filter outputs are incremented by mN.
k=O
I VI
t b-2,1^ ,I b-LJ^ I 1^ b0.J^^1 ,
I bJ1,i^ I -^ b0.l^ - I^ bi,i^ I one symbol interval (n Fig. 3. Illustration of channel output samples r[k], which are used to compute the estimated symbol bo, 1. The contribution to the channel output from the desired user and the jth interferer are shown. For the delay T~ shown in the figure, the estimate 6 0 , 1 depends on the interference symbols b-1, 3 , b , j , h , j , and b - 1 , 1.
B. Over-Sampling The CSFB scheme is similar (but not equivalent) to sam-
pling the output of the single matched filter f a = 01 D times
per symbol period. Over-sampling the output of the matched filter in this context is analogous to the fractionally spaced equalizer for single-user channels [7]. Denote the matched
filter output at time N - 1 - ZA as
N - 1 v, = a1 [k]r[k - iA] (45) k=O where A = [N/DJ is the interval between successive sam-
ples. The decision rule for b a , ~ is then given by b a , ~=
sgn(aTw) where wT = (v,,...,vg,...,~g-~--l),_ m is a
phase offset, and a E R D is chosen to minimize the MSE.
In general, m can also be selected to minimize the MSE; however, the numerical results in the next section assume that m = [(D - l)/2J. It is shown in Appendix E that the vector w can be expressed as shown in (6) where p l ,... , p L are chosen appropriately. The performance of this scheme can therefore be evaluated by using the formulas in Section 111.
From (45) it is apparent that samples r [ k ] for k = -(D -
m -! ) A ,. -. , N - 1 + m A are used to compute the esti- mate bo, 1. As illustrated in Fig. 3, depending on the relative delay rJ, the jth interfering signal during this interval may contain segments from three different symbol intervals. The
bits associated with these different segments are bo, 3 , b-1, 3 ,
and either b-2, or bl, J. This is in contrast to the CSFB and N-tap MMSE schemes, in which the jth interfering signal contributes segments from at most two successive symbol
intervals with associated bits and b - l , J. Also, for the
over-sampling scheme adjacent symbol intervals of the desired transmission, associated with bits b - 1 , 1 and b l , 1, act as additional interference, which is not present in the CSFB scheme. For the same value of D the performance of the over-sampling scheme is therefore expected to be, on average, somewhat worse than that of the CSFB scheme. According to the approximate rule stated at the end of Section 111-B, since each asynchronous interferer can effectively contribute three interference vectors, the over-sampling scheme should perform significantly better than the matched filter detector when the number of strong asynchronous interferers is less
than or equal to (D - 1)/3.
Both the over-sampling and CSFB schemes require D filter
taps. However, the CSFB scheme requires D samplers (as
MADHOW AND HONIG: MMSE INTERFERENCE SUPPRESSION 3185
compared to one for the over-sampling scheme), together
with D connections between sampled outputs as shown in
Fig. 2(b). The increase in complexity required by the CSFB scheme relative to the over-sampling scheme seems quite modest so that the CSFB scheme may be preferable in many situations.
V. NUMERICALRESULTS
We now compare numerically the performance of the in- terference suppression schemes discussed in preceding section with the matched filter receiver for a specific example. All of
the following results assume that the processing gain N =
31, and that the interferers are asynchronous, but have the same signature sequence, which is different from the desired signature sequence. Both the desired and interference signature sequences are Gold sequences taken from [3,^ Table^ VI.^ The corresponding interference vectors are linearly independent, due to the different delays of the interferers, and the fact that distinct shifts of a PN sequence are linearly independent. The delays of the interferers relative to the desired signal, as
multiples of the chip interval T,, are denoted by vg, .... v ~.
In order to reduce the computation required to average over all delays, we impose the relation v3 = vg + ( j - 2)X (modulo N )
for 3 5 j 5 K , for fixed A, which fixes the delay between user
j , j > 3, and user 2. The following examples are averaged over v2. Fig. 4 shows a plot of near-far resistance, as defined in Section 111 versus the delay vg for the N-tap, CSFB, and over- sampling schemes. (The near-far resistance of the matched
filter is zero.) The parameters are K = 3 users, X = 5.5, and
D = 7 for the CSFB and over-sampling schemes. According to
the discussion in Section IV, the minimum number of taps for which the near-far resistance can be expected to be positive is
D = 5 for the CSFB scheme and D = 7 for the over-sampling
scheme. In fact, for D = 5 (not shown here), and for nearly all
delays u2, the over-sampling scheme does have zero near-far resistance, and the near-far resistance of the CSFB scheme is quite small. Fig. 4 shows that the near-far resistance of each of the simpler schemes is very sensitive to delay, and can be quite small. In contrast, the near-far resistance of the N-tap MMSE detector is much less sensitive to delay, and stays relatively high. It is interesting that there are several delays at which the over-sampling scheme has greater near-far resistance than the CSFB scheme. However, the succeeding sets of numerical results, which are averaged over delays, show that the CSFB scheme offers a modest performance improvement relative to the over-sampling scheme. We now consider the situation in which S (strong) inter- ferers each have power P , assumed to be large, relative to
the desired user, and the remaining K - 1 - S interferers
have the same power as the desired user. Figs. 5 and 6 show SIR and average error probability, respectively, as a function
of P, which varies from 0 to 10 dB, relative to the desired
user. The performance measures are averaged over the relative delay vz^ (quantized to multiples of^ T,/4).^ The signal-to-noise ratio for the desired user in the absence of multiple-access interference is fixed at 20 dB. The curves for the CSFB
0 5 10 15 20 25 30
Delay Fig. 4. Near-far resistance of MMSE detectors versus delay vz ( N = 31, A’ = 3).
15
10 % s 5
0
.......................... /”................................................. N-tap MMSE (S=K-l=lO) ............................ CSFB^ (D=7) .....
.....
over-sampling(D=5) Matched Filter I I I I I I 0 2 ‘ 4 6 8 10
Relative power of strong interferers Fig. 5. SIR versus the relative power of strong interferers. For the N-tap MMSE detector ( N = 31), K = 11 with 10 strong interferers, and for the simpler schemes, A = 5 with two strong interferers.
scheme, over-sampling scheme, and matched filter receiver
assume that K = 6 and S = 2, whereas the curves for the
N-tap MMSE detector assume that K = 11 and S = 10. That is, for the simpler schemes there are five interferers, two of which are strong, and for the N-tap MMSE detector there are 10 interferers, all of which are strong. Ignoring the potential problem of slow adaptation speed, Figs. 5 and 6 show that the N-tap MMSE detector essentially eliminates the need for power control. Despite their relative simplicity, the CSFB and over-sampling schemes perform substantially better than the matched filter receiver, and would likely loosen the requirements for power control in this con- text. These results also show that for the same D the CSFB scheme performs somewhat better than the over-sampling scheme. We add that the degradation in the performance of the MMSE schemes with respect to single-user performance
MADHOW AND HONIG: MMSE INTERFERENCE SUPPRESSION 3187
C. Derivation of (32)-(35) It is convenient to rewrite (16) as
(C.
where gives
denotes the MMSE from (12).^ Substituting from (
L L L
j = 2 k = 2 k= and taking the inner product of each side with pm for 2 5
m 5 L gives (32)-(33). From (12) and (15) we have that
Substituting (17) for d l , and (31) and (32) for (t?)T gives
which implies (34) and (35).
D. Complexity Reduction for CSFB: General Case
We show that the vector y can be generated with D - 1 filters
of length A, and one of length A + S where S = N - DA.
The output of f i at time N - 1 is
N - 1 yi = x a l [ ( k + i A ) m o d N ] r [ k ] k=O
k=Om=O 0-2A-
k=i m=O A - 1 + 6
m=O
. ~ [ m + (D - 1 - i)A + SI,
for 0 5 i 5 D - 1. Define the subfilter eo-1 by
The subfilters e o , “. , e D - - 2 are again defined by (43). Let &[m] denote the output of the filter e k at time m. From (D. 1) we have that
i-
k=O 0 - 2 + x. l t k [ ( k - i + 1)A - 11 + y ~ - 1 [ N - 1 - ZA] (^) (D.3) k=i
which expresses yi as the sum of the outputs of the filters e l ,... , e o - 1 at appropriate chip times.
E. Over-Sampling Solution and Pegormanee The aperiodic crosscorrelation between the signature se- quences a k and a; is denoted as
From (45) and (5) w^ can be written as
for 1 5 j 5 K where
the index i takes on values -m,...,O,..., D - m - 1,
q = 61 = 0, and PI = 1. Comparing (E.2) with (6), the
vector s 1 , o is the desired vector, and the remaining vectors are interference vectors (including s-1,1 and s 1 , 1 from user 1). Note that for each 2 5 j 5 K , either S I , j or s-2, j must be
zero. The Gaussian noise vector w has zero mean and D x D
covariance matrix U with components
Note that the performance of the CSFB scheme can similarly be expressed in terms of crosscorrelations between interfering signature sequences and cyclic shifts of the desired signature sequence.
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Upamanu Madhow received the B.S. degree in electrical engineering from the Indian Institute of Technology, Kanpur, in 1985. He received the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign in 1987 and 1990, respectively. From August 1990 to July 1991, he was a Visiting Assistant Professor at the University of Illinois. From August 1991 to July 1994, he was a research scientist at Bell Communications Research. Since August 1994, he has been with the University of Illinois at Urbana-Champaign, where he is currently an Assistant Professor. His current research interests are in communication systems and networks for wireless mobile communications, and in high speed computer communication networks. Dr. Madhow was awarded the President of India Gold Medal for graduating at the top of his undergraduate class. He was the recipient of a University of Illinois fellowship from 1985 to 1986, and a Schlumberger fellowship from 1987 to 1988.
Michael L. Honig (S’&M81SM’92) was bom in phoenix. AZ, in 1955. He received the B.S.degree in electrical engineering from Stanford University in 1977, and the M.S. and h. D. degrees in electrical engineering from the University of^ California.^ Berkeley,^ in^ 1978 and 1981.^ respectively. He subsequently joined Bell Laboratories. Holmdel, NJ, where he worked on local area netwollrs, adaptive filtering. and voiceband data transmission. In 1983 he joined the Systems Principles Research Division at Bellcore. where he worked on Digital Subscriber Lims and wireless cunmunications. He has also been a visiting lecturer at Princeton University. Since Fall 1994, he has been with Noahwestem University, where he is currently the Ameritech Professor in Information Technology in the Electrical Engineering and Computer Science Department.
