Multistage Nonlinear Blind Interference Cancellation for DS-CDMA Systems.
In this paper we propose a multistage nonlinear blind interference cancellation (MS-NL-BIC) receiver for direct-sequence code-division multiple-access (DS-CDMA) systems. The receiver uses higher order statistics of the received baseband signal. Specifically, we use the second and fourth moments of the received signal to determine a component of the received vector that has significant mean energy and low variability of the energy, both of which are favorable characteristics for application in an interference cancellation scheme that uses hard decisions. The structure of the receiver is multidimensional and can be viewed as a matrix of receivers. Each row in the matrix consists of receivers that perform (hard decisions) cancellation of successive components that have significant mean energy and low variability of the energy. The columns of the matrix essentially resemble multistage receivers that iteratively refine performance from earlier stages. Simulation results show that unlike linear receivers, the MS-NL-BIC is exceptionally efficient in systems with strong and highly correlated interferers, as may be the case in overloaded DS-CDMA systems.
Introduction
One of the driving forces of the next generation of wireless communications is the demand for
higher data rates and higher capacity of wireless systems. Primary applications of higher data rates
seem to be in the downlink direction (for example, typical internet data tra#c such as downloading
of web pages). Furthermore, wideband direct-sequence code-division multiple-access (DS-CDMA)
technology has emerged as one of the most promising candidates for future wireless systems (e.g.,
third generation systems [1, 2]). It is therefore of great interest to investigate the performance of
these systems and its viability for higher data rates envisioned in the future.
In DS-CDMA systems, in general, crosscorrelations between signature (spreading) sequences are
nonzero. This results in the near-far e#ect where the multiple-access interference (MAI) can disrupt
reception of a highly attenuated desired user signal [3]. Baseband signal processing techniques such
as multiuser detection and interference cancellation have the potential to combat this problem and
provide higher performance at the cost of increasing complexity of the receiver. Rapid progress in
the area of semiconductor technology has resulted in a significant increase in the processing speeds
of core technologies (DSP, FPGA and ASIC device). Advances in VLSI technology, and design of
the algorithms that are optimized with respect to a specific implementation platform, are further
narrowing the gap between the complexity of the algorithms and processing speeds (e.g., solutions
that are presented in [4, 5]). These and other developments suggest that the transceivers in future
wireless systems will employ some form of interference mitigation.
Several multiuser receivers have been proposed (for example, see [6-9]). These receivers are denoted
as centralized because they require knowledge of parameters (signature sequences, amplitudes
and timing) for all users in the system. Therefore, they are more suitable for processing at the base
station. For the downlink, it is desirable to devise decentralized receivers that exploit the knowledge
of the desired user parameters only. The use of short signature sequences simplify the task of
multiuser detection and interference cancellation, since a receiver can adaptively learn (estimate)
the structure of the MAI [10]. Decentralized receivers may be further classified into data aided and
nondata aided receivers. Data aided adaptive multiuser detection is an approach which does not
require a prior knowledge of the interference parameters. But, it requires a training data sequence
for every active user. For example, adaptive receivers in [7,11,12] are based on the MMSE criterion,
and the one in [13] is based on minimizing probability of bit-error. More recently, decision feedback
detectors using the MMSE criterion have been proposed [14,15]. Unlike data aided receivers, blind
(or nondata aided) multiuser detectors require no training data sequence, but only knowledge of
the desired user signature sequence and its timing. The receivers treat MAI and background noise
as a random process, whose statistics must be estimated. Majority of blind multiuser detectors
are based on estimation of second order statistics of the received signal. In [16], a blind adaptive
MMSE multiuser detector is introduced (proven to be equivalent to the minimum output energy
(MOE) detector). A subspace approach for blind multiuser detection is presented in [17]; where
both the decorrelating and the MMSE detector are obtained blindly. Further, adaptive and blind
solutions are analyzed in [18], with an overview in [10]. A blind successive interference cancellation
(SIC) scheme, which uses second order statistics, is proposed in [19, 20].
In this paper we propose a novel blind interference cancellation receiver, which assumes knowledge
of only the desired user's signature sequence. The receiver is based on determining that
component of the received signal that has significant mean energy and low variability in the energy.
It applies the minimum variance of energy and maximum mean energy criterion (MVE-MME),
which is described in Section 3. Furthermore, we analyze a relationship between the above criterion
and Godard's dispersion function [21] and constant modulus (CM) criterion [22]. In Section
4, using the MVE-MME criterion, we derive a nonlinear multistage blind interference cancellation
receiver. The structure of the NL-MS-BIC receiver is multidimensional and can be
viewed as a matrix of IC stages. Each row in the matrix consists of IC stages that perform the blind
(hard decisions) successive interference cancellation. The columns of the matrix essentially resemble
multistage receivers that iteratively refine performance from earlier stages. This particular multi-stage
structure of the receiver allows concurrent (parallel) execution of the IC stages which makes
it very suitable for implementation using multiprocessor DSP and/or FPGA (or ASIC) platform.
Simulation results are presented in Section 5, and we conclude in Section 6.
Background
The received baseband signal, r(t), in a K-user asynchronous DS-CDMA additive white Gaussian
noise (AWGN) system is
where A k is the received amplitude, b k (i) # {-1, +1} is binary, independent and equiprobable data,
is the signature sequence which is assumed to have unit energy, # k is the relative time o#set,
all for the k th user. T is the symbol period and n(t) is AWGN with unit power spectral density,
with # being square root of the noise power. is the number of data symbols per user per
frame.
It is well known that an asynchronous system with independent users can be analyzed as synchronous
if equivalent synchronous users are introduced, which are e#ectively additional interferers
[3]. In this paper we consider the received signal r(t) over only one symbol period that is
synchronous to the desired user 1). The discrete representation for the received signal in (1)
can be written in vector form as
where the number of the interferers doubled due to the equivalent synchronous
user analysis. r, s k and n are vectors in # M , where M is the number of chips per bit.
Consider the nonlinear centralized SIC scheme which is presented in [5, 8]. We now present
a brief outline of the above scheme because its approach to nonlinear interference cancellation is
generalized in this paper and later applied in a blind interference cancellation scheme (Section 4). In
the nonlinear centralized SIC scheme [5, 8] it is assumed that the signature sequences are perfectly
known (centralized approach). The basic operations of the SIC algorithm are (see Figure 1):
1. Detect one user with the conventional detector, i.e., matched filter (MF).
2. Regenerate the baseband signal (vector) for this user.
3. Cancel the regenerated signal (vector) from the received baseband signal.
Then, this operation is repeated successively for all the users in the system. The idea is that
successive cancellations result in reduced MAI for the remaining users. The received vector r j+1
after stage j of the cancellation is given by
where r j is the received vector, -
A j and - b j are the corresponding estimates of the amplitude and the
bit, respectively, all for stage j 1 . The above implementation of the SIC algorithm is nonlinear in
that it uses hard decisions ( - b j # {-1, 1}) in successive stages. A primary reason why the nonlinear
centralized SIC cannot achieve performance of the single user lower bound (SULB) is due to
erroneous estimate of bit decision (i.e., - b j #= b j ). When an error happens, it causes the SIC scheme
to double the interference, which is, of course, undesirable. Furthermore, the doubled interference
propagates through the following IC stages, which degrades the overall performance of the receiver.
For the same reason, this receiver also is not near-far resistant [3]. Further, imperfections in amplitude
and delay estimates can lead to the non ideal regeneration and cancellation. Accordingly, to
obtain best results, the user with the highest signal to interference ratio (SIR) should be cancelled
first. This condition is usually relaxed and the user with highest received power is cancelled first,
followed by the second strongest, and so forth [8, 9]. Thus, it is desirable to identify users (or
signature sequences) that have significant power (energy). Note that the SIC scheme requires amplitude
estimates for the users, which implicitly requires low variability of the amplitude estimates
for perfect cancellation.
Let us now generalize the nonlinear cancellation given by (3). In equation (3), let us replace s j
with a vector v j (not necessarily a signature sequence). Furthermore, we replace -
A j with the square
root of the estimate of # Thus, the nonlinear cancellation in
the j th stage is executed as
In the following, we propose favorable characteristics of the vector v j to be successfully applied
in the above scheme. We now analyze the estimate of the energy # j of the received signal in the
z denotes an estimate of z
direction of the vector v j . To estimate # j we use sample statistics as
where N is the size of the averaging window (number of samples), and n and m are time indices
(will be omitted in the following). It is well known that the error of the estimate in (5) is directly
related to the variance of x . Using the Chebyshev inequality [23] it can be shown that as
the variance of x j gets lower, the accuracy (mean square error) of the energy estimate is improved:
We can say that for the vector v j that corresponds to x j with lowest variance (among all vectors
in # M ), the estimate of the energy is the most reliable, i.e., the mean square error of the energy
estimate is the lowest. Note that the variance of x j is the variance of the energy of the received
vector r in the direction of the vector v j (i.e., variance of squared projection of the vector r onto
the vector v j ).
The above analysis leads us to believe that the vector v j which corresponds to the low variability
of the energy and significant mean energy of the vector r in the direction of v j , is desirable for the
nonlinear cancellation given by (4). These characteristics of the vector v j o#er reliable estimates
of the corresponding energy # j and sign of r # v j . In the following we present a scheme that blindly
determines (estimates) the vector v j and further applies this vector to realize a multistage nonlinear
interference cancellation scheme.
3 MVE-MME Optimization Criterion
We now present an optimization criterion which is used in deriving a nonlinear blind adaptive
interference cancellation scheme. According to the analysis in Section 2, the goal of the optimization
approach is to determine a component of the received vector r that has low variability in the energy
and significant mean energy. We consider the squared output of the projection of r onto a vector
. The vector v is obtained from the following nonlinear procedure which is
where is subject to u # 1. The function # 1 (u) denotes the variance of
the squared output r # u and is given as
The function # 2 (u) in (7) denotes the square mean energy given as
Consider the function # 1 (u). We now present the following proposition that gives an intuitive
description of the minimum variance of energy criterion, which is obtained by minimizing the
expression in (8).
Proposition 1 For the synchronous antipodal DS-CDMA system (described in (2)), with zero
linearly independent signature sequences {s i } L
, the solutions for
arg min u constrained as u # are classified in two groups:
(a)
k=1,k #=i and s # i d i #= 0. Actually, d i is the modified
matched filter that corresponds to decorrelating detector for user i [6].
(b) w is any vector from the noise subspace.
Further, the above solutions correspond to the absolute minimum of # 1 (u) where it is zero.
We present a proof of the above proposition in Appendix A. Let us now compare # 1 (u) with the
well known, Godard's dispersion function [21]:
Where # is a real constant, and p is an integer. For
2, the cost function
in (10) is directly proportional to # 1 (u). In other words, # 1 (u) penalizes dispersions of the
squared output (r # u) 2 away from the constant E h (r # u) 2
. Furthermore, the well studied constant
modulus (CM) cost function is defined as a special form of the function in equation (10), where
2. The CM cost function is widely used for blind equalization (see [22]
and references therein). Later in this paper, # 1 (u), which may be viewed as a slightly modified
form of the CM cost function, is applied for blind interference cancellation in DS-CDMA systems.
Let us now consider the function # 2 (u). It can be shown that the vector v
constrained as u # is equal to the vector that also maximizes the mean energy E h (r # u) 2
It is shown in [19, 20] that v max is the eigenvector that corresponds to the largest eigenvalue of the
input covariance matrix R Instead of the mean energy E h (r # u) 2
is
applied in (7) such that both terms are of the same order (i.e., fourth order). Based
on the above, the vector v, which is defined in the equation (7), corresponds to that component
of the received signal r that has low variability in the energy and significant mean energy. As discussed
in Section 2, these characteristics are favorable for application of the vector v in a nonlinear
interference cancellation scheme. The parameter - is used to control which of these two characteristics
(low variability of the energy or significant mean energy) is dominant. For example, if
the optimization in (7) is equivalent to minimum variance of energy (MVE), and for
equivalent to maximum mean energy (MME) optimization criterion. Therefore, we refer to (7) as
the minimum variance of energy and maximum mean energy (MVE-MME) optimization criterion.
Note that in Subsection 3.2 we revisit issues related to the parameter - and propose its design.
3.1 Adaptive Solution
We now present an adaptive algorithm that solves (7). We exploit some properties of the functions
given in (8) and (9). Let us assume that the input process r is wide sense stationary (WSS) and
also that
where n and m are time indices, and n #= m. In other words, we assume that the energy of r in
direction of the vector u is uncorrelated in di#erent symbol (bit) intervals. Using the properties of
WSS processes and (11) we can show that (8) can be written as
for all integer n and m, n #= m. Similarly, the expression (9) can be written as
According to (12) and (13), and using sample statistics, the function f(u) is defined as an approximation
of #(u) as
f(u, n) =F
F
where F is a number of consecutive symbols used for the approximation. Gradient of f(u, n) is
defined as
F
We can use a stochastic gradient algorithm [24] that solves (7) as
where l is the index of the iteration step, and 0 < # < 1 is a certain scalar which defines the length
of adaptation step. The constraint |-v l+1
forced after every iteration, where - v l stands for
estimate of v in l th iteration step.
3.2 Choice of Parameter -
As addressed earlier, the parameter - is used to control which of the two characteristics of v (low
variability of the energy or significant mean energy) is dominant. We choose - as
Note that the above definition is similar to the inverse of the normalized kurtosis
[23]), but further analysis of this relationship is beyond the scope of this paper. Furthermore, as an
approximation of the above definition, we set
in equation (15). Considering characteristics of the parameter - that is defined by (17), it can be
shown that
1. If r # u is a real-valued Gaussian random process, - is 1/3.
2. Let -M denote - corresponding to r # u, which is a uniform discrete real-valued M-ary random
process, i.e., r # u # {a i | a A is the
maximum absolute value of r # u. Based on the above definition, it can be shown that

Figure

2 depicts the parameter - as a function of the alphabet size of a uniform, real-valued M-ary
random process. As a reference, we present - that corresponds to a continuous uniformly
distributed random process (denoted as CU), and a Gaussian random process (denoted as
GP). Note that the function is decreasing with alphabet size M , or in other words,
Furthermore, we may note that - is maximum at real-valued
bipolar random process. In addition, we note -M in equation (19) converges towards - that
corresponds to the continuous uniformly distributed random process (CU in Figure 2).
From the above properties of the parameter -, we may draw the following conclusions. When
the received signal of the output of the correlator, r # u, is a real-valued Gaussian random process
(i.e., u lies in the noise subspace of the received vector r), then - takes a value close to its minimum
thereby steering the MVE-MME criterion towards minimizing variance of energy (MVE). On the
other hand, when the output r # u is a close to a discrete-valued random process (as in the case when
MAI dominates), - approaches its maximum value thus steering the MVE-MME criterion towards
maximizing mean energy (MME). In the course of adaptation, the value of - given in equation (18)
changes according to the projection r # u, i.e. u being in the noise (Gaussian) part of the signal
subspace or the interference (discrete-valued random process) subspace.
4 Application of the MVE-MME Criterion in the Multi-stage
Nonlinear Blind IC Receiver
We now present a multistage nonlinear blind interference canceler, denoted as MS-NL-BIC. The
structure of the receiver is multidimensional and can be viewed as a matrix of receivers (i.e., matrix
of IC stages). The MS-NL-BIC receiver consists of P rows and Q columns, where each entry of the
matrix corresponds to an interference cancellation stage denoted as IC ij
The following steps are executed in the IC ij stage (where r ij is the input vector to that stage):
1. Add back x (i-1)j as
where x (i-1)j is a portion of the received signal that is cancelled in the IC i-1j stage. Note that
the IC i-1j stage is the same column, but earlier row of the matrix. For the first row (i = 1),
because no cancellation is performed prior to this row.
2. Use adaptation rule in (16) (r # ij replaces r) to estimate v ij as - v ij (see Figure 3). Note that the
vector further processed in the very same manner as an interferer signature sequence in
the case of the nonlinear centralized SIC scheme (see Section 2).
3. Estimate the energy
Note that the estimation should be reliable because
as a component of the vector r # ij , has low variability in the energy (due to the term
4. Detect the sign of r # ij
Note that detection should be reliable, because the component
has significant mean energy (due to the term -# 2 (u) in (7)) and low variability.
5. Perform nonlinear cancellation as
where (see Figure 4)
The above procedure is executed successively (within the i th row of the matrix), where for the
new stage IC ij+1 , the input vector is r ij+1 (see equation (22)). The structure of the i th row (i.e.,
horizontal topology) is depicted in Figure 5. From the above, each row may be viewed as a blind
equivalent to the nonlinear centralized SIC scheme, where the components replace the actual
signature sequence. After su#cient number Q of the stages in the i th row, cancellation is repeated
in the (i th row (see Figure 6). The input vector r i+11 of the (i th row is r iQ+1 . The
stage IC i+1j is used to iteratively refine the cancellation which is executed in the earlier stage IC ij
appropriate delay, the vector x ij , that is canceled in the stage IC ij is added
back (step 1), and within the stage IC i+1j processing is performed again (steps 2 to 5).
In Section 5, Q is selected to be equal to the number of dominant interferers, but in the more
general case, this number might not be known at the receiver. A number of di#erent schemes can
be employed in order to determine the number of IC stages within each row of this receiver. Here,
we propose the following simple scheme. In the first row the stage IC 1j may be determined as
the last stage in the row if the estimate of the energy -
# 1j drops below a certain threshold
e . In other words
This simple scheme assumes that -
# ij+1 (i.e., the energy estimate is decreasing with column
index j). In addition, the scheme is based on the assumption that the component that corresponds
to the mean energy
which is below the threshold T e , is not relevant for
the cancellation. Furthermore, the number of the rows P is directly related to the performance of
the receiver. Thus, the trade-o# in performance versus complexity can be controlled by the number
P . After su#cient number P of the rows, detection of the desired user is performed using a linear
detector (e.g., matched filter).
Note that implicit in the above algorithm is the assumption that the interferers are stronger
than the desired user. If the desired user is strong, then additional processing is required to ensure
that the desired signal is not canceled out before the detection. Briefly, we propose a corresponding
scheme which is based on a threshold rule. In last row,
is projected onto the desired user signature sequence s 1 . The absolute value of the projection
is compared against a predefined threshold T h (0 < T h < 1). If the absolute value exceeds the
threshold T h ,
the vector x P j (defined in equation (23), where should be added back to r PQ+1 . Having
inspected all the vectors -
the addition is performed as follows
where the corresponds to all vectors -
that have met the criterion in (25). Further, the
linear detection of the desired user is performed using r # PQ+1 as input signal.
Note that a centralized multiuser detection scheme, which is proposed in [2,25], applies a similar
iterative (recursive) refinement approach that is presented above. That particular scheme executes
centralized SIC and iterative refinement in order to improve channel estimates for the users in the
system. Unlike the MS-NL-BIC receiver, the scheme in [2,25] assumes the knowledge of all signature
sequences of the users in the system (i.e., it is not blind). Further, the multistage structure of the
receiver allows concurrent (parallel) execution of the IC stages. This inherent parallelism of the
algorithm is a favorable characteristic for its implementation using multiprocessor DSP and/or
FPGA (or ASIC) platform (see [4, 5] and references therein).
5 Simulation Results
We consider a synchronous AWGN DS-CDMA system using randomly generated signature sequences
with processing gain 8. The users are independent and the following cases are analyzed:
1. System with users (fully loaded), and equal-energy interferers: A 2
2. System with users, and equal-energy interferers: A 2
3. System with users (overloaded system); three strong equal-energy interferers: A 2
4, and eight interferers with the same energy as the desired user: A 2
The crosscorrelation profile of the users with respect to the desired user is depicted in Figure 7.
Note that in this particular example the crosscorrelations are very high, except for users 5, 10
and 12 which happen to be orthogonal to the desired user. In the case 1, the system has users
and in the case 2 only In all our results the input sample covariance
matrix is estimated according to
R r (i) =N
where N is the size of the averaging window (number of samples), and i is the time index (will be
omitted in the following text). Performance of the conventional matched filter (MF), the centralized
MMSE receiver (denoted as MMSE), the blind MMSE receiver (BMMSE) (detector
and the single user lower bound (SULB) are used as benchmarks for evaluation of the MS-NL-
BIC receiver. The centralized MMSE assumes perfect knowledge of all the signature sequences,
amplitudes and the variance of the AWGN. Performance of the MS-NL-BIC is evaluated for MF
(MS-NL-BIC-(MF)) and the blind MMSE (MS-NL-BIC-(MMSE)); where these linear detectors are
used for detection of the desired user after the cancellation (after P rows and Q IC stages within
each row). The MS-NL-BIC-(MMSE) uses the sample covariance matrix of the output signal of the
last IC stage (IC PQ ). Note that the desired user energy is set to be much lower than the energy
of the interferers, and, as discussed in Section 4, the prevention of the excessive cancellation of the
desired user is not performed. In each IC stage, the performance is measured after 1000 symbols
used for the estimation in (16) and (27), and (18). Regarding the parameter
-, we apply the approximation given by (18). We assume the knowledge of the number of dominant
interferers Q, which is the number of columns of the receiver matrix.
For the case 1, Figure 8.a depicts bit-error rate (BER) as a function of signal to background
noise ratio (SNR) (with respect to the desired user). The results are obtained after a total of
rows and which is where the BER reaches minimum. Additional IC stages do not
introduce any improvement for this particular example. For versus total number
of IC stages is presented in Figure 8.b. Note that in this example the MS-NL-BIC(MMSE) performs
better than the MS-NL-BIC(MF). In this case, after the last IC stage (IC 47 ) MAI is still present,
therefore, BMMSE detector can further improve the performance of the MS-NL-BIC receiver in the
case 1.
Equivalent results, for the case 2, with are shown in Figures
9.a and 9.b, respectively. In Figure 9.b, note that the MS-NL-BIC(MMSE) converges faster with
respect to number of IC stages, but, at the end the MS-NL-BIC(MF) o#ers lower BER for this
particular example. In this case, after the last IC stage (IC 43 ), MAI is almost completely removed.
Introduction of the BMMSE as linear detector in the MS-NL-BIC receiver, may cause a drop in the
performance due to estimation errors of the covariance matrix (in (27)) which is used to derive the
BMMSE detector (this particular topic is analyzed in [20]).
We consider the performance of our receiver in the case 3, which is an overloaded DS-CDMA
system. Figure 10 depicts BER versus SNR (with respect to the desired user). The same figure
presents the performance of the matched filter (denoted as MF-8) for the system without the strong
interferers (only the desired user and eight equal-energy interferers, which is identical to ideal
cancellation of the users 2, 3 and 4). The receiver with used. From the
results in Figure 10, we note that the MS-NL-BIC completely cancels the strong users i.e, the MS-
NL-BIC(MF) performance is identical to the MF-8 performance (in Figure 10 their characteristics
overlap).
From these results, it is seen that the MS-NL-BIC outperforms linear receivers (MF and BMMSE),
significantly. The performance of the linear receivers is expected because it is well known that they
do not perform well in the case of the systems with strong and highly correlated interferers (with
respect to the desired user signature sequence) [3], as may be the case in overloaded systems.
But, these results suggest that the MS-NL-BIC may be applied as a blind solution in the case of
overloaded systems with strong interferers.
Let us now study the characteristics of the estimates In all
cases that we have observed, as the processing progresses from row to row, estimates
each column of the receiver matrix, approach one of the actual signature sequences (with sign
ambiguity). Each column corresponds to a di#erent signature sequence. For example, in the case 1,
we observe how the estimates approach the actual signature sequences
respectively. The absolute value of the crosscorrelation - l is depicted in Figure 11,
where the abscissa represents index of the row 4). The results appear to be similar
for all columns 7. Note that in the last row, the estimates are practically identical to
the signature sequences i.e., the absolute value of the crosscorrelation is 1. We have consistently
observed these results in our simulations, and a detailed mathematical analysis and explanation of
the above phenomenon is of future interest.
6 Conclusion
We have introduced the MVE-MME optimization criterion which is then used to implement the
MS-NL-BIC receiver. The receiver is based on determining the component of the received vector
that has significant mean energy and low variability in the energy. The MS-NL-BIC consists of
multiple IC stages, and can be viewed as a matrix of IC stages. The columns of the matrix
resemble multistage receivers that iteratively refine performance from earlier stages, while each
row corresponds to a blind equivalent to the nonlinear centralized SIC scheme. The ability of the
receiver to exceed the performance of the linear receivers is confirmed via the simulation results. It
is seen that this scheme is particularly e#ective for systems with very strong interferers which are
strongly correlated with the desired user signature sequence. Therefore, this may be a very viable
solution for implementation in the case of overloaded systems with strong interferers.
A
Proof of Proposition 1
Consider a synchronous antipodal DS-CDMA system, with zero AWGN linearly
independent signature sequences. Let us denote a component of the signature sequence s i as d i
which is orthogonal to other users' signature sequences, or in other words, d i #
k=1,k #=i and s # i d i #= 0, which can be written as
where |c i | # 1 and c i #= 0, and # ij is the Kronecker delta function. Projection of the received vector
r (see equation (2)) on d i yields (in the absence of additive background noise):
where A i and b i are the amplitude and bit, respectively, all for user i. From (29), it follows that
which results in
Note that
Therefore, the absolute minimum of # 1 (u) is zero. Based on (30) and (31), # 1 (u) reaches the
absolute minimum for d i Using the above approach, the same result can be obtained
for -d i This proves part (a) of Proposition 1.
Further, if N denotes the noise subspace and by definition of the noise subspace [26]
follows that
Consequently
which proves that for any vector u from the noise subspace, # 1 (u) reaches the absolute minimum
of zero. This concludes proof for part (b) of Proposition 1.



--R

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--TR
blind interference cancellation;successive interference cancellation;iterative refinement