Applicability of blind beamforming techniques to
FMA-1 with spreading
Alle-Jan van der Veen
Delft University of Technology, Dept. Electrical Engineering/DIMES, 2628 CD Delft, The Netherlands
Abstract— The purpose of adaptive antenna techniques is to separate
and equalize superpositions of source signals impinging on a phased an-tenna array. The required space-time equalizer coefficients are obtained from deterministic properties of the signals such as known training bits, known spreading codes, Toeplitz structures and/or constant modulus prop-erties. Alternatively, we can assume a parametric channel model and estimate directions-of-arrival and propagation delays. Within the Eu-ropean ACTS program, the FRAMES project is responsible to define a candidate multiple access system for UMTS. We will look at the pro-posed FMA-1 modulation format with spreading, discuss the properties that are available for estimating the equalizer coefficients, and indicate generic algorithms that exploit them.
1. INTRODUCTION
In the context of array signal processing, beamforming is concerned with the reconstruction of source signals from the outputs of an sensor array. This can be done either by co-herently adding the contributions of the desired source, or by nulling out the interfering sources. The latter is an instance of the more general problem of source separation.
Classically, beamforming requires knowledge of a look di-rection, which is the direction of the desired source. Cur-rently, “blind beamforming” is a hot topic within signal pro-cessing: the idea is to separate sources without direction in-formation, relying instead on various structural properties of the problem. Even more recent is “semi-blind beamforming”, in which the structure provided by known training sequences is also taken into account.
The first blind beamforming techniques proposed were based on direction finding. The direction of each incoming wave-front is estimated, at the same time producing a beamformer to recover the signal from that direction. This requires at least that the antenna array is calibrated. If a source comes in via several directions (coherent multipath), then direction find-ing is more complicated. Dependfind-ing on the situation, we also need to consider delay spread. Thus, the applicability of these techniques is much dependent on the channel conditions and in general requires a small number of well defined propaga-tion paths per source (a parametric channel model).
More recently, new types of blind beamformers have been proposed that are not based on specific channel models, but instead exploit properties of the signals. A striking example is the constant modulus algorithm (CMA), which separates sources on the fact that their baseband representation has a constant amplitude, such as is the case for FM or phase mod-ulated signals. A prime advantage is that these beamformers are not dependent on channel properties or array calibration, which makes them very robust. Several other properties are often available, for example cyclostationarity caused by the bauded nature of digital communication signals or introduced by small differences in carrier frequencies.
In this paper, we will look at the applicability of such tech-niques to the FRAMES Multiple Access format, mode 1 (TDMA).
This is a rather diverse format. To limit the discussion, we will look only at the FMA-1 mode with spreading, which is most interesting from a signal processing point of view. In this mode, the slots have fixed size, up to 8 cochannel users are allowed in a single slot, and the users modulate their sym-bols with short orthogonal codes of length 16. The symbol al-phabet is either 4-PSK or 16-QAM. A linearized GMSK pulse shape provides for some constant modulus structure (limited by the symbol alphabet), and training symbols are available as well. The delay spread necessitates equalization of up to 2-3 symbols, or 33 chips. Altogether, a rich structure is available. The objective is to find space-time beamforming coefficients to jointly separate the 8 cochannel users and to equalize them.
2. FMA1-SPREAD MODULATION FORMAT
The FRAMES Multiple Access format mode 1 with spread-ing is described in document [1]. For our purposes here, it can be summarized as follows.
FMA1-spread is a TDMA scheme with 8 time slots per frame. The length of a time slot is 577µs. The symbol pe-riod is Ts= 7.39µs. Each symbol is modulated with a code of length Q= 16. Thus, the chip period is Tc= 0.46µs (more precisely, the slot has length 1250 chips, approximately 78 symbols). There are 16 different orthogonal codes, but only 8 are used, to accommodate 8 synchronous cochannel users. The time slot consists of 2∗28 symbols, and a training mi-damble of 296 chips (an equivalent of 18.5 symbols). The
re-maining part of the slot is used as a guard period. The training sequences of the 8 users are selected in a special way, which allows channel estimation using properties of cyclic matrices [2].
The delay spread is considered to be less than 15µs, corre-sponding to 32 or 33 chips, or 2 symbol periods. As in GSM, the doppler shift is sufficiently low in comparison to the length of the time slot so that the channel can be regarded stationary within the slot, but not from one frame to the next.
The pulse modulation function c0(t) is a linearized GMSK
pulse of length 5Tc[1], see figure 1. The chips in a chip se-quence are chosen alternatingly real (±1) and imaginary (±j): for the i-th user,
ci= [ci0 · · · ci15] = [±1 ±j · · · ±1 ±j] ci(t) = 15
∑
k=0 cikδ(t−kTc) The symbol modulating function gi(t) is thusgi(t) = ci(t)∗c0(t)
and it has approximately a constant modulus over a single symbol period.
0 1 2 3 4 5 0
0.5 1
mntime [Tc]
Fig. 1. Linearized GMSK pulse shape function
The symbol alphabet is either 4PSK, i.e.,Ω={±1,±j}, or 16-QAM, i.e.,Ω={±1,±3}⊕{±j,±3 j}. The transmitted
sig-nal for the i-th user then has the form
zi(t) = di(t)∗ci(t)∗c0(t) , where di(t) = N
∑
k=1 dikδ(t−kTs) , dik∈ ΩiHere, dik denotes the k-th symbol of the i-th user and N is the total number of transmitted symbols in a data block. (In fact, this is not an entirely correct representation because the training sequence does not have the same form: it is an “ar-bitrary” sequence of length 296 chips, alternatingly real (±1) and imaginary (±j).)
3. CHANNEL MODEL
The propagation of signals through a radio channel is fairly complicated to model. A correct treatment would require a complete description of the physical environment, not very suitable for the design of signal processing algorithms. To ar-rive at a more useful parametric model, we have to make sim-plifying assumptions regarding the wave propagation. Pro-vided this model is reasonably valid, we can, in a second stage, try to derive statistical models for the parameters to obtain agreement with measurements.
In the present section, we will look at some general data models as used in the signal processing community. In the next section, we will then give an overview of the types of structure that may be available and (blind) beamforming al-gorithms that use this structure. We will then come back to the FRAMES modulation format and discuss how the general techniques apply to it.
3.1. Instantaneous mixtures
Assume that d source signals s1(t), · · · , sd(t) are transmitted from d independent sources at different locations. If the delay spread is small, then we will receive a simple linear combina-tion of these signals:
x(t) = a1s1(t) + · · · + adsd(t)
where x(t) is a stack of the output of the M antennas. Suppose
we collect a batch of N samples, then
X= AS , A= [a1· · · ad] ,
where X= [x(0) · · · x(N−1)] and S = [s(0) · · · s(N−1)]. The
resulting[X = AS] model is called an instantaneous
multi-input multi-output model, or I-MIMO for short. It is a generic
of the dominant rays is much smaller than the inverse band-width of the signals.
The objective of beamforming for source separation is to construct a left-inverse W of A, such that WA= I and hence
W X= S: see figure 2(a). This will recover the source signals
from the observed mixture. It immediately follows that in this scenario it is necessary to have d≤M to ensure interference-free reception, i.e., not more sources than sensors. If we know already (part of) S, e.g., because of training, then W= SX†,
where X† denotes the Moore-Penrose pseudo-inverse of X. Blind beamforming is to find W with knowledge only of X.
If each source is received from only a single direction (no multipath), then the columns of A can be described by the ar-ray response vector a(θ). E.g., for a uniform linear array and
a single source, x(t) = 1 θ .. . θM−1 s(t) = a(θ)s(t) , θ= ej2π∆sin(α)
whereαis the direction of the source and∆is the spacing between the elements of the array (in wavelengths). Without multipath, the columns of A lie on the array manifold{a(θ) :
|θ|= 1}.
3.2. Convolutive mixtures
An often-used parametric channel model that is valid for wideband sources is x(t) = " r
∑
1a(θi)βig(t− τi) #
∗s(t) = h(t)∗s(t) .
Here, it is assumed that the source is digital (more precisely, a dirac-pulse sequence), linearly modulated by a pulse shape function g(t). The channel is supposed to be a simple
multi-path propagation channel, consisting of r distinct multi-paths, each parametrized by a directionθi, a relative path delayτi, and a complex amplitude (fading)βi. The channel has finite length L symbols.
Suppose that the pulse-shape function g(t) has support (length)
Lg, augmented with zeros to length L, and that we sample at a rate P times the symbol rate. We can then define the temporal signature vector g(τ) := g(0− τ) g(P1− τ) .. . g(Lg−P1− τ) .
It is thus seen that h(t) =∑r
1a(θi)βig(t− τi) has structure: let gi= g(τi), ai= a(θi), then
h := h(0) h(1 P) .. . h(Lg−1P) = [g1⊗a1, · · · , gr⊗ar] β1 .. . βr .
The combined vector g(τ)⊗a(θ) is the space-time response
mnxM(t) mnbeamformer mn[s1]k mn[sd]k mn[s1]k mnW mn[sd]k mnx1(t) mn: delay 1 P mnspace-time eq. mnxM(t) mn[s1]k mnW mnx1(t) mng(t) mn[s1]k mn[sd]k mnP mn: downsampling P:1 mn[sd]k
Fig. 2. (a) Spatial beamformer with an I-MIMO channel. (b) Space-time equalizer with an FIR-MIMO channel.
After collecting data samples during N symbol periods, the convolutive model x(t) = h(t)∗s(t) can be written in matrix
form as X= HSL, where X = x(0) x(1) · · · x(N−1) x(1 P) x(1+ 1 P) · .. . ... x(P−1 P ) · · · · · H = h(0) · · · h(L−1) h(1 P) · .. . ... h(P−1 P ) · · · h(L− 1 P) SL = s0 s1 . .. sN−2 sN−1 s−1 s0 . .. . .. sN−2 . .. . .. . .. . .. . .. s−L+1 . .. . .. . .. sN−L .
This factorization is typical for a linear FIR channel (single-source). Note in particular thatSL has a Toeplitz structure (constant along diagonals), which is a consequence of the time-invariance of the channel. With multiple sources, a similar model holds, butSLbecomes a block-Toeplitz structure with vector-entries sk.
An equalizer in this context can be written as a vector w which combines the rows of X to generate an output y= w∗X. In the model so far, we can only equalize among the antenna outputs (simple beamforming) and among the P samples within one sample period (polyphase combining). More in general, we would want to filter over multiple sample periods, leading to a space-time equalizer. For an equalizer length of m symbol periods, we have to augment X with m−1 horizontally shifted copies of itself: Xm= x0 x1 . . . xN−m x1 x2 . . . . .. . .. . .. . .. xN−2 xm−1 . . . xN−2 xN−1 mMP×(N−m+1)
Each column ofXmis a regression vector: the memory of the filter. UsingXm, a general space-time equalizer can be written as y= w∗Xm, which combines mP snapshots of M antennas: see figure 2(b). The augmented data matrixXmhas a
factor-ization Xm = HmSL+m−1 =
0
H . .. . .. H H0
sm−1 ... sN−2 sN−1 ... ... ... s N−2 s−L+2 s−L+3 ... ... s−L+1 s−L+2 ... sN−L−m+1 (1)whereH = Hmhas size mMP×(L + m−1) and the m shifts of H to the left are each over 1 position. H has a block-Hankel
structure: it is constant along antidiagonals. SL+m−1has the
same structure asSL. A necessary condition for space-time equalization (y is equal to a row ofS) is that H is tall, which
gives minimal conditions on m in terms of M, P, L [3, 4].
Un-like spatial beamforming, it will not be necessary to findH†:
it suffices to reconstruct a single block row ofS, which can be
done with d space-time equalizers wi. Other equalizer struc-tures than FIR filters are possible, e.g., by using feedback, but are not discussed here.
4. PRINCIPLES OF BLIND BEAMFORMING
A summary of the data model developed so far is I-MIMO: X= AS , A= [a(θ1), · · · , a(θd)]
FIR: X= HS , h = vec(H)
= [g1⊗a1, · · · , gr⊗ar]βββ. The first part of these model equations is generally valid for LTI channels, whereas the second part is a consequence of the adopted multiray model.
Based on this model, the received data matrix X has several structural properties. In several combinations, these are often strong enough to allow to find the factors A (or H) and S (or
S), from knowledge of X alone. A number of properties are
discussed below. 4.1. Training
If training symbols are present, then a number of columns of S orS are known. This number should be such that this
known submatrix St is a wide matrix, in which case it gen-erally has a right inverse S†t. This directly allows estimation of A or H as XtSt†, where Xt is the corresponding window of the data matrix. With A or H known, there are a large number of suitable space-time equalizers (e.g., zero-forcing, MMSE, decision-feedback), differing in performance, complexity and symbols/noise assumptions. The literature is abundant; see e.g., [5, 6] and references therein.
4.2. Toeplitz structure
The fixed baud rate of communication signals, along with time invariance, result in the fact that X has a factorization in
SIGNAL-CHANNEL STRUCTURAL PROPERTIES
Properties A orH S orS
I/FIR-MIMO
matrix H block Hankel S block Toeplitz training block subspace col(A) = col(X) row(S) = row(X)
col(H) = col(X ) row(S) = row(X ) modulation FA, CM, independence, · · ·
temporal CDMA codes, temp. nonwhite
spectral cyclostationarity, res. carriers
parametric
spatial known a(θ) temporal known g(τ) spectral residual carriers
whichS is block Toeplitz. This is a strong property, and
al-lows e.g., the blind equalization of unknown channels carry-ing unknown digital signals with equal baud rates [7–9,3,10]. It cannot be used for source separation, but it is useful for re-ducing X= HS to X = AS. One limitation is that it requires a
low rank data model (X instead of X); this can almost always
be obtained by shifting and stacking the data, but the result-ing matrices may be large if the delay spread is large. This ef-fect is mitigated by using several antennas (M large), which can drastically improve the performance. Alternatively, we can use oversampling (P large), but the effectiveness of this is limited by the Nyquist rate [4].
4.3. Signal modulation structure
The signal modulation structure relates to the instantaneous amplitude of the modulated signal and includes the symbol constellation. Some typical modulation structures are listed below.
Constant modulus. In many wireless applications, the trans-mitted waveform has a constant modulus (CM). This occurs e.g., in FM modulation, or in phase modulation as in GSM. So-called constant modulus algorithms can separate arbitrary linear superpositions of such signals, by finding out which lin-ear combinations of the antenna outputs give back signal that have the CM property. This property is robust and can be used for blind equalization as well [11–15].
Finite alphabet. Another important structure in digital com-munication signals is their finite alphabet (FA). The modu-lated signal is a linear or nonlinear map of an underlying fi-nite alphabet, e.g.,{+1,−1}for signals with a BPSK constel-lation. As with the constant-modulus property, it is possible to separate arbitrary linear combinations of FA signals in a more or less unique way [16–19].
Distributional properties and independence. More in general, if the source distribution is known and not Gaussian, separa-tion is possible by restoring the distribusepara-tion funcsepara-tions at the output of the beamformer, e.g., using maximum-likelihood techniques. Even if the distributions are not known, we can restore distributional properties expressing the independence of sources. This is a vast area of research with many direc-tions. Algebraic methods are possible by using higher-order stochastic moments and functions thereof, such as cumulants; see e.g., [20, 21].
The temporal structure relates to s(t) as well, but now with
regard to its temporal properties. These can include knowl-edge of its pulse shape function and, in the case of CDMA signals, knowledge of the source codes, but also certain sta-tistical properties for sources that are temporally non-white. CDMA codes. In direct-sequence CDMA, the emitted ‘chip symbols’ skare in fact modulations of low-rate source sym-bols dnby known code vectors c of length Q:
[sk· · · sk+Q−1] = dnc= dn[c1· · · cQ] , n= bk/Qc . (The codes vectors are different for each source.) Because the only unknowns are the dn, this reduces the number of un-knowns in S by a factor Q. The source symbols can e.g., be re-covered by row span template matching techniques [22–24], essentially straightforward least squares algorithms.
Temporally non-white and independence. If the sources are independent and temporally non-white, separation is possible by using the fact that the cross-covariance and cross-cumulants of the signals at the output of the beamformer should be zero for all time lags. This allows to separate sources, but in this form cannot be used to equalize them. Often, already the second-order conditions are sufficient to find the beamformer; an example of an algebraic technique is [25].
Cyclostationarity. Many signals exhibit cyclo-stationary prop-erties, i.e., their cyclic autocorrelation function Rαx(τ) = E(x(t)x(t−
τ)∗e−j2παt) is wide-sense stationary and has spectral lines at
selective lagsτand frequenciesα. This is typically caused by periodicities such as the symbol rate in bauded communi-cation signals, or residual carrier frequencies. If two sources have spectral peaks for different(α,τ), then they can be
sep-arated based on this [26]. It is usually required that these pa-rameters are known, although they can be estimated in spe-cific cases.
For digital communication signals, a straightforward way in which the cyclostationarity property can be expressed is by oversampling the antenna outputs. The samples obtained during one symbol period presumably give independent lin-ear combinations of the same transmitted bits, just as antennas give independent linear combinations from sampling in space. This fact was noted first in [7] and has stirred a lot of inter-est since; see e.g. [8, 9, 3]. Although initially called a second-order technique, the Toeplitz structure is a deterministic rather than a stochastic property.
It is also possible to design modulation formats specifically to contain cyclo-stationary properties. There are several pos-sibilities, e.g., by simply repeating the data twice, or inserting zero symbols at periodic locations [27–30].
4.5. Parametric properties
Parametric properties relate to the multipath model that we have derived, and extensions of this. It makes sense to use such models if the number of parameters is much smaller than, e.g., the number of coefficients in an unstructured FIR model. The spatial manifold. In the I-MIMO model, each column of A is a linear combination of array response vectors{a(θi)}, each of which is on the array manifold. If the array manifold is known, e.g., by calibration or from structural considerations, then we can try to fit the column span of X (hence A) to the
of rays is not large and if the calibration data is reliable. For this purpose, various direction finding techniques have been proposed, notably ESPRIT [31].
The temporal manifold. Similarly, h is a linear combination of vectors of the form{g(τi)⊗a(θi)}, where g(τ) is the temporal
manifold function, the sampled response to an incoming pulse g(t−τ). The temporal manifold is usually known to a good
ac-curacy. There are several options to use this information. It is rather straightforward to restrict the channel estimates to be in the subspace spanned by this manifold function [32]. This can be done for any FIR model, and no delay estimation is neces-sary. Alternatively, if the specular multipath model holds true and the number of rays is not large, the received signal is con-structed from several delays of g(t), hence can be viewed as
superpositions of a number of vectors g(τi), and we can try to
find the delay parameters{τi}. With knowledge of both the spatial and temporal manifold, we can also attempt to do a joint estimation of all angles and delays [33–36]. An appli-cation of space-time manifolds to CDMA is described in [37, 38, 23]. These results are limited to cases where the channel impulse response is (much) shorter than the symbol period.
Residual carriers. Independent narrow band sources modu-lated at high frequencies rarely have exactly the same carrier frequency. Consequently, after demodulation, the co-channel sources have unequal residual carriers, with only partially over-lapping spectra. If the spectral properties of the sources are known or if we sample sufficiently fast so that we can use sta-tionarity properties of the sources, the residual carriers can be estimated and the sources can be separated, even if the array manifold is unknown. This can be regarded as a special case of cyclostationarity. See [39, 40].
5. APPLICABILITY TO FMA1-SPREAD
We now return to the proposed FMA-1 modulation scheme in section 2, and discuss what structural properties are avail-able in this format, and which of the algorithms in section 4 would apply.
5.1. FMA1-spread convolutive model
In FMA1-spread, the channel length is 2 symbols, or 32 chips. Thus, we need a convolutive[X = HS] model. The
lin-earized GMSK pulse modulation waveform c0(t) has a length
of 5 chips, so that the actual overall channel length has length 35 chips. However, it seems that the intention in the defining document [1] is to arrive at a channel length of 33 chips, so we will adopt that number.
Let us define some notation by which to express the data model. The i-th user transmits a data sequence[di0· · · di,N−1]
consisting of N symbols in the alphabetΩi. The data sequence is expanded by a chip sequence ci= [ci0· · · ci,15] consisting of
16 chips. In continuous time, these are represented by delta-sequences,
data sequence: di(t) =∑Nk=1dikδ(t−kTs) chip sequence: ci(t) =∑15k=0cikδ(t−kTc) chipped data sequence: si(t) = di(t)∗ci(t) .
transmitted signal: zi(t) = di(t)∗ci(t)∗c0(t) = di(t)∗gi(t) symbol pulse shape: gi(t) = ci(t)∗c0(t) .
The data is transmitted and subsequently received by M anten-nas. The outputs of these antennas are stacked in M-dimensional vectors x(t). For a linear channel, x(t) is the superposition of
each of the d= 8 user responses xi(t). If the corresponding physical channel impulse response is denoted by hpi(t), then
received signal: xi(t) = di(t)∗ci(t)∗c0(t)∗hip(t)
= si(t)∗hi(t)
= di(t)∗bi(t) where
chip channel: hi(t) = c0(t)∗hip(t) is the channel as seen by the chipped data sequence, and
symbol channel: bi(t) = gi(t)∗hip(t) = ci(t)∗hi(t) is the channel for the symbol sequence.
If the channel can be represented by a discrete multipath model consisting of ripaths with anglesθik, attenuationsβik and delaysτik, then we further have
bi(t) = ∑rki=1a(θik)βikgi(t− τik)
hi(t) = ∑rki=1a(θik)βikc0(t− τik) .
5.2. Data matrix factorization
The above convolutive models for x(t) provide several
pos-sible factorizations of the data matrices.
The physical channel length is 2 symbols, but every sym-bol is spread by the code over the full symsym-bol period, so the effective channel length is L= 3: every received sample at
the antennas is some linear combination of (at most) 3 sym-bols per user, hence 3d= 24 symbols in total. Suppose that
we sample at the chip rate.1Then we can collect a data matrix
X= x(0) x(Ts) · · · x(Tc) x(Ts+ Tc) · · · .. . ... x(15Tc) · · · · .
If we sample during N symbol periods, then X has size 16M× N. It has a factorization X= BD where B = B(0) B(Ts) B(2Ts) B(Tc) · · .. . ... ... B(15Tc) · B(2 + 15Tc) 16M×3d D = d0 d1 d2 · . .. ∗ d0 d1 d2 . .. ∗ ∗ d0 d1 . .. 3d×N
1More in general, we would sample at P times the chip rate to increase di-mensionality and alleviate synchronization requirements. Typical would be
dk
the k-th symbol of each user sequence, and
B(t) = [b1(t) · · · b8(t)]M×d collects the symbol channels in a matrix function.
With d = 8 users and M = 1 antenna, B has size 16×24. For equalization, we need B to be tall (since an equalizer is obtained from a left inverse of B). We can achieve this by us-ing M= 2 antennas, oversampling (P = 2), or constructing an
augmented data matrix by taking a shift of the data (m= 2), as
in equation (1): this leads to a similar data modelX = BD4,
where X = x(0) x(Ts) · · · .. . ... x(15Tc) · · · · x(Ts) x(2Ts) · · · .. . ... x(Ts+ 15Tc) · · · · 32M×(N−1) B = 0 B B 0 32M×4d (2) D4= d1 d2 d3 · . .. d0 d1 d2 d3 . .. ∗ d0 d1 d2 . .. ∗ ∗ d0 d1 . .. 4d×N
With M= 1 antenna, B has size 32×32 and is (presumably) invertible;B−1contains equalizer coefficients for all 8 users
at 4 different delays. The fact that no oversampling or mul-tiple antennas are needed to reach the required dimension is because the code length is 16 and there are only 8 users.
5.3. Alternative data matrix factorization
An alternative representation similar to X= BD is obtained
by stringing all observed data in a single (block) vector ¯X of size M×16N,
¯
X= [x(0) x(Tc) x(2Tc) · · · ] (3) which has a factorization ¯X= HS,
H = [H1· · · Hd]M×264, S = S1 .. . Sd 264×16N Hi = [hi(0) hi(Tc) · · · hi(2Ts)]M×33 Si = si0 si1 si2 · . .. ∗ si0 si1 · . .. ∗ ∗ si0 si1 . .. 33×16N sik= si(kTs) .
This representation is at the chip level rather than at the sym-bol level. Sihas additional structure: groups of 16 consecu-tive bits in a row have the form dik[ci0 · · · ci,15].
5.4. Structure of B
Since bi(t) = ci(t)∗hi(t), B has structure as well: for each user, vec(Bi) = Hici= Cihi, (i = 1, · · ·, d) (4) Hi= hi(0)
0
hi(Tc) . .. .. . hi(0) hi(2Ts) ... . .. ...0
hi(2Ts) , ci= ci0 .. . ci15 Ci= ci00
ci1 . .. .. . ci0 ci15 ... . .. ...0
ci15 ⊗IM, hi= hi(0) hi(Tc) .. . hi(2Ts) Here, it should be noted thatCiand ciare known.
Similarly, since bi(t) = hip(t)∗gi(t) =∑kri=1a(θik)βikgi(t− τik) we can write
bi = [a(θi1)⊗g(τi1) , · · · , a(θir)⊗g(τir)]βββi
= [A(θθθi) G(τττi)]βββi
( denotes a column-wise Kronecker product) where the
tem-plate vectors a(θ) and g(τ) are known vector functions: the
spatial and temporal response vectors, respectively. A simi-lar factorization holds for hi, now using samples of c0(t) as a
template vector. In summary, we have
X = BD
vec(Bi) = Hici = Cihi
hi = [A(θθθi) G(τττi)]βββi
D block Toeplitz
Ci, ci known (codes)
A(θθθ), G(τττ) known parametric matrix functions 5.5. Applicability of blind algorithms
At this point, we can discuss the structure that is available in the data models, and how it can be used to recover the data symbols.
Training symbols. The midamble of the data block consists of 296= 33 · 8 + 32 known chips. This is geared towards
equa-tion (3), where it means that there is a known block of size 264×264, T say, in the center of the data matrixS. This can
be inverted and applied to the left hand side (a submatrix of ¯X) to recover the chip channel coefficients H. In fact, the train-ing block is fixed and can be inverted off-line. Moreover, in the proposal [1], the training sequences are constructed such that the matrix to be inverted has a cyclic Toeplitz structure. This means that T has a factorization into T= F−1ΛF, where
F is the matrix representation of a Fourier transformation, and Λis a diagonal matrix. Using FFTs, application of T or its in-verse to a vector can be done more efficiently, in order n log(n)
rather than n2, where n= 264. It has to be applied to M
vec-tors.2
2If the channel length would be defined as 32 chips, then we would obtain
hi
can compute Bi= Cihiand construct the block Hankel matrix
B in (2). The required space-time equalizers which separate
the signals and equalize them are left inverses ofB: e.g., the
zero-forcing equalizer is the Moore-Penrose pseudo-inverse. Training symbols, alternative. Another way to estimate the channels from training is to use the equation X= BD. This is
possible if the training would consist of known symbols mod-ulating the known codes (this would require small changes in the definition of the training block). Since B has 3 · 8= 24
rows, we need 24 consecutive columns of D to be known, which implies 24+ 3−1= 26 known symbols (this is more
than the 296/16 = 18.5 equivalent symbols required in the
previous scheme). Another difference is that, now, to estimate the channels, we need to invert and apply only a 24×24 ma-trix. It is still possible to design the training symbols such that this matrix has a cyclic Toeplitz structure, although the dimen-sions are almost too small to effectively use this. This time, the inverted matrix has to be applied to 16M vectors to esti-mate the channels. This makes the overall complexity similar to the previous scheme. However, the computational latency can be significantly less.
An improved accuracy of B can be obtained if we further constrain the solution to satisfy the coding structure vec(Bi) =
CihiwhereCiis known. This constraint is not hard to apply (it involves a projection onto the column span ofCi, separately for each user).
It should also be possible to directly estimate a suitable left inverseB†ofB, i.e., to directly estimate the space-time
equal-izers without first estimating the channels.
Blind equalization. The block Toeplitz structure ofD enables
blind equalization. Several schemes are available; they are usually limited in either the assumption that the channel re-ally has length L= 3 and not 2 or 1, or that the rows of D are
sufficiently orthogonal to each other. Blind equalization does not provide separation of the 8 users yet: we end up with an instantaneous MIMO model X= AD. For separation, the
fi-nite alphabet structure of the symbols can be used, but since d= 8 users is rather large, this will be computationally
com-plex and not reliable without good initialization. It is more at-tractive to use the code structure provided by vec(Bi) = Cihi for separation. Such schemes are possible but have not been investigated yet.
Semiblind. A semiblind approach would use both the training block and force the Toeplitz structure ofD, and perhaps the
finite alphabet structure as well. Such schemes are promising but at this point the techniques are immature.
Constant modulus algorithms. Some constant modulus struc-ture is present: gi(t)∗ci(t) has a constant modulus over its
support. If a BPSK alphabet would be used, then the modu-lated signal has constant modulus. In the present case, a 4PSK alphabet is used, and the modulated signal can be viewed as a sum of two constant-modulus signals, which by itself does not have constant modulus, and standard techniques do not apply. It seems more natural to regard gi(t)∗ci(t) as a known pulse shape function. Hence, constant-modulus algorithms are not likely to play a role.
CDMA code structure. Techniques such as [22–24] use the model ¯X= HS in (3), and force the template structure that
strings of 16 bits in the rows ofS have the form dik[ci0· · · ci,15]
torization ¯X= HS is low-rank, so that the row span of S can
be estimated from that of ¯X. However, in the present situa-tion, ¯X has M rows, butS has 264. Hence, the factorization is
certainly not low rank. This aspect can be improved by aug-menting ¯X with m−1 shifts (regressions) of itself, but every shift will also add 8 new rows toS, and the augmented matrix
will have an excessively large number of rows. Thus, these algorithms are not directly applicable.
Known pulse shape function. Since bi(t) = gi(t)∗hip(t) and the pulse shape function gi(t) is known, we can constrain B to be of the form vec(Bi) = Gihip, with similar structure as in (4). Similarly, H can be constrained as well. This constraint can be implemented by a projection onto the column span of
Gi.
Known array manifold and pulse shape function. If the array manifold is known, we can try to estimate directions of mul-tipath rays. Since every user can have several directions and the number of directions that can be estimated is limited by the number of antenna elements, it is not possible to do this jointly for all users. Thus, DOA techniques by themselves can not be used to separate users, they have to be combined with other properties. However, after users have been sepa-rated (e.g., based on training), we can estimate the DOAs for each individual user from the estimated channel of that user, and thus force the columns of the estimated channel to lie on finite linear combinations of vectors on the array manifold.
More ambitious would be to assume a parametric multipath model and constrain the channel estimates to lie on the plane spanned by finite combinations of parametric vectors of the form a(θ)⊗g(τ). Again, the only techniques presently known
for this use a two-step approach: start from a channel estimate (e.g., from training), then estimate the angle-delay parame-ters, and use the refined channel estimate to construct a more accurate beamformer [36]. This is suboptimal. A simplifica-tion is possible in those CDMA systems where the channel length is much shorter than a single symbol period (this hap-pens for long codes), and most existing algorithms make use of this. In FMA-1, these techniques are not directly applica-ble.
Accurate results can only be obtained if the channel really consists of a small number of discrete paths, and if the ar-ray manifold is well known. This makes DOA or DOA-delay algorithms somewhat unreliable as channel estimation tech-niques in general scenarios with diffuse scattering. Nonethe-less, they can be applied for user localization, and estimation of directional transmit beamforming vectors in FDD schemes.
6. CONCLUSIONS AND REMARKS
– There are several interesting possibilities for blind or semi-blind beamforming in FMA1-spread. A significant amount of structure is present. Existing algorithms are not directly applicable but some can be extended.
– It is probably worthwhile to investigate if the currently proposed training block (296 known chips) should be mod-ified to be of the form [known symbols modulated by the user codes]. Although the latter scheme requires 7 ad-ditional training symbols, it leads to more elegant signal processing structures, smaller matrices and faster imple-mentations.
ing overhead is unnecessarily large. The training required to estimate the channels of d cochannel users is(d + 1)L
(where L is the channel length in symbols). Thus, to ac-commodate 8 users, we need 9/2 times the usual amount
of training. It should be possible to omit the training since the codes are known and have length 16 for only 8 users: this implies that there is a redundancy by a factor of 2 already available for channel estimation. In fact, blind equalization schemes exploit precisely this redundancy, but the existing algorithms need to be extended to also ex-ploit knowledge of the codes and perhaps the pulse shape function.
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