o Layered Space-Time Architecture for
`Wireless Communication in a Fading
`Environment When Using Multi-Element
`Antennas
`
`Gerard J. Foschini
`
`This paper addresses digital communication in a Rayleigh fading environment when
`
`the channel characteristic is unknown at the transmitter but is known (tracked) at
`
`the receiver. inventing a codec architecture that can realize a significant portion of
`
`the great capacity promised by information theory is essential to a standout long-
`
`term position in highly competitive arenas like fixed and indoor wireless. Use (n r, n3)
`
`to express the number of antenna elements at the transmitter and receiver. An (n, n)
`
`analysis shows that despite the it received waves interfering randomly, capacity
`
`grows linearly with n and is enormous. With n = 8 at 1% outage and 21-dB average
`
`SNR at each receiving element, 42 bisle is achieved. The capacity is more than 40
`
`times that of a (l, i) system at the same total radiated transmitter power and band-
`
`width. Moreover in some applications, n could be much larger than 8. in striving for
`
`significant fractions of such huge capacities, the question arises: Can one construct
`
`an (n, n) system whose capacity scales linearly with n, using as building blocks n sep-
`
`ara tely coded one-dimensional (l-D) subsystems of equal capacity? With the aim of
`
`leveraging the already highly developed i-D codec technology, this paper reports
`
`just such an invention. in this new architecture, signals are layered in space and time
`
`as suggested by a tight capacity bound.
`
`Introduction
`
`This paper describes a new point-to-point com-
`
`processing higher dimensional signals with the aim of
`
`munication architecture employing an equal number
`
`leveraging the already highly developed one-dimen-
`
`of antenna array elements at both the transmitter and
`
`sional (1-D) codec technology. Note that in this con-
`
`receiver. The architecture is designed for a Rayleigh
`
`text, “higher dimensional” refers to space. (Generally, a
`
`fading environment in circumstances in which the
`
`bandwidth-efficient l-D code involves many dimen-
`
`transmitter does not have knowledge of the channel
`characteristic. This new communication structure,
`
`sions over the temporal domain. 1-D refers to a complex
`
`alphabet which is, of course, 2-D in terms of reals.)
`
`termed the layered space-lime architecture, targets appli-
`
`cation in future generations of fixed wireless systems,
`
`As the paper points out, the capacity that this
`architecture enables is enormous. At first, the number
`
`bringing high bit rates to the office and home. The
`
`of bits per cycle might seem too great to be meaning-
`
`architecture might also he used in future wireless local
`
`ful. The capacity is achieved, however, in terms of n
`
`area network (LAN) applications for which it promises
`
`equal lower component capacities, one for each
`
`extraordinarily high bit rates.
`
`antenna at the receiver {or transmitter}. A form of the
`
`The architecture is a method of presenting and
`
`new architecture attains a capacity equal to a tight
`
`Copyright 1996. Lucent Technologies Inc. All rights reserved.
`
`Bell Labs Technical Journal 0 Autumn 1996
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`Panel 1. Abbreviations, Acronyms, and Terms
`
`AWGN—additive white Gaussian noise
`BER—bit error rate
`codec—coderldecoder
`
`iid—independent identically distributed
`LAN—local area network
`
`
`
`MEA—multi-element array
`MMSE—minimum mean square error
`SNR—signaI-to-noise ratio
`
`lower bound on the capacity attainable using multi-
`
`element arrays (MEAs) with an equal number of ele-
`ments at both ends of the link. The next section
`
`describes this lower bound on capacity. Subsequently,
`
`the layered space-time architecture is discussed.
`
`Although additional background details are avail-
`
`able,1 this paper provides a self-contained description
`
`of the architecture. The perspective is one of a com-
`
`plex baseband view of signaling over a fixed linear
`matrix channel with additive white Gaussian noise
`
`(AWGN). Time proceeds in discrete steps, normalized
`
`so that t = 0, l, 2,
`
`. The following notation and basic
`
`assumptions should be reviewed:
`
`0 Number ofantennas. The MBA at the transmit-
`
`ter has nT. The MBA at the receiver has nR. For
`
`convenience, the pair {nT, nR} denotes a com-
`
`munication system with nT transmit elements
`
`and nR receive elements. Figure 1 illustrates
`the notation.
`
`- Transmitted signai sfi). This signal has fixed nar-
`
`row bandwidth. The total power is constrained
`to Is regardless of r1T (which is the dimension
`of sttn. The bandwidth is narrow enough that
`
`the channel frequency characteristic can be
`treated as flat across the band.
`
`0 Noise at receiver v(i). This is the complex nR-
`
`dimensional AWGN. The components are sta-
`
`tistically independent and of identical power N
`
`at each of the nR antenna outputs.
`
`- Received signal rft). At each point in time, this is
`
`an nR-dimensional signal. There is one com-
`
`”average" meaning spatial average.
`
`0 Average signal—to—noise ratio (SNR) at each receive
`
`antenna. This is p = PIN, independent of nT.
`
`- Matrix channel impulse response gm. This matrix
`
`has nT columns and nR rows. The notation htt)
`
`is used for the normalized form of g{t). The
`
`normalization is such that each element of hm
`
`has a spatial average power loss of unity.
`
`The basic vector equation describing the channel
`
`operating on the signal is
`
`(1)
`r=gas+v,
`where ”a" means convolution. These three vectors are
`
`complex nil-dimensional vectors {ZnR real dimen-
`
`sions). Because of the narrowband assumption, the
`channel Fourier transform G is treated as a matrix
`
`constant over the band of interest. Thus, g is written
`
`for the nonzero value of the channel impulse
`
`response, thereby suppressing the time dependence of
`
`g(t). The same is true for h and its Fourier transform
`
`H. Thus, in normalized form, (1) becomes
`
`r=_/pinT-h-s+v,
`
`(13}
`
`The random channel model we use is the Rayleigh
`channel model. Assume that the MEA elements at
`
`each end of the communication link are separated by
`
`about half a wavelength. At 5 GHz, for example, half a
`
`wavelength measures only about 3 cm, so many
`
`antenna elements are often possible. (Additionally,
`
`there are two states of polarization [see Figure 2]}.
`With a half-wavelength separation, the Rayleigh
`
`model for the r1R x nT matrix H representing the chan-
`
`nel in the frequency domain is approximated by a
`
`matrix having the following independent identically
`
`distributed {iid}, complex, zero-mean, unit-variance
`entries:
`. Hij 2 Normal (all-‘5)
`+ —1- Normal (0,1i J3).
`
`0
`
`2
`IHiJ-I2 is a chi-squared variate with two degrees
`of freedom denoted by X; but normalized so
`
`plex vector component per receive antenna.
`
`Capacity
`
`With each transmit antenna transmitting
`power
`f5! [1 ' P denotes the average power at
`the output of each receiving antenna, with
`
`The viewpoint assumed in treating capacity is dis-
`
`cussed next. We stress that capacity is a limit to error-
`
`free hit rate that is provided by information theory,
`
`42
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`Transmitter
`
`Receiver
`
`
`
`Figure 1.
`Shorthand notation (n7, n9) for the number of transmit antennas n,- and receive antennas nR (dipole antennas shown).
`
`and this limit can only be approached in practice with
`
`that is nearly always available.
`
`the advance of technology: any working system can
`
`In a given system, not all communication bursts
`
`only achieve a bit rate (at some desired small bit error
`
`are successful. As explained below, for some small per-
`
`rate [BERD that is only a fraction of capacity. In what
`
`centage of instances of H, the transmitter’s assumed
`
`follows, the term “capacity" will often be used as an
`indicator of some smaller deliverable bit rate.
`
`Long-Burst Perspective
`
`Communication in long bursts means bursts hav-
`
`ing many symbols—so many that an infinite time
`
`horizon information-theoretic description of commu-
`
`nication portrays a meaningful idealization. Yet bursts
`
`are assumed to be of short enough duration that a
`
`channel is essentially unchanged during a burst. The
`channel is assumed to be unknown to the transmitter
`
`capacity value may be too optimistic. In such cases,
`
`delivering the bit rate at the desired BER required of a
`
`successful burst may be impossible. When it is impossi-
`
`ble, 3 channel outage is said to have occurred and the
`channel is considered to be in the OUT state.
`
`Outage is dealt with probabilistically because H is
`
`random: thus, capacity is a random variable. The
`
`channel is random even though the base and user in
`an office LAN environment or the communication
`
`sites in fixed wireless applications may be “fixed.” In
`
`but learned (tracked) by the receiver. The channel
`
`actuality, the reason for this is that such sites are only
`
`might change considerably from one burst to the next.
`
`nominally fixed because perturbations of the commu-
`
`By a channel being unknown to the transmitter,
`
`we mean that the realization of H during a burst is
`
`nicators and the communication medium are possible.
`For indoor LANs—even for a user at a desk—
`
`unknown. Actually, the average SNR value and even
`
`some motion in and around the workspace is likely.
`
`r1R might not be known to the transmitter.
`
`Not only people but various (especially metal} objects
`
`Nonetheless, for purposes of this discussion, these two
`
`could be moving in the propagation path. For pre-
`
`parameters are considered to be known. The reason
`for this is that at the transmitter, one assumes that
`
`dominately outdoor fixed wireless links, weather-
`related motion of antenna structures occurs, as well as
`
`communication is taking place with a user for which
`
`significant channel changes due to, say, vehicles and
`
`at least a certain nR and average SNR are available.
`
`foliage. Half-wavelength movements can be impor-
`
`These minimum values represent what the transmit-
`
`tant. Thus, assuming that the channel is fixed during
`
`ter conservatively uses to determine a capacity value
`
`a burst, the channel may vary from burst to burst,
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`Bell Labs Technical Journal 0 Autumn 1996
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`layer
`
`Detail shows monopole antenna
`elements for two polarization states.
`
`Dielectric
`su rface
`
`Metal
`surface
`
`Dielectric
`
`Section of available
`surface of a base
`
`'1" V
`
`T’l‘l—‘V
`I
`
`For example, at 5 GHz, M2 a 3 cm
`
`Figwe 2.
`Section of easing paved with half-wavelength lattice.
`
`and one might be interested in the capacity that can
`
`the end of the "Conclusion” section). In very severe
`
`be attained in nearly all transmissions (for example,
`
`situations, provision for movement of the receive
`
`95 to 99% or even higher).
`
`Complementary capacity distributions, discussed
`
`antenna may be desirable to avoid the risk of being
`stuck with an undesirable H for excessive time.
`
`below, focus on the high-probability tail. {In special
`
`Deployment of a relay site is yet another alternative.
`
`applications like very large file transfers, however,
`
`Fading correlation time and its incorporation into
`
`maximum attainable throughput over long time dura-
`
`more refined performance criteria is an interesting
`
`tions may be a preferable figure of merit.) A compan-
`ion articlel mentions that based on the results of
`
`research}:3 the transmitter can use a single code even
`
`though the specific value of the H matrix is unknown.
`
`The distribution of capacity is derived from an
`
`ensemble of statistically independent Gaussian nR >< nT
`H matrices (the aforementioned Rayleigh model). In
`
`this paper, the system is considered to be either OUT
`or NOT OUT for each realization of H. As mentioned
`
`earlier, the OUT state corresponds to the event that a
`
`prespecified capacity level (for example, X) cannot be
`
`met. For instance, given a 1% outage level, one would
`
`say a certain capacity can be assured at that level.
`
`By employing MEAs, capacity tail probabilities can
`
`be
`
`significantly improved. The
`
`subsection
`
`“Opportunity for Enormous Bit Rates” below discusses
`
`the great capacity available and how the tail probabil-
`
`ity improves with larger and larger n.
`
`{For cases in which the time constant of channel
`
`change is very large, extra receive antenna elements
`
`may be needed to ensure that outage is minimal {see
`
`subject for future investigation in measurement and
`
`analytical studies.)
`
`Key Capacity Expressions
`A generalized capacity formula and a capacity
`
`lower-bound formula are referred to below.I The gen-
`eralized formula is derived from other basic formu-
`
`las.4‘5 Additionally, the capacity formula for optimum
`
`ratio combining is needed.
`
`The generalized capacity formula for the general
`
`(nT, nR) case is
`
`C=log2det[In +(plnT)-HHl]bein.
`
`{2}
`
`In this equation, “det” means determinant, In is the
`nR X nR identity matrix and “t” means trahspose
`conjugate.
`
`The capacity lower bound for the (n, n) case in
`
`terms of n independent chi-squared variates is
`
`C > 210g: [1+ (p i‘ n)- 1;] bisi'Hz _
`k=l
`
`(3}
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`99 percentile {Prob[OUTAGE] = .01}.
`Average signal—to—noise ratio at each
`receive antenna is 0, E, 12, 18, and 24 dB.
`
`99 percentile (ProbEOUTAGE] = .01.
`Average signal—to-noise ratio at each
`receive antenna is —24, —18, ~12, HE}, and 0 dB.
`
`
`
`Capacity(bisin)
`
`
`
`Capacity(bisin)
`
`
`
`
`
`(a)
`
`99 percentile (Prob[OUTAGE] = .01).
`Average signal—to—noise ratio at each
`receive antenna is 0, 6, 12, 18, and 24 dB.
`
`99 percentile (ProblOUTAGE] = .01).
`Average signal—to—noise ratio at each
`receive antenna is —6, —12, —18, and —24 dB.
`
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`
` Capacity(bisinidimension)
`
`
`
` Capacity(bisinidimension)
`
`
`
`(bi
`
`The number of transmit antennas equals the number of receive antennas.
`
`— Capacities
`— Capacity lower bounds
`
`Figure 3(a).
`Capacity in his/Hz versus the number of antenna elements at each site.
`Figure 3(b).
`Capacity in b/s/Hz/dimension versus the number of antenna elements at each site.
`Figure 3(c).
`Capacity in b/s/Hz versus the number of antenna elements at each site.
`Figure 3(d).
`Capacity in h/s/Hz/dimensian versus the number of antenna eiements at each site.
`
`Notice that some nonstandard notations have been
`
`later in an asymptotic sense. the bound in {3} for large
`
`to denote directly a chi-
`used—for example, X;
`squared variate with 2k degrees of freedom. Because
`
`p and n is quite tight. While (3} was initially derived
`
`elsewhere,‘ the subsection “A (6, 6) Example of
`
`the entries of H are zero mean unit variance compiex
`
`Processing at the Receiver” below includes a rederiva-
`
`Gaussians, the mean of this variate is k. As discussed
`
`tion of (3) that is constructive. That is, the right-hand
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`Primitive
`data strea m
`
`
`
`l
`
`Layer 1
`(modlcode)
`i
`
`Layer n
`Layer 2
`(modicode) 1' ' ' (modlcode)
`l
`1
`Mod ulo-n shift of layer-antenna
`correspondence every 1: seconds
`
`Antennas \\
`
`\1‘
`
`\k
`
`I
`
`noted from the ordinate, which ranges to 300 blsi'Hz.
`
`the capacities are enormous. For example, at a lZ-dB
`SNR and even for modest numbers of antenna ele-
`
`ments like eight or 12, significant capacity is avail-
`
`able—that is, about 2] and 32 bi'sle, respectively.
`
`Even at a O-dB SNR, very significant capacity exists.
`
`About 25 blsle is available for, say, n = 32.
`
`From the standpoint of signal constellations, one
`
`must be concerned with per-dimension capacities.
`
`Figure 3(b) shows the same capacity results as Figure
`
`3{a) but they are expressed in terms of the bi'slelsig-
`
`nal dimension. In preparing Figure 3(b), the cases
`
`11 = 1, 2, 4, 8, 16, 32, and 64 were actually computed
`
`and the remaining cases interpolated. Note that even
`
`when the overall capacities are great, the per-dimen-
`
`sion capacities can be reasonable. Figures 3(a) and 3(b}
`
`depict the capacity {bold curves), as well as the capac-
`
`ity lower bound (light curves). The capacity lower
`
`bound is quite tight at the higher p values.
`As the antennas increase in number, saturation of
`
`|
`
`Fading noisy matrix channel
`
`\
`
`Figure 4.
`Trammission process using space- time layering.
`
`side will be associated with the capacity of a limiting
`
`the lower bound on the per-dimension capacity
`
`form of a communication architecture using 1-D
`codecs.
`
`becomes evident as Figure 3(b} indicates. This asymp-
`
`totic behavior can be explained by looking at the
`
`A special application of (2)—important to this dis-
`
`right-hand side of the inequality in (3). For the right-
`
`cussion—is the case of optimum combining, which
`
`corresponds to (1, r1} receive diversity. 0C(n) is writ-
`
`ten to convey a system employing optimum combin-
`
`ing with n-fold receive diversity.
`
`The capacity formula for optimum ratio combin-
`
`ing or receive diversity (nT = l, nR = n) is
`(4)
`C = log2 [1+ p- xi, bis! Hz .
`The lower-bound (3} suggests that in some sense, one
`
`might be able to embed n 0C(k} systems (with
`
`k = l, 2,
`
`n) in an (n, n) system. The argument of the
`
`logarithm in (3} suggests that each of the 11 single-
`
`transmit antenna systems would have transmit power
`ls i n so that each receive array element has an aver-
`age SNR of pin. As explained below, such embedding
`
`is indeed possible.
`
`hand side divided by n, the large n asymptote is'
`1
`Iologz (1+ px) dx
`_1
`= (1+p )-log2 (1+p)—log2 e.
`
`(5}
`
`Note that in the limit of large p, the dominant term in
`
`the last expression is log2 {pie}. For example, as Figure
`
`3(b) shows, for an SNR of 24 dB and n = 64, an
`
`asymptotic value of about 6.5 blsr‘Hzfdimension and
`
`indeed log2 (102-4) as 6.5. The "Asymptotic Optimality
`
`of the layered Architecture” subsection later on con-
`
`cludes that not just the lower bound but also the
`
`capacity per dimension or Cln converges to log2 (pie)
`
`as p and 11 increase without bound.
`
`Figures 3{c) and 3(d} correspond to Figures 3(a)
`
`and 3(b) but only for SNRs that are negative when
`
`Opportunity for Enormous Bit Rates
`
`expressed in dB. Note that even at negative SNRs,
`
`Before demonstrating how to do the embedding.
`
`the great capacity at stake is worth reviewing.
`
`interesting capacity levels are possible. The tightness of
`the lower bound deteriorates more and more, how-
`
`Figure 3(a) shows the capacity 99 percentile {1% out-
`
`ever, as p is lowered. The ”Related Options” section
`
`age} for SNRs of 0 dB to 24 dB in steps of 6 dB. As
`
`later in the paper further discusses these aspects.
`
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`
`
`Each of the above rectangles corresponds to a
`finite sequence of noisy received 6—D vectors as
`illustrated by the right—hand figure for the eighth
`rectangle. Six identical copies of these rectangles
`Space are stacked on top of each other as shewn below.
`
`
`
`EEENEEHE
`HEENEEEE
`EHHEHHEE
`HHEHEEEH
`RENEEEEH
`NHNEHEHN
`
`Nominal processing timeline through
`space—time (shown as thin solid directed line)
`
`O.
`
`Rectangles in
`the same column
`
`. are distinguished
`by hovur the
`6-D vectors are
`processed.
`
`
`
`HC(
`
`UE2f
`
`
`
`
`l)5-—
`0)
`
`EEWCT
`
`ULH
`‘O
`‘DH
`.9U
`0mm
`c:
`
`Figure 5.
`Flow of nominal processing time for a received signal.
`
`Layered Space-Time Architecture
`This section provides a high-level description of a
`
`form of the new architecture having an equal number
`
`of antenna elements at both ends of the link. Its capac-
`
`ity is associated with the lower bound (3) on the
`
`1 S k S n + 1, lei Him.
`satisfying
`spanned by the column vectors H_J.
`k S j S n. Because no such column vectors exist when
`
`denote the vector space
`
`is simply the null space.
`k = n + 1, the space H[n+l.n|
`Note that the joint density of the entries of H is a
`
`capacity achievable with MEAs for the higher SNR val-
`
`spherically symmetric (complex) nil-dimensional
`
`ues indicated in Figure 3(a). A simple description of
`
`Gaussian. This makes it possible to state that, with
`
`the architecture is provided following some brief
`
`mathematical background information, which is
`needed to establish that the architecture indeed offers
`
`tremendoris capacity. The previously mentioned
`
`”Related Options" section provides some advantageous
`variations on the architectural theme.
`
`Mathematical Background
`
`probability one, Him is of dimension n — k + 1.
`Furthermore, with probability one, the space of vec-
`
`tors perpendicular to HM] denoted as
`
`is k — 1
`H; n]
`n, 11j is defined as the pro-
`dimensional. Forj = 1, 2,
`J.
`.
`.
`.
`ii+|.n|'
`jECIlOII of H11.
`into the subspace H
`As explained next, with probability one, each TI] is
`essentially a complex j-dimensional vector with iid
`
`The following linear algebra helps clarify the
`
`N (0,1) components (11,, is just H,I1 }. Strictly speaking,
`
`architecture. Let H_i with l S] S n denote the n
`columns of the H matrix ordered left to right so
`that H=(HII,H_2, mH...) . For each k such that
`
`an le is n dimensional. When viewing le using an
`orthonormal basis—with the first basis vectors being
`
`those spanning Hl“m“ and the remaining vectors
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`(6, 6) example: layers labeled a, b, c, d, e, and f;
`detection of first complete a layer.
`
`Already
`detected
`
`Detect Detect
`now
`
`Space l
`(associated
`transmitter
`element)
`
`from transmitter antenna:
`
`0 Reconstruct signals for five underlying layers
`and then subtract off interference from the
`previously detected bits.
`0 Avoid interference from five overlying layers
`by forming decision statistics that avoid
`interference from those overlying layers.
`
`61:
`
`Time —--
`
`Demoda
`waveformon:
`t£t<21
`2tst<31
`3tst<4t
`4t£t<5t
`5t£t<51l
`5t$t<7t
`
`Avoiding interference
`
`Figure 6.
`Space-time layering in reception.
`
`being those spanning H-|_i+l,n] —the first j components
`
`primitive data stream is demultiplexed into n data
`
`of nj are iid complex Gaussians while the remaining
`components are all zero. Looking at these projections
`
`streams of equal rate. Each data stream is encoded in
`
`some unspecified way except to say that the encoders
`
`in the order r1“, 11,14,
`
`11] shows that the totality of
`
`can proceed without sharing any information with
`
`the n x (n+l}l'2 nonzero components are all iid stan-
`
`dard complex Gaussians. Consequently, the ordered
`
`each other. Rather than committing each of the n-
`the bit-
`encoded streams
`to an antenna,
`
`sequence of squared lengths are statistically indepen-
`
`streamrantenna association is periodically cycled. The
`
`dent chi-squared variates having 2n, 2{n — 1},
`
`dwelling time on any association is 1: seconds so that a
`
`degrees of freedom, respectively. With the choice of
`
`full cycle takes 11 x T seconds. The n-encoded sub-
`
`normalization presented in this paper, the mean of the
`
`streams, then, share a balanced presence over all 1]
`
`squared length of le is j.
`Transmission
`
`In a spectrally economical system, the layered
`
`paths to the receiver. Therefore, none of the individual
`
`substreams is hostage to the worst of the n paths.
`With communication structured in this balanced
`
`space-time architecture described here would be
`
`way, each subchannel has the same capacity. This
`
`employed in conjunction with an efficient 1-D code.
`
`setup serves to “uniforrnize” the multiplexingtdemulti-
`
`The form of the code employed in a specific instance of
`
`plexing and codingrdecoding processes—that is, all n
`
`the architecture is not within the scope of this paper.
`
`constituents are rendered virtually identical in struc-
`
`For expositional simplicity, however, it is best to begin
`
`the description by considering some nonspecific block
`
`code rather than a convolutional code implementa-
`tion.
`
`ture. Of course, because the balance makes it possible
`to use the same constellation for each subchannel, the
`
`lowest maximum number of constellation points per
`
`subchannel is obtained. Each channel is essentially the
`
`Figure 4 illustrates the transmission process. A
`
`same regarding the opportunity for coding.
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`48
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`
`
`Set layer index to initial
`
`—i—
`Detect bits in current layer
`
`—i—
`Reconstruct ideal received
`version of signal in current layer
`—i—
`Remove all interference stemming
`from signal in current layer
`
`
`
`layer index
`
`ls
`layer index
`equal to
`final?
`
`Increment
`
`Figwe 7.
`Temporal View of the processing of successive
`space-time layers.
`
`sequence of rectangles. Each rectangle in this linear
`
`sequence symboiizes a sequence of 1 6—D received vec-
`
`tors. The right side of the figure illustrates the succes-
`
`sion oft 6—D vectors {with complex components)
`
`corresponding to the eighth rectangle. These are the t
`
`vectors arriving on the time interval [71, 81:}. The
`
`heavy dots in the planes {circular sections shown} rep-
`
`resent the complex received signal components that
`
`can be seen for the first and last of this sequence of 1:
`
`vectors. Each of these vectors includes noise plus n-
`
`interfering transmitted signals from the transmit
`antennas.
`
`For clarification, visualize constructing a stack of
`
`six identical copies of this aforementioned sequence of
`
`rectangles, one atop the other as shown in the figure.
`
`This stack is a visual aid for explaining how the
`
`received signals are preprocessed. A spatial element is
`
`associated with each rectangle on this "wall" of rectan-
`
`gles—specifically, a transmitter antenna—depending
`on the ordinate value. These elements are numbered
`
`1, 2,
`
`6 as are the ordinate values to which they cor-
`
`respond. The resulting rectangular partition of space-
`
`time must be understood figuratively as both space
`
`In the next subsection, it will be seen that within
`
`and time are discrete. The duration t can span any
`
`each substream, ”good" symbols (those with a high
`
`number of time units and, as previously mentioned,
`
`SNR} can compensate for ”bad" symbols (those with a
`
`low SNR) through coding. The subsection will help
`
`each space element is associated with the single trans-
`mitter element as indicated on the left of the stack.
`
`clarify that in a certain sense, the capacity obtained is
`
`Note that for the same rectangle base interval of
`
`the sum given by the right-hand side of {3).
`
`duration 1, the very same ’t-consecutive vector sig-
`
`As pointed out in the "Related Options” section
`
`nals with complex components are associated with
`
`below, advantages can be achieved by allowing an
`
`each of the six vertically stacked rectangles. The six
`
`optional second stage of multiplexingfdemultiplexing.
`
`rectangles having the same T-duration base interval
`
`For now, however, this discussion assumes only one
`
`will be distinguished by the preprocessing applied to
`
`stage of multiplexingldemultiplexing as indicated in
`
`the vector signals. The transmitter antenna associa-
`
`Figure 4.
`
`A (6, 6} Example of Processing at the Receiver
`In describing processing at the receiver, a {6, 6)
`
`tions were made because a rectangle at ordinate i will
`
`play a key role in the process of extracting the signal
`
`radiated by the i"I transmitter. As explained later,
`
`example is used. The extension to arbitrary (n, n) is
`
`besides relating to the nature of information to be
`
`immediate. A training phase (not described here) is
`
`assumed to be already completed. During this start-up
`
`extracted, the ordinate also determines the way in
`which interference must be handled in the course of
`
`phase, known signals were transmitted and processed
`
`extracting infonnation.
`
`at the receiver to expedite the H matrix becoming
`
`Processing time is distinct from signal reception
`
`accurately known to the receiver. The transmitter,
`however, does not know the channel.
`
`time. Figure 5 illustrates the flow of time in processing
`
`the received signal. As processing time passes, process-
`
`The top of Figure 5 shows the first eight of a finite
`
`ing proceeds top to bottom along a succession of con-
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`Sampled n—D vector signal
`from receiver front end
`
`
`l
`
`First: top of stack
`
`Second: next to top
`
`Last: bottom of stack
`
`Seeking signal from
`transmitter n
`
`Seeking signal from
`transmitter n—1
`
`Seeking signal from
`transmitter 1
`
`Linear
`combination
`avoiding no
`interference
`
`Li near
`combination
`avoiding
`interference
`from transmit
`antenna n
`
`
`Linear
`combination
`avoiding
`interference
`from transmit
`antennas n, n—1
`
`2
`
`Sum of
`Sum of
`n—2 unavoided
`n—1 unavoided
`
`interferences interferences
`
`o o 0
`
`
`
`n scalar signals for
`further processing
`
`n spatial coordinates for a
`fixed time interval of duration t
`{unavoided interferences can be subtracted only
`after the signals from which they stem have been detected)
`
`Figwe 8.
`Spatial view of receiver processing.
`
`secutive space-time layers (diagonals), moving left to
`
`that need not be nulled are those that will be sub-
`
`right as indicated by the thin solid directed line (really
`
`tracted out. Of course, when nulling interferers, any
`
`an ordered sequence of directed lines}. The time flow
`
`possible enhancement of the noise caused by the inter-
`
`in Figure 5 is only nominal for two reasons. First, with
`
`ference nulling process must be carefully assessed. As
`
`block coding where we assume that a layer is synony-
`mous with a code block, no time arrow need be associ-
`
`explained later in reference to the mathematical setup
`
`that has been carefully tailored for capacity analysis,
`
`ated with the processing of the Symbols of a block.
`
`the noise assessment will be easy to do.
`
`Second, for convolutional coding, which has a definite
`
`Figure 6 illustrates additional details of the steps
`
`time direction, a significant modification of the obvi-
`
`required for proceeding along iterated diagonal layers.
`
`ous time progression within each full layer might have
`
`For expositional convenience, a repetitive abcdef label-
`
`an advantage as explained later.
`
`ing on the stack is included. Detection of the first com-
`
`The central theme of the architecture is interference
`
`plete diagonal a layer through which is drawn a
`
`avoidance, and this discussion assumes that interfering
`
`dashed diagonal line is described. Other layers, includ-
`
`signals will be nulled out. (As discussed later, instead
`
`ing boundary layers, are handled similarly. Boundary
`
`of nulling, SNR could be maximized. In such a case,
`
`layers are those layers involved with where a burst
`
`“noise" means including not just AWGN but all inter-
`
`starts or ends (those having fewer than six rectangles).
`
`ferers not yet subtracted out.) Fewer interferers must
`
`The first complete a layer comprises six parts,
`
`be nulled at the higher stack levels. The interferences
`
`Ell-Tit} (j = l, 2,
`
`6), in which the subscript indicates at
`
`50
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`Primitive
`bit stream
`
`n equal rate
`substreams
`
`‘
`‘
`Periodic
`t-varymg
`vector
`
`Symbol
`stream
`
`Interference
`vector from
`periodically
`varying set of
`n—t antennas
`
`
`
`Interference-free
`
`
`encoded stream-
`
`Notationally, "<-, ->" means complex scalar product.
`AWGN -Aclditive white Gaussian noise
`
`* Channel knowledge required.
`
`Figure 9.
`System diagram of the processing involved at the receiver (discrete time haseband perspective).
`
`what point in time the part that lasts for 1: time units
`
`simultaneously transmitted from antennas other than
`
`begins. All layers relatively disposed to be located par-
`
`the k’”. The interference stemming from signals that
`
`tially underneath this a layer are assumed already suc-
`
`were simultaneously transmitted from antennas
`
`cessfully detected while all layers disposed to be
`
`l, 2,
`
`k—l are inconsequential because these signals
`
`partially above the a layer are yet to be detected. The
`
`are assumed to have been already perfectly detected
`
`capacity associated with this case will also be found,
`
`and subtracted out. What is necessary is to null out the
`
`and then the capacity associated with the (n, n} case
`
`interference from yet undetected signals—namely,
`
`will be apparent. (With block coding, a full layer could
`
`those simultaneously transmitted from antennas
`
`correspond to exactly one block, although as pointed
`
`k+1, k+2,
`
`n. This is exactl