`
`V-BLAST:An Architecture for Realizing Very High Data Rates
`Over the Rich-Scattering Wireless Channel
`
`P.W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela
`Bell Laboratories, Lucent Technologies, Crawford Hill Laboratory
`791 Holmdel-Keyport Rd., Holmdel, NJ 07733
`
`ABSTRACT
`
`2. SYSTEM OVERVIEW
`
`Receni information theory research has shownthat the
`rich-scattering wireless
`channel
`is
`capable
`of
`enormous theoretical capacities if the multipath is
`properly exploited.
`In this paper, we describe a
`wireless
`communication
`architecture
`known
`as
`vertical BLAST (Bell Laboratories Layered Space-
`Time) or V-BLAST, which has been implemented in
`realtime in the laboratory. Using our laboratory
`prototype, we have demonstrated spectral efficiencies
`of 20 - 40 bps/Hz
`in an indoor propagation
`environment at realistic SNRs and error rates. To the
`best of our knowledge, wireless spectral efficiencies of
`this magnitude
`are
`unprecedented,
`and
`are
`jurthermore unattainable using traditional techniques.
`
`1. INTRODUCTION
`
`A high-level block diagram of a BLAST system is
`shown
`in Fig.l.
`A single
`data
`stream is
`demultiplexed into M substreams, and each substream
`is then encoded into symbols and fed to its respective
`transmitter.
`(The encoding process is discussed in
`more detail below.) Transmitters 1 — M operate co-
`channel
`at
`symbol
`rate 1/7 symbols/sec, with
`synchronized symboltiming. Each transmitteris itself
`an ordinary QAM transmitter. The collection of
`transmitters comprises,
`in effect,
`a vector-valued
`transmitter, where components of each transmitted
`M-vector
`ate
`symbols
`drawn
`from a QAM
`constellation. We assume that the same constellation
`is used for each substream, and that transmissions are
`organized into bursts of L symbols. The power
`launched by each transmitter is proportional to 1/M so
`that
`the
`total
`radiated power
`is
`constant
`and
`independent of M,
`
`
`
`environment
`
`
`
`
`
`
`In the past few years, theoretical investigations have
`revealed that the multipath wireless channel is capable
`of enormous capacities, provided that
`the multipath
`scattering is sufficiently rich and is properly exploited
`D>
`through the use of
`an
`appropriate processing
`£=
`D
`architecture [1-4]. The diagonally-layered space-time
`RX
`architecture proposed by Foschini [1], now known as
`3s
`Vector
`7x
`processing:
`dita
`Eotmate «
`
`2a
`diagonal BLAST (Bell Laboratories Layered Space-
`and decode
`Time) or D-BLAST,is one such approach. D-BLAST
`utilizes multi-element
`antenna
`arrays
`at
`both
`transmitter and receiver and an elegant diagonally-
`layered coding structure in which code blocks are
`dispersed across diagonals in space-time.
`In an
`independent Rayleigh scattering environment, this
`processing structure leads to theoretical rates which
`grow linearly with the number of antennas (assuming
`equal numbers of transmit and receive antennas) with
`these rates approaching 90% of Shannon capacity.
`from
`However,
`the diagonal
`approach suffers
`certain implementation complexities which makc it
`inappropriate for initial implementation,
`In this paper,
`we describe a simplified version of BLAST known as
`vertical BLAST or V-BLAST, which has been
`implemented in realtime in the laboratory. Using our
`laboratory prototype. we have demonstrated spectral
`efficiencies of 20 - 40 bps/Hz at average SNRs
`ranging from 24 to 34 dB. Although these results were
`obtained in a rclatively benign indoor environment,
`webelieve that spectral efficiencies of this magnitude
`are
`unprecedented,
`regardless
`of
`propagation
`environment or SNR, and are simply unattainable
`using traditional techniques.
`
`data|eneoder 1’
`
`
`
`
`
`
`
`Notation:
`F
`B Vector symbol: a = Ce
`B Numberof transmitters: M
`; Numberof receivers: N
`
`Figure 1: V-BLASThigh level system diagram
`
`The essential difference between D-BLAST and V-
`BLAST lics in the vector encoding proccss.
`In D-
`BLAST,
`redundancy between the
`substreams
`is
`introduced through the use of specialized imter-
`substream block coding. The D-BLAST code blocks
`are organized along diagonals in space-time.
`It is this
`coding that
`leads
`to D-BLAST’s higher
`spectral
`efficiencies for a given number of transmitters and
`receivers. In V-BLAST, however, the vector encoding
`process is simply a dernultiplex operation followed by
`independent
`bit-to-symbol mapping
`of
`each
`substream. No intcr-substream coding, or coding of
`any kind, is required, though conventional coding of
`the individual substreams may certainly be applied.
`For the remainder of this paper, we will assume for
`
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`
`
`the substreams comprise uncoded,
`simplicity that
`independent data symbols.
`individually, conventional
`Receivers
`1
`AN are,
`QAM receivers. These receivers also operate co-
`channel, each receiving the signals radiated from all M
`transmit antennas. For simplicity in the sequel, flat
`fading is assumed, and the matrix channel
`transfer
`function is H’”™, where h,, is the (complex) transfer
`function from transmitter j to receiver i, and M<N.
`We take the quasi-stationary viewpoint
`that
`the
`channel time variation is negligible over the L symbol
`periods comprising a burst, and that the channel
`is
`estimated accurately, e.g. by use of a
`training
`sequence embedded in each burst, thus, for brevity in
`the remainder of the paper, we will not make the
`distinction between H andits estimate.
`Although V-BLAST,as shown above, is essentially
`a single-user system which uses multiple transmitters,
`one can naturally ask in what ways the BLAST
`approach differs
`from simply using
`traditional
`multiple access techniques in a single-user fashion,i.e.
`by driving all the transmitters from a single user’s data
`which has been split into substreams. Some of these
`differences are worth pointing out: First, unlike code-
`division or other spread-spectrum multiple access
`techniques, the total channel bandwidth utilized in a
`BLAST system is only a small fraction in excess of
`the symbol rate, ic. similar to the excess bandwidth
`required by a conventional QAM system. Second,
`unlike FDMA, each transmitted signal occupies the
`entire system bandwidth, Finally, unlike TDMA, the
`entire system bandwidth is used simultaneously by all
`of the transmitters all of the time.
`together are
`Taken together,
`these differences
`precisely what give BLAST the potential to realize
`higher spectral efficiencies than the multiple-access
`techniques. In fact, an essential feature of BLAST is
`that no explicit orthogonalization of the transmitted
`signals is imposed by the transmit structure at all.
`Instead, the propagation environment itself, which is
`assumed to exhibit significant multipath, is exploited
`to achieve the signal decorrelation necessary to
`separate the co-channel signals. V-BLAST utilizes a
`combination of old and newdetection techniques to
`separate the signals in an efficient manner, permitting
`operation at significant
`fractions of the Shannon
`capacity and achieving large spectral efficiencies in
`the process.
`
`3. V-BLAST DETECTION
`In what follows, we take a discrete-time baseband
`view of the detection process for a single transmitted
`vector
`symbol,
`assuming
`symbol-synchronous
`receiver
`sampling
`and
`ideal
`timing.
`Letting
`a= (d},49, °°: dy)" denote
`the
`vector
`of
`transmit symbols, then the corresponding received N-
`vectoris
`
`r, = Ha + v
`
`()
`
`where V is a noise vector with components drawn from
`TID wide-sense stationary processes with variance 0°.
`One way to perform detection for this system is by
`using conventional adaptive antenna array (AAA)
`techniques,
`i.c.
`linear combinatorial nulling [6]:
`Conceptually, each substream in turn is considered to
`be the desired signal, and the remainder are considcred
`as
`"interferers". Nulling is performed by linearly
`weighting the received signals so as to satisfy some
`performance-related criterion,
`such
`as minimum
`mean-squared error (MMSE)or zero-forcing (ZF).
`For example, zero-forcing nulling can be performed
`by choosing weight vectors w;, = 1,2, ---,.M,
`such that
`
`w) (HD, = 3i
`
`(2)
`
`is the j-th column of H, and 3 is the
`where (H);
`Kronecker delta. Thus,
`the decision statistic for the
`i-th substreamisy; = wir;.
`This linear nulling approach is viable, but superior
`performance is obtained if nonlinear techniques are
`used. One particularly attractive nonlinear alternative
`is to exploit the timing synchronism inherent in the
`system model
`(the
`assumption
`of
`co-located
`transmitters makes this completely reasonable) and
`use symbol cancellation as well as linear nulling to
`perform detection. Using
`symbol
`cancellation,
`interference from already-detected components of a is
`subtracted out
`from the received signal vector,
`resulting in a modified received vector
`in which,
`effectively,
`fewer
`interferers are present. This
`is
`somewhat
`analogous
`to
`decision
`feedback
`equalization.
`the order in
`When symbol cancellation is used,
`which the components of a are detected becomes
`important to the overall performance of the system.
`Later, we will show how to determine a particular
`ordering which is optimal in a certain sense; for now,
`we first discuss the general detection procedure with
`respect to an arbitrary ordering.
`Let the ordered set
`
`S= {hy kp, - ++, Rag}
`
`(3)
`
`imtegers 1,2, ---,M
`the
`a permutation of
`be
`specifying the order
`in which components of the
`transmitted symbol vector a are extracted. The
`detectton process proceeds generally as follows:
`Step 1: Using nulling vector w,,,
`form decision
`Statistic Vp,
`
`7
`_
`Ye, = Wet
`
`(4)
`
`Step 2: Slice y,. to obtain a,:
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`
`
`ay=On) (5)
`
`(slicing)
`quantization
`the
`denotes
`where Q(:)
`operation appropriate to the constellation in use.
`Step 3: Assuming that @,, = d,,, cancel a,, from the
`received vector r,,
`resulting im modified received
`vector 4:
`
`wi, (HDs,
`
`=~
`
`19
`1
`
`|
`4
`
`IL
`
`7
`
`
`particular w,
`is constrained to be orthogonal to, the
`larger its norm, and thus, according to (8), the smaller
`its post-detection SNR. Whenusing cancellationthen,
`the px, are lower bounded (with equality only for px,)
`by their corresponding nulling-onlyp,.
`The importance of ordering is simply that it permits,
`during the detection of the i-th component, a choice as
`to which subset of M — i rows that w,, should be
`(6)
`ry = 1) ~ ay, (H),,
`constrained by; different choices lead to different p,.
`
`
`
`
`For an M=3>system,example, in detecting
`where (H),, denotes the &,-th column of H. Steps 1-
`component | first (in the presence of 2 and 3) will, in
`3 are then performed for components ky, +--+ , ky by
`general, result in a different p,
`than if component 2
`operating in turn on the progression of modified
`was detected first (in the presence of 1 and 3), With
`received vectors 2,13, °°°, ly.
`pure nulling, each component is always detected in the
`Thespecifics of the detection process depend onthe
`presenceof ali the others, so ordering does not matter,
`criterion chosen to compute the nulling vectors wy.
`Now recall that all components of a are assumed to
`thc most common of these being MMSE or ZF. The
`utilize the same constellation. Under this assumption,
`detection process is described here with respect to the
`the component with the smallest p,, will dominate the
`ZF critcrion since it is somewhat simplerto state. The
`error performance of the system. Thus, an obvious
`k;-th ZF-nulling vector
`is defined as
`the unique
`figure of merit for this system - though not the only
`minimum norm vector satisfying
`one possible - is the maximization of the worst,
`i.e.
`the minimum, of the p, over all possible detection
`orderings. A surprising result - and one which we
`believe has not been previously appreciated -
`is that
`simply choosing the best p,, a! each stage in the
`detection pracess leads
`to the globally optimum
`ordering, Sop:
`in this maximin scnsc. The proof is
`given in the appendix.
`We remark that
`this optimality result may have
`wider applicability to multi-user canccllation-based
`detection
`as well. Although
`the
`“best
`first”
`cancellation approach is widely known within the
`multi-user community [7-8], essentially being the
`defacto approach, we are not aware of any previous
`proofofits optimality in the sense given here.
`The tull ZF V-BLAST detection algorithm can now
`be described compactly as a recursive procedure,
`including determination of the optimal ordcring, as
`follows:
`
`is orthogonal to the subspace spanned by
`Thus, w,,
`the contributions to r; due to those symbols not yet
`estimated and cancelled,
`It is not difficult to show that
`the unique vector satisfying (7) is just the &;-th row of
`Hy, where the notation Hj; denotes the matrix
`obtained by zeroing columns k,,k2, °°:
`,k; of H
`and * denotes the Moore-Penrose pseudoinverse [5].
`The post-detection SNR for
`the &,-th detected
`componentof a is easily obtained by substituting (1)
`and (7) into (4), and taking expected values,i.e.
`<a,[?>
`a?|| wz, |I°
`
`Px,
`
`(4)
`
`where the expectation in the numerator is taken over
`the constellation set.
`
`3.1 OPTIMAL DETECTION ORDERING
`As mentioned earlier, when symbol cancellation is
`uscd, the system performanceis affected by the order
`in which the components of a are detected, whereas it
`does not matter when pure nulling is used. In order to
`appreciate this, first consider why it
`is that nulling
`with cancellation performs better than pure nulling,
`regardless of ordering.
`When nulling alone is used, each uulling vecwur is
`required, according to (2), to be orthogonal w M — 1
`rows of H. However, when symbol cancellation is
`employed in addition to nulling, w,, is required to be
`orthogonal only to the M — i undetected components
`av per (7) A simple consequence of the Gauchy-
`Schwartz inequality is that the more rows of H that a
`
`297
`
`initialization:
`ie i
`.
`G, =H
`ky
`= argmin| (Gj)
`recursion:
`
`We = (Gis,
`Ye = WEE
`a, = O(n)
`Pei
`= FT ay, (HDs,
`Gio = He
`Rist = argmin | (Gie all
`if itl
`
`(9a)
`(9b)
`(Ye)
`
`(9d)
`(9e)
`(Of)
`(9g)
`(9h)
`(91)
`(9)
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`

`
`
`are used for training. In this experiment, each of the
`eight substreams utilized uncoded 16:QAM,i.e. 4
`bits/symbol/Lransmitter, so that the payload block size
`is 8x4x80 = 2560 bits. The raw spectral efficiency
`of this configuration is thus
`Eo
`=
`(8xmtrs )x(4b/sym/xmtr) x(24,3 ksym/s)
`“
`30kHz
`
`= 25.9 bps/Hz
`
`and the payload efficiency is 80% of the above, or
`20.7 bps/Hz, corresponding to a payload data rate of
`621 kbps in 30 kHz. bandwidth.
`
`M=8, N=12 8 x 16QAM = 26 bps/Hz
`
`O nutionty
`© Null + optimized cancel
`
`
`
`19°
`
`1077
`
`iq?
`
`au
`
`s a"
`
`10%
`
`
`———
`20
`22
`24
`26
`28
`SNR (dB)
`Figure 2: Single-position performance
`
`The upper curve in Fig.2 shows performance
`obtained when conventional nulling is used. The lower
`curve
`shows
`performance
`using mulling
`and
`optimally-ordered cancellation. The average difference
`is about 4 dB, which corresponds to a raw spectral
`efficiency differential
`(for
`this
`configuration) of
`around 10 bps/Hz.
`
`BLER ond BER at 24 dB SNR vs. position
`ToT
`oT TT TT
`.
`
`+
`
`
`
`-
`
`o
`
`1
`
`L
`2
`
`.
`
`P
`}
`2
`
`.
`
`e
`
`*
`
`soe oe -
`
`
`4
`ae
`
`1
`L
`6
`5
`4
`Position Number
`
`I
`7
`
`4
`
`1
`
`l
`8
`
`|
`9
`
`a
`
`a (10
`197!
`-1
`-3
`
`se
`+ 10
`nN
`© 10
`wi
`a tot
`107
`me
`ge We
`1977
`
`=A
`
`Figure 3: Multiple-position performance
`
`Figure 3 shows performanceresults obtained using
`the
`same BLAST system configuration (M = 8,
`N = 12, 16-QAM) when the receive array was left
`fixed and the transmit array was located at different
`positions throughout the environment. In each case,
`
`(9c,i)
`is the j-th row of G;. Thus,
`where (G;);
`determine the elements of 5,the optimal ordering;
`(9d-f) compute respectively the ZF-nulling vector, the
`decision statistic, and the estimated componentof a;
`(9g) performs cancellation of the detected component
`from the received vector, and (9h) computes the new
`pseudoinverse for the next iteration. Note that
`this
`new pscudcinverse is based on a "deflated" version of
`H,
`in which columns k,,k2,-°--,k; have been
`zeroed. This is because these columns correspond to
`components of a which have already been estimated
`and cancelled, and thus the system becomes equivalent
`to a “deflated” version of Fig, 1 in which transmitters
`k,, ka. --+, k; have been removed, or equivalently, a
`system in which ay, = --- = dy, = 0.
`
`4. LABORATORY RESULTS
`
`A laboratory prototype of a V-BLAST system has
`been constructed for the purpose of demonstrating the
`feasibility of the BLAST approach. The prototype
`Operates at a carrier frequency of 1.9 GHz, and a
`symbol rate of 24.3 ksymbols/sec, in a bandwidth of
`30 kHz. The receiver processing is similar to that
`shownin (9).
`The system was operated and characterized in the
`actual laboratory/office environment, not a test range,
`with transmitter and receiver separations up to about
`12 meters. This environment is relatively benign in
`that the delay spread is negligible, the fading rates are
`low, and there is significant neat-field scattering from
`nearby equipmentand office furniture. Nevertheless,it
`is a representative indoor lab/office situation, and no
`attempt was made to “tune”
`the system to the
`environment, or to modify the environment in any
`way.
`The antenna arrays consisted of A/2 wire dipoles
`mounted im various arrangements. For the results
`shown below,the receive dipoles were mounted on the
`surface of a metallic hemisphere approximately 20 cm
`in diameter, and the transmit dipoles were mounted on
`a flat metal sheet, in a roughly rectangular array with
`about 2/2 inter-element spacing.
`In general,
`the
`system performance was
`found
`to
`be nearly
`independent of small details of the array geometry.
`Fig. 2 shows results obtained with the prototype
`system, using M = 8
`transmitters
`and N = 12
`receivers. In this experiment, the transmit and receive
`arrays were each placed at a single representative
`position within the environment, and the performance
`characterized.
`The horizontal axis
`is
`spatially
`averaged received SNR,i.c., I &¥ SNR ;, where SNR;
`i=l
`is the the ratio of received signal power (from all M
`transmitters) to noise power at the i-th receiver. The
`yertical axis is the block error rate, where a "block"is
`defined as a single transmission burst. In this case, the
`burst length 1 is 100 symbol durations, 20 of which
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`
`
`the transmit power was adjusted so that the avérage
`received SNR was 24+0.5 dB. Nulling with optimized
`cancellation was used.
`spectral
`this
`It can be seen that operation at
`efficiency is reasonably robust with respect to antenna
`position. In all positions,
`the system had at
`least 2
`orders of magnitude margin relative 10 107* BER.
`For a completely uncoded system, these are entirely
`reasonable error rates, and application of ordinary
`error correcting codes would significantly reduce this.
`At 34 dB SNR, spectral efficiencies as high as 40
`bps/Hz have been demonstrated at similar error rates,
`though with less robust performance.
`to be
`We believe
`these
`spectral
`efficiencies
`It
`is
`unprecedented
`for
`the wireless
`channel.
`worthwhile to point out that spectral efficiencies of
`these magnitudes are essentially impossible to obtain
`using traditional
`approaches
`in which a
`single
`transmitter
`is used,
`simply because the required
`constellation
`loadings would
`be
`immense.
`For
`example, to obtain the equivaicnt of the 32 bits per
`vector symbol in the experiments above, but using a
`single transmitter, would require a constellation with
`2°? or more than a billion (10°) points, which seems
`well outside of the realm of practicality, regardless of
`SNR.
`
`SUMMARY AND CONCLUSIONS
`We have described V-BLAST,a wireless architecture
`capable of realizing extraordinary spectral efficiencies
`over the rich-scattering wireless channel. The general
`BLASTapproach and the V-BLASTdetection scheme
`were motivated and described in detail, and an
`interesting
`new
`optimality
`result
`regarding
`cancellation-based detection (which may have wider
`applicability to multi-user detection as well) was
`reported. Early results with our V-BLAST realtime
`laboratory prototype have proven the feasibility of the
`concept,
`and we
`have
`demonstrated
`spectral
`efficiencies of 20 - 40 bps/Hz under real-world indoor
`conditions, exceeding any results that we are aware of
`using traditional modulation techniques. Although
`these resulls were obtained in a relatively benign
`environment,
`we
`are.
`nevertheless
`strongly
`encouraged, and believe that
`the BLAST approach
`mayeventuallylead to significantly improved spectral
`efficiencies in wireless systems.
`
`APPENDIX: PROOF OF THE OPTIMALITY OF
`ORDERING IMPOSEDBYEQ.(9)
`
`Definitions and notation:
`ordering
`detection
`For
`a
`given
`5 = {8,,8_, +++ ,Sy} detine the constraini set of
`S;
`to be the set {Siat, Si4a, 00° ou), or the null
`set
`if
`¢ = M. The constraint
`set
`is
`just
`those
`gomponents cf a whieh have net yet been detected and
`cancelled.
`
`Let Sbe a detection ordering. Then define p g, 10 be
`thc post-detection SNR at
`the i-th stage of the
`detection process whenusing this ordering,ie. ps is
`the post-detection SNR when detecting as, according
`to (e).
`Let £2 {L,,L2, ---,Ly} be
`optimum ordering obtained using (9).
`The following trivial
`lemmas are used in what
`follows and are stated here withoutproof:
`
` locally-
`
`the
`
`Lemma I: Let A and B be two distinct orderings. If
`Ay = By, and the constraint sets of A, and By,
`consist of identical elements (regardless of their
`order), then py, — Pz,-
`
`Lemma 2; Let A and 8 be two distinct orderings. If
`Ay = By, and the constraint set of A, is a subset of
`the constraint set of #,, thenp,, 2 pz,.
`
`Proof:
`arbitrary
`an
`Let Q= (Q1,@Q2,°°::,Qy)} be
`ordering distinct from 4. Let d be the indexof thefirst
`Ueftmost) element for which £ and Q differ. Let r be
`the index for which QO, = L,. (Note thatr > d, since
`£ and Q have commonelements up to index d—-1.)
`By Lemma1,
`
`PL. = Po,
`
`l<i<d-1.
`
`(Al)
`
`Now define Q’ to be a perturbation of Q obtained by
`moving Q, from index r to index d, and "squcezing”
`the rest of Qso that the elements of Q’ are
`
`Q = {Q1, Q2, 1+ ,Og-1, Q,; Qasr. Qu}
`
`where it is understood that the sequence above Q, is
`actually “missing” the repositioned Q, element. Note
`that Q’ matches £ in the first d positions, whercas Q
`matches £ only in the first d—1 positions.
`Now consider the post-dcetection SNRs that would
`result from using Q ins:cad of @
`By Lemma
`1, Po, =Po,, Pe, = Pai. >
`Po... = Pg’... since tiese eleménis have the same
`constraint sets.
`By either Lemma 1 or Lemma 2, py, SPqg,,>
`Pon, SPO.» Po, SPQ’, since these elements
`either have the same constraint sets, or the constraint
`set of the Q’ elements are subsets of the constraint sets
`of the corresponding Qelcments.
`Finally, Pg, S Pg, since Pg, = pz, and p,, is,
`by virtue of the local naaximization procedure (9), at
`least as large as any other choice in that position.
`Thus,
`
`min
`1 Pa
`
`< minpy:
`u Pe,
`
`A.2)
`
`(
`
`that by
`allows
`induclive argument
`An obvious
`successive similar perturbations, Q can be transformed
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`5
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`5
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`into £, while maintaining at each perturbation an
`inequality analogous to (A.2). Thefinal result is that
`minpg, $ min, «
`(A.3)
`
`Since Qis any arbitrary ordering distinct from £ the
`steps leading to (A.3) are valid for all possible
`orderings, and thus no ordering does better than £.
`
`REFERENCES
`{1] G. J. Foschini, “Layered Space-Time Architecture
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`in
`a
`Fading
`Environment When Using Multiple Antennas",
`Bell Laboratories Technical Journal, Vol. 1, No.
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`.
`[2] G. G. Raleigh, and J. M. Cioffi, “Spatio-Temporal
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`(31 G, J. Foschini and M. J. Gans, “On Limits of
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`Wireless Personal Communications, Vol. 6, No. 3.
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`[4] G. G. Raleigh, and J. M. Cioffi, "Spatio-Temporal
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`{5] G. H. Golub and C. F. Van Loan, “Matrix
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`[6] R. L. Cupo, G. D. Golden, C. C. Martin, K. L.
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`(7] C. ¥. Yoon, R. Kohno, H.
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`6
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`6
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