T28.1
`
`MIMO systems with antenna selection - an
`.
`overview
`
`Andreas F. Molisch, Senior Member, IEEE
`
`Abm-act- We consider multiple-input - multiple-output
`(MIMO) systems with reduced complexity. Either one, or both,
`link ends choose the "best" L out of N available antennas. This
`impliES that only L instead of N transceiver chains have to be built,
`and also the signal prooes,,mg can be simplified, We show that in
`ideal channels, full diversity can be achieved, and a1so the number
`of independent data streams for spatial multiplexing can be main•
`tained if certain conditions on L are fulfilled, We then discuss the
`impact of system nonidealities, like noisy channel estimation, cor(cid:173)
`relations of the received signals, etc.
`
`Fig. l. Blockdiagram of the considered sys1em.
`
`In this paper, we describe the performance that can be
`achieved with such a system. We furthermore describe how the
`"best" antennas can be selected in an efficient manner, and what
`nonidealities have a significant effect on the performance. The
`paper gives an overview of the results in the literature; more
`details can be found in the cited papers.
`Notation: in this paper, a vector is denoted by an arrow, a!,
`a matrix by underline Ll_. Superscript * denotes complex conju(cid:173)
`gation; superscript H denotes the Hermitian transpose.
`
`I. INTRODUCTION
`MIMO (multiple•input - multiple output) wireless systems
`are those that have antenna arrays .at both transmitter and re(cid:173)
`ceiver. First simulation studies that reveal the potentially large
`capacities of those systems were already done in the 1980s
`[l], and later papers explored the capacity analytically [2], [3].
`Since that time, interest in MIMO systems p.as exploded. Lay•
`ered space-time (ST) receiver structures and coding strategies
`allow to approach the theoretical capacities; such systems have
`become known as "spatial multiplexing" or "BLAST" systems
`II. SYSTEM MODEL
`[4]. An alternative way for exploiting the multiple antenna el(cid:173)
`Figure 1 shows the generic system that we are considering. A
`ements is the use of transmit and receive diversity purely for
`hit stream is sent through a vector encoder and modulator. This
`link-quality improvement, exploiting the diversity effect. It has
`encoder converts a single bitstream into Lt parallel streams of
`been shown that with N, transmit and N, receive antennas, a
`diversity degree of N1N, can be ac~eved [5].
`complex symbols. These streams can contain all the same infor•
`Regardless of the use as "BLAST" or as "diversity" system, mation (e.g., for a simple transmit diversity system with chan(cid:173)
`the main problem of any MIMO system is the increased com-
`nel knowledge), can all have independent symbol streams (e.g.,
`plexity, and thus cost, due to the requirement of N, (N,) com-
`in V-BLAST spatial multiplexing), or have partially correlated
`plete RF chains. There are numerous situations where this high
`data streams. Each modulated symbol stream is multiplied by
`a complex weight u whose actual value depends on the current
`degree of hardware complexity is undesirable - this is especially
`channel realization. If the channel is unknown at the transmit(cid:173)
`important for the mobile station (MS). Additional antenna ele-
`ments (patch or dipole antennas) are usually cheap, and the ad-
`ter, all weights are set to unity. Subsequently, a multiplexer
`switches the modulated signals to the best L, out of Mt avail(cid:173)
`ditional digital signal processing is becoming less of a burden
`as digital processing becomes ever more powerful. However,
`able antennas.
`RF elements like low-noise amplifiers, downconverters, and
`In a real system, the signals are subsequently upconverted to
`analog-to-digital converters are a significant cost factor. Due passband, amplified by a power amplifier, and filtered. For the
`performance computations, these stages, as well as their cor•
`to the reason, there is now great interest in so-called hybrid-
`selection schemes, where the "best" Lout of N antennas are
`responding stages at the receiver, are usually omitted, and the
`chosen (either at one, or at both link ends), downconverted, whole problem is treated in equivalent baseband. Note, how(cid:173)
`ever, that exactly these stages are the most expensive and make
`and processed. This reduces the number of required RF chains
`from N to L, and thus leads to significant savings; this comes
`the use of'antenna selection desirable.
`at the price of a"(llsually small) performance loss compared to
`Next, the signal is sent over a quasi-static flat-fading channel.
`the full-complexity system. In the case that the multiple an- We denote the N, x N, matrix of the channel as H. The output
`tennas are used for diversity purposes, the approach is called of the channel is polluted by additive white Gaussian noise. At
`"hybrid-selection - maximum ratio combining (HS-MRC), or
`the receiver, the best L, of the avaj]able JV, antenna elements are
`sometimes also '~eneralized selection combining" [6]; if they
`selected, and downconverted for further processing (note that
`only L, receiver chains are available). This further processing
`are used for spati,tl multiplexing. the scheme is called hybrid-
`can consist of weighting with complex weights w• (where )
`selection/MIMO (HS-MIMO).
`.
`.
`.
`.
`0 A F M li h
`and linear combining (if the transmitter uses simple transmit
`.
`. o sc
`d de odi
`11; with Mitsubishi Electnc Research Labs, Cam-
`di
`• )
`·
`·
`brid~e. MA USA, and the Department of Electrosctence, Lund Uni-
`vers1ty , or space-time-processmg an -
`c
`ng.
`vers1t,y, Sweden. Email: ~dreas.Mohsch@1eee.or,.Jl,
`Unless otherwise stated, we assume in the following that
`0-78U3-7829-6/03/:ti17.00 © 2003 lt:.EE
`167
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`1) The fading at the different antenna elements is indepen-
`dent, identically distributed Rayleigh fading.
`2) The fading is frequency flat.
`3) The receiver have perfect knowledge of the channel.
`4) The channel is quasi-static. The capacity thus becomes a
`random variable, rendering the concept of a "capacity cu(cid:173)
`mulative distribution function" and "outage capacity [2]
`a meaningful measure.
`The input-output relationship can thus be written as
`
`(1)
`where 7 is the transmit signal vector, and rt is the noise vector.
`
`III. PERFORMANCE RESULTS
`
`·
`
`A. Diversity
`For the case of pure transmit diversity with channel knowl(cid:173)
`edge, 7 ="it• s, wheres is a (scalar) symbol. This means that
`we are just transmitting a single symbol, differently weighted,
`over the different antenna elements. Similarly, at the receiver,
`we obtain a "soft" symbol estimate as -:;Jul•, which is then
`processed ( decoded and demodulated) in the usual way.
`References [7], [8] analyze the case where there is an(cid:173)
`tenna selection only at one link end, while the other one uses
`~um-ratio combining. Define a set of matrices H, where
`His created by striking N, -_Lt _columns from H, and S(H)
`den~s the set of all possible H, whose cardinality is (f'). The
`achievable SNR 'Y of the reduced-complexity system is now
`'Y = m~ ( m~cX;))
`
`S(H)
`
`'
`
`(2)
`
`where the A; are the singular values of H. The papers give
`analytical expressions for upper and lower bounds on the SNR.
`as well as Monte Carlo simulations of the exact results for the
`SNR as weU as the BER and capacity derived from it. The mean
`SNR E { 'Y} is computed in [9].
`Figure 2 shows the cwnulative distribution of the capacity
`for different values of Lt, with mean SNR f = 20 48, N, = 2.
`N, = 8. The capacity obtained with L, = 3 is already very
`close to the capacity of a full-complexity scheme. For compari(cid:173)
`son, we also show the capacity with pure Mirr. Actually, it can
`be shown that the diversity degree obtained with antenna selec(cid:173)
`tion is proportional to N, not to L. Also for a space-time-coded
`system, where the transmitter has no channel knowledge, and
`the :receiver perfonns antenna selection, the achievable diversity
`is NtNr, while a coding gain decreases by up to 10log(Nr/ L),
`see [10].
`In a highly correlated channel, no diversity gain can be
`achieved, but all gain is due to improvment of the mean SNR.
`Thus antenna selection is ineffective, and the SNR gain is only
`influenced by the number of actually used antenna elements.
`
`,
`u O.& ·i .. ·H;=l
`~ ~:-·-.tt
`o.2+i·
`s
`
`!
`
`i
`
`,o
`capacity C [blts/s/1-12]
`
`2
`
`15
`
`1 u o.s -N,=
`16 0.6
`2
`u 0.4.. 3
`0.2 4
`
`5
`
`,o
`capacity C [bits/s/Hz]
`
`15
`
`FiJ. 2. Upper figure: Capacity of a system with H-SIMKI' at the tnmsmitler
`and MRC at the receiver fm various values of Li wilh N1 = 8, N, = 2,
`SN R = 20 d8- Lower figure: capacity of a system wilh MRT at transmilter
`aml MRC at receiver for ....rious values of N, and N, = 2, SN R = 20 dB.
`From [8].
`
`to unity. The receiver now selects those antennas that allow
`a muimization of the capacity. As shown in Ref. [2], the
`capacity is linearly proportional to min(N., N1}. Any further
`increase of either N, or N, while keeping the other one fixed
`only increases the diversity degree, and consequently allows a
`logarithmic increase of the capacity. Thus, if the number of
`antennas at one link end is limited e.g. due to space :restric(cid:173)
`tions, a further increase in the antenna number at the other link
`end does not allow to add statistically independent transmission
`channels (which would imply linear- increase in system capac(cid:173)
`ity), but only provides additional diversity. But we have already
`seen that antenna selection gives good diversity degree. We can
`thus anticipate that a hybrid scheme with N, > L, = N, to give
`good perfonnance. This line of argument can be quantified by
`performance bounds [11], [12).
`Figure 3 shows the cwnulative distribution function (cdf) of
`capacity for N, = 8, N, = 3, and various L,. With full ex(cid:173)
`ploitation of all available elements, an outage capacity of 21.8
`bit/s/Hz can be achieved at 20dB SNR. This number decreases
`gradually as the number of selected elements L, is decreased,
`reaching 18.2 bit/s/Hz at L, = 3. For L, < N., the capac(cid:173)
`ity decreases drastically, since a sufficient number of antennas
`to provide N, independent transmission channels is no longer
`available.
`At low SNRs, diversity can give higher capacities than spatial
`multiplell.ing when antenna selection is employed, [13], similar
`results also hold in the case of strong interference [14].
`
`C. Space-time coded systems
`Next, we consider the problem of space-time coded systems
`with transmit and receive antenna selection, where the transmit(cid:173)
`B. Spatial Multiplexing
`ter has knowledge about the statistics of the fading. The chan(cid:173)
`nelshows correlation, and the correlation matrices are known
`For spatial multiplexing, different data streams are transmit-
`at the TX. Then, we introduce a modified correlation matrix .H
`ted from the different antenna elements; in the following, we
`consider the case where the TX, which has no channel knowl- which is the submatrix of the total correlation matri,i: H corre(cid:173)
`sponding to the selected antennas. The pairwise error proba(cid:173)
`edge, uses all antennas, while the receiver uses antenna selec-
`tion [11]; all (linear) weights -:;t, w in Fig. I are set equal
`bility (i.e., confusing codeword c(i) with codeword 5Ul) for a
`168
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`3
`
`V. EFFECT OF NONlDEALITIES
`A. Low-rank channels
`Previously, we have assumed that the channel is i.i.d. com(cid:173)
`plex Gaussian, or shows some correlation at the transmitter
`and/or receiver. However, in all of those cases is the chan(cid:173)
`nel matrix full-rank, and the goal of the antenna selection is
`to decrease complexity, while keeping the performance loss as
`small as possible. There are, however, also propagation chan(cid:173)
`nels where the matrix lJ.. has reduced rank [22]. Under those
`circumstances, antenna selection can actually increase the ca(cid:173)
`pacity of the channel [23].
`
`·
`
`B. Frequency-selective channel
`In frequency-selective channels, the effectiveness of antenna
`selection is considerably reduced. For different (uncorrelated)
`frequency bands, different sets of antenna elements are opti(cid:173)
`mum. Thus, in the limit that the system bandwidth is much
`larger than the coherence bandwidth of the channel, and if the
`number of resolvable multipath components is large, all pos(cid:173)
`sible antenna subsets become equivalent. This can also be in(cid:173)
`terpreted by the fact that such a system has a very high diver(cid:173)
`sity degree, so that any additional diversity from antenna se(cid:173)
`lection would be ineffective anyway. However, for moderately
`frequency-selective channels, antenna selection still gives sig(cid:173)
`nificant benefits. A precoding scheme for CDMA that achieves
`such benefits is described in [24].
`
`' 10
`
`20
`
`30
`
`capacity C (bits/s/Hz]
`
`Fig. 3. Opacity for a spatial mulliplex.ing sysiem with N, = 8, N, = 3,
`SNR= 20 dB, and L = 2,3, .... 8.
`
`space-time coded system is derived in [9], (15], (16]. The opti(cid:173)
`mum antenna selection is thus the one that maximizes the deter(cid:173)
`minants of .B_. Assume further that the so-called "K.ronecker(cid:173)
`model" [17], [18] is valid, so that the total correlation fan be
`described by the correlation matrices at TX and RX, Jk and
`Hr. In that case, the selection at the transmitter and the receiver
`can be done independently.
`
`IV. ANTENNA SELECTION ALGORITHMS
`
`requires on the order of 1;
`
`~: computations of determi-
`
`The only mechanism for a truly optimum selection of the
`antenna elements is an exhaustive search of all possible combi-
`nations for the one that gives the best SNR (for diversity) or ca-
`pacity (for spatial multiplexing). However, for HS-MIMO, this
`(
`) (
`)
`
`c. Channel estimation errors
`We next investigate the influence of erroneous antenna se-
`lection on the capacity of the system [25]. We assume that
`in a first stage, the complete channel transfer matrix is esti-
`nants, which quickly becomes impractical. For this reason, var- mated. Based on that measurement, the antennas that are used
`ious simplified selection algorithms have been proposed. Most
`for the actual data transmission are selected, and the antenna
`of them are intended for systems where the selection is done at weights are determined. Consider now the following cases:
`only one link end.
`(i) perfect choice of the antennas and the antenna weights, (ii)
`The simplest selection algorithm is the one that chooses the
`imperfect antenna selection, but perfect knowledge of.the an(cid:173)
`antenna elements with the largest power, i.e., the largest Frobe-
`tenna weights, (iii) imperfect choice of the antennas, as well as
`of the antenna weights at the transmitter, and perfect antenna
`nius column (or row) norm. For the diversity case, this algo-
`rithm is quite effective. However, for spatial multiplexing, this weights at the receiver, and (iv) impelfect choice of the antenna
`approach breaks down. Only in about 50% of all channel real- weights at transmitter and receiver. The errors in the transfer
`izations does the power-based selection give the same result as · functions are assumed to have a complex Gaussian distribution
`the capacity-based selection. This behavior can be interpreted with SN Ri,itot, which is the SNR during the transmission of the
`in geometric terms because the phase shifts between the an-
`pilot tones. In our example, the capacity starts to decrease sig(cid:173)
`tenna elements are the decisive factors for capacity, and are far
`nificantly only when the pilot tone SNR is smaller than the SNR
`more important than the instantaneous SNR [11].
`for the actual data transmission, see Figure 4.
`An alternative class of algorithms has been suggested by
`Another type of channel estimation error can be caused by a
`[19]. Suppose there are two rows of the H which are identi-
`limit on the number of bits for the feedback of antenna weights
`cal. Clearly only one of these rows should be selected in fI.
`to the TX. This problem is especially important for the W(cid:173)
`Since these two rows carry the same information we can delete CDMA standard. Attempts to send the full transmit weight
`information then has to result either in a very coarse quantiza(cid:173)
`any row of these two rows without losing any information about
`the transmitted vector. ln addition if they have different powers
`lion, or the feedback information has to be sent of many slots,
`{i.e. magnitude square of the norm of the row), we delete the
`so that - in a time-variant environment - the feedback infonna(cid:173)
`lower power row. When there are no identical rows we choose
`tion might be outdated by the time it arrives at the transmitter.
`next two rows for the deletion whose mutual information is the Thus, the attempt of getting full channel state information to
`next highest. In this manner we can have the c]lannel matrix iI
`the transmitter carries a penalty of its own. The use of hybrid
`whose rows have minimum mutual information and have max-
`antenna selection might give better results in this case, since it
`imum powers. This method achieves capacities within a few
`reduces the number of antennas for which channel information
`tenths of .i bit/s/Hz of the capacities with ideal selection. Other has to be transmitted. An algorithm for optimizing the "effec-
`tive" SNR is discussed in [26].
`algorithms are derived in Ref. [20] and [21].
`169
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`discussions. Part of this work was supported by the "double(cid:173)
`directional channel model" INGVAR project of the Swedish
`Strategic Research Foundation.
`
`4
`
`[10]
`
`[12)
`
`[131
`
`[14]
`
`[15)
`
`[16)
`
`[17)
`
`[18)
`
`(19]
`
`[20]
`
`Fig. 4. Impact of errors in the estimation of transfer function rnatri~ H. Cdf of
`the capacity for (i) ideal channel knowledge al TX and RX (solid), (ii) imperfect
`antenna sele.ction, but perfect an1enna weights. ( dashed). (iii) imperfect antenna
`weights at TX only (dotted). and (iv) impcrfoct antenna weights at TX and RX
`(dash-dotted). SN Rp,1o, = 5 dB. From [25].
`
`D. Hardware aspects
`
`Finally, we consider the effects of the hardware on the per(cid:173)
`formance. In all the previous sections, we had assumed "ideal"
`RF switches with the following properties:
`• they do not cause any attenuation or additional noise in the
`receiver
`• they are capable of switching instantaneously
`• they have the same transfer function irrespective of the
`output and input port, and should be linear
`Obviously, those conditions cannot be completely fulfilled in
`practice. The attenuation by the switches is the most critical
`issue. In the TX, the attenuation by the switch must be com(cid:173)
`pensated by using a power amplifier with higher output power.
`At the receiver, the attenuation of the switch plays a minor role
`only if the switch is placed after the low-noise receiver ampli(cid:173)
`fier (LNA). However, that implies that N, instead of Lr receive
`amplifiers are required, eliminating a considerable part of the
`savings of antenna selection.
`
`VI. SUMMARY AND CONCLUSIONS
`
`[21l
`
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`
`This paper presented an overview of MIMD systems with an-
`tenna selection. Either the transmitter, the receiver, or both use
`only the signals from a subset of the available antennas. This
`allows considerable reductions in the hardware expense. We
`found that antenna selection retains the diversity degree ( com-
`pared to the full-complexity system), both for linear diversity
`systems with complete channel knowledge, and for space-time
`coded systems. However, there is a penalty with respect to the
`average SNR. For spatial multiplexing systems (BLAST), an-
`tenna selection at the receiver gives a capacity comparable to
`the full-comple.x:ity system as long as Lr 2: Nt (and similarly
`for the selection at the transmitter). Thus antenna selection is an
`extremely attractive scheme for reducing the hardware expense
`in MIMO systems.
`Acknowledgement: The author would like to thank Prof.
`Moe Win, Dr. Makoto Miyake, Dr. Jin Zhang, Ms. Xinying
`Zhang, Prof. S. Y. Kung, and Dr. Jack Winters for helpful
`170
`
`[22]
`
`[231
`
`r241
`
`[25]
`
`[l6]
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