Analog Beam Forming in MIMO Communications With Phase Shift Networks and Online
Embed Size (px)
Transcript of Analog Beam Forming in MIMO Communications With Phase Shift Networks and Online
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010 4131
Analog Beamforming in MIMO CommunicationsWith Phase Shift Networks and Online
Channel EstimationVijay Venkateswaran and Alle-Jan van der Veen, Fellow, IEEE
Abstract—In multiple-input multiple-output (MIMO) systems,the use of many radio frequency (RF) and analog-to-digital con-verter (ADC) chains at the receiver is costly. Analog beamformersoperating in the RF domain can reduce the number of antennasignals to a feasible number of baseband channels. Subsequently,digital beamforming is used to capture the desired user signal.In this paper, we consider the design of the analog and digitalbeamforming coefficients, for the case of narrowband signals.We aim to cancel interfering signals in the analog domain, thusminimizing the required ADC resolution. For a given resolution,we will propose the optimal analog beamformer to minimize themean squared error between the desired user and its receiverestimate. Practical analog beamformers employ only a quantizednumber of phase shifts. For this case, we propose a design tech-nique to successively approximate the desired overall beamformerby a linear combination of implementable analog beamformers.Finally, an online channel estimation technique is introduced toestimate the required statistics of the wireless channel on whichthe optimal beamformers are based.
Index Terms—ADC power consumption, analog beamforming,matching pursuit, passive RF phase shifters.
M ULTIPLE-INPUT multiple-output (MIMO) and multi-sensor communication systems employ multiple receive
antennas to exploit selection diversity and improve multiplexinggains. The aim is to achieve reliable communication close totheoretical limits . However, the introduction of multiple an-tennas at the receiver leads to separate radio frequency (RF)front ends and analog to digital converter (ADC) units, i.e., in-creased circuit size and power consumption.
The implementation of digital baseband algorithms followsMoore’s law, resulting in a power reduction by a factor of 32for every ten years. In contrast, ADC power was reduced onlyby a factor of 10 in the past decade . In existing multiantennareceivers, an ADC operation requires the same power as that ofhundred thousands of logic gates . To enable the full poten-
Manuscript received November 19, 2009; accepted April 07, 2010. Date ofpublication April 15, 2010; date of current version July 14, 2010. The associateeditor coordinating the review of this manuscript and approving it for publi-cation was Dr. Martin Schubert. This research was supported by Senter-Novemunder the IOP-Gencom program (IGC-0502B). Parts of this paper was presentedat the International Conference on Acoustics, Speech and Signal Processing(ICASSP), Dallas, TX, March 15–19, 2010 .
The authors are with the Delft University of Technology, Faculty of the Elec-trical Engineering, Mathematics and Computer Science, 2628 CD Delft, TheNetherlands (email: [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2010.2048321
tial promised by MIMO capacity theory to become reality, thereis a need for novel RF architectures with digital assistance. Oneparticular option is to consider a multiuser cellular/WLAN sce-nario, where the bandpass RF signals contain contributions fromthe desired user, noise and interfering users. In the presence ofstrong interferers, the ADCs are forced to spend a significantpart of their dynamic range on digitizing the unwanted inter-ferers and noise. If we are able to cancel most of the interferencebefore it reaches the ADC we can use lower resolution ADCs,which directly translates into reduced power consumption.
If the number of receiver antennas is given, one well knownsuboptimal technique to reduce the number of RF and ADCchains is to use antenna/diversity selection. Basically, we selectthe antenna with the highest signal energy . This techniquedoes not enable interference cancellation before the ADC.
An advancement over antenna selection is the use of analogpreprocessing networks (APNs) for linearly combining the an-tennas, i.e., beamforming. Current hardware developments offermany possibilities. In –, a phase shift preprocessor is im-plemented, which uses active and passive weighting elements tocombine signals from the antenna array in the RF domain. Ha-jimiri et al.  propose a design where the required phase delaysare implemented in the RF to baseband demodulation step, byusing a bank of several phase-shifted local oscillators. Typicallythese designs provide about 16 possible phases (4-bit phase res-olution) and no variable amplitude, thus implement a poorlyquantized set of possible beamformers. These papers focus onthe hardware design and only briefly touch upon the questionhow these beamformer coefficients should be selected. E.g., in, a set of beamforming vectors is precomputed to steer beamsin predefined directions, with a resolution of about 22 . Thisonly allows to select the direction with highest energy, whichdoes not necessarily result in the desired signal in the presenceof multipath and interference.
Improvements are possible by considering multiple outputstreams. Shown in Fig. 1 is an architecture where an analog pre-processing network (APN) directly operates on the RF signals,mapping antenna array signals to receiver chains(i.e., ADCs). A digital beamformer subsequently combinesthe ADC outputs to generate the desired user estimate. Zhang etal.  considered such an architecture, and proposed severalMIMO transmitter/receiver beam steering techniques.
Problem Statement: Our aim in this paper is to design an op-timal APN beamforming matrix. Our focus is to minimize theinterference at the input of the ADCs, so that reduced resolutionis possible, leading to reduced power consumption. Some de-sign issues are (1) to choose , (2) to select the beamforming
1053-587X/$26.00 © 2010 IEEE
4132 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
Fig. 1. Proposed receiver architecture: the analog preprocessing network (RF beamformer) cancels interference and reduces the number of antenna signals to asmaller number of ADC chains.
coefficients, and (3) to determine how many bits are needed ineach of the ADCs. A design constraint is the poor resolutionof the APN coefficients. Design criteria are the mean-squareerror (MSE) at the output of the digital beamformer and theADC power consumption.
As an example, consider a wireless channel with one inter-fering user transmitting signals with the same energy as that ofthe desired user, with antennas and receiverchains. It will be seen from the simulation results in Section VI,that RF interference cancellation with an APN can reduce theADC power consumption by half for the same MSE at the re-ceiver output.
In the array signal processing literature, several types ofpreprocessing matrices have been designed to reduce thenumber of receiver chains. One related context is “beamspacearray processing,” where a preprocessing is done on the receiveantennas to reduce dimensionality; see, e.g., . In earlierwork, this is called “partially adaptive beamforming,” wherethe preprocessor is fixed and the digital beamformer is adaptive;see, e.g., . In this literature, the design of the beamspacetransformation matrix is based on prior knowledge of the loca-tion of the signal of interest and/or on the interference scenario.Reduced dimension transformations, to cancel interferers usingstatistics from the desired user, have also been proposed .In the present paper, we aim to design the APN using feedbackfrom the baseband processor, so that it can be optimized for theactual situation on a block-by-block basis.
Transformation preprocessors have also been pursued in di-rection-of-arrival (DOA) estimation problems, e.g., . In thatpaper, a set of preprocessors is applied over time, and the re-sults are combined to estimate the DOA. In contrast, our aim isto minimize interference and reconstruct the signal of interest;however, we will apply techniques from  to estimate the re-quired channel parameters.
In practice, the APN coefficients are quantized. There existsa significant amount of literature on (adaptive) beamformingusing variable phase only (cf. equal gain combining). Most lit-erature considers a single weight vector (with variable phasesonly) that should be designed to match certain performancecriteria, e.g., , , and . The paper  considers anAPN with multiple outputs, and it is shown that any desiredweight vector can always be obtained by linearly combiningtwo phase-only beamformers, thus is sufficient. Thework of  has generated some follow-up work, focusing on
phase shift beamforming at the transmitter and receiver to lin-early combine the signals. However, these approaches do notemphasize on interference cancellation nor on implementableAPN weights. We will consider a more restricted case wherethe APN weights and ADC taps are severely quantized.
B. Contributions and Outline
In the paper, we progressively study various aspects of theAPN beamformer design. In Section II, the system setup andthe data model is specified. In Section III we consider the casewhere the APN and ADC have sufficiently high resolution. Wewill aim to design an preprocessing matrix that min-imizes the MSE. This leads to a non-unique design. To make itunique, we also take the ADC quantization error into accountand we design the APN to maximize the signal to quantizationnoise ratio (SQNR). For an APN with infinite precision, we willderive that it is sufficient to consider only output chain.
However, as mentioned earlier, in practice the APN is a pro-grammable discrete phase shifter with a coarse quantization.In this case, using allows us to extract different lowresolution streams, some representing the user of interest andsome the interferers, followed by digital combining. Thus, inSection IV, we study the case where the APN quantization isthe limiting factor in the design. To minimize the MSE, we tryto match an optimal beamformer to a linear combination ofvectors from the set of available quantized beamformers; for thiswe propose a quantized version of the matching pursuit (MP)algorithm , . All the above mentioned designs dependon knowledge of the channel statistics: the antenna covariancematrix and the antenna cross-correlation vector with the desireduser signal. Note that the digital receiver does not have directaccess to the antenna signals. In Section V, we provide an algo-rithm to deduce this information from various observations ofbeamformed outputs (the algorithm is related to that of Shein-vald et al. ). The results are combined in the digital beam-former to obtain a high resolution estimate and illustrated usingsimulation results.
Notation: Vectors and matrices are represented in lower andupper case bold letters. , , and represent transpose,complex conjugate transpose and pseudo inverse, respectively.
and represent Kronecker product and Frobenius norm.The operation transforms a matrix to a vectorby stacking its columns, while does the opposite.Continuous time signals are represented with round braces as in
and sampled signals as .
VENKATESWARAN AND VAN DER VEEN: ANALOG BEAMFORMING IN MIMO COMMUNICATIONS 4133
II. SYSTEM SETUP AND DATA MODEL
A. RF Data Processing
Consider an RF signal received at an antenna. Assumingsuitable bandpass prefiltering, only a narrow frequency bandaround a carrier frequency is of interest, and we can write
where is the complex envelope or baseband signal. In thereceiver, the “RF to baseband” processing block recoversusing quadrature demodulation. This signal subsequently entersthe ADC unit. Here it is sampled at time instants (where
is the sampling period) leading to and quantized usingbits, leading to . We will always assume that the Nyquistcondition holds. The ADC unit includes an automatic gain con-trol (AGC) that scales the input signal such that its amplitudematches the range of the ADC without overload.
If the RF signal is delayed by , we obtain
The approximation is valid if for all fre-quencies in the bandwidth of , i.e., the “narrowband con-dition”. After RF to baseband conversion, the delayed basebandsignal is and the sampled signal is .
If we have an array with receive antennas, it will be con-venient to stack all signals into vectors , , and
B. Received Data Model
Consider now a communication setup, where user signalsare transmitted over the same carrier
, propagate over a multipath channel, and are received by thearray with antennas. Without loss of generality, letis the desired user signal, and the other signals are consideredinterferers. Assume that the narrowband condition holds for allpropagation delays (except for a bulk delay that we will ignorehere) so that they can be represented by phase shifts. We canwrite the equivalent discrete time data model as
where is an vector ofuser signals, and is an vector of noise signals.is a matrix denoting the MIMO channel response withcomplex entries , representing the channel coefficients forthe propagation of the th user signal to the th receive antenna,which includes the transmit/receive filters, array response, am-plitude scalings, and phase delays.
Throughout the paper, we will make the following standardassumptions on this model.
• The user signals are modeled as random processesthat are zero mean, independent, wide-sense stationary,with equal powers normalized to 1.
• The noise signal vector is sampled from an i.i.d.Gaussian process, zero mean, with unknown covariancematrix .
Fig. 2. (a) Proposed receiver architecture with RF beamformer. (b) Discretetime equivalent.
C. High-Resolution Digital Beamforming
Given observations , our goal is to obtain an estimateof the desired user signal . We will consider linear
beamforming and a minimum mean-square error (MMSE) cri-terion. Thus, let be an weight vector, then the digitalbeamforming output is
and the MMSE beamformer is obtained as the solution of
As is well known, the solution is given by the Wiener beam-former 
where and . Esti-mates of and are obtained from the sample covariancematrix and sample cross-correlation vector; this requires accessto all antenna signals and the availability of a reference signal(training sequence) for the desired user.
The above solution (2) will serve as our reference design.In its derivation, the effect of the quantizer operator wasignored.
D. Analog Preprocessing Network
As mentioned in the Introduction, it is expensive to insert acomplete RF receiver chain and high-rate high-resolution ADCunit for each antenna. We thus consider an APN inserted inthe RF domain immediately after the low-noise amplifiers andbandpass filter [see Fig. 2(a) and its discrete time equivalentFig. 2(b)]. Although there are various implementations, we willmodel the APN as an analog beamformer that constructs linearcombinations of slightly delayed antenna signals . This re-sults in output signals , , where the numberof outputs . As before, the output signals are stackedinto vectors , downconverted (leading to basebandsignals ), sampled and quantized (leading to ). The thADC has resolution bits.
The effect of the APN on the baseband signal is modeledusing a discrete time equivalent matrix operation
4134 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
where is a matrix of size andis a vector. Each entry cor-
responds to the phase delay introduced by the APN for the threceive antenna and the th output signal. Practical implemen-tations limit to a small set of possible phase shifts, perhaps8 to 16 choices (3 to 4 bits). Amplitude changes are usually notpossible. The weights can be controlled by the basebandprocessor but note that is not directly available at the pro-cessor, making the design of a challenge.
The digital baseband signals are subsequently combinedusing a digital beamformer , resulting inthe output signal
To obtain an estimate of , can be designed as the MMSEbeamformer solving
leading to a Wiener beamformer specified interms of correlations of where and
. For given , is known and thisproblem can be solved, hence is a function of .
E. Problem Formulation
Our aim in this paper is to design the APN . After that,is fixed and the design of is relatively straightforward. For thedesign, there are a number of side conditions or assumptions:[A1] The APN circuits consist of a limited number of phase
shift combinations, hence the elements of are se-lected from a finite set, denoted as a dictionary .
[A2] Each ADC performs uniform quantiza-tion with a resolution of bits, hence
.The ADC power consumption can be approximated as
, where is the sampling frequency (in ourcase the Nyquist rate) and is the ADC resolution in bits.We wish to minimize the number of bits in the ADC, sincethis is directly related to the power consumption in the analogsection of the receiver. The number of bits is determined by therequired dynamic range, which is partially controlled by .Indeed, if more interference cancellation is performed, fewerbits are needed for the same MSE performance.
Design objectives are 1) minimizing the MSE at the outputof the digital beamformer, including the quantization noise, and2) minimize the energy consumption in the ADCs, representedby . Several problems can be formulated using theseobjectives, but they do not all have feasible solutions.
We therefore approach the APN design in the following order:
[P0] We initially relax [A1] and [A2] and assume a perfectand continuous APN, and high resolution ADCs.What are then the constraints on the design of ? (Itwill follow that is not unique.)
[P1] Assuming low-resolution ADCs, each withbits, how does the design change? Can we compute aunique ?
[P2] Now considering the discrete nature of the APN, selectfrom a fixed set of discrete phase shifts present in
, such that the overall MSE is minimized. Here it isassumed that the ADC resolution is not limiting.
The above design problems [P1] and [P2] form the core of thispaper and are covered respectively in Sections III and IV.
The design techniques will assume knowledge of theantenna array covariance matrix and a
cross covariance vector . Note thatthe introduction of the APN implies that this covariance matrixis not available in the digital part of the receiver. In Section V,we explain a technique to estimate and from a set of low-rate beamformers in digital baseband, assuming that a trainingsequence of the desired user is available.
III. PREPROCESSOR DESIGN-APN NOT QUANTIZED
In this section, we consider problem [P1]: design the APNconsidering only the quantization by the ADCs, and design thenumber of bits which each ADC should use. We do not considerthe limited choice of phase shifts for the APN—this case is de-ferred to Section IV.
A. Conditions on to Minimize the MSE
Let us also ignore the quantization operation by the ADCsfor the moment, i.e., consider problem [P0]. The problem is todesign
We have , and the output shouldbe close to the desired signal in MMSE sense, i.e., werequire
Thus, implements a rank reduction on the space spannedby . Some common but suboptimal designs based on rankreduction are listed in Appendix I.
Instead, we will now consider which conditions has to sat-isfy such that we can optimize the MSE. Note that, once isspecified, we know , and if we want to mini-mize the MSE of the output , we know we willselect
Thus, the MSE is a function of only. We can see immediatelythat will not be unique: e.g., we could choose
(for ), or (for), among many other possibilities.
Define the “whitened” correlation matrices
The following lemma characterizes all solutions that lead toMMSE-optimal beamformers .
Lemma 1: Consider the scenario [P0]: the APN is not quan-tized, and the quantization error of the ADCs is ignored. Then
VENKATESWARAN AND VAN DER VEEN: ANALOG BEAMFORMING IN MIMO COMMUNICATIONS 4135
all beamformers that lead to the MMSE-optimal solutionare characterized by the condition
Proof: implies that is inthe column span of . Hence, a necessary and also sufficientcondition on is .
An alternative proof is as follows. Define the orthogonal pro-jection matrix
For any , and corresponding optimal, the MSE is given by
Thus, the MMSE solution satisfies
Clearly, this only specifies that , and therecan be many solutions.
B. Maximizing the SQNR
The next question is whether we can narrow down the set ofavailable solutions for , by satisfying additional design ob-jectives. Our approach is to incorporate the power consumptionor ADC resolution in (7), and minimize the MSE while max-imizing the signal to quantization noise ratio (SQNR) of theoutput estimate. This leads to a design for , and also to cri-teria on the number of bits which each ADC should use.
We thus consider problem [P1]. Incorporating the effect ofthe ADC on the signal at the output of the analog beamformer,we have
The effect of the quantizer will be modeled by an addi-tive noise vector , i.e.,
As usual, is modeled as uniformly distributed noise, entry-wise independent, and uncorrelated to . The correspondingcovariance matrix of is [replacing (4)]
where is a diagonal matrix whose diagonal entries representthe quantization noise variance. Suppose that the th ADC has
a resolution of number of bits. We assume that an automaticgain control (AGC) is used such that the dynamic range of theADC is optimally used. The quantization noise variance
will then depend on the signal variance at theinput of the ADC, with some abuse of notation1 denoted as .Using the well-known Lloyd–Max equation , we have
where is an AGC scaling factor that models the differencebetween the average input power and the peak input power.
To reach a feasible optimization problem, we will limit our-selves to the case where all ADCs use an equal number of bits,
. The noise covariance matrix can then be expressed as
Given , the optimal digital beamformer, acting on , is still. At the output of the beamformer, the average
energies of the desired user signal and quantization noise are,respectively
and the corresponding SQNR of the output signal is
where and are functions of as specified in (8). Theobjective is to design , first of all, to minimize the MSE asdiscussed in the previous subsection, and this leads to designfreedom, which is used to maximize the SQNR.
Regarding the minimization of the MSE, we can follow thederivation that led to (6), however, is now slightly differentsince it also involves the quantization noise . For a reasonablenumber of bits, we will have that
. In that case, we can ignore the influence ofon , (6) still holds, and the MMSE is obtained for anysatisfying (7),
Using the whitened quantities (5), the numerator of the SQNRexpression can be written as
and the denominator (up to scaling by ) as
1since � ��� denotes the signal at the output of the ADC in this subsection.
4136 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
where . Subject to (10), the numeratoris equal to , which is a constant independent of . Itsuffices to minimize the denominator. It further follows from theexpression of the SQNR that the scaling of is not important.
The SQNR optimization problem (subject to optimal outputMSE) becomes
We will solve this problem in closed form for the case .Theorem 1: Consider the scenario [P1]: the APN is not quan-
tized, the ADCs are quantized at bits. Assume . Thenthe optimal APN that minimizes the MSE and maximizes theSQNR (subject to optimal MSE) is obtained if all columns of
are equal to the MMSE beamformer, , up to scalingand certain linear transformations.
Proof: To solve (11), we first parametrize such that theconstraint is satisfied. Thus, let
where is a 2 2 unitary matrix since , andis an matrix such that
(Note that .) Then
Introduce a sufficiently general parametrization foras
where , and are in the range . (A completelygeneral parametrization would also have two complex phasefactors at the right, but one phase can be absorbed in , andthe other can be extracted to multiply the complete matrix ;
Fig. 3. ���� as a function of � � �� � and �.
the form of the cost function shows that that phase will cancel.)Then
where ; note that . The cost function(11) becomes
Since , minimizing the cost function will requirechoosing at extremes,
Subsequently, optimal values for will follow as well, as a func-tion of . The location of the minima of isshown in Fig. 3. The value of the minimum is 0.5. Althoughthere are multiple minima, in any case, we will have :the correlation coefficient between and has absolute value1, which implies that and are equal, up to a scaling andphase rotation.
In summary, we derived that, given a specific resolution of theADCs and the number of ADCs, the optimal approach to mini-mize the MSE and maximize the SQNR is to choose the columnsof all parallel to . Translated to , it means that eachbeamformer in the APN is parallel to the Wiener beamformer
. In actuality, they should differ by at least a phase shiftsuch that the quantization noise on the beamformer outputsbecomes uncorrelated. The digital beamformer will simply av-erage the results, i.e., average out the additive noise and quanti-zation noise.
The result was obtained for ; however, it seems rea-sonable that it generalizes to larger . Also, the result was ob-tained for all ADCs having the same resolution , but thesame result will follow also for unequal resolutions. In this casethe diagonal terms in in will have unequal scaling,but it can be seen that the optimization still leads to , sothat the same conclusion follows.
VENKATESWARAN AND VAN DER VEEN: ANALOG BEAMFORMING IN MIMO COMMUNICATIONS 4137
Finally, let us consider the effect of the APN on the powerconsumption by the ADCs. It is clear that by inserting an APN,interference cancellation becomes possible, leading to reducedrequirements on ADC resolution and hence enabling power re-duction. As function of the interference power, the benefits canbe arbitrarily large compared to a setup without APN.
Next, we compare to , while keeping aconstant output MSE after digital beamforming. For the optimalAPN, the digital beamformer will simply be averaging the out-puts of the ADCs, so that the quantization noise power at theoutput is halved. Quantization noise of one channel is propor-tional to . Thus, for and the same SQNR, eachADC needs half a bit less than for . However, powerconsumption is also proportional to . For two ADCs, eachwith half a bit less, the total power consumption is constant.Thus, there is no particular advantage to choose fromthis perspective.
More generally, allows us to use multiple ADCs withlower resolution in situations where high rate high resolutionADCs cease to exist due to fundamental limitations .
IV. PREPROCESSOR DESIGN-APN WITH
DISCRETE PHASE SHIFTS
In the previous section, we did not take the quantization of theAPN coefficients into account. In practice, the elements ofcan only be selected from a discrete alphabet, usually only froma set of possible phase shifts. We will now study this case, mean-while ignoring the quantization of the ADCs, i.e., assuming theirresolution is high enough such that it does not dominate thedesign.
A. Matching the Cross-Correlation Vector
Since the elements of are quantized, let represent theset of all possible phase shift vectors that a column incan take. The entries of are denoted as , where
is the size of the dictionary. If the APN taps are quantized bybits then . Typically, would be 2 to 4 bits.
Similarly, call the set of all possible APN matrices .The APN design is now transformed into a problem of se-
lecting such that the MSE distortion is kept at aminimum,
Lemma 2: The MMSE beamformer solving (13) is obtainedas , where
and where .Proof: Following (7), solving (13) is equivalent to solving
This is equivalent to
which is equivalent to (14), since for given the optimalchoice for is .
Thus, the problem becomes to match in least squaressense to linear combinations of columns of , each of whichcan assume only values in a discrete set. Equivalently, thecolumns of should span a subspace to which is close.The selection complexity is exponential in , , and .
B. Quantized Matching Pursuit
To reduce the complexity, the columns of are selectedone-by-one. The matching pursuit (MP) technique  is agreedy technique that recursively chooses the dictionary ele-ments to obtain the best approximation of an input vector, inthis case . Indeed, write
In the greedy approach, we first solve
Given , the optimal solution for is, so that the problem reduces to
The solution requires a search in the dictionary, at a complexityexponential in and . To facilitate the search, we can firstnormalize the vectors in to unit norm, and then search for thevector with maximal correlation to .
After selecting and , we compute the residual vectorand proceed similarly as (15),
After selecting and , we could continue the processwith the residual vector
However, and are not orthogonal, and better coefficientsand can be computed at this point, leading to a smaller
residual. This requires to solve
4138 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
TABLE IQUANTIZED MATCHING PURSUIT (QMP) ALGORITHM
Define and introduce a QR factorization
where is a orthonormal matrix and a 2 2upper triangular matrix. The solution to (16) is
and the corresponding (smaller) residual is
which is the projection onto the orthogonal complement of thecolumn span of . The recursion follows in an obvious way.Note that the QR factorization (17) is easily updated once newvectors are added, and that, in fact, it is not necessary to ex-plicitly compute at intermediate steps. The algorithm is sum-marized in Table I.
Further refinements of this algorithm are possible. The updateof the QR factorization and the computation of the residualcan be integrated into a single update step. In principle, a betterselection of could be obtained by computing the residualsfor all possible from the dictionary and selecting the onethat gives smallest residual; however, the complexity of this isprobably too high.
In practical implementations, the beamforming weights areusually quantized phase shifts with unit amplitude. In this case,it may be more accurate to first split a desired weight vectorinto two weight vectors and , each with entries on the unitcircle, such that a linear combination of them gives the desired
. Zhang et al.  showed that such a partitioning is alwayspossible. Subsequently, the phase vectors and are eachquantized into discrete phase shifts, resulting in the “greedy”
Fig. 4. (a) Architecture 1, where each antenna has its own ADC. (b) Architec-ture 2, containing time varying beamformers � � � � ��� � � � � �� for differenttraining periods. (c) Architecture 3 with � low resolution beamformers.
selection, now consisting of a pair of vectors. The process con-tinues as before to recursions.
In this section, APN has been designed from (13) assumingthat the ADC quantization is negligible. It is later shown usingsimulations in Section VI that for a small (as is the case inpractice) and bits, the MSE is dominated by the quanti-zation of the .
V. ONLINE CORRELATION ESTIMATION
The APN phase shift design requires the knowledge ofand . However, since the digital baseband processor has onlyaccess to the beamformed outputs and not to the individualantenna signals , it is not possible to directly compute thesecorrelations from the available observations (and, in the case of
, a training signal ). In the context of direction of arrivalestimation, a similar problem was studied by Sheinvald et al. and Tabrikian et al. .
Regarding system architectures, there are a number of optionsas enumerated in Fig. 4.
1) Each antenna has its own ADC, operating at a low rate andlow resolution. In this way, full (but noisy) informationis available and estimates of can be computedstraightforwardly.
2) During a training phase, a range of beamformersare applied instead of the optimal , resulting
in output sequences , for .Here is a matrix with and is a
vector denoting the beamformer output during thetraining phase. The corresponding correlation statistics of
are computed and the statistics of areinferred, as discussed below. Note that the automatic gaincontrols may have to readjust because the interfering sig-nals will not be suppressed during the estimation phase.
3) A separate APN and bank of low-rate/low-resolutionADCs is used to monitor the inputs, resulting in outputsequences , for . We referto this setup as low resolution beamformers (LRB’s) andthe quantization is typically with 1–2 bit ADCs. This is
VENKATESWARAN AND VAN DER VEEN: ANALOG BEAMFORMING IN MIMO COMMUNICATIONS 4139
quite similar to case 2, except that there is more flexibilityin the number of APN outputs (can be different than )and ADC resolutions. The APN could simply be a set ofswitches, making a selection of the antennas towards alow number of ADCs.
The choice of system architecture depends on various criteria,such as avaliable space for additional hardware, the station-arity of the received signals, and the duration and density ofthe training periods in the desired signal. Regarding the trainingsignal, there are also issues related to acquisition and synchro-nization to the desired signal; we will not discuss this further.
Without loss of generality, we will consider case 3 and dis-cuss how and are inferred. Let be the number ofAPN outputs (dimension of each ). From the LRB outputsequences , we will be able to form esti-mates of , of size , with model
As in , we subsequently stack the columns of each of thesematrices into vectors , with model
where the identity was used.Stacking these vectors results in the model
Assuming has a left inverse, we can estimate the data co-variance matrix using Least Squares2 as .The complexity of this step is in the order of .
The Hermitian property of can be exploited by intro-ducing a vectorization operator “ ” that separatelystacks the real and imaginary components of the upper tri-angular (respectively, strictly upper triangular) part of itsargument. Similar to , , this refinement reduces thecomputational complexity and ensures that the resulting esti-mate is Hermitian.
A necessary condition for the left invertibility of is thatthis is a “tall” matrix, or . Once this condition is sat-isfied, simple designs for the are already sufficient to obtaininvertibility. E.g., for , , , matrices of theform
with distinct and , lead to a full rank . Such lowcomplexity selection matrices have also been used for DOA ap-plications in .
2The paper  proposes to use a weighted Least Squares, but it can be shownthat, for this unparametrized estimate of � , the weight does not change any-thing.
If only switches are used, then (for ) each givesaccess to one cross-correlation entry in . For thereare 6 such entries, and a minimal design is
The vector can be estimated in a similar way from esti-mates of via the model equations
This requires the matrix in the RHS to be tall andfull column rank, which is a milder condition than what we hadfor the estimate of . As mentioned, we need the desired userto be synchronized to the receiver and the receiver must haveknowledge of the training sequence transmitted at the start ofthe packet.
VI. SIMULATION RESULTS
To assess the performance of the proposed algorithms, wehave applied it to a multiuser/antenna setup and computer gen-erated data. We present simulation results that incorporate theimpairments, discrete design and channel parameter estimationas covered in Sections III–V.
The input SNR is the signal to noise power ratio for the de-sired signal and the noise as received at antenna 1; it is thesame for all antennas. The input SIR is the signal to interferencepower ratio for the desired signal and the sum of all interferencesignals as received at antenna 1; it is the same for all antennas.All users transmit QPSK signals, with zero mean and unit vari-ance as assumed in Section II-B and the interferers have equalpowers. The performance indicators are usually as follows:
1) SINR at the first ADC input—a high SINR indicates thatless power is spent in quantizing the interferers for a givenADC resolution;
2) MSE—observed at the output of the digital receiver.All results are obtained by averaging 100 Monte Carlo runs,
each with independent Rayleigh fading channel realizationsand independently generated data signals. Each run transmitsdata packages of size 8192 symbols, as in a WLAN transmis-sion packet. The QMP algorithm we proposed in Table I andSection IV is used to design the APN weights from the trainingsequence. Unless specified otherwise, we used 4 trans-mitters, 4 receive antennas, and 2 ADC receiverchains. The ADC resolution is 10 bits and transmit SIRis 5 dB. The APN is represented by a dictionary with4 bits. In the cases where the receiver is based on estimatedchannel coefficients, these are estimated from a training se-quence of length 256 symbols incorporated in the data packet.
4140 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
Fig. 5. Performance comparison of the APN setup for different � and � �
�� as function of transmit SNR: (a) SINR at the input of first ADC; (b) MSE atthe output of baseband receiver.
A. Finite Sized APN Dictionaries
Fig. 5(a) and (b) shows the SINR at the input of the firstADC and the MSE at the receiver output respectively. Atraining sequence of length 256 is used for the design of theAPN. We show curves for varying dictionary size , andfixed ADC resolution of 10 bits. Consider the SINR plot[Fig. 5(a)], curve 1 corresponds to a case with no APN and
4 ADCs operating with float precision and curves 2–6show the performance of the APN setup for increasing .Comparing curves 1–4, the results show that the introductionof the APN with dictionary size bits improves theSINR at the first ADC input up to a factor of 20 dB. Forincreasing SNR, the performance saturates: it is limited by theresidual interference power. The performance can be furtherimproved by increasing . Consider Fig. 5(b) comparing theMSE at the receiver. For an MSE 0.05, the setup with4 bits and 2 ADCs each with 10 bits (curve 4)performs 2 dB worse than the optimal Wiener beamformerwith ADCs and float precision (curve 1).
Fig. 6. MSE performance comparison at the output of the baseband receiveras a function of transmit SNR (a) for varied � and (b) for various numbers andresolutions of ADCs.
B. Effect of the ADC Resolution
Fig. 6(a) and (b) shows the MSE performance at the receiverfor a similar setup as in Section VI-A, where the APN resolutionis 4 bits. In Fig. 6(a), curve 1 corresponds to a case with
4 ADCs, with float precision and curves 2–5 correspondto 2 ADCs and varying ADC resolution bits.The transmit SIR is 5 dB. We observe that for 6 bits,MSE curves overlap and the finite precision APN leads to anerror floor. In other words, for higher resolution ADCs, the APNresolution is the limiting factor.
Fig. 6(b) gives an idea of ADC power savings with the intro-duction of an APN, for an APN designed with 5 bits andvarying ADC resolution . Comparing curves 2–3 and 4–5, re-spectively, we see that the introduction of an APN with2 ADCs operating with and 6 bits results in a sim-ilar MSE values as that of a receiver without APN, 4ADCs with precision 4 and 6 bits followed by op-timal Wiener beamformer. Since ADC power consumption isrelated to , this suggests that the use of an APN canreduce the ADC power consumption by half.
VENKATESWARAN AND VAN DER VEEN: ANALOG BEAMFORMING IN MIMO COMMUNICATIONS 4141
Fig. 7. MSE performance comparison at the output of the baseband receiverfor different � .
C. Effect of the Number of APN Outputs
Fig. 7 shows the MSE performance at the output of the base-band receiver for a similar setup as in Section VI-A, where thenumber of ADCs is varied, 4–5 bits and ADC reso-lution . Typically, for APN with perfect interferencecancellation, . However, fixed precision APNand interfering users limit the performance gains. From curve 3,we see that even with , the MSE curves lead to an errorfloor and this suggests that the APN with might beill-conditioned. Increasing the APN Resolution to bitsleads to improved MSE performance as is obvious from curves4 and 5. However, to limit the APN circuit size it is suggestedto choose and .
D. Effect of Source Spacing
Fig. 8(a) and (b) shows the SINR and MSE performance asa function of the spacing between two adjacent sources. Thesimulations consider line of sight simulations without multipath,and results are observed for with the desired usertransmitting from an angle say 0 . The MSE is a functionof transmit 20 dB, transmit 5 dB, and angularspacing between the desired user and interferers. We considertwo interferers equidistant from the desired user and in oppositedirections transmitting from angles and .From Fig. 8(a) we see that for 20 , the APN improves theSINR at the first ADC by a factor of 15 dB. In both the cases,we see that the APN performs poorly for angular spacing 5 ,and the performance can be improved by increasing the number
of the antennas.
E. Communication Setup and Channel Estimation With LRB’s
The previous sections have given indications on the improve-ments in SINR, power consumption and MSE for phase shifterbased APN as functions of and . Here we focus onchannel estimation using LRBs as specified in Section V. Weselect the architecture type 3 in Section V and choose 6LRBs with 2 outputs. As specified in that section, switches
are used to estimate .
Fig. 8. Performance comparison for � � � setup as a function of spacingbetween desired user and two interferers: (a) SINR at the input of the first ADC;(b) MSE at the output of the baseband receiver.
Fig. 9(a) and (b) compares the MSE and BER performanceof the fixed precision 4, 2 APN estimated usingLRBs as a function of training lengths. The results are comparedwith the reference 4 ADC case and beamformer designedusing true channel parameters. The ADC resolution is kept at
6 bits.In Fig. 9(a), the curves 1 and 3 show that going from
4 to 2 ADCs leads to performance degradation of 2 dBat MSE of 0.05. For the sake of completeness, we also show theBER performance between the transmitted and received QPSKsignals and see that curves 1 and 3 in Fig. 9(b) show that goingfrom to 2 ADCs APN leads to performancedegradation of 2 dB at BER .
Considering that the above setup reduces the interferer con-tributions at the ADCs, these results suggest that architectureswith reduced RF chains, limited by RF imperfections can per-form close to theoretical MIMO while consuming a fraction ofthe power.
VII. CONCLUDING REMARKS
In this paper, we have proposed a MIMO receiver employingan analog preprocessing network (APN) or multichannel
4142 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 8, AUGUST 2010
Fig. 9. Performance comparison, when � and � are estimated for 2-bitLRBs and varied channel lengths: (a) MSE; (b) BER.
beamformer in the RF domain, followed by a digital beam-former in baseband. The prime advantage of this architectureis that it reduces the number of antenna elements to a smallernumber of mixers and ADC chains. Further, it can reduce theinterference at the inputs of the ADC, so that less dynamicrange and fewer bits are required. Overall, significant powersavings are possible.
An optimal preprocessor to minimize the MSE and maximizethe desired user SQNR at the digital baseband was derived. Itwas shown that, if the APN quantization is very fine, it is suffi-cient to consider only 1 analog beamforming output. Inpractice the quantization is poor, and a larger number of out-puts is required such that the cross-correlation vector iswell approximated. Further research is needed in the followingdirections.
• In practice, the APN coefficients have poor accuracy; im-plementation errors of up to 7% of phase have been re-ported . This affects both the channel estimation andthe APN design. How can this effect be modeled and in-corporated into the design?
• Initially, we are not synchronized to the source of interest.It may then be complicated to estimate and design theAPN; subsequently, the interference may overwhelm the
ADCs and make acquisition impossible. What is a goodinitialization strategy?
An alternative to using an APN to reduce ADC power, isto exploit spatial and temporal oversampling with a predictiveSigma-Delta ADC as in .
APN DESIGN USING CROSS SPECTRAL PROJECTIONS
To obtain some intuition on the APN design problem [P0] asspecified in Section III-A, we first consider a few suboptimaltechniques before deriving a closed form APN in Section III-B.Introduce an eigenvalue decomposition of :
where is an unitary matrix con-taining the eigenvectors, and is a diagonal matrix containingthe eigenvalues of , sorted from large to small. can bechosen as the dominant eigenvectors, corresponding to the
largest eigenvalues, . This is somewhatsimilar to one of the approaches proposed in . In this way,computation of retains the components withthe largest energy and drops the components with less power.However, this does not distinguish between desired and inter-fering users. If the interferers are strong, the desired user couldbe projected out.
This is avoided in the following design, based on “cross spec-tral projections.” Instead of selecting dominant eigenvec-tors, selecting the eigenvectors that contain a large correlationwith the desired user array response given by results in abetter approximation. This approach is similar to one techniquein  where the authors sort the basis vectors based on thecross spectral norm, defined as
More precisely, let with . Thenwe can write
and obtain the energy of the output signal
where and is the -th eigenvalue in. Selecting the columns of corresponding to the largest
(weighted) cross spectral norm terms, i.e., the largest termsamong
would lead to the “best” representation of the desireduser signal, in the sense of maximizing the output energy of thedesired user.
Collect the selected columns of into a matrix , andlikewise for . Although it seems natural to chooseand , the above technique does in fact not prescribe apartitioning of into and . Moreover, by selecting columns
VENKATESWARAN AND VAN DER VEEN: ANALOG BEAMFORMING IN MIMO COMMUNICATIONS 4143
of we have also limited our choice: is projected on asubspace of eigenvectors. This is not necessarily optimal.
It will be shown later in Lemma 1 that the APN leading toMMSE has to satisfy the condition .Choosing equal to the dominant eigenvectors of isoptimal if is in this subspace, which occurs only if there areat most sources in white noise. This also implies that, forthe same scenario, the solution from cross spectral projections,where consists of eigenvectors maximizing the crossspectral norm would be optimal only if the dominant eigen-vectors are selected.
The authors would like to thank Prof. J.P. Linnartz,Prof. P. Baltus, and J. H. C. van den Heuvel for stimulatingdiscussions especially for detailing the possibility of usingpassive weighting elements in RF, and Prof. A. Leshem for hiscontribution to the problem formulation.
 V. Venkateswaran and A. J. van der Veen, “Optimal phase shifter de-sign to cancel RF interference in multi-antenna systems,” in In Proc.Int. Conf. Acoustics, Speech, Signal Processing (ICASSP), Dallas, TX,Mar. 2010.
 D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, “From theoryto practice: An overview of MIMO space-time coded wireless sys-tems,” IEEE J. Sel. Areas Commun., vol. 21, pp. 281–302, Mar. 2003.
 R. Walden, “Analog to digital converter survey and analysis,” IEEE J.Sel. Areas Commun., vol. 17, pp. 539–550, Apr. 1999.
 B. Murmann, “Digitally assisted analog circuits,” IEEE Micro, vol. 26,no. 2, pp. 38–47, Feb. 2006.
 S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,”IEEE Commun. Mag., vol. 42, no. 10, pp. 68–73, Oct. 2004.
 S. Denno and T. Ohira, “Modified constant modulus algorithm for DSPadaptive antennas with microwave analog beamforming,” IEEE Trans.Antennas Propag., vol. 50, no. 6, pp. 850–857, Jun. 2002.
 P. Baltus and R. Dekker, “Optimizing RF front ends for low power,”Proc. IEEE, vol. 88, no. 10, pp. 1546–1559, Jan. 2000.
 R. Eickhoff, R. Kraemer, I. Santamaria, and L. Gonzalez, “Developingenergy-efficient MIMO radios,” IEEE Veh. Technol. Mag., vol. 4, no.1, pp. 34–41, Mar. 2009.
 A. Hajimiri, H. Hashemi, A. Natarajan, X. Guang, and A. Babakhani,“Integrated phased arrays in silicon,” Proc. IEEE, vol. 93, no. 9, pp.1637–1655, Sep. 2005.
 X. Zhang, A. F. Molisch, and S. Y. Kung, “Variable phase shift basedRF baseband codesign for MIMO antenna selection,” IEEE Trans.Signal Process., vol. 53, no. 11, pp. 4091–4103, Nov. 2005.
 M. D. Zoltowski, S. D. Silverstein, and C. P. Mathews, “Beamspaceroot MUSIC for minimum redundancy linear arrays,” IEEE Trans.Signal Process., vol. 41, no. 1993, pp. 2502–2507, Jul. 1993.
 B. D. van Veen and R. Roberts, “Partially adaptive beamformer designvia output power minimization,” IEEE Trans. Signal Process., vol. 11,pp. 1524–1532, Nov. 1987.
 J. S. Goldstein, I. S. Reed, and L. L. Scharf, “A multistage representa-tion of the Wiener filter based on orthogonal projections,” IEEE Trans.Inf. Theory, vol. 44, no. 7, pp. 2943–2959, Nov. 1998.
 J. Sheinvald and M. Wax, “Direction finding with fewer receivers viatime varying preprocessing,” IEEE Trans. Signal Process., vol. 47, no.1, pp. 2–9, Jan. 1999.
 S. T. Smith, “Optimum phase-only adaptive nulling,” IEEE Trans.Signal Process., vol. 47, no. 7, pp. 1835–1843, Jul. 1999.
 G. M. Kautz, “Phase-only shaped beam synthesis via technique of ap-proximated beamaddition,” IEEE Trans. Antennas Propag., vol. 47, no.5, pp. 887–894, May 1999.
 S. Mallat and Z. Zhang, “Matching pursuits with time-frequency dic-tionaries,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3415,Dec. 1993.
 V. K. Goyal, M. Vetterli, and N. T. Thao, “Quantized overcompleteexpansions in � : Analysis, synthesis and algorithms,” IEEE Trans.Inf. Theory, vol. 44, no. 1, pp. 16–31, Jan. 1998.
 L. L. Scharf, Statistical Signal Processing. Reading, MA: Addison-Wesley, 1991.
 A. Gersho and R. Gray, Vector Quantization and Signal Compres-sion. Norwell, MA: Kluwer, 1994.
 D. Tabrikian and A. Faizakov, “Optimal preprocessing for source local-ization by fewer receivers than sensors,” in Proc. 11th IEEE StatisticalProcessing Workshop, Aug. 6–8, 2001, pp. 213–216.
 A. J. van der Veen and A. Paulraj, “An analytical constant modulusalgorithm,” IEEE Trans. Signal Process., vol. 44, no. 5, pp. 1136–1155,May 1996.
 N. Fourikis, Phased Array-Based Systems and Applications. NewYork: Wiley, 1997.
 V. Venkateswaran and A. J. van der Veen, “Multichannel Sigma-DeltaADCs with integrated feedback beamformers to cancel interferencein MIMO communications,” IEEE Trans. Signal Process., Apr. 2010,submitted for publication.
 J. S. Goldstein and I. S. Reed, “Reduced rank adaptive filtering,” IEEETrans. Signal Process., vol. 45, no. 2, pp. 492–496, Feb. 1997.
Vijay Venkateswaran received the M.S. degree in electrical engineering fromthe University of Arizona, Tempe, in 2003. He is currently working towards thePh.D. degree from the Delft University of Technology, Delft, The Netherlands.
From 2003 to 2005, he worked as a member of technical staff at Sony Cor-poration, Tokyo, Japan, and in 2002 as an intern at Seagate LLC. His researchinterests are in the general area of signal processing advancements in wirelesscommunications and data storage, and, in particular, joint RF-baseband inter-ference cancellation techniques.
Alle-Jan van der Veen (F’05) was born in TheNetherlands in 1966. He received the Ph.D. degree(cum laude) from Delft University of Technology(TU Delft), The Netherlands, in 1993.
Throughout 1994, he was a postdoctoral scholar atStanford University, Stanford, CA. Currently, he is aFull Professor in signal processing at TU Delft. Hisresearch interests are in the general area of systemtheory applied to signal processing, and in particularalgebraic methods for array signal processing, withapplications to wireless communications and radio
astronomy.Dr. van der Veen is the recipient of a 1994 and a 1997 IEEE Signal Pro-
cessing Society (SPS) Young Author Paper Award. He was an Associate Editorfor IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1998 to 2001, Chairmanof IEEE SPS Signal Processing for the Communications Technical Committeefrom 2002 to 2004, a Member-at-Large of the Board of Governors of the IEEESPS, Editor-in-Chief of the IEEE Signal Processing Letters from 2002 to 2005,and Editor-in-Chief of the IEEE TRANSACTIONS ON SIGNAL PROCESSING. Hecurrently is Technical Co-Chair of the International Conference on the Interna-tional Conference on Acoustics, Speech and Signal Processing (ICASSP) 2011in Prague, a Member of the IEEE SPS Awards Board and Fellow ReferenceCommittee, and the IEEE TAB Periodicals Review and Advisory Committee.