IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 2, FEBRUARY 2014 325

Matrix Approximation Based Design of a CombinedAnalog and Digital Beamformer in Frequency-Selective Fading Channels

Seung Joon Lee, Senior Member, IEEE

Abstract—In this letter, a combined analog and digital beamform-ing system is considered in frequency-selective fading channels.Analog beamforming matrices (mapping between RF chainsand antennas) should be constant over frequency, while digitalbeamforming matrices (mapping between spatial data streamsand RF chains) can be adapted to frequency. More RF chainsthan the number of spatial data streams are employed to enablea combined analog and digital beamformer to be adapted tofrequency. A novel method of designing analog beamformingmatrices is proposed by using the Eckart-Young-Mirsky matrixapproximation theorem. The proposed method is optimal ina sense that for a given number of RF chains, the weightedsum of the squared distances between the combined analog anddigital beamforming vectors (comprising a beamforming matrix)derived by the proposed method and the ideal beamformingvectors (singular vectors of the channel matrix at each frequency)is minimized, where the weights are the singular values corre-sponding to the singular vectors. The proposed beamformer isshown to significantly outperform a conventional one.

Index Terms—60 GHz, beamforming, matrix approximationtheorem, millimeter-wave, MIMO, spatial multiplexing.

I. INTRODUCTION

M ILLIMETER (mm)-wave communication technologieshave attracted attention for high bandwidth short range

wireless and also mobile data services [1], [2]. In mm-wave communication systems, antenna array has played animportant role in reducing the size and power consumptionof the communication devices, as easily seen in almost allmm-wave communication standards [3]–[6]. Also, a techniqueof beamforming combined with spatial multiplexing has beenemployed in [6].

If there is a non-blocked line-of-sight path between a trans-mitter (TX) and a receiver (RX) and we are only concernedwith a single spatial stream (without spatial multiplexing), theoptimal beamforming vector is nearly frequency-flat over arange of a few GHz bandwidth. Otherwise, on the other hand,the optimal beamforming vector (for a single spatial stream)or matrix (for multiple spatial streams) may be frequency-selective, as in [7] and will be shown in this letter.

Digital beamforming (or precoding and postcoding) is dif-ferentiated from analog beamforming in that the former isperformed in a digital baseband block while the latter is per-formed in an analog RF block. The antenna array relies on theanalog beamforming. Subcarrier-dependent MIMO precodersemployed in mobile WiMAX [8] and LTE [9] imply the digitalbeamformers.

Manuscript received October 26, 2013. The associate editor coordinatingthe review of this letter and approving it for publication was A. Conti.

S. J. Lee is with the Department of Electronics Engineering, KangwonNational University, Chuncheon 200-701, Korea (e-mail: s.j.lee@ieee.org).

Digital Object Identifier 10.1109/LCOMM.2014.010314.132404

Combination of digital and analog beamformers has alsobeen considered in [6, Figs. 19 & 20] and [10]–[12]. In [10]and [11], a hybrid adaptive antenna array structure whereanalog subarrays are followed by a digital beamformer wasproposed to overcome the large complexity of a fully digitalbeamformer. There the analog and digital beamforming vec-tors were determined based on an estimated angle-of-arrivalinformation. The structure of the hybrid antenna array in [10],[11] corresponds to the split antenna architecture of [6, Fig.19], where each antenna belongs to only one analog RF chain.

The combined analog and digital beamformer in [12] isa kind of the shared antenna architecture of [6, Fig. 20],where each antenna belongs to all RF chains. The problemconsidered for designing analog and digital beamformingvectors in [12] is different from that of [10] and [11]. In afrequency-selective fading channel, the optimal beamformingvector depends on frequency (or a subcarrier in OFDM),but the analog beamformer cannot vary with frequency. Thusmultiple (instead of one) RF chains were employed in [12]so that the combined analog and digital beamformer could beadapted to frequency. The performance of such adaptivity isexpected to depend on the number of RF chains. However,the allowable number of RF chains is limited due to theimplementation complexity.

In this letter, we also consider the same problem as in[12], but we propose a new beamformer outperforming that of[12]. The rest of this letter is organized as follows. In SectionII, we describe the model for a combined analog and digitalbeamforming system. A new beamforming scheme is proposedin Section III. The performance of the proposed beamformeris evaluated in Section IV. Conclusion is drawn in Section V.

II. A MODEL FOR THE COMBINED ANALOG AND DIGITAL

BEAMFORMING SYSTEM

A combined analog and digital beamformer under consid-eration is shown in Fig. 1. There are MTX TX and MRX RXantennas. Also, there are NTX TX and NRX RX RF chains.The number of spatial data streams (spatial multiplexing gain)to be transmitted is denoted by K .

The precoding (TX digital beamforming) matrix represent-ing a mapping from K spatial data streams to NTX TX RFchains is denoted by PTX of dimension NTX × K and thepostcoding (RX digital beamforming) matrix representing amapping from NRX RX RF chains to K received spatial datastreams is denoted by P†

RX of dimension K × NRX. Let thechannel matrix be H (of dimension MRX ×MTX) and the TXand RX analog beamforming matrices be WTX (dimensionMTX×NTX) and W†

RX (dimension NRX×MRX), respectively.If we let the noise signal vector at the RX antennas be

denoted by n (dimension MRX ×1) and the transmitted signal

1089-7798/14$31.00 c© 2014 IEEE

326 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 2, FEBRUARY 2014

Pre-codingPTX[s]

(NTX×K)

TX analog beamforming

WTX

(MTX×NTX)

Channel

H[s](MRX×MTX)

x[s]

n[s]

y[s]†RX

RX RX

RX analogbeamforming

N MW †

RX

RX

Post-coding

K NP

Fig. 1. Transmission structure for combined analog beamforming and digital pre/post-coding.

for K data streams by x, the signal y for the received K datastreams is written as

y[s] = P†RX[s] ·W†

RX (H[s] ·WTX ·PTX[s] · x[s] + n[s]) (1)

where s (s= 0, 1,· · ·, S−1) denotes a frequency (subcarrierin OFDM) index. In (1), frequency-dependent variables areindicated by attaching them with ‘[s]’. It is noted that theanalog beamformers cannot be frequency-depedent, while theprecoder and postcoder can be. When both matrices P†

RX[s]

and W†RX are comprised of orthonormal rows, the transmission

system of (1) has the channel capacity given by (2) shown atthe bottom of the page, where Es/N0 denotes the symbolenergy over one-sided noise power spectral density [13].

If the analog beamforming matrices WTX and WRX areonce obtained, the frequency-dependent precoding and post-coding matrices PTX[s] and PRX[s] maximizing the capacityof (2) can be derived by using the result of [13]. Thus achallenging issue is how to design the frequency-independentmatrices WTX and WRX in order for the combined analogand digital beamformer to perform well in frequency-selectivefading channels.

In [12] where the authors tried to maximize the average(averaged over frequency s) SNR in a single spatial datastream transmission, they derived WTX and WRX that max-imize the average sum of the all eigenvalues of the matrix(W†

RXH[s]WTX) · (W†RXH[s]WTX)

†, although the averagemost dominant eigenvalue of the matrix should have beenmaximized for obtaining the maximal average SNR. They didso because the latter is mathematically intractable.

Throughout this letter, SL [A]1:n is defined as a matrixwhose columns are comprised of the most dominant n leftsingular vectors of matrix A and SR [A]1:n is defined as amatrix whose columns are comprised of the most dominantn right singular vectors of A. If a column is placed to theleft of another column within each of those matrices, then theformer is considered to be more dominant than the latter.

III. PROPOSED SCHEME

First we consider imaginary, ideal analog beamformingmatrices which can be adaptive to frequency. Also, assumethat NTX = NRX = K and the ideal analog beamformingmatrices are given by

W(ideal)TX [s] = SR [H[s]]1:K (3a)

W(ideal)RX [s] = SL [H[s]]1:K . (3b)

Then the capacity is maximized when the precoding matrixperforms the water-filling operation [13]. The postcodingmatrix may be just the identity matrix. The capacity of thesystem with the above imaginary, ideal analog beamformingmatrices and the water-filling precoding matrix is denoted asC(ideal), henceforth.

Now let’s come back to a real situation where the ana-log beamforming matrices should be frequency-independent.Even in this case, if NTX = NRX = SK , then WTX =[W(ideal)

TX [0] W(ideal)TX [1] · · · W(ideal)

TX [S − 1]]

and WRX =[W(ideal)

RX [0] W(ideal)RX [1] · · · W(ideal)

RX [S − 1]]

can achieve the

same C(ideal) by choosing PTX[s] and PTX[s] such that the di-agonal entries of the ((s− 1)K)-th to ((s− 1)K +K − 1))-th columns are set to non-zeros and all the other entries arezero in the matrices, where the non-zero values in PTX[s]are derived from the water-filling principle and the non-zerovalues in PRX[s] are set to 1.

However, a large number NTX < SK or NRX < SK ofRF chains are not feasible. So we should consider NTX <SK and NRX < SK where C(ideal) may not be achieved. Wetry to design analog beamforming matrices WTX and WRX

such that the resulting WTXP′TX[s] and WRXPRX[s] look as

similar as possible to W(ideal)TX [s] and W(ideal)

RX [s], respectively,where P′

TX[s] is related to PTX[s] by PTX[s] = P′TX[s]D,

and where D is a diagonal matrix corresponding to the water-filling operation and the non-zero columns of P′

TX[s] have aunit norm.

The optimization problem for the analog TX beamformingis formulated as

QTX = arg minQ,rank(Q)=NTX

∥∥∥W(Σ)TX −Q

∥∥∥ (4)

where ‖ ‖ denotes the Frobenius norm, W(Σ)TX is given by

(5) shown on the next page, λk,s denotes the k-th singularvalue of H[s], and QTX is related to the optimal WTX byW(opt)

TX = SL [QTX]1:NTX. In (5), it is noted that the columns

of SR [H[s]]1:K are weighted by their own correspondingsingular values, in order that a singular vector of H[s] with alarger singular value makes more effect on the solution of theoptimization problem of (4).

From the Eckart-Young-Mirsky matrix approximation the-

C[s] = log2

∣∣∣∣IK×K +Es/No

K

(P†

RX[s]W†RXH[s]WTXPTX[s]

)(P†

RX[s]W†RXH[s]WTXPTX[s]

)†∣∣∣∣ [bits/symbol] (2)

LEE: MATRIX APPROXIMATION BASED DESIGN OF A COMBINED ANALOG AND DIGITAL BEAMFORMER IN FREQUENCY-SELECTIVE FADING . . . 327

0

1

2

3

4

5

6

7

8

9

10

2 4 6 8 10 12

Ave

rage

cap

acity

[bits

/sym

bol]

Number of TX or RX chains,

IdealConventional

Proposed

NT = NR

K=1 K=2

Fig. 2. Capacity comparison for K = 1 and 2 with SNR = 0 dB in a 6independently faded subcarrier channel.

orem [15, p. 31], the optimization problem (4) is solved as

QTX = W(Σ)TX YTXY

†TX (6a)

YTX = SR

[W

(Σ)TX

]1:NTX

. (6b)

In a similar way, the optimal WRX is obtained as

W(opt)RX = SL [QRX]1:NRX

(7a)

QRX = W(Σ)RX YRXY

†RX (7b)

YRX = SR

[W

(Σ)RX

]1:NRX

(7c)

where W(Σ)RX is given by (8) shown at the bottom of the page.

IV. PERFORMANCE EVALUATION WITH NUMERICAL

EXAMPLES

We present the performance simulation results for twocases: a simple imaginary channel model and a practicalchannel model. We consider MTX = MRX = 16 for bothcases.

In the first model, we assume there are 6 subcarriers(S = 6), all the channel gains experience Rayleigh fading, andthe channel gains are mutually independent between differentsubcarriers or different antennas. Fig. 2 shows the comparedcapacity with varying number of TX and RX RF chains atSNR=0 dB for K = 1 and 2, where we see that the proposedscheme has large gains over the conventional one of [12]1.

1 In [12] only one spatial data stream was considered, but the same WTXand WRX derived for a single spatial data stream is used for two spatialstreams in Fig. 2.

Rx array

10 d m

Tx array

2y

x

y

1y

1x

2x

Fig. 3. Floor plan of a conference room STA-STA channel model.

0

1

2

3

4

5

6

7

8

59.6 59.8 60 60.2 60.4

Cap

acity

[bits

/sym

bol]

Frequency [GHz]

(a)(b)(c)

(d)(e)

Fig. 4. Comparison of frequency-selective capacities for a single spatialstream in the conference room STA-STA channel model: (a) an ideal beam-former, (b) a beamformer with NT = NR = 1 optimized only at 60 GHz,(c) a beamformer with NT =NR =1 optimized only at 60.1 GHz, (d) theproposed beamformer with NT =NR=1, and (e) the proposed beamformerwith NT =NR=2.

As expected, the proposed scheme has the same capacity asthe ideal system when NTX = NRX = SK (= 6 and 12 forK = 1 and 2, respectively), while the conventional one doesnot.

Next we consider a practical channel model based on the 60GHz conference room STA-STA model of [16]. In this model,we assume that the TX and RX antenna arrays are located asshown in Fig. 3 with x1 = 4.5 m and x2 = 5.5 m. We considervertical polarization but we don’t consider reflection fromceiling since its effect is minor in vertical polarization [17]. Wealso consider the antenna spacing of 3.54 mm2 and the line-

2This value is chosen because the frequency selectivity of the channel looksprominent at this value.

W(Σ)TX =

[SR[H[0]]1:K diag (λ1,0, · · · , λK,0) · · · SR[H[S − 1]]1:K diag (λ1,S−1, · · · , λK,S−1)

](5)

W(Σ)RX =

[SL[H[0]]1:K diag (λ1,0, · · · , λK,0) · · · SL[H[S − 1]]1:K diag (λ1,S−1, · · · , λK,S−1)

](8)

328 IEEE COMMUNICATIONS LETTERS, VOL. 18, NO. 2, FEBRUARY 2014

(a)(b)(c)

(d)(e)

0

2

4

6

8

10

59.6 59.8 60 60.2 60.4

Cap

acity

[bits

/sym

bol]

Frequency [GHz]

Fig. 5. Comparison of frequency-selective capacities for two spatial streamsin the conference room STA-STA channel model: (a) an ideal beamformer,(b) a beamformer with NT = NR = 2 optimized only at 60 GHz, (c) abeamformer with NT =NR=2 optimized only at 60.1 GHz, (d) the proposedbeamformer with NT = NR = 2, and (e) the proposed beamformer withNT =NR=4.

4.6

4.8

5

5.2

5.4

5.6

5.8

6

6.2

6.4

6.6

1 2 3 4 5 6 7 8

Ave

rage

cap

acity

[bits

/sym

bol]

Number of TX or RX chains,

IdealProposed

K=2

IdealProposed

K=1

NT = NR

Fig. 6. Average capacity with varying number of RF chains in the conferenceroom STA-STA channel model.

of-sight (LoS) path blocked3. We evaluate the capacity overthe frequency range of 59.5 to 60.5 GHz. For the proposedscheme, we consider H[s] = H(f = 59.5 + 0.1s GHz),s = 0, 1, · · · , 9, are available, where H(f) denotes the channelstate information at frequency f .4

Fig. 4 shows the compared capacities for a single spatialdata stream when y1=5.5 m and y2=4.5 m. There the graph(a) denotes the capacity of the ideal system, the graphs (b) and(c) denote the capacities of the systems with NTX =NRX =1where WTX and WRX are optimized at frequency 60 GHz

3This is assumed since the channel with the LoS path for a single spatialstream is always nearly frequency-flat.

4However, {H[s], s=0, 1, · · · , 9} are expected not to be directly availablein practice. Instead, other learning methods will be necessary for obtaining(approximate) W

(Σ)TX and W

(Σ)RX , which is beyond the scope of this letter.

and 60.1 GHz, respectively, but they are used over the entirerange of frequencies, and the graphs (d) and (e) denote thecapacities of the proposed scheme with NTX =NRX = 1 and2, respectively. Fig. 5 shows the frequency-selective capacityfor K = 2 when y1 = 5.3 m and y2 = 4.7 m. Fig. 6 showsthe average (averaged over the range from 59.5 to 60.5 GHz)capacity of the proposed scheme with a varying number ofRF chains.

V. CONCLUSION

A novel combined analog and digital beamformer has beenproposed for a frequency-selective fading channel. Beamform-ing vectors derived by the proposed scheme are optimizedin a sense that the sum of the squared distances of themto the singular vectors of the channel matrix at each fre-quency weighted by their singular values is minimized. Thecapacity of the proposed beamformer has been shown to besignificantly larger than the conventional one by computersimulations, where the IEEE802.11ad 60 GHz WLAN channelmodel was considered as well as a simple channel modelcomprising 6 independent subcarriers.

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