Interpolation Based Transmit Beamforming for MIMO ... - CiteSeerX

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO. XX, XXX 200X

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Interpolation Based Transmit Beamforming for MIMO-OFDM with Limited Feedback Jihoon Choi, Member, IEEE, and Robert. W. Heath, Jr., Member, IEEE

Abstract— Transmit beamforming and receive combining are simple methods for exploiting spatial diversity in multipleinput multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) system. Optimal beamforming requires channel state information in the form of the beamforming vectors for each OFDM subcarrier. This paper proposes to a limited feedback architecture that combines beamforming vector quantization and smart vector interpolation. In the proposed system, the receiver sends a fraction of information about the optimal beamforming vectors to the transmitter and the transmitter computes the beamforming vectors for all subcarriers through interpolation. A new spherical interpolator is developed that exploits parameters for phase rotation, to satisfy the phase invariance and unit norm properties of the transmitted beamforming vectors. The beamforming vectors and phase parameters are quantized at the receiver and the quantized information is provided to the transmitter. The proposed quantization system provides only a moderate increase in complexity versus over comparable approaches. Numerical simulations show that the proposed scheme performs better than existing diversity techniques with the same feedback data rate.

Index Terms — MIMO-OFDM, transmit beamforming, spherical interpolation, antenna diversity, mutual information. I. I NTRODUCTION Multiple-input multiple-output (MIMO) systems, which use multiple antennas at both transmitter and receiver, provide spatial diversity that can be used to mitigate signal-level fluctuations in fading channels [1]–[3]. In narrowband channels, MIMO systems can provide a diversity advantage in proportion to the product of the number of transmit and receive antennas. When the channel is unknown to the transmitter, diversity can be obtained by using space-time codes [1]–[3]. When channel state information (CSI) is available at the transmitter, however, diversity can be obtained using a simple approach known as transmit beamforming and receive combining [4]–[11]. Compared with space-time codes, beamforming and combining achieves the same diversity order as well as additional array gain, thus it can significantly improve system performance. This approach, however, requires knowledge of the transmit beamforming vector at the transmitter. When the uplink and downlink channels are not reciprocal (as in

This material is based in part upon work supported by the Texas Advanced Technology Program under Grant No. 003658-0614-2001, the National Instruments Foundation, the Samsung Advanced Institute of Technology, and the Post-Doctoral Fellowship Program of Korea Science and Engineering Foundation (KOSEF). J. Choi is with Media Lab. 1, Telecommunication Network Business, Samsung Electronics, 416, Yeongtong-gu, Suwon-city, Gyeonggi-do, Korea 442-600, (e-mail: [email protected]). R. W. Heath, Jr. is with Dept. of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712 USA, (e-mail: [email protected]).

a frequency division duplexing system), this necessitates that the receiver informs the transmitter about the desired transmit beamforming vector through a feedback control channel. The beamforming techniques proposed for narrowband channels can be easily extended to frequency selective channels by employing orthogonal frequency division multiplexing (OFDM). The combination of MIMO and OFDM, known as MIMO-OFDM, converts a broadband MIMO channel into a series of parallel narrowband MIMO channels, one for each OFDM subcarrier [12]–[14]. Transmit beamforming with receive combining can be performed independently for each subcarrier of MIMO-OFDM. In non-reciprocal channels, this requires that the MIMO-OFDM receiver computes and sends back to the transmitter the optimal beamforming vector for every active subcarrier. Practically, the feedback rate can be managed by using limited feedback techniques where the beamforming vectors are quantized using a beamforming codebook designed for narrowband MIMO channels (see e.g., [15]–[18]). Unfortunately, even with quantization, the feedback requirements generally grow in proportion to the number of active subcarriers. In an attempt to reduce the feedback, in this paper we consider a new approach to transmit beamforming that combines quantized beamforming with interpolation of the beamforming vectors. The idea of our approach is to exploit the correlation between the beamforming vectors at neighboring subcarriers. In OFDM, the length of the cyclic prefix (CP) is designed to be much less than the number of subcarriers to increase spectral efficiency (see e.g., [19], [20]). Accordingly, the subchannels obtained from the discrete Fourier transform (DFT) of the sampled time-domain channel are significantly correlated. As a result, in MIMO-OFDM the beamforming vectors determined by the subchannels are also substantially correlated. Using this fact, we propose a new beamforming scheme that sends back only a fraction of beamforming vectors to the transmitter and reconstructs the beamforming vectors for all subcarriers through interpolation at the transmitter. We assume that the transmit power is equally assigned to all subcarriers. Power allocation can be included in the system model with additional feedback but we defer this to future work. Since the beamforming vectors have unit norm by the transmit power constraint, they lie on the unit sphere. Consequently it is natural to look toward spherical interpolators to perform interpolation between beamforming vectors [21]–[25]. The nonuniqueness of optimal beamforming vectors [17], however, compromises the performance of standard spherical interpolators. In this paper, we propose a new spherical interpolator based on a modification of Watson’s interpolator [23]. To resolve the ambiguities in the optimal beamforming vector, our


interpolator employs a parameter for phase rotation, denoted as ejθ , which is chosen by optimizing a cost function based on the channel gain or the mutual information. Because the optimization is difficult to solve in closed form, we solve for the phase rotation by performing either a numerical grid search or by solving a closed form solution with an approximate cost function. To enable practical implementation with limited feedback, we combine our new spherical interpolator with beamforming vector quantization as proposed in [15]–[18]. With our approach, the receiver quantizes the beamforming vectors for a fraction of the subcarriers by selecting the best element from the codebook. Using the quantized vectors, the receiver then selects a near-optimal quantized phase θ for each quantized vector based on the phase cost function. After being informed about the quantized beamforming vectors and phases, the transmitter uses the proposed spherical interpolator to reconstruct the beamforming vectors for all the subcarriers. We analyze the computational complexity and feedback requirements for the proposed method and compare with several related diversity techniques. Computer simulations show that the proposed scheme outperforms existing diversity techniques with comparable complexity and feedback bits and performs close to the ideal beamforming with full feedback information. Compared with prior work that deals with unquantized partial channel information [26]–[29], our approach focuses on methods for reducing feedback requirements by subsampling in frequency and quantizing separately the optimal beamforming vectors computed from the instantaneous channel state (versus the first or second order statistics of the channel) at each subcarrier. This paper is organized as follows. Section II reviews MIMO-OFDM communication with beamforming and combining. The proposed interpolation based beamforming scheme is presented in Section III, while the complexity and feedback requirements for the proposed scheme are analyzed in Section IV. Through numerical simulations, the proposed beamforming method is compared with existing diversity techniques in terms of bit error rate (BER) and mutual information in Section V. Finally, we provide some conclusions in Section VI.

Fig. 1. Block diagram of a MIMO-OFDM system with Mt transmit antennas, Mr receive antennas, and N subcarriers. The feedback channel conveys a fraction of information about {wi (k); 1 ≤ i ≤ Mt , 1 ≤ k ≤ N }, the coefficient of the beamforming vector for subcarrier k on the i-th antenna.

where n(k) is the Mr -dimensional noise vector whose entries have the independent and identically distributed (i.i.d.) complex Gaussian distribution with zero mean and variance N0 . We assume that the power is allocated equally across all subcarriers thus E[|s(k)|2 ] = Es is a constant and kw(k)k = 1 where k(·)k means 2-norm of (·)) to maintain the overall power constraint. Since the signal model in (1) is identical to that of a narrowband MIMO system, w(k) and z(k) can be chosen to maximize the signal to noise ratio (SNR) using the results on beamforming and combining for narrowband MIMO systems [4], [6]. Without loss of generality we can fix kz(k)k = 1. Then the SNR for subcarrier k can be written as γ(k) =

Es |zH (k)H(k)w(k)|2 Es = Γ(k) N0 N0

A MIMO-OFDM system with transmit beamforming and receive combining, using Mt transmit antennas and Mr receive antennas, and N subcarriers is illustrated in Fig. 1. At the transmitter, the k − th subcarrier modulates the symbol s(k) using the beamforming vector w(k) = [w1 (k), w2 (k), · · · , wMt (k)]T and launches the resulting signal into the propagation environment. Assuming that the sampled impulse response of the channel is shorter than the CP, the channel for the k − th subcarrier after the DFT can be described by a Mr -by-Mt channel matrix H(k) whose entries represent the complex channel gains experienced by subcarrier k. After processing with the combining vector z(k) = [z1 (k), z2 (k), · · · , zMr (k)]T , the combined signal at subcarrier k can be expressed as (1)

(2)

where Γ(k) = |zH (k)H(k)w(k)|2 is the effective channel gain. The SNR maximizing solution request choosing the transmit beamforming vector w(k) and receive combining vector z(k) based on H(k) to maximize the SNR of the subcarrier k in (2). For any w(k), under the unit norm constraint on z(k), it is possible to show that the SNR maximizing solution uses maximum ratio combining (MRC) with z(k) =

II. S YSTEM OVERVIEW

r(k) = zH (k){H(k)w(k)s(k) + n(k)}, 1 ≤ k ≤ N

2

H(k)w(k) . kH(k)w(k)k

(3)

Since the combining is performed digitally, it is reasonable to use optimum MRC processing instead of a suboptimum technique such as antenna selection. Choosing w(k), however, depends in part on how much CSI is available at the transmitter. For example, selection diversity transmission chooses w(k) from the set of the columns of the Mt -by-Mt identity matrix IMt thus requiring only log2 Mt bits of information of the CSI per subcarrier at the transmitter. Maximum ration transmission (MRT), on the other hand, chooses w(k) as the right singular vector of H(k) corresponding to the largest singular vector of H(k) [6], [8] for every k. In Rayleigh fading channels, both approaches achieve full diversity but MRT also obtains the maximum amount of array gain. Unfortunately, MRT’s improved performance comes at the expense of more extensive CSI at the transmitter in the form of the complete channel


impulse responses or equivalently the optimum beamforming vectors {w(k), k = 1, 2, · · · , N }. In this paper, we consider a burst-mode communication system where data is transmitted in frames that consist of multiple MIMO-OFDM symbols from the system described in Fig. 1. We assume that the channel is fixed in a frame but varies randomly between frames (commonly known as the block fading model). Further we assume that full CSI is not available to the transmitter, but there exists a low-rate, error-free, zero-delay feedback link from the receiver back to the transmitter. To accommodate the low-rate requirement, we assume that the number of feedback bits per frame is limited to a constant. The receiver uses CSI (assumed perfect in this paper) obtained from a training phase at the beginning of the frame to determine the feedback bits corresponding to the set of beamforming vectors {w(k), k = 1, 2, · · · , N } for all OFDM symbols in the frame. To respect the feedback constraints, in this paper we develop a quantization strategy for transmit beamforming in MIMOOFDM communication systems. A conventional vector quantization approach would be to quantize the impulse response of the channel in the time domain. The amount of feedback, though, can be substantial since for a MIMO impulse response with length L, the number of coefficients to be quantized is LMt Mr . Additionally, a high resolution quantization is required because we have to take the DFT and then the eigendecomposition of the quantized channel. In our case we are primarily interested in the dominant singular vector of each H(k) thus we do not need to inform the transmitter about the complete channel. Consequently we develop a method that allows us to inform the transmitter about the desired transmit beamforming vectors {w(k), k = 1, 2, · · · , N }, within the constraints of the limited feedback channel. We argue in Section III that the beamforming vectors are correlated, thus the amount of feedback data is proportional to only LMt (where L is normally much less than N ) by sending back only a fraction of vectors combined with intelligent interpolation at the transmitter. We combine this result with beamforming vector quantization [15]–[17] to reduce the feedback required and provide a specific recipe that can be followed for practical implementation in MIMO-OFDM systems. III. I NTERPOLATION BASED B EAMFORMING FOR MIMO-OFDM In this section we present the motivation and detailed description of our approach for interpolation based beamforming in MIMO-OFDM systems. First we show that the correlation between the beamforming vectors is similar to the subchannel correlation with one receive antenna. Next we use these results to motivate subsampling the beamforming vectors in frequency followed by beamformer interpolation across subcarriers. We introduce a beamforming vector interpolator based on spherical linear interpolator with additional parameters for phase rotation, which takes the invariance of the optimal beamforming vectors into account. We optimize the phase rotation parameters in the sense of maximizing the minimum effective channel gain or mutual information. Finally, the

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proposed beamforming method is combined with quantization of beamforming vectors for practical implementation. A. Correlation Between Beamforming Vectors For notational convenience, we define the (Mt Mr )dimensional vector representing the channel gains for subcarrier k as h(k) = vec(H(k)) (4) where vec(X) is the vector obtained by stacking the columns of X one on top of another and 1 ≤ k ≤ N . Generally, the correlation between subchannel k and subchannel (k + d) (assuming modulo N addition) is defined as E[hH (k + d)h(k)]

ρh (d) = p

E[kh(k + d)k2 ]E[kh(k)k2 ]

.

(5)

Similarly, the correlation between w(k + d) and w(k) can be expressed as ρw (d) = E[wH (k + d)w(k)].

(6)

The definition in (6), however, is not a good correlation indicator for beamforming vectors due to the nonuniqueness of the optimal beamformers, which we discuss more in Section III.B. Briefly, when w(k) is the optimal beamforming vector for the subcarrier k, ejφ w(k) is also optimal from (2). In practice, the arbitrary phase φ is determined by the algorithm that computes the beamforming vector. The phase ambiguity means that ρw (d) can take a different value for the same channel because the expectation of (6) is dependent on the arbitrary phase φ. For this reason, we measure correlation using a different definition of the beamformer correlation which is independent of φ: ηw (d) = E[|wH (k + d)w(k)|2 ].

(7)

To make our comparison we also consider a similar modification to the subchannel correlation: ηh (d) =

E[|hH (k + d)h(k)|2 ] . E[kh(k + d)k2 ]E[kh(k)k2 ]

(8)

Now we compare the subchannel correlation with the beamformer correlation through computer simulations. For illustration purposes we generated the time domain channel impulse response following the ETSI/BRAN Channel Model A, B, and C [30]. Note that the root mean square (RMS) delay spread of Channel Model A, B, and C is 50ns, 100ns, and 150ns, respectively. It was assumed that channel impulse responses between different transmit and receive antenna pairs are independent. Fig. 2 shows ηw (d) when Mt = 4, Mr = 2, and N = 64. For comparison, ηh (d) given by (8) were plotted for Mt = 4, Mr = 1, and N = 64. The amount of correlation for ηw (d) and ηh (d) was the greatest for Model A because the coherence bandwidth of the channel is inversely proportional to the RMS delay spread. Of particular interest is that the correlation between the beamforming vectors with two receive antennas is very similar to the correlation between the channel vectors with one receive


not preserve the unit norm constraint on w(k). An alternative approach is to use spherical interpolation algorithms (see e.g. [21]–[23]) that naturally enforce the unit norm constraint by construction. One such algorithm, introduced in [23], is to weight average of points on a sphere in Euclidean space and then renormalize the result back to the unit sphere. For example, a p-th order interpolator computes Pp bi vi v ˆ = Pi=0 (9) p k i=0 bi vi k

1

Correlation

0.8

0.6

η (d) Model A h η w(d) Model A η h(d) Model B η w(d) Model B η (d) Model C h η (d) Model C

0.4

0.2

0 −8

w

−6

−4

−2 0 2 Distance between subcarriers (d)

4

4

6

8

Fig. 2. Comparison between ηh (d) with one receive antenna and ηw (d) with two receive antennas when Mt = 4, N = 64, and L = 8 or 16. This figure shows the similarity between ηw (d) and ηh (d).

antenna. This is a trend we also observed for other combinations of receive antennas and transmit antennas. One explanation is given in [17], where it was shown that the distribution of the optimal beamforming vector in a narrowband independent identically distributed complex Gaussian matrix channel is independent of the number of receive antennas. Based on these observations, we will use the coherence bandwidth predicted by ηh (d) with one receive antenna to estimate the coherence of the beamforming vectors. B. Proposed Interpolation Based Beamforming Method A simple method for exploiting the correlation of neighboring beamforming vectors is to combine adjacent subcarriers into a cluster and use the beamforming vector corresponding to the center subcarrier in the cluster. This method will be referred to as clustering 1 . If we combine K subcarriers into one cluster, the amount of feedback is reduced by 1/K. As the cluster size K increases, however, the performance of beamforming degrades because of the distortion experienced by subcarriers near the cluster boundary. This limits the cluster size K and the consequent reduction of feedback information is restricted. As an alternative, we consider an interpolationbased beamforming method where we send back a subsampling of the beamforming vectors and use a smart interpolation at the transmitter to fill in the gaps. Suppose that N is divided by the subsampling rate K. The receiver evaluates the optimal beamforming vectors for the selected subcarriers and sends the subsampled beamforming vectors {w(lK + 1), 0 ≤ l ≤ N/K − 1} to the transmitter. The transmitter reconstructs the beamforming vectors for all subcarriers by interpolating {w(lK+1), 0 ≤ l ≤ N/K−1} for the missing subcarriers. Conventional interpolators, however, are not suitable for beamformer interpolation because they do

1 We derive this terminology based on work in [31] where clustering was used to reduce the required feedback for an OFDM system with adaptive modulation.

where kvi kP = 1 and {bi } are weights values such that each p bi ≥ 0 and i=0 bi = 1. When p = 1, the interpolated vector v ˆ is a point on the spherical line between points v0 and v1 . Unfortunately, the algorithm in (9) (and generally the class of spherical interpolation algorithms) cannot be applied to our approach due to the lack of uniqueness of the optimal beamforming vector as we discussed in Section III.A. To emphasize this point, recall the effective channel gain Γ(k) = kH(k)w(k)k2 in (2). When w(k) is the optimal beamforming vector maximizing the effective channel gain, ejφ w(k) also maximizes the effective channel gain. In other words, the optimal beamforming vector is not a unique point but represents a complex line. As a consequence, algorithms that are used to compute the optimal beamforming vector, e.g. the singular value decomposition (SVD), typically choose the phase to force the first coefficient of each w(k) to be real. The phase, however, has a dramatic impact on the resulting interpolated vectors as calculated through (9) as well as by other spherical interpolators in [21]–[23]. This makes it difficult to directly apply spherical interpolation to perform beamformer interpolation. To solve this problem, we propose a new algorithm through modification of the interpolater in (9). For notational convenience, let us define wl = w(lK + 1). Given {wl , 0 ≤ l ≤ N/K − 1}, the proposed interpolator computes for subcarrier lK + k w(lK ˆ + k; θl ) =

(1 − ck )wl + ck ejθl wl+1 k(1 − ck )wl + ck ejθl wl+1 k

(10)

where ck = (k−1)/K is the linear weight value, wN/K = w0 , 1 ≤ k ≤ K, and θl is a parameter for phase rotation. Note that {w(k; ˆ θN/K−1 ), N − K + 1 ≤ k ≤ N } are obtained by wN/K−1 and ejθN/K−1 wN/K (equivalently ejθN/K−1 w0 ). While the spherical interpolator in (9) with p = 1 only utilizes wl and wl+1 , the proposed interpolator evaluates the beamforming vector from wl and ejθl wl+1 . Essentially the role of θl is to remove the distortion caused by the arbitrary phase rotation of the optimal beamforming vectors. Notice that the parameters {θl , 0 ≤ l ≤ N/K − 1} are conveyed to the transmitter along with the selected beamforming vectors {w(lK + 1), 0 ≤ l ≤ N/K − 1}. C. Phase Optimization for the Proposed Interpolator To maximize the performance of the proposed interpolator, the receiver evaluates the optimal phases {θl , 0 ≤ l ≤ N/K − 1} based on a communication-theoretic performance metric. Ideally the phase parameter θl would be determined


to minimize the average BER performance of the proposed interpolator. This criterion, however, makes it complicated to define the cost function with respect to θl especially when modulation, coding, and interleaving are taken into account. Instead, we propose to find θl by maximizing the minimum of the effective channel gains for all subcarriers. We find through simulations that this criterion tends to decrease the average BER. For capacity comparisons, we find the optimal {θl } in the sense of maximizing the mutual information. In both cases we find the optimal solution through a numerical grid search and/or mathematical analysis after some approximations of the cost function. In (10), θl is only used for computing {w(k; ˆ θl ), lK + 1 ≤ k ≤ (l + 1)K}, and thus the optimal θl maximizing the minimum effective channel gain can be found as θl = arg max θ

min

lK+1≤k≤(l+1)K

kH(k)w(k; ˆ θ)k2

(11)

where 0 ≤ l ≤ N/K − 1. Due to the normalization factor in w(lK ˆ + k; θ), it is difficult to solve for (11) analytically. Instead, we use a numerical grid search by uniformly quantizing the phase and modifying (11) as θl = arg max

min

θ∈Θ lK+1≤k≤(l+1)K

kH(k)w(k; ˆ θ)k2

(12)

2(P −1)π 4π where Θ = {0, 2π }, and P is the number P , P ,··· , P of quantized levels which determines the performance and complexity of the search. As an alternative to the grid search, we simplify (11) through an approximation and finally derive a closed-form solution. The advantage of a closed form solution is that we can use it to reduce search time or search complexity, especially for large P . On average, w(lK ˆ + K/2 + 1) suffers from the largest distortion induced by interpolation, because the subcarrier (lK + K/2 + 1) is the furthest from the subcarriers (lK + 1) and ((l + 1)K + 1) whose beamforming vectors are available through feedback. Since the instantaneous channel gain is a function of the channel matrix and the beamforming vector, though, the minimum channel gain can happen at any subcarrier in the interpolation interval. Yet we assume that the subcarrier (lK + K/2 + 1) has the worst effective channel gain to simplify the cost function in (11). From this assumption, (11) is approximated to

θl

=

arg max kH(lK + K/2 + 1)w(lK ˆ + K/2 + 1; θ)k2

=

arg max

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H where α1 = (wlH Rwl+1 )(wlH wl + wl+1 wl+1 ) − H H H H (wl Rwl )(wl wl+1 ) − (wl+1 Rwl+1 )(wl wl+1 ), α2 = H 2(wlH Rwl+1 )(wl+1 wl ), and Imag(·) is the imaginary part of (·). By Theorem 1 in the Appendix, |α1 | ≥ |Imag(α2 )|, thus there always exists θ that satisfies (14). Suppose that sin−1 (x) ∈ [−0.5π, 0.5π) and let denote αi = |αi |ejφi . Then the solutions for (14) are given by

ϑ1

= −φ1 + ² ½ −φ1 + π − ² = −φ1 − π − ²

ϑ2

d j{α1 ejθ + (α2 − α2∗ ) − α1∗ e−jθ } = −2 Real(α1 ejθ ) (17) dθ where Real(·) is the real part of (·). Since (17) decides the sign of the second derivative of the cost function, the solution maximizing the cost function becomes one of the solutions ϑ1 and ϑ2 such that −2 Real(α1 ejθl ) is negative. This solution will be referred to as the closed-form solution. Alternatively, θl can be determined to maximize the sum rate of all the OFDM subcarriers. This is useful for information theoretic comparisons. When waterfilling is considered, the optimal phase maximizing the sum rate is found by

θ

½

(l+1)K

θl = arg max θ

X

log2

k=lK+1

P (k)kH(k)w(k; ˆ θ)k2 1+ N0

¾ ,

(18) where P P (k) is the transmit power allocated to the subcarrier N k and k=1 P (k) = N Es by the power constraint. Since the optimal P (lK +k) determined by waterfilling across subcarriers is a function of θl , it is complicated to find the optimal θl . For example, the numerical grid search computes the effective channel gain and evaluates P (k) through waterfilling for each candidate of θl , thereby requiring an extensive computational cost. Also, waterfilling requires additional feedback information about power allocation among subcarriers. To overcome these difficulties, we assume that the transmit power is equally assigned to all subcarriers. When the grid search is used, (18) is simplified to (l+1)K

(13)

(16)

´ ³ |α2 | sin φ . Now, the derivative of (14) where ² = sin−1 − |α 2 1| is denoted as

θ

(wl + ejθ wl+1 )H R(wl + ejθ wl+1 ) kwl + ejθ wl+1 k2

(15) if ² ≥ 0 if ² < 0

θl = arg max θ∈Θ

X

k=lK+1

½ ¾ Es kH(k)w(k; ˆ θ)k2 log2 1 + . (19) N0

where R = HH (lK + K/2 + 1)H(lK + K/2 + 1). Note that this cost function does not guarantee that we maximize the minimum channel gain over all subcarriers and thus is suboptimal relative to (11). Our simulations, though, show that the cost function in (13) performs comparably to the grid search in (12). By differentiating the cost function in (13) with respect to θ, the optimal solution satisfies

Following the approach in (13), θl can be approximately found by solving ½ θl = arg max log2 1 + θ ¾ Es kH(lK + K/2 + 1)w(lK ˆ + K/2 + 1; θ)k2 . (20) N0

j{α1 ejθ + (α2 − α2∗ ) − α1∗ e−jθ } =Imag(α1 ejθ + α2 ) =0 (14)

From the concavity of the log function, this equation is equivalent to the solution to (13).


D. Interpolation Combined with Beamformer Quantization For practical implementation, we propose to quantize the beamforming vectors using the codebooks designed for narrowband MIMO systems [16]–[18]. These codebooks are specially designed to take into account the phase invariance of the optimal beamforming vector. Given {H(k)}, the quantized beamforming vectors are determined by selecting the element of the codebook that maximizes the effective channel gain as follows. wQ (lK + 1) = arg max kH(lK + 1)xk x∈W

(21)

where W is the codebook including all possible beamforming vectors and 0 ≤ l ≤ N/K − 1. The quantized beamforming vectors are used to find the optimal θl and θl is uniformly quantized on [0, 2π). As an aside, instead of separately quantizing w((l + 1)K + 1) and θl , the whole vector ejθl w((l + 1)K + 1) can be quantized by spherical quantization based on the Euclidean distance [32], [33]. This approach, however, requires almost the same amount of feedback as with separate quantization, yet makes it more complicated to determine the phase θl . Therefore, we only consider the separate quantization of w((l + 1)K + 1) and θl . When quantized beamforming vectors are used, the interpolated vector w ˆ Q (lK + k; θl ) is evaluated by replacing w(lK + 1) with wQ (lK + 1) in (10). Since we are quantizing θl , it is natural to use the grid search to find the best θl . By modifying (12) and (19), we obtain the following equations that optimize θl for the quantized beamforming vectors in the sense of maximizing the minimum effective channel gain θl = arg max

min

θ∈Θ lK+1≤k≤(l+1)K

kH(k)w ˆ Q (k; θ)k2

6

TABLE I (a) Required number of complex multiplications for beamforming in MIMO-OFDM.

Selection Diversity Clustering Ideal Beamforming Proposed Grid Search (P = 4, 8, 16, · · · )

# of Multiplications N Mt log2 N 2 N Mr ( 12 log2 4N + Mt ) N Mt log2 N 2 (Mt +1)|W| 1 } N Mr { 2 log2 4N + K N Mt log2 N 2 N Mr { 12 log2 4N + (Mt + 1)|W|} N Mt log2 N + 2N Mt 2 (Mt +1)|W| N Mr { 21 log2 4N + } K N (2Mt Mr +1) 3 +( 2 P − 2)N + K

(b) Required number of complex additions for beamforming in MIMO-OFDM.

Selection Diversity Clustering Ideal Beamforming Proposed Grid Search (P = 4, 8, 16, · · · )

Tx Rx Tx Rx Tx Rx Tx Rx

# of Additions N Mt log2 N N Mr (log2 2N + Mt ) N Mt log2 N M |W| N Mr (log2 2N + tK ) N Mt log2 N N Mr (log2 2N + Mt |W|) N Mt log2 N + N Mt M |W| N Mr (log2 2N + tK ) t Mr + 52 N P + 2N M K

TABLE II R EQUIRED NUMBER OF COMPLEX OPERATIONS WHEN Mt = 4, Mr = 2, N = 64, K = 8, P = 4, AND |W| = 16.

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or maximizing the mutual information ½ ¾ (l+1)K X Es kH(k)w ˆ Q (k; θ)k2 θl = arg max log2 1 + . θ∈Θ N0

Tx Rx Tx Rx Tx Rx Tx Rx

Selection Diversity Clustering Ideal Beamforming Proposed

# of Multiplications Tx Rx 768 1024 768 1792 768 10752 1280 2184

# of Additions Tx Rx 1536 1408 1536 1920 1536 9088 1792 2688

k=lK+1

(23) IV. C OMPUTATIONAL C OMPLEXITY AND F EEDBACK R EQUIREMENTS This section compares the complexities and feedback requirements for implementing selection diversity for each subcarrier, clustering, the proposed grid search with beamformer quantization, and ideal beamforming with full feedback of the quantized beamforming vectors. In the complexity analysis, we roughly calculate the number of complex multiplications and the number of complex additions. The computation at the transmitter includes the inverse DFT, transmit beamforming, and beamformer interpolation for the proposed algorithm. The computation at the receiver includes the DFT, receive combining, quantization of beamforming vectors, and determination of the phase rotation parameters for the proposed algorithm. For simplicity, the computational load for synchronization and channel estimation is not considered as it is likely to be comparable for all approaches. The DFT or inverse DFT can be implemented using fast Fourier transform (FFT) algorithms such

as the power of two decimation-in-time and decimation-infrequency algorithms, which require (N/2) log2 N complex multiplications and N log2 N complex additions [34]. The computational load for MIMO-OFDM is multiplied by Mt and Mr at the transmitter and the receiver, respectively. When quadrature amplitude modulation (QAM) is used, the complexity for transmit beamforming is almost negligible. Receive combining needs N Mr complex multiplications and additions, respectively. The beamforming vectors are quantized at the receiver by (21) performing N (Mt + 1)Mr |W| multiplications and N Mt Mr |W| additions per subcarrier, where |W| is the cardinality of the chosen beamforming codebook. In the proposed algorithm, beamformer interpolation requires 2N Mt complex multiplications and N Mt complex additions. The phase rotation parameter is determined by (22), which t Mr +1) complex multiplications requires ( 23 P − 2)N + N (2MK 2N Mt Mr 5 and 2 N P + additions. K The complexity calculations at the transmitter and receiver for various diversity techniques are summarized in Table I. Selection diversity necessitates the least operations because


7

TABLE III F EEDBACK REQUIREMENTS OF VARIOUS DIVERSITY TECHNIQUES FOR MIMO-OFDM WHEN Mt = 4, N = 64, K = 8, P = 4, AND |W| = 16.

6

5.5

5

4.5 Mutual Information (bps/Hz)

Selection Diversity Clustering Ideal Quantized Beamforming Proposed

Feedback bits per frame N log2 Mt = 128 N log2 |W| = 32 K N log2 |W| = 256 N (log2 |W| + log2 P ) = 48 K

−1

10

3.5

3

2.5

Clustering Grid search ( P=4 ) Closed−form solution Grid search ( P=16 ) Ideal beamforming

−2

4

2

10

1.5

1 −5 10

5

10

BER

Fig. 4. Mutual information without quantization of beamforming vectors when Mt = 4, Mr = 2, N = 64, and K = 8. Grid search in (19) and closed-form solution in (20) were used.

−4

10

−5

V. S IMULATION R ESULTS

10

−6

−4

0 SNR (dB)

−3

10

Ideal beamforming w/ waterfilling Ideal beamforming w/o waterfilling Closed−form solution Grid search w/ waterfilling ( P=4 ) Grid search w/o waterfilling (P=4) Clustering

−2

0

2

4 Eb /N0 (dB)

6

8

10

12

Fig. 3. Uncoded BER performance without quantization of beamforming vectors when Mt = 4, Mr = 2, N = 64, and K = 8. Grid search in (12) and closed-form solution in (13) were used.

it does not require the quantization of beamforming vectors. Ideal beamforming quantizes the beamforming vectors for all subcarriers, thus it needs extensive computational load. Compared to clustering, the proposed method requires additional operations to interpolate beamforming vectors at the transmitter and find the phase rotation parameters at the receiver: in Table II, the number of complex multiplications and the number of additions are increased by 35.3% and 29.6%, respectively. Table III compares the required number of feedback bits per frame for various diversity techniques. The information required for antenna selection is log2 Mt bits per subcarrier. When the transmit antenna with the maximum channel gain is selected independently for each subcarrier, the total feedback information becomes N log2 Mt bits. The beamforming vectors are quantized by using the codebook W, thereby requiring log2 |W| feedback bits per beamforming vector. The N log2 P additional bits of feedback proposed algorithm needs K to transfer the information about the phase rotation parameters. Note that the proposed grid search requires much less feedback than selection diversity and ideal beamforming. For example, in Table III the proposed method needs only 48 feedback bits per frame, which is about one third of feedback for selection diversity and about one fifth of feedback for the ideal beamforming.

To illustrate the performance of the proposed approach, we present Monte Carlo simulations for a system with parameters: Mt = 4, Mr = 2, N = 64, K = 8, and CP length of 16. We assumed that the discrete-time channel impulse response was generated according to the ETSI/BRAN Channel Model B in [30]; the channels between different transmit and receiver antenna pairs were independent; the channel was fixed for a frame and randomly varied between frames in Figs. 3–7; n(k) was i.i.d. complex Gaussian with zero mean; the transmit power was equally allocated to all subcarriers; QPSK was used for the BER simulations; and the receiver used MRC with perfect channel knowledge. In all simulations except Fig. 6, it was assumed that the feedback channel had no delay and no transmission error. Every point of the simulation results was obtained by averaging over more than 3000 independent realizations of the channel and the noise. Simulation results are presented in the following two subsections. In the first part, we simulate the transmit beamforming schemes without quantization to provide an upper bound on the performance with quantization. Next, we consider the quantization of beamforming vectors and θl shown in Section III-D. A. Unquantized Beamforming We compare the proposed beamforming scheme with clustering and ideal beamforming with feedback of all beamforming vectors. The dominant right singular vector was used for the beamforming vector for MRT/MRC and quantization was not considered. Fig. 3 compares the uncoded BER performance for various beamforming methods. In the figure, “grid search” and “closed-form solution” describe the proposed interpolators using the grid search in (12) and the closed-form solution in (13), respectively. The grid searches with P = 4 and P =


8

−2

10

Theoretical lower bound of OSTBC Clustering w/ 32 bit feedback Selection diversity w/ 128 bit feedback Proposed w/ 48 bit feedback Ideal beamforming w/ 256 bit feedback

−1

10

Theoretical lower bound of OSTBC Clustering w/ 32 bit feedback Selection diversity w/ 128 bit feedback Proposed w/ 48 bit feedback Ideal beamforming w/ 256 bit feedback

−2

10

−3

10

BER

BER

−3

10

−4

10

−4

10

−5

10

−5

10 −6

10

−4

−2

0

2

4 6 Eb/N0 (dB)

8

10

12

14

Fig. 5. Uncoded BER performance with quantization of beamforming vectors when Mt = 4, Mr = 2, N = 64, and K = 8. Channel was fixed in a frame and the grid search in (22) was used for the proposed interpolator.

16 both performed comparably to the closed-form solution. The proposed methods outperformed clustering, yet exhibited slight diversity loss compared to the ideal beamforming. Fig. 4 presents the mutual information for the whole OFDM symbol obtained using various beamforming techniques. Because we only consider beamforming, the gains in sum rate are much smaller than if used multiple eigenvectors of the channel. For comparison, we plotted the ideal beamforming and the proposed grid search with waterfilling across the subcarriers while all other curves were obtained without waterfilling. The proposed techniques had higher mutual information than the clustering and performed within 0.3 dB of the ideal beamforming without waterfilling. Since the diversity provided by multiple antennas reduces channel fluctuation between subcarriers, the mutual information gain by waterfilling was small. Notice that waterfilling requires additional feedback information for power assignment among subcarriers. Figs. 3 and 4 show that the grid search with P = 4 provides a good tradeoff between the performance and complexity. Thus the grid search with P = 4 will be used in the following simulations. B. Quantized Beamforming The beamforming vectors were quantized by using the codebook designed according to equation (22) in [17]. We used the codebook with |W| = 16 that requires 4 bits of feedback per beamforming vector. θl was uniformly quantized by the grid search with P = 4. For comparison, we simulated antenna selection diversity [7], clustering, and ideal beamforming. Also, the theoretical bound of orthogonal spacetime block codes (OSTBC)2 [2] was considered. In selection diversity, the transmit antenna was selected independently for each subcarrier. The feedback requirements for the diversity techniques under consideration are provided in Table II.

2 It was shown that there are no complex OSTBC with full rate when M ≥ t 3, but the theoretical performance bound can be obtained for any Mt .

−20

−18

−16

−14

−12 −10 −8 MSE of channel predictor (dB)

−6

−4

−2

0

Fig. 6. Uncoded BER performance with quantization of beamforming vectors when Eb /N0 = 7 dB, Mt = 4, Mr = 2, N = 64, and K = 8. Channel prediction error was considered and the proposed interpolator is the same in Fig. 5.

Fig. 5 shows the uncoded BER performance when the beamforming vectors were quantized. The proposed method outperformed clustering and the theoretical lower bound of OSTBC, yet exhibited slight loss in diversity order compared to the ideal beamforming. Selection diversity, as expected, achieves full diversity order but exhibits some array gain loss. Thus, the proposed method performs comparably to selection diversity. Note that the proposed scheme requires only 48 feedback bits per frame, which is 38 of the feedback required for selection diversity. To study the effects of channel estimation error and/or feedback delay, we employ the first order autoregressive model given by p ˜ i,j (k) = γhi,j (k) + 1 − γ 2 ui,j (k) h (24) ˜ i,j (k) is the predicted channel at the transmitter, where h hi,j (k) is the (i, j)-th element of H(k), ui,j (k) is i.i.d. complex Gaussian noise with zero mean and unit variance, and 0 ≤ γ ≤ 1. Then the mean square error (MSE) between ˜ i,j (k) and hi,j (k) is denoted as h ˜ i,j (k) − hi,j (k)|2 ] = (1 − γ)2 + 1 − γ 2 = 2 − 2γ. (25) E[|h Fig. 6 shows the BER degradation caused by the channel prediction error at Eb /N0 = 7 dB. Similar to Fig. 5, the proposed method performed better than clustering and selection diversity. Note that the proposed beamforming method is better than the theoretical bound of OSTBC as long as the MSE ≤ −7.5 dB. This gives an indication when estimation error and/or delay are significant enough to mitigate the advantages of closed-loop diversity techniques over open-loop techniques. Fig. 7 compares the mutual information when quantized beamforming vectors are used. As before, the proposed method outperformed clustering, selection diversity, and the theoretical upper bound of OSTBC. In addition, the proposed


VI. C ONCLUSIONS In an attempt to reduce the feedback information required for closed-loop MIMO-OFDM, a new transmit beamforming scheme with limited feedback and beamformer interpolation was proposed. In the proposed scheme, the receiver sends back only a fraction of information about the beamforming vectors along with additional parameters for phase rotation used in the interpolation. The transmitter calculates the beamforming vectors for all the subcarriers using a new spherical interpolator. It was demonstrated that the proposed interpolator can be optimized by adjusting the parameters for phase rotation depending on different error rate or sum rate inspired criteria. Through computer simulations, it was shown that the proposed method has considerable performance benefits with modest feedback and complexity requirements compared with other closed-loop diversity techniques. For the beamformer interpolation, we only considered the spherical linear interpolator. In fact, it is apparent that higher

3 In [20], only 48 subcarriers among 64 are used for data transmission. To use all the subcarriers for our simulation, the index of the interleaver and deinterleaver were extended.

7

6

5 Mutual Information (bps/Hz)

performed very close to the ideal beamforming: the SNR loss was less than 0.2 dB. The uncoded BER performance in time-varying channels is shown for various Doppler frequencies in Fig. 8 where fd Ts denotes the Doppler frequency normalized by the OFDM symbol duration Ts . In the figure, it was assumed that all the beamforming methods have the same feedback ratio over data bits. Taking into account the feedback requirements shown in Table II, the frame length was set as 40, 60, 160, 320 symbols for clustering, the proposed, selection diversity, and the ideal beamforming, respectively. The proposed method had the lowest BER when 3.5 × 10−4 ≤ fd Ts ≤ 3.1 × 10−3 . The BER performance of ideal beamforming increases rapidly with fd Ts because it has the longest framelength and thus largest beamformer update period. OSTBC performed best as long as fd Ts ≥ 3.1 × 10−3 since its performance is irrelative to fd Ts . This confirms the intuition that when the Doppler is large enough, there is not a good reason to use closed loop diversity modes. The coded BER values in time-varying channels with fd Ts = 0.0005 are presented in Fig. 9. For channel coding, we used a convolutional code with generator polynomials g0 = 1338 and g1 = 1718 with coding rate 1/2, along with the interleaver and deinterleaver defined in [20]3 , and soft Viterbi decoding. The frame length was the same as that in Fig. 8. Each OFDM symbol transmitted 64 bits which is a half of the uncoded case and the feedback bits are just 1.25 % of data bits for all cases. The proposed scheme outperformed other diversity techniques and exhibited a 2.5 dB gain over selection diversity and a 2.8 dB gain over the ideal beamforming at BER= 10−5 . Note that the proposed scheme had huge performance gain over the OSTBC. Overall, these results demonstrate that the proposed beamforming vector quantization and interpolation scheme is advantageous in terms of BER and mutual information compared with existing closed-loop diversity techniques.

9

4

3

2

Ideal beamforming w/ 256 bit feedback Proposed w/ 48 bit feedback Clustering w/ 32 bit feedback Selection diversity w/ 128 bit feedback Theoretical upper bound of OSTBC

1

0 −5

0

5 SNR (dB)

10

15

Fig. 7. Mutual information with quantization of beamforming vectors when Mt = 4, Mr = 2, N = 64, and K = 8. The grid search in (23) was used for the proposed interpolator.

order interpolation can decrease the distortion of the beamforming vectors, however, the determination of θl becomes more complicated because the higher order interpolation requires joint optimization of {θ0 , θ1 , · · · , θ N −1 }. Therefore, K the use of a higher order interpolation requires a practical method for finding the optimal phases {θl } with reasonable complexity. Our work proposed herein is limited to a MIMO system that transmits a single data stream per subcarrier. The capacity of MIMO-OFDM, however, can be increased by supporting the transmission of multiple data streams. One solution is to combine spatial multiplexing with special per-tone unitary precoding matrices along the lines of [35]. An extension of our proposed approach is to evaluate the precoding matrices for the subcarriers through interpolation. While the proposed interpolator for transmit beamforming could be used, it does not retain the orthogonality between the column vectors of the interpolated matrix. Developing an interpolator for spatial multiplexing remains a future research topic.

A PPENDIX Theorem 1: For any Mt -dimensional complex vectors a, b, c, d, let us define β1 and β2 as β1 β2

= aH b(cH c + dH d) − (aH a + bH b)cH d (26) = (aH b)(dH c) − (bH a)(cH d). (27)

Then, β1 and β2 satisfy |β1 | ≥ |β2 |. In addition, let us define a = H(lK +K/2+1)w1 , b = H(lK +K/2+1)w2 , c = w1 , and d = w2 . From (14), we get β1 = α1 and β2 = Imag(α2 ). Therefore, Theorem 2 implies that |α1 | ≥ |Imag(α2 )|.


−2

10

2ab. Thus,

10

Theoretical lower bound of STBC Clustering Selection diversity Proposed Ideal beamforming

|β1 |2 − |β2 |2 ≥ 2|aH bdH c|(aH a + bH b)(cH c + dH d) − (aH a + bH b)(cH c + dH d){aH bdH c + bH acH d}

−3

10

− 2|aH bdH c|2 + {(aH bdH c)2 + (bH acH d)2 } BER

= {(aH a + bH b)(cH c + dH d) − 2|aH bdH c| − aH bdH c − bH acH d}

−4

10

· {2|aH bdH c| − aH bdH c − bH acH d} ≥ {2|aH bdH c| − aH bdH c − bH acH d}2 ≥ 0.

(29)

−5

10

From (29), we get |β1 | − |β2 | ≥ 0. −4

−3

10

−2

10 fd⋅ Ts

10

R EFERENCES Fig. 8. Uncoded BER performance with quantization of beamforming vectors when Eb /N0 = 8 dB, Mt = 4, Mr = 2, N = 64, and K = 8. Channel was time-varying within a frame and beamforming vectors were periodically updated through feedback. The proposed interpolator is the same in Fig. 5.

−1

10

Theoretical lower bound of OSTBC Ideal beamforming Clustering Selection diversity Proposed

−2

10

−3

BER

10

−4

10

−5

10

−6

10

−5

−4

−3

−2

−1

0 1 Eb /N0 (dB)

2

3

4

5

6

Fig. 9. BER performance with quantization of beamforming vectors when Mt = 4, Mr = 2, N = 64, and K = 8. Channel was time-varying within a frame with fd Ts = 0.0005. A convolutional code with rate 1/2 was used along with interleaving and the proposed interpolator is the same in Fig. 7.

Proof: Let us consider

|β1 |2 − |β2 |2 = |aH b|2 (cH c + dH d)2 + (aH a + bH b)2 |cH d|2 − (aH a + bH b)(cH c + dH d){aH bdH c + bH acH d} − 2|aH bdH c|2 + {(aH bdH c)2 + (bH acH d)2 }. (28)

For any nonnegative real values a and b, we have a2 + b2 ≥

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Jihoon Choi received the B.S., M.S., and Ph.D. degrees from the Korea Advanced Institute of Science and Technology (KAIST), Korea, in 1997, 1999, and 2003, respectively. From 2003 to 2004, he was with Department of Electrical and Computer Engineering, the University of Texas at Austin, where he performed research on MIMO-OFDM systems with limited feedback. Since 2004, he has been with the Samsung Electronics, Korea. In Samsung, he has participated in Portable Internet (PI) and Enhanced Broadcast Multicast Service (BCMCS) projects. His research interests include MIMO, multicarrier modulation, equalization, and space–time coding for wireless communications.

Robert W. Heath, Jr. Biography text here.

PLACE PHOTO HERE