Transmit Equal Gain Precoding in Rayleigh Fading Channels

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Transmit Equal Gain Precoding in Rayleigh Fading Channels

Shang-Ho Tsai

Abstract—Precoding with limited feedback information can achieve

satisfactory performance while the amount of feedback information is kept small. In this paper, we analyze the theoretical performance of equal gain precoder and find that its performance is at most 1.049 dB worse than the optimal precoder no matter how the number of transmit antennas increases. Moreover, we analyze the performance degradation of the equal gain precoder due to scalar quantization theoretically. The result shows that 2–3 bits per transmit antenna (excluding the first antenna) can achieve 0.5–0.25-dB performance gap compared with the same scheme without quantization. Furthermore, we found that the equal gain precoder in general can achieve comparable performance with the Grassmannian precoder in the same moderate feedback bits. Simulation results are provided to corroborate the theoretical results.

Index Terms—Beamforming, equal gain precoding, limited feedback,

MIMO, precoding, scalar quantization.

I. INTRODUCTION

MIMO techniques are widely used in current wireless communica-tion standards such as IEEE 802.11n and IEEE 802.16. Among the MIMO skills, precoding/beamforming can provide full diversity order and additional precoding gain. Such nice properties can greatly im-prove system performance.

If complete channel formation is known to the transmitter, we can jointly design precoders and decoders by optimizing several parameters such as MMSE, maximizing information rate, or maximizing SNR (see [2], [11], and [13]). Although such precoding schemes can achieve op-timal performance using different design criterion, the hardware com-plexity and the amount of feedback information are high.

To overcome the drawbacks of the optimal precoders, research has been directed to the precoding schemes with limited feedback recently. The precoder design with the channel mean and the covariance matrix available at the transmitter was studied in [8] and [15] from the view point of channel capacity. In [6], the precoding schemes that achieve the minimum outage probability was studied. Examples of codebook construction were also given. In [3], equal gain precoders with different combining methods were shown to achieve full diversity order. In [4], Grassmannian precoding was proposed. The Grassmannian precoder has been shown to have good performance in practical communica-tion systems. The Grassmannian precoding was extended to space–time block code by the same authors in [5]. In [7], the authors analyzed the capacity loss of equal gain precoder due to both vector and scalar quan-tization. An optimal bit allocation for equal gain precoder with scalar quantization was proposed in [19].

In this paper, we analyze the theoretical performance of the equal gain precoder in multiple-input single-output (MISO) channel environ-ments and found several interesting results as follows.

First, the bit error probability (BEP) performance gap between the equal gain precoder and the optimal precoder varies from 0.5 to

Manuscript received December 29, 2008; accepted March 30, 2009. First pub-lished April 21, 2009; current version pubpub-lished August 12, 2009. The associate editor coordinating the review of this manuscript and approving it for publica-tion was Prof. Gerald Matz. This work is supported by the Napublica-tional Science Council (NSC), Taiwan, Cooperative Agreement 97-2221-E-009-071.

The author is with the Department of Electrical Engineering, National Chiao-Tung University, Hsinchu 300, Taiwan (e-mail: shanghot@mail.nctu.edu.tw).

Digital Object Identifier 10.1109/TSP.2009.2021498

1.049 dB as the number of transmit antennas grows from 2 to infinity. This is interesting since the performance of equal gain precoder is at most 1.049 dB worse than the optimal precoder, no matter how the number of transmit antennas increases.

Second, we analyze the performance loss due to scalar quantization for the equal gain precoder theoretically. We found that in the interested range of the number of transmit antennas, e.g., four antennas with RF (radio frequency) for Wi-Fi and Wi-MAX in current standards, perfor-mance loss is within 0.5 dB by using six total feedback bits in the scalar quantized equal gain precoder.

Finally, we found from simulation results that the scalar quantized equal gain precoder can achieve comparable performance with the Grassmannian precoder in the same moderate number of feedback bits. This result shows several advantages of the equal gain precoder. First, the codeword determination of the equal gain precoder is simple, since in MISO channel environments the optimal solution for the equal gain precoder is actually the channel vector. Hence, there is no need to perform exhaustive search to determine the codeword as Grassmannian does. Second, for the equal gain precoder, when the feedback bits are less than two per transmit antenna, there is no need to perform multiplications in the transmitter side since the codeword elements in this case are61 or 6j. Please note that for small size Grassmannian precoder, operation without multiplications is also possible. Third, the equal gain precoder can be easily extended to arbitrary transmit antenna number since it does not need to construct the codebook in advance.

Notations: fxg is the expectation of x. A3andAtare the con-jugate and transpose ofA, respectively. Ay is the conjugate-transpose ofA. <fxg denotes the real part of variable x. 2_xis the variance ofx.

II. SYSTEMMODEL ANDPERFORMANCEANALYSIS Let the number of transmit antennas beNt. First, one symbolx is sent toNtbranches and each symbol in different branch is multiplied by a different phase rotation, i.e.,ej =pN_t. After the precoding, the symbol vector to be transmitted iss = (s1s2 1 1 1 sN)t= 1=pNtpx, wherep is a N_t 2 1 vector and its ith element is ej . Then,s is transmitted to the channel. At the receive, the received symbolr is r = ht_{s+n, where h is a N}

t21 channel vector and its ith coefficient ishi, andn is a noise scalar. To achieve the best performance, we use MRC [3] in the receiver and this leads to

z = py_h3_{r = 1}_p

Nt x + p

y_h3_n ₍₁₎

where = pyh3htp is a gain effect (including diversity gain and precoding gain) due to the precoding.

Now, we analyze the average SNR performance and it terms out that this average SNR performance is highly correlated to the bit error prob-ability performance (see [1] and [4]). From (1), for a given channel re-alization (channel is deterministic), the instantaneous SNR of the equal gain precoder is e= _N t 2 x 2_n: (2)

From (2), for a given channel realization and_x2=2_n, the instanta-neous SNR is determined by . Let us look at more detailed. From (1) and due to phase mismatch, can be upper bounded by

(2)

Let the phase ofh_ibe_i. The equality of (3) holds when(_j0 _i) = 0(j0 i): It is intuitive to choose

i= 0i; 1 i Nt: (4) Using the solution in (4), all the transmit antennas require to multiply individual phases. However, we notice that multiplying an extra arbi-trary phase rotation in both the transmitter and the receiver sides simul-taneously does not change . In this case, the equality also holds when i= 0 if i = 1; i= 0(i0 1) otherwise: (5) From (5), we do not need to perform precoding for the first transmit antenna and thus the feedback information can be reduced. The above results were also shown in [3] and [19].

From (2) and (3), for a specified_x2=_n2, the average SNR can be upper bounded by

hfeg 2 x

2_n h jhij2 + (Nt0 1) hfjh3ihjjg : (6) Without losing the generality, let us assume that both the real part and the imaginary part ofhihave unit variance. In this case, h jhij2 = 2

h = 2. Next, let us see how to obtain hfjh3ihjjg.

Lemma 1: Assume thathiis complex Gaussian with zero mean and unit variance in both the real part and the imaginary part. The proba-bility density function (PDF) ofjh3_ihjj is given by

fh(x) = x₂(Ko(x) + K2(x)) 0 K1(x) (7) whereK(x) is the modified Bessel function of the second kind.

Proof: Following the conditions forhi, the cumulative distribu-tion funcdistribu-tion (CDF) ofjh3_ih_jj is F_h(x) = 1 0 xK₁(x) (see [16]). From [17], we have @K(x)=@x = 01=2 (K01(x) + K+1(x)). Thus, using the fact thatf_h(x) = @F_h(x)=@x; we can obtain (7).

Lemma 2: Assume thathiis complex Gaussian with zero mean and unit variance in both the real part and the imaginary part. The mean of the random variablejh3_ih_jj can be calculated as

hfjh3ihjjg = 1:5708: (8) Proof: From Lemma 1, we have

hfjh3ihjjg = 1 01xfh(x)dx = 1 0 x2 2K0(x)dx + 1 0 x2 2K2(x)dx 0 1 0 xK1(x)dx (9) where we have used the fact thatK(x) = 0, for x < 0. According to [18], we have ₀1x01K(x)dx = 2020 ( 0 =2) 0 ( + =2) ; where0(x) is the gamma function. Hence, from (9), we can have (8). Please note that without using the PDF ofjh3_ih_jj, we can still derive hfjh3ihjjg as follows: Since hi and hj are independent, we have hfjh3ihjjg = hfjhijg hfjhjjg = ( hfjhijg)2 = =2, where we have used the property that the expectation of the Rayleigh random variable is =2[14, pp. 279–280].

Theorem 1: Let the channel coefficienthibe complex Gaussian with zero mean. Without quantization, the SNR gap of the optimal precoder and the equal gain precoder in a MISO channel is given by

hfog hfeg =

Nt

0:7854Nt+ 0:2416: (10)

Fig. 1. Average SNR of the optimal and the equal gain precoders, where (a) is without taking dB and (b) is taking dB.

Proof: In the MISO case, the optimal precoder is actuallyhy. Its average SNR can be shown to be

hfog = 2 x 2_n h N i=1 jhij2 = Nt 2 x 2_n h jhij2 : (11) To have a fair comparison, we also assume that the channel coefficients in the optimal precoder have unit variance in both the real and the imag-inary parts. Thus, _h jh_ij2 = 2. From (11), we have _hf_og = 2

x=2n2Nt: From (6) and (8), for the equal gain precoder without quan-tization, we have

hfeg = 2 x

2_n(2 + (Nt0 1)1:5708): (12) Hence, we can obtain the result in (10). It is worth to emphasize that since this is a fair comparison for the two precoders, there is no need to constrain the variance ofh_iin Theorem 1.

Please note that hfeg can also be directly derived from [19, Eq. (29)], without obtaining the PDF ofjh3_ih_jj. From Theorem 1, when Nt 1, the ratio approximates 1=0:7854 = 1:049 (dB), which is a constant performance gap. This is an interesting result because it means that the performance loss due to the use of phase alone is at most around 1 dB, despite the increase of the transmit antennas.

1) Example 2: Constant Performance Gap Between the Optimal and the Equal Gain Precoders: Lethibe complex Gaussian with unit variance in both the real and the imaginary parts. The average SNR of both the optimal and the equal gain precoders as a function of the number of transmit antennas are shown in Fig. 1. We observe that when Nt= 4, the two precoders have performance gap around 0.75 dB and approximate to 1.049 dB whenNt> 8. These results show that there may be no need to use complicated precoding schemes such as the op-timal precoder if we are capable to sacrifice around 1-dB performance. Take IEEE 802.11n and IEEE 802.16e-2005 for instance, the antenna number (with RF) is at most four in these standards. Under such situa-tions, the performance loss is at most 0.75 dB.

III. SCALARQUANTIZATION

Let us consider the scalar quantization effect for the equal gain pre-coder. Since we assume the channel coefficients are complex Gaussian, the phase is uniformly distributed in[0 ]. In this case, let us use equal space quantization and quantize the phase to the closet avail-able value. For instance, if the bit number,b, to represent the phase per transmit antenna is 2, the available values can be 0,=2, or

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0=2. Please note that with scalar quantization, the total required bits isb(Nt0 1). For vector quantization, it does not have such limitation and it considers the optimal solution for a given bit budget, e.g., [5] and [7].

Now consider how the quantization effect will deteriorate the av-erage SNR. Let ^ibe the quantized phase. From (3), the average SNR due to scalar quantization is given by

h e = 2 x 2_n h jhij2 + c1 h < h31hjej^ + c2 h < hi3hjej(^ 0^ ) (13) wherec₁ = 2(N_t0 1)=N_t and c₂ = (N_t0 1)(N_t0 2)=N_t. In addition, the reason that we separate the terms fori = 1 and i 6= 1 is because ^1 = 0 from (5). Define the quantization error of the phase asi = ^i0 i. Sincei = 0(i0 1) according to (5), we have ^_i = 0(_i0 ₁) + _i. Assuming that the quantization error of the phase is independent of channel (see [9]) and using the fact that h3

ihje0j( 0 )is real, we can rewritten (13) as h e =

2 x

_n2 h jhij2 + c1 h jh31hjj h cos (j) + c2 h jh3ihjj h cos (j0 i) : (14) Lemma 3: Assume that the phase is uniformly distributed in[0 ]. Let the number of bits to represent the phase per transmit antenna (ex-cluding the first antenna) beb. Using equal space quantization, we have the following result

hfcos (j0 i)g = 2 2b

22 1 0 cos 2₂b : (15) Proof: Let us use_ito denote the phase error of theith transmit antenna. According to [12], if the phase is uniformly distributed in [0 ], the phase error iis uniformly distributed in 0=2b=2b . Now let us find the distribution ofi0 j. From [10], the distribution of0_jis still uniformly distributed in 0=2b=2b . It is known that the PDF of the random variablez = x + y, where x and y are inde-pendent, can be obtained by convolutingfx(x) with fy(y)[10]. Thus, the distribution of_i0 _j = _i+ (0_j) is the linear convolution of the two identical PDFs with uniformly distribution in 0=2b=2b . Hence, lettingz = _i0 _j the PDF ofz can be shown to be

fz(z) = 2 4 z +22; 022 z 0 02 4 z +22 ; 0 < z 22 0; otherwise. (16)

From (16), we know thatfz(z) cos(z) is an even function. According to [10], we have the property: fg(x)g = ₀₁1 g(x)f_x(x)dx; where g(x) is function of x. Thus, we have

hfcos (j0 i)g = 2 =2 02 =2fz(z) cos(z)dz = 2 2=2 0 0 2 2b 42z + 2 b 2 cos(z)dz: (17)

Using integration by part: z cos(z)dz = z sin(z) + cos(z) + C. Thus, we can rewrite (17) as

h cos (j0 i) =2 0 22b 42(z sin(z)+cos(z)) 2=2 0 + 2b 2sin(z) 2=2 0 : (18) Manipulating (18), we can obtain the result in (15).

Similarly, the expectation value ofcos (_i) can be calculated as

hfcos (i)g = =2 0=2 cos () 2 b 2d = 2 b sin 2b : (19) Assumehiis complex Gaussian with zero mean and unit variance in the real and imaginary parts. Using (14), (19), and Lemma 3, we have

hfeg = 2 2 x _n2 1 + 0:7854(N t0 1) Nt 2b 2 sin ₂_b + 2₂b(Nt0 2) 1 0 cos 2₂_b : (20) Moreover, from Theorem 1, we know _hf_eg. Hence, we have the following theorem.

Theorem 2: For the equal gain precoder (with solution_igiven in (5)) in a MISO channel with complex Gaussian distribution, the SNR degradation due to scalar quantization is given by (21), shown at the bottom of the page.

The above result is obtained by settingiaccording to (5), i.e.,1 = 0 and there is no information feedback for 1. It is interesting to ask: When we feed back bits for1, will the performance due to scalar quan-tization improve? To answer this question, let_ibe set according to (4). From (3), we have h e = 2 x 2 n h jhij 2 _{+ (N} t0 1) 2 h < hi3hje0j( 0 ) h ej( 0 ) = x2 2 n h jhij 2 _{+ (N} t0 1) 2 h jh3ihjj h cos (j0 i) : (22) Using (14), (22), and Lemma 3, we can calculate f_eg. Hence, we have the following corollary.

Corollary: For the equal gain precoder (with solution_igiven in (4)) in a MISO channel with complex Gaussian random distribution, the SNR degradation due to equal scalar quantization is given by

feg feg=

2 + 1:5708(Nt0 1) 2 + 1:5708(Nt0 1) 1 0 cos(2₂ ) ₂2

: (23)

1) Example 3: Scalar Quantization for Equal Gain Precoder (The-oretical Result): Let us see the quantization effect for1 = 0 first. Using (21), we plot the performance loss due to quantization in Fig. 2. We see that forNt= 2, 3 and 4, the performance loss for b = 2 is less than 0.5 dB. Moreover, forN_t < 16, the performance loss for b = 3

hfeg hfeg = 0:7854Nt+ 0:2146 1 + 0:7854(N 01) N 2 2 sin(2) +22 (Nt0 2) 1 0 cos(22 ) : (21)

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Fig. 2. The gap of average SNR between the equal gain precoders with and without quantization.

Fig. 3. Comparison of SNR degradation due to quantization for = 0 and = .

is less than 0.25 dB. These theoretical results can provide useful de-sign references to determineb for the equal gain precoder with scalar quantization.

Next, let us answer the question: will the performance improve by usingi= i, i.e., extra feedback in1. Fig. 3 shows the performance loss due to quantization for₁= 0 and ₁= ₁according to (21) and (23), respectively. It is interesting to note that the performance with extra feedback for₁ is worse than that without extra feedback. The reason is explained as follows: the two solutions in (4) and (5) are ac-tually equivalent without quantization. However, with quantization, we need to quantize₁as well in (4). This demands extra bits to represent 1. If we have infinite bits to represent1, the two solutions would lead to the same performance. However, with limited feedback, solution in (4) is worse than that in (5). Please note that if we consider the optimal bit allocation for scalar quantization to each antenna as in [19], more bit budget will lead to better performance.

IV. SIMULATIONRESULT

The simulation is conducted using the following parameters: The channel is quasi-stationary and the channel coefficients are i.i.d. com-plex Gaussian random variables with zero mean and unit variance. The

Fig. 4. Performance comparison of the optimal precoder and the equal gain precoder without quantization.

Fig. 5. Scalar quantization effect of the equal gain precoders.

modulation level is 16-QAM. More than 60 000 channel realizations are used. We use the solution in (5). For description convenience, we letb be the number of bits to represent the phase of each transmit an-tenna (except the first anan-tenna) for the equal gain precoder. We letB be the number of total feedback bits. Thus, we have the relationship that B = (Nt0 1)b for the equal gain precoder.

1) Example 4: Comparison of the Optimal and the Equal Gain Pre-coders: To see the best performance that the optimal and the equal gain precoders can achieve, we do not quantize the precoding vectors p in this example. Fig. 4 shows the bit error probability (BEP) perfor-mance of these two precoders without quantization. We observe that the optimal precoder outperforms the equal gain precoder around 0.5 dB whenNt = 2. When Nt = 8, the gap is around 0.9 dB and when Nt > 16, the gap is around 1 dB. This result shows that the perfor-mance gap between the optimal and the equal gain precoders is around 1 dB, which corroborates the theoretical result in Theorem 1.

2) Example 5: Quantization Effect of the Equal Gain Precoder: Fig. 5 shows the BEP performance of the equal gain precoder due to quantization effect. We observe several interesting results: First, asN_t increases,b needs to be increased to achieve comparable performance without quantization. For instance, whenNt = 2, b = 2 can achieve comparable performance with its corresponding performance without

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Fig. 6. Performance comparison for STBC, equal gain (EG), Grassmannian (GS) and antenna selection (AS) precoders.

Fig. 7. Performance comparison for the (EG) precoder, and the Grassmannian (GS) precoder.

quantization, but whenNt = 8, b = 3 is required to achieve compa-rable performance. Moreover, we found the performance gap with and without quantization in general matches the results in Theorem 2 (also see Fig. 2). The exception is whenN_t= 4 and 8 with b = 1, where the assumption that the quantization error and the precoding coefficients are independent may no longer be valid.

3) Example 6: Comparison of Various Precoders With Quantiza-tion Effect: We compare the performance of the equal gain precoder, the Grassmannian precoder [3] and the antenna selection precoder. The performance comparison is shown in Fig. 6. For fair comparison, total required bits are shown for all precoders. To evaluate the performance improvement due to precoding, we also include the 22 1 STBC per-formance as shown in the solid-square curve. From the figure, we see that the three precoders have the same diversity gain and hence their slopes are the same. However, the Grassmannian and the equal gain precoders can achieve a better performance than the antenna selection precoder. Moreover, we see in this simulation case that with the same required total bitsB, the equal gain precoder can achieve comparable performance with the Grassmannian precoder (less than 0.2-dB perfor-mance gap).

To have a more detailed comparison between the Grassmannian and the equal gain precoders, let us use all available Grassmannian code-words found in http://cobweb.ecn.purdue.edu/~djlove/grass. The

per-formance curves for these two precoders are shown in Fig. 7. We see that forb = 1, i.e., B = Nt0 1, the equal gain precoder seems not perform well whenN_t > 2. For instance, when N_t = 3 and b = 1, i.e.,B = 2, the performance gap between these two precoders is up to 2.5 dB (see star and circle curves). However, asb 2, the equal gain precoder improves rapidly and can achieve comparable performance with Grassmannian. Although forb = 1 the equal gain precoder may not achieve comparable performance with the Grassmannian in some cases, it has several advantages as we mentioned in the introduction. Moreover, the equal gain precoder can be easily extended to arbitrary number of transmit antennas (see curves forNt = 16), where con-structing the codebook of the Grassmannian precoder may not be easy in this case [4].

ACKNOWLEDGMENT

The author would like to thank the anonymous reviewers for their constructive suggestions, which have significantly improved the quality of this work.

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