Minimum Euclidean Distance Based Precoders for ... - IEEE Xplore

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 60, NO. 3, MARCH 2012

Fig. 9. Comparison between theoretical tracking error estimates (solid lines) and real GPS L1 C/A measurements (markers) for two loop configurations for a range of SNR values.

A slight discrepancy between measured and theoretical curves is observed in the B! = 2.0 Hz case, for high SNRc values and is believed to be the result of residual local oscillator induced frequency instability. This extra source of error is not accounted for in the theoretical curve, yet will, of course, appear as a tracking error when a narrow (2.0 Hz) FLL cannot adequately track this frequency variation. V. DESIGN CONSIDERATIONS In light of the results presented in Section IV some performance enhancing design guidelines can be inferred. These guidelines strive to sustain loop performance in spite of the SNRc -induced effects presented in Section III by manipulating the loop parameters: TI and F (z ). First, gain compensation should always be employed. Specifically, the loop filter F (z ) should always be scaled by a factor of K1 . Failure to do so, especially under weak-signal conditions, can lead to unpredictable and degraded loop performance. From Fig. 8, it is clear that, at a certain value of SNRc , the performance undergoes a sudden degradation (approximately 13 and 12 dB for the four-quadrant arctangent and arctangent discriminators, respec[1] tively). This corresponds to the sudden reduction in KD and 0 R R [0] ! (Figs. 4 and 6), and the increase in Var(n ) (Fig. 5). For a given receives signal strength and noise floor, a designer can manipulate SNRc by varying TI , as shown in (5). The value of TI , therefore, should be chosen to provide a sufficiently large SNRc (greater than 13 dB, for example) to sustain high receiver performance. From a dynamic performance perspective, of course, a lower value of TI is attractive as it provides superior tracking capabilities and facilitates higher tracking bandwidth loops. Perhaps, then, TI may be chosen to situate SNRc at the knee of the variance curve of Fig. 8. When the above choice of TI is not possible, a designer may utilize (19) and (17) to predict the thermal noise-induced tracking error variance. The loop filter F (z ) may then be modified to maintain acceptable tracking error. In general, when the FLL operates in the low-SNRc region, tracking bandwidth must be sacrificed to curtail excessive tracking error. One final, perhaps obvious, remark is that, when operating in the low-SNRc region, use of the four-quadrant arctangent discriminator (if possible) will provide superior tracking performance, as compared to the arctangent discriminator.

REFERENCES [1] P. Misra and P. Enge, Global Positioning System, Signals, Measurements and Performance. Lincoln, MA: Ganga-Jamuna Press, 1996, vol. 2, 0-9709544-1-7.

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[2] H. Meyr and G. Ascheid, Synchronization in Digital Communication, ser. Wiley Series in Telecommunications and Signal Processing. New York: Wiley, 1990, vol. 1, Phase-, Frequency-Locked Loops, and Amplitude Control. [3] J. Curran, D. Borio, G. Lachapelle, and C. Murphy, “Reducing front-end bandwidth may improve digital GNSS receiver performance,” IEEE Trans. Signal Process., vol. 58, no. 4, pp. 2399–2404, Apr. 2010, 1053-587X. [4] A. J. Van Dierendonck, Global Positioning System: Theory & Applications. Reston, VA: AIAA, 1996, vol. 1, Progress in Astronautics and Aeronautics, ch. 8, pp. 329–408, 1-56347-106-X. [5] Understanding GPS: Principles and Applications, E. D. Kaplan, Ed. Norwood, MA: Artech House, 2006, vol. 1, ch. 5, pp. 179–194, 1-58053-894-0. [6] A. Razavi, “Carrier loop architectures for tracking weak GPS signals,” IEEE Trans. Aerosp. Electron. Syst., vol. 44, pp. 697–710, Apr. 2008. [7] F. D. Natali, “AFC tracking algorithms,” IEEE Trans. Commun., vol. COM-32, no. 8, pp. 935–947, Aug. 1984. [8] F. D. Natali, “Noise performance of a cross-product AFC with decision feedback for DPSK signals,” IEEE Trans. Commun., vol. 34, no. 3, pp. 303–307, Mar. 1986. [9] W. Hagmann and J. Habermann, “On the phase error distribution of an open loop phase estimator,” in IEEE Int. Conf. Commun. (ICC). Digital Technology—Spanning the Universe Conf Rec., Jun. 1988, vol. 2, pp. 1031–1037.

Minimum Euclidean Distance Based Precoders for MIMO Systems Using Rectangular QAM Modulations Quoc-Tuong Ngo, Olivier Berder, and Pascal Scalart

Abstract—From the feedback of the channel state information (CSI), precoding techniques improve the performance of multiple-input multiple-output (MIMO) systems by optimizing various criteria. In this correspondence, an efficient precoder that maximizes the minimum distance ( ) of two received vectors is studied. This criterion leads to a nondiagonal precoding scheme and allows achieving a full diversity order. However, the optimized solution for MIMO systems using a high-order QAM modulation is rather complex and changes for different constellations. Therefore, we propose herein a general form of minimum Euclidean distance based precoders for all rectangular QAM modulations. It is shown that the new solution optimizes the distance for small and large dispersive channels. Index Terms—Channel state information (CSI), linear precoder, minimum Euclidean distance, multiple input multiple output (MIMO), ML receiver, spatial multiplexing.

I. INTRODUCTION In recent years, multiple transmitter and receiver antennas are employed in wireless communications systems to adapt various demands of high-speed wireless links. These systems obtain large capacity and diversity gains in comparison with single transmitter and single receiver systems [1]. One of the multiple-input multiple-output Manuscript received January 21, 2011; revised June 14, 2011, October 07, 2011, October 31, 2011; accepted November 02, 2011. Date of publication December 02, 2011; date of current version February 10, 2012. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. TaSung Lee. The authors are with the University of Rennes I/IRISA, Lannion 22305, France (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this correspondence are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2011.2177972

1053-587X/$26.00 © 2011 IEEE

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(MIMO) techniques is spatial multiplexing which consists of transmitting simultaneously multiple independent data-streams. Through a feedback link, the channel state information (CSI) is readily available at the transmitter, and a precoding matrix can be designed to improve the performance of MIMO systems [2]. First, the MIMO channel is decoupled into many independent and parallel data-streams by using the singular value decomposition (SVD). The transmitted data vectors are then premultiplied by a precoding matrix. Various criteria can be used for channel characteristics optimization such as, for example, minimizing the mean-square error (MSE) [3], maximizing the received signal-to-noise ratio (SNR) [4], or maximizing the minimum singular value [5]. These optimized precoding matrices are diagonal in the virtual channel representation and belong to an important set of linear precoding techniques named diagonal precoders. The authors in [6] presented a nondiagonal structure that optimizes the Schur-convex functions of MSE for all channel substreams. Another efficient nondiagonal precoder which minimize the upper bound of pairwise error probability (PEP) when using arbitrary STBC over correlated Ricean fading channels is also illustrated in [7]. In the present correspondence, a nondiagonal linear precoder which maximizes the minimum Euclidean distance (max 0dmin ) between two received data vectors [8]–[10] is studied. This max 0dmin precoder obtains a large performance improvement in term of bit error rate (BER) compared to diagonal precoders. Unfortunately, the max 0dmin solution is only available for two independent data-streams with a loworder QAM modulation. That is due to the expression of the distance dmin that depends on the number of data-streams, the channel characteristics, and the modulation. The authors in [11] presented a design of a max 0dmin precoder which allows transmitting more than two independent data-streams, and increasing the 4-QAM alphabet to 16-QAM or 64-QAM modulations. However, this precoding technique is only suitable to quasi-stationary MIMO channels where a suboptimal solution is proposed by considering a block-Toeplitz form. We present, herein, an idea not only to reduce the complexity of the max 0dmin precoder but also to provide a significant enhancement of the minimum distance for all rectangular QAM modulations. For a two independent data-streams transmission, the MIMO channel is diagonalized by using a virtual transformation and the precoding matrix is obtained by optimizing the minimum distance on both virtual subchannels. Then, the optimized expressions can be reduced to two simple forms: the precoder F1 pours power only on the strongest virtual subchannel, and the precoder F2 uses both virtual subchannels to transmit data symbols. These precoding matrices are designed to optimize the distance dmin whatever the dispersive characteristics of the channels are. The expression of F1 depends on the order of the rectangular QAM modulation, while that of F2 does not change for all of the modulations. It can be demonstrated that these expressions optimizes the minimum distance for small and large dispersive channels. Furthermore, by decomposing the propagation channel into 2 2 2 eigenchannel matrices, and applying the new max 0dmin precoder for independent pairs of data-streams, a suboptimal solution for large MIMO systems (i.e. using more than two data streams) can be obtained [12].

where y is the b 2 1 received symbols vector, s is the b 2 1 transmitted symbols vector, is the nR 2 1 additive Gaussian noise vector, H is the nR 2 nT channel matrix, F is the nT 2 b precoding matrix, and G is the b 2 nR decoding matrix. When full CSI is available at both the transmitter and receiver, the channel can be diagonalized by using the virtual transformation [8]. The precoding and decoding matrices are then decomposed as F = Fv Fd and G = Gd Gv . The virtual MIMO channel representation is therefore

y = Hv Fd s + v

where Hv = Gv HFv is the b 2 b virtual channel matrix, v = Gv is the b 2 1 transformed additive Gaussian noise vector. In the correspondence, a maximum likelihood (ML) detection is considered at the receiver, and then, the decoding matrix Gd has no effect on the performance. Hence, Gd is consequently assumed to be an identity matrix of size b. Thanks to the singular value decomposition, the virtual channel matrix is diagonal and defined by Hv = diag(1 ; . . . ; b ), where i stand for every subchannel gain and are sorted in decreasing order. The precoding matrix Fd is designed under the power constraint trace

fFd F3d g = Es

(3)

where Es is the average transmit power. B. Minimum Euclidean Distance Based Precoder Firstly, let us denote S as the set of all possible transmitted symbols. The minimum squared Euclidean distance between two symbols at the receiver is then

2 dmin

=

min

s ;s 2S

kHv Fd (sk 0 sl )k2 = min kHv Fd xk2 x2X

Fd

min :

= arg max d F

(5)

The optimized solution is only available for a small number of data-streams because the received constellation size and the space of solution increase exponentially with b. The optimal expressions of max 0dmin precoders for two independent data-streams are presented in [8] and [9]. In this case, the virtual channel matrix can be parameterized as

Hv =

1 0

0

2

=

cos 0

0

A. Virtual Channel Representation The MIMO system with nT transmit, nR receive antennas and b independent data-streams over Rayleigh fading channel can be modeled as [3], [8] (1)

(6)

sin

where = 12 + 22 is the channel gain, and = arctan is the channel angle. One should note that 0 4 . Due to the symmetries of rectangular QAM modulation, the precoding matrix Fd can be represented as

p

Es

cos

0

0

sin

cos

sin

1

0 sin cos 0 with 0 , ' 2 and 0 4 . The parameter

II. SYSTEM OVERVIEW

(4)

is a difference vector defined by x = sk 0 sl with sk 6= sl . where x The precoding matrix Fd is obtained by optimizing the criterion

Fd

y = GHFs + G

(2)

0

ei'

(7)

controls the power allocation while and ' correspond to scaling and rotation of the received constellation, respectively. This precoder performs a significant BER improvement in comparison with other traditional precoders. Unfortunately, the exact expression of Fd is only available for BPSK, QPSK, and 16-QAM modulations [8], [9]. In the next section, we will present the simplified general form of the max 0dmin precoder for all rectangular QAM modulations.


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Fig. 2. Normalized minimum Euclidean distance. Fig. 1. Received constellation of the precoder F .

III. GENERAL EXPRESSION OF

4

= dx2 = dx2 , we obtain 1p 'F = arctan 2N + 3 F = arctan(2sin 'F ) :

By considering dx2

max 0dmin PRECODER

For a rectangular k -QAM modulation, the transmitted symbols belong to the set

1 fa + bi; a 0 bi;0a + bi; 0a 0 big M

Sp

= (4 0 1)

A. Precoder

2 (1 3 . . . 2 0 1)

F1

max0

When the dmin precoder pours power only on the first virtual . The precoding matrix in subchannel, it means that the angles (7) is, then, simplified as

=0

F1 =

pE cos sin ei' s 0 0

:

(9)

Fig. 1 shows the received constellation on the first virtual subchannel provided by a form of the precoder F1 . We observe that it can be divided into four regions with four corner points named as A, B, C, and D. When the angles and ' in (9) vary, these four regions are scaled and rotated, respectively. The distance dmin is optimized such that the nearest neighbors have the same distance. In other words, the triangle (C, D, E) which is created by three transmitted vectors

p10Mi ; 0Np+MNi

T

,

0p10i ; Np+Ni M M

The minimum Euclidean distance obtained by the precoder then

dF2

(8)

2 k and a; b ; ; ; k . where M 3 The linear precoder, which maximizes the distance dmin for two independent data streams, can be classified into two categories: pouring power only on the strongest or on both virtual subchannels. Let us denote these precoders as F1 and F2 , in respectively.

T

, and

(11)

2 = Es 2 M4 N 2 +cosp3N + 2 :

F1 is (12)

For a beamforming precoder, which has the same bit rate, i.e., M 0

2 (42k 0 1), the minimum distance is given by 3 dF2

2 = Es 2 M4 0 cos2 = Es 2 M4 N 2 +cos2N + 2 :

=

(13)

It is observed that the precoder F1 provides a slight improvement in term of dmin in comparison with the beamforming design. The normalized distance of our new precoder is plotted in Fig. 2, and its performance will be discussed in Section IV. B. Precoder F2

max 0

The authors in [9] presented the optimized dmin solution for a 16-QAM modulation. It has many expressions, and each expression corresponds to different interval of the channel angle . Let us consider the last expression, i.e., the precoder FT . The optimized expression is , ' , and depending on . This precoder is obtained with 4 4 denoted as F2 , in the correspondence, and is expressed as

=

p

=

F2 = 2Es cos0

0 sin

p2 1 + i 0p2 1 + i

:

(14)

approach shows that the optimized dmin provided by = 2 01, is equilateral. The normal- F2Aisnumerical always obtained by two difference vectors: difference vectors ized distances, E d (T ) , of2 the three corresponding 2 1 ; x5 = p2 1 : T 2 2 x1 = pM (0; i) , x2 = pM (1; 0N ) and x3 = pM (1; 0N + i)T , x4 = p 0 01 +i M M are given by The corresponding normalized distances are defined by dx2 = cos2 sin2 dx2 = cos2 [cos2 0 2cos sin N cos ' + N 2 sin2 ] 2 2 + 1 sin2 sin2 (10) dx2 = 12 cos 2 dx2 = cos2 cos2 0 2cos sin (N cos ' + sin ') p2 cos 2 2 2 + 2+p2 sin2 sin2 : 2 0 2 2 d = cos

cos x +(N + 1)sin : 2 2

0p10i ; N +(pN 02)i M M

T

, where N

k

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By solving the equation d2x = dx2 , we obtain F = arctan

p2 0 1 tan

:

(15)

We present, herein, an idea not only simplifying the expression of max dmin precoder, but also improving the performance in term of dmin . That is, for small dispersive channels, only precoding matrix F2 in (14) is used to transmit signals on both virtual subchannels. The minimum distance obtained by F2 is

0

dF2

p 2 2 2 4 (2 0 2) cos sin p = Es : M 1 + (2 0 2 2) cos2

First, we demonstrate that the precoder

(16)

F2 optimizes the distance

dmin when there is no dispersion between both virtual subchannels. Indeed, the minimum Euclidean distance of

= 4 is given by

F2 at the channel angle

4 1 dF2 j ==4 = Es 2 = Es 2 =M: M4

(17)

Proposition 1: When the channel angle = 4 , the maximum value = 4 or of dmin is given by EM , and obtained if and only if = 4 . Proof: See Appendix A. Fig. 2 shows the normalized minimum distance, i.e. dmin = 4Es 2 =M , obtained by the precoder F2 . It is observed that the maximum value of dF is obtained with = = max . The exact value of max can be determined by using the following calculus:

@ 2 d = 0; @ F

and

@2 2 d < 0: @ 2 F

cos

1

max = p ; 2

or

1

max = arccos p : 2

(19)

dF2 j =

In the case of large MIMO channels (b > 2), we can extend the previous solutions by decomposing the channel into 2 2 eigenchannel matrices and optimize the distance dmin for each pair of data-streams. The extension is split into four steps [12]: 1) Obtain the virtual diagonal matrix v by using a virtual transformation. 2) Associate 2b couples of singular values following the combination to obtain 2b 2-D virtual sub(1 ; b ); (2 ; b01 ) . . . ; systems. 3) Apply the optimal 2-D max dmin solution on each subsystem with the power constraint is equal to 1. 4) Allocate the power of each subsystem by the coefficient 7i such that

2

H

0

2

Proposition 2: For every channel angle max , the distance dmin provided by a precoding matrix Fd cannot exceed the distance dF in (16). Proof: See Appendix B. In other words, the optimized minimum distance, for every channel angle max , is only provided by the precoder F2 . Furthermore, the distance defined in (20) is the maximum value that a linear precoder can obtain (see Appendix C). These properties reaffirm that the proposed precoding matrix F2 is suitable to optimize the distance dmin on both virtual subchannels, especially when the channels SNRs are small dispersive.

C. Channel Threshold 0 Fig. 2 illustrates the normalized minimum distance, i.e.

p2 0 1 p : tan2 0 = p 2 p 2N + 6N + 2 0 1

i

7i = Es

(21)

k=1

01

1 k2

for i = 1; . . . ; b=2

(22)

where i is the minimum Euclidean distance of the subsystem #i given in Step 3. Note that the error probability can be approximated as [13]

Nd2

Pe

2 dmin 4N0

erfc

(23)

is the average number of all nearest neighbors, and N0 where Nd is the variance of the noise . Furthermore, the distance dmin provided by the proposed precoder is bounded as 2

2 dmin (max 0dmin ) Es 2 1

2

(24) 2

4p M (N + 3N +2) ,

where 1 = cos , 1 = and 2 = M . By using the upper and lower bounds of the largest eigenvalue 1 3 = [12], [14], we can demonstrate that of the Wishart matrix the diversity order of the max dmin precoder for large MIMO chanb b nels is equal to nT 2 + 1 nR 2 + 1 . For b = 2 data-stream transmission, the proposed precoder obtains a full diversity order, i.e., nT nR .

W HH 0

0

0

0

IV. PERFORMANCE OF max dmin PRECODER A. Comparison of Minimum Euclidean Distance First, we indicate the improvement of our new precoder in term of minimum Euclidean distance. For diagonal precoders, the distance dmin is obtained when two transmit vectors and are different from only a symbol. The minimum Euclidean distance between two symbols in (4) is then simplified as

s

dmin = 4Es 2 =M , obtained by two new precoding matrices F1 and F2 . It is observed that the precoder F1 provides the optimized dmin for small , while precoder F2 is valid for high values of the channel 2 = dF2 in (12) angle . For a given modulation, by considering dF and (16), we obtain the value of the optimal channel threshold 0 such that

2

Es 1 1 (20)

'

D. Extension for Large MIMO Channels

The minimum distance provided by F2 at max is then

p 4 4 201 p2 : = Es 2 sin2 max = Es 2 M M

'

'

(18)

By solving the equation of the first derivative and verifying the second derivative test, we obtain 2

When < 0 , the precoder F1 is used and the signal is transmitted over the strongest virtual subchannel. On the contrary, when

0 , the precoder F2 is chosen and both virtual subchannels are used to transmit signal. It is obvious that the higher the modulation order is (N increases), the less we use the precoder F1 , or in other words, the 17.28 for QPSK, 0 8.09 for 16-QAM, smaller 0 is (e.g., 0 and 0 3.95 for 64-QAM modulation).

2 dmin =

min

s;r2S;s6=r

Es

= Es min i fi2 i=1...b

=

b i

r

i fi2 jsi 0 ri j2 min

s;r2S;s6=r

4 Es min i fi2 i=1...b M

jsi 0 ri j2 (25)

F W HH

where f1 ; . . . ; fb are diagonal elements of matrix d , and 1 3 = . . . . b are the ordered eigenvalues of

2


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TABLE I COMPARISON OF THE MINIMUM EUCLIDEAN DISTANCES

Fig. 4. BER performance for a 64-QAM modulation.

Fig. 3. Normalized minimum Euclidean distance for 64-QAM.

Table I shows the distance dmin obtained by diagonal precoders in 0dmin precoder. The normalized distances, comparison with our Es 2 =M , for each precoder in the case of 64-QAM modi.e. dmin = ulation are illustrated in Fig. 3. One should note that, for diagonal schemes (e.g., minimum mean-square error (MMSE) [3], 0min [5], and water-filling [15]), the average transmit power is chosen large enough such that the power is allocated on both virtual subchannels. It 0dmin precoder brings a slight is observed that when 0 , the enhancement compared to beamforming design. This improvement remains constant for every channel angle and reduces for higher order 0min solution is better modulations. The performance of the than the water-filling and MMSE ones, but it is really outperformed by the proposed precoder.

4

max

max

max

max

B. Bit-Error-Rate Performance

max =2

This section illustrates the BER improvement of the new 0dmin precoder in comparison with other traditional precoding strategies. Let transmit and nR receive us consider a MIMO system with nT antennas. In this system, the symbols are separated into two independent data streams. The channel matrix is independently and independent identically distributed (i.i.d.) zero-mean complex Gaussian, while is zero-mean additive white Gaussian noise. Given the enhancement of the minimum Euclidean distance, we can expect a gain of our 0dmin precoder in term of BER compared to diagonal precoding strategies. Fig. 4 illustrates the BER performance with respect to SNR for a 64-QAM modulation. It is obvious that the 0dmin precoder obtains a significant BER improvement in comparison with diagonal precoders. This result clearly demonstrates that

=2

H

max

max

Fig. 5. BER performance for large MIMO systems.

our new precoder is particular suited for reducing BER when a ML detection rule is considered at the receiver. C. Performance for Large MIMO Systems In this section, we compare the performance of the proposed precoder with other sophisticated transceivers such as the linear precoder using decision feedback equalization (DFE) transceiver [16], the linear transceiver with bit allocation [17], the vector perturbation precoding scheme [18], and the minimum BER block design for ZF equalization [19]. Since the extension for large MIMO channel is obtained by decomposing the channel into 2 2 2 eigenchannel matrices and optimizing the distance dmin for each pair of data streams, 2b M 2 ML tests are implemented to optimize the minimum distances of 2b subsystems. In comparison with the sophisticated transceivers above, the ML complexity of our proposed precoder is higher (i.e., 2b M 2 compared to bM ). How0dmin precoder exhibits a higher ever, the extension of the 2-D diversity order than the other precoding strategies. For a given number of data-streams or equivalently for similar diversity order, the BER en0dmin precoder is significant. hancement obtained by our new Fig. 5 illustrates the BER performance for MIMO (4,4) systems using 16-QAM modulation. The comparison of our proposed precoder and

max

max

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other schemes shows that the BER performance is significantly enhanced at high SNR. A gain of about 4 dB is observed at high SNR in comparison with other precoding schemes. V. CONCLUSION In this correspondence, we introduced a new general expression of the minimum Euclidean distance based precoder for MIMO systems using rectangular QAM modulations. For two independent 0dmin precoder has two expressions: data-streams, the proposed F1 pours power only on the strongest subchannel, and F2 uses both virtual subchannels. It is demonstrated that our general form provides the optimized minimum distance for small and large dispersive channels. The distribution of both precoding matrices depends on the channel characteristics and the number of antennas used at the transreceiver. The more dispersive the virtual subchannels are, the less the precoder F1 is used. By decomposing the channel into 2 2 2 eigenchannel matrices, a suboptimal solution for large MIMO systems is proposed by using our new 0dmin precoder. Simulation results show that the proposed precoder offers a large improvement in term of BER compared to other precoding strategies.

max

max

;

+ sin2 sin2 ] 0 [cos2 cos2 0 sin2 sin2 ]2 sin cos cos ' p [cos2 cos2 + sin2 sin2 ] 0 2 db2 0 sin2 sin2 p p cos2 cos2 + ( 2 + 1)sin2 sin2 0 2dF2 : p One should note that ( 2 + 1)sin2 cos2 sin2 , for all

values of in the range of the following: a) if > 2

4 max . For this reason, we obtain

da2 + db2 < cos2 cos2

2 + sin2 sin2 2 = 2dF2

;

2 p p 2 2 dc < cos cos2 2 + ( 2 + 1)sin2 sin2 2 = ( 2 + 1)dF2 p201 where 2 = arctan tan

is the power allocation parameter of the precoding matrix F2 . In conclusion, the distances da2 , db2 , and dc2 cannot be all greater than dF2 .

b) if