- Open Access
Throughput analysis of transmit-nulling SDMA with limited feedback
© Mun and Jo; licensee Springer. 2013
- Received: 26 March 2013
- Accepted: 6 November 2013
- Published: 20 November 2013
We recently proposed a precoder codebook for a transmit-nulling space-division multiple access (TN-SDMA) to share spectrum with existing wireless services. Since a portion of the spatial subspaces of a multiantenna broadcast channel is used to eliminate the interference to coexisting systems, TN-SDMA could benefit the efficiency of spectrum usage from the coexistence of different systems in the same band while it always yields lower throughput per unit bandwidth than the orthogonal SDMA (called per user unitary and rate control (PU2RC)) that utilizes all the spatial subspaces for data transmission. This study aims to theoretically quantify the throughput loss of TN-SDMA relative to PU2RC and to analyze the effect of the main system parameters (signal-to-noise ratio (SNR) and the numbers of transmit antennas, users, and feedback bits) on the throughput loss. We derive the theoretical upper bound of the throughput loss of TN-SDMA relative to PU2RC, with the same feedback bits (codebook size). The throughput loss is lower with more transmit antennas, fewer users, lower SNR, or fewer feedback bits. It is interesting to note that the throughput loss converges to an upper limit with an increase in the SNR, which indicates that the SNR has a comparatively minor effect on the throughput loss in the high-SNR region. We also derive the required additional number of feedback bits for TN-SDMA to achieve the throughput of PU2RC (i.e., zero throughput loss). We find that the throughput achieved is feasible at the cost of a practically small number of additional feedback bits.
- Spectrum Sharing
- Complementary Cumulative Distribution Function
- Precoding Matrix
- Codebook Size
The economical use (or reuse) of the radio spectrum is increasingly essential as the number of radio spectrum shortages has risen because of an explosive growth in traffic . In the background, high spectral efficiency (in bps/Hz) and smart spectrum sharing are becoming the key requirements of emerging (or future) wireless networks such as cognitive radio networks, femtocell networks, small cell networks, and International Mobile Telecommunications (IMT)-Advanced networks [2–5]. The use of multiple antennas in wireless networks has been of worldwide interest, and the resulting innovative techniques such as beamforming, single-user multiple-input multiple-output (MIMO), and multiuser MIMO (also named space-division multiple access (SDMA)) have been developed. Null-steering beamforming is prevalent for interference suppression in wireless communication and radar applications. An interest in SDMA is increasing because of its advantages over single-user MIMO , and furthermore, SDMA is considered as a high-data-rate solution for 3GPP Long-Term Evolution (LTE) and 3GPP LTE-Advanced . Therefore, this paper focuses on SDMA that shares the spectrum with other coexisting systems.
1.1 Multiple antennas for spectrum sharing and SDMA
Spectrum sharing is possible by the sufficient separation of radio resource dimensions in time, frequency, and space; for example, a wireless communication system adjusts system resources such as the transmit power [8, 9], operating frequency , and time of transmission . Furthermore, by not radiating the interference in a known direction of the coexisting systems, null steering can protect coexisting systems without additional radio resources in time or frequency [12, 13]. However, when a base station performs null steering without any use of multiple-antenna techniques for higher throughput, no downlink throughput gain caused by the usage of multiple antennas is observed owing to their focusing on mitigating interference toward the coexisting system.
Dirty paper coding (DPC) is non-causal and thus impractical, although it achieves the MIMO broadcast channel capacity . This clear finding has inspired the engineers in the field of wireless communications to develop numerous practical algorithms for SDMA [15–20]. In the industry, a codebook-based orthogonal beamforming SDMA has been proposed for the 3GPP-LTE standard  under the name per-user unitary rate control (PU2RC) and has been included in the 3GPP2-Ultra Mobile Broadband (UMB) standard . In this scheme, on the basis of limited feedback information on the preferred precoding matrix within a codebook and the corresponding signal-to-interference-and-noise ratios (SINRs), a multiuser precoding matrix is selected within a codebook to maximize the sum throughput. In , the performance of PU2RC is intensively analyzed and compared with that of zero-forcing SDMA. The orthogonal beamforming of PU2RC focuses on throughput improvement by reducing the inter-user interference in a homogeneous system. However, the SDMA systems that share the spectrum with other wireless systems require suppression of the interference between heterogeneous systems as well as the inter-user interference in a homogeneous system.
As introduced above, employing multiple antennas is desirable for both spectrum sharing and throughput improvement. A multiple-antenna technology that simultaneously accomplishes null-steering and orthogonal beamforming could achieve both a high data rate and spectrum sharing. More specifically, it could be considered that a part of orthogonal spatial subspaces provided by multiple antennas is dedicated to spectrum sharing, and the remainder (i.e., the corresponding null space) is allocated for data transmission. In [22, 23], we realized this concept explicitly by designing a transmit-nulling SDMA (TN-SDMA) codebook satisfying both null-steering and orthogonality constraints, where each precoding matrix comprises mutually N−1 (N denotes the number of transmit antennas) orthonormal vectors that are orthogonal to the array steering vector in the direction of a coexisting system. The codebook design proposed in [22, 23] ensures low complexity and small overhead as compared with the well-known Gram-Schmidt process.
In , simulation results show that the throughputa of TN-SDMA is always lower than that of PU2RC, with the same feedback bits, which is natural because not all of the orthogonal spatial subspaces of a broadcast channel are used for simultaneous data transmission in TN-SDMAb. However, it is still of importance to theoretically quantify (1) how large the throughput loss of TN-SDMA is relative to PU2RC; (2) how the throughput loss is affected by the signal-to-noise ratio (SNR) and the numbers of antennas, users, and feedback bits; and (3) how many additional feedback bits are necessary for TN-SDMA to achieve the PU2RC throughput, i.e., zero throughput loss, all of which are addressed in this paper.
We derive the throughput loss of TN-SDMA relative to PU2RC, with the same number of feedback bits in Theorems 1 and 2. TN-SDMA uses a codebook comprising multiple sets of orthonormal vectors and PU2RC scheduling with limited feedback. In this sense, our analysis is in the same spirit as the work of . However, we deterministically generate the precoding matrices with a systematic rule, whereas  randomly generates the matrices. This results in different statistics for the channel-shape quantization error from  as well as random vector quantization [24, 25]. Additionally, we adopt a precoding matrix comprising N−1 mutually orthonormal column vectors in for a transmitter with N antennas, thereby having a different received SINR from that in .
In Theorem 3, we further derive the required number of feedback bits for TN-SDMA to achieve the throughput of PU2RC, for a given number of feedback bits in PU2RC. From such a derivation, by varying the number of antennas, we examine the possibility of a practically small number of feedback bits that yields the throughput achievement of TN-SDMA. It provides design insights into feedback channels to handle specific overhead signaling requirements as well as throughput improvements.
For spectrum sharing (interference mitigation) of TN-SDMA, a subspace of the vector space of the MIMO broadcast channel is not used to transmit data, but the corresponding null space is only allocated for data transmission. Thus, the proposed analytical framework could be widely applied (or extended) to the cooperative (or non-cooperative) wireless networks that simultaneously transmit data through the null space of the interference channel matrix. The interference could include intra-cell or inter-cell interference in cellular networks or inter-system interference in heterogeneous (or cognitive radio) networks.
where is the channel gain vector with zero mean unit variance and z k is the complex additive Gaussian noise with unit variance. The channel gain vector h k has uncorrelated complex Gaussian entries. On the other hand, the highly correlated channel from a transmitter to other coexisting systems is assumed on the basis of the high line-of-sight probability between them. The highly correlated channel facilitates the mitigation of the interference to the coexisting systems by construction of a transmit null at the azimuth direction angle of the coexisting systems (we name the angle ‘AOC’). AOC is relative to the array broadside, as shown in Figure 1. , and s= [ s1…sN−1]T, where (·)T represents the transpose matrix operation, is an uncoded symbol vector that satisfies . The total transmit power P is equally allocated over N−1 scheduled users. The N−1 precoding vectors (beams) are selected within the TN-SDMA codebook with a size M=G(N−1), which consists of G orthonormal matrices . The beam and user selection algorithm is described in Section 2.2.
2.1 Systematic codebook
In this section, we briefly review the design of the TN-SDMA codebook presented in [22, 23] and its characteristic. Furthermore, we newly derive the exact value of the common null points that the codebook forms. Our design objective is to construct a codebook that satisfies two constraints: (1) construction of a transmit null at AOC and (2) orthogonal beamforming. The transmitter obtains AOC by adopting a popular spatial-spectrum estimation direction-finding method [26, 27] or from a database with information concerning the AOC. The AOC is then sent to all K users via a downlink control channel.
This means that the codebook size of is M=G N.
where c is any integer except multiples of N. Figure 2a shows an example of the transmit gain of with a common null point 41.8°. From here on, we omit the superscript (n) in to simplify the notation.
Next, we finally design the desired matrix from using the matrix that steers a transmit null in the direction of ϕ g to AOC and maintains the orthogonal beamforming in the following proposition, which is also presented in [22, 23].
where and .
where (a) and (b) are obtained from (3). □
The method ensures low complexity owing to the simple matrix product. Note that R g preserves orthogonality in contrast with conventional null steering . The codebook size of is M=G(N−1)=2 B , and each user feeds back B bits quantization of the channel. Figure 2b shows an example of the transmit power gain of N1(30°) where a transmit null at 41.8° is shifted to AOC 30°.
2.2 Beam and user selection with limited feedback
Here, we assume that the SINR is reported to the transmitter without quantization and is invariable during beam and user selection (i.e., no CQI delay) as ,, to investigate the effects of the quantized channel shape on the sum throughput.
In this section, we analyze the sum throughput of TN-SDMA. We first present preliminary calculations for the throughput analysis, on which the throughput loss of TN-SDMA relative to PU 2 RC is derived.
3.1 Preliminary calculations
As defined in Section 2.2, the quantization error of the k th user’s channel direction is, where and, respectively, are the original and quantized channel directions of the k th user. The quantization error of the codebook composed of multiple unitary matrices is well studied in . Whereas each unitary precoding matrix is independently and randomly generated in , TN-SDMA employs the precoding matrices designed systematically. Therefore, we get a different approach and result from those of [x.
See Appendix Appendix 1: proof of Lemma 1. □
where η=0.5772… denotes Euler’s constant,, and.
See Appendix Appendix 2: proof of Lemma 2. □
Note that in the proof of Lemma 2, we propose a tighter lower bound than the bound used in the proof of Lemma 3 in , which subsequently results in the tighter bound in Theorem 2.
3.2 Main results
and are assumed to have the same codebook size, M, i.e., the same number of feedback bits, B=⌈log2M⌉; thus, their numbers of precoding matrices ( for PU2RC and for TN-SDMA) are not equal. The throughput loss is derived for the three SNR regimes named high- (or interference-limited), low- (or noise-limited), and normal-SNR regimes. We now derive the throughput loss for the normal-SNR regime where SNR values are so moderate that both interference and noise are considerable.
See Appendix Appendix 3: proof of Theorem 1 and Corollary 1.
In (26), is a decreasing function of the number of transmit antennas N. Therefore, Theorem 1 states that the throughput loss decreases with an increase in N. This is because the ratio of the number of data streams in TN-SDMA to PU2RC,, increases to 1 as N increases. Theorem 1 also shows that a higher SNR P causes an increase in the throughput loss. This can be explained by the fact that PU2RC sends one more data stream than TN-SDMA, and the throughput of the additional data stream increases with P. For low P, the upper bound of the throughput loss is derived analogously as the following corollary.
Note that the throughput loss is still an increasing function of P.
See the last paragraph of Appendix Appendix 3: proof of Theorem 1 and Corollary 1. □
The throughput loss of the interference-limited system is given by the following theorem.
where η=0.5772… denotes Euler’s constant.
See Appendix Appendix 4: proof of Theorem 2. □
Theorem 2 states that the throughput loss eventually converges to an upper limit as SNR P increases, while the throughput loss increases with a codebook size M (or the number of feedback bits). This conclusion, in contrast with Theorem 1, is because at high P both the throughput of TN-SDMA and PU2RC in (28) depend on the quantization error sin2θ k (which is dependent on M) but not P. We also note that the throughput loss decreases with an increase in N, as shown in Theorem 1.
Theorems 1 and 2 are obtained for TN-SDMA and PU2RC using the same number of feedback bits, which always causes a throughput loss in TN-SDMA relative to PU2RC. Clearly, more feedback bits for TN-SDMA than PU2RC yield zero throughput loss. We quantify how many feedback bits are required for the zero throughput loss of TN-SDMA in the interference-limited or high-SNR regime. We let M E and M N denote the codebook sizes of PU2RC and TN-SDMA, respectively. They are given as M E =G E N and M N =G N (N−1), where G E and G N are the numbers of precoding matrices in PU2RC and TN-SDMA, respectively. The numbers of feedback bits are then given as (PU2RC) and (TN-SDMA).
and c=η log2e=0.8327….
See Appendix Appendix 5: proof of Theorem 3. □
From the above results, it is found that the upper bounds in Theorems 1 and 2 provide the relation for the throughput loss with P, N, and K accurately. In addition, the results indicate that the throughput loss of TN-SDMA relative to PU2RC is smaller at larger N, lower P, or smaller K. It should be noted that although TN-SDMA always has a lower throughput as compared to PU2RC in the non-coexistence scenario, it offers an opportunity for reusing the spectrum already allocated to coexisting systems. Consequently, when a gain of data rate (in bps) due to such a larger bandwidth transmission is superior to a loss of data rate relative to PU2RC, TN-SDMA will provide higher data rate than PU2RC does.
We have derived theoretical upper bounds of the throughput loss of TN-SDMA relative to PU2RC. On the basis of the bounds, we also quantify the number of feedback bits of TN-SDMA required to achieve the throughput of PU2RC. We find the resulting design fundamentals as follows. First, in terms of minimizing the throughput loss, TN-SDMA is better for point-to-multipoint communication with more transmit antennas, fewer receivers (users), lower SNR, or fewer feedback bits. Second, given a fixed number of transmit antennas and users, the throughput is affected by SNR to a greater extent than by the number of feedback bits in the normal (or low)-SNR region and vice versa in the high-SNR region. Therefore, using more feedback bits is recommended to increase the throughput of high-SNR users. Third, adding feedback bits in only single figures is sufficient for TN-SDMA to achieve the throughput of PU2RC in high-SNR or interference-limited networks. Further extension of this approach could include downlink network MIMO with limited feedback that mitigates inter-cell interference by transmitting data through the null space of an inter-cell interference channel matrix.
a Here, the throughput means spectral efficiency which is numerically expressed in bits per second per hertz or nats per second per hertz (1 nps/Hz = 1.44 bps/Hz).
b We need to note that in spite of the lower spectral efficiency, TN-SDMA is desirable in order to keep existing systems in operation.
c 3GPP LTE-Advanced base station is designed to support up to eight antennas.
Appendix 1: proof of Lemma 1
where (a) follows from (see the right figure of Figure 3) and (b) follows from (32).
Appendix 2: proof of Lemma 2
Combining the left inequality in (45) with (41), we obtain the desired lower bound.
where . This gives the desired upper bound.
Appendix 3: proof of Theorem 1 and Corollary 1
where (a) follows the asymptotic behavior of in [16, (A10)] and (b) is given on large K assumption.
where (a) follows from
and (b) is given from the last inequality in the proof of Proposition 1 in , where we use and the substitution of N t =N−1,, U=K, . (c) follows from the large K assumption, where and . From (48) and (5), we obtain the desired result in Theorem 1.
from which we obtain the desired result in Corollary 1.
Appendix 4: proof of Theorem 2
where (a) is obtained from the upper bound in Lemma 2 and (b) is given on large K assumption.
where (a) follows from the large K assumption. From (53) and (57), we obtain the desired result.
Appendix 5: proof of Theorem 3
Obviously, the zero upper bound of is sufficient for the zero throughput loss. Therefore, TN-SDMA with feedback bits given in (61) yields zero throughput loss, i.e., achieving the throughput of PU2RC.
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2013R1A1A1005731), and by the Ministry of Science, ICT & Future Planning (MSIP), Korea, in the ICT R&D Program 2013.
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