Email: {kallu, vur}@hyd.hellosoft.com. AbstractâLow complexity decoding schemes are presented for combined space time block coding and V-BLAST options.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.
Low Complexity Decoders for Combined Space Time Block Coding and V-BLAST G. Sandeep and C. Ravi Teja
G. Kalyana Krishnan and V.U. Reddy
International Institute of Information Technology Gachibowli, Hyderabad 500032, India Email: {sandeep gogineni, ravitejachinta} @students.iiit.ac.in
Hellosoft India Pvt. Ltd., 8-2-703, Road No.12, Banjara Hills Hyderabad 500034, India Email: {kallu, vur}@hyd.hellosoft.com
Abstract— Low complexity decoding schemes are presented for combined space time block coding and V-BLAST options in Enhanced Wireless Consortium draft [1] for IEEE 802.11n. We exploit the structure of the transmitted data in developing the schemes. They are introduced for a frequency flat fading scenario and the diversity orders they yield are analyzed. The techniques are extended to orthogonal frequency division multiplexing mode of baseband modulation, and the impact of spatial correlation among the antennas is studied. Simulations are used to validate the predicted performance of the proposed schemes, predicted from the diversity order analysis. Also, their performance is compared with that of Zero Forcing Successive Interference Cancellation (ZF-SIC). Computational complexities of the proposed as well as that of ZF-SIC are evaluated. The results show that one of the proposed schemes performs like ZF-SIC while its computational complexity is about one-half of that of ZF-SIC. Simulations with TGn channels [2], which are frequency selective channels and have spatial correlation, show that the performance degrades with increasing correlation.
I. I NTRODUCTION Information theoretic results show that Multiple Input Multiple Output (MIMO) systems achieve high spectrum efficiency in rich multi-path scattering environment [3]. Vertical Bell Labs Layered Space Time (V-BLAST) [4] is a MIMO spacetime architecture which achieves a significant portion of this capacity, and therefore, is expected to be core of upcoming high data rate wireless communication systems. In a pure VBLAST, uncoded parallel data streams are transmitted from different transmit antennas simultaneously. Low complexity receivers such as Zero Forcing with Successive Interference Cancellation (ZF-SIC) [4] and Minimum Mean-Square-Error with SIC (MMSE-SIC) [5] were proposed for decoding VBLAST (see also [6]). The fundamental phenomenon which makes reliable wireless communication difficult is time-varying multipath fading. One way to reduce the effect of multipath fading is to provide antenna diversity. This is easier to provide at the transmitter, typically a base station. Combining multiple antennas with space time coding gives transmit diversity which can be gainfully exploited to improve the signal quality at the receiver. Recently, a simple transmit diversity scheme was proposed by Alamouti [7]. Enhanced Wireless Consortium (EWC) draft [1] for IEEE 802.11n defines Space Time Block Coding (STBC) options
combined with V-BLAST at the transmitter. In one such option consisting of three transmitting antennas, the data stream to be transmitted is first demultiplexed into two streams. One stream is encoded using Alamouti space time code [7] and mapped to first two transmit antennas, while the other stream is directly mapped to the third transmit antenna. We consider the down-link case where the mobile station (MS) is receiving. To reduce the cost and power required by MS, the number of antennas in the MS is kept to minimum possible, which is 2. Thus, we have a system with 3 transmit antennas (Mt = 3) and 2 receive antennas (Mr = 2). We develop decoding schemes for this important scenario assuming flat fading, analyze the diversity orders they yield, and suggest a modification, at a cost of marginal increase in complexity, for improved performance. Instead of using post-detection Signal to Noise Ratio (SNR) ordering [4], we exploit the structure of the transmitted data, or equivalently, of the channel that inherits this structure, and use predetermined ordering which helps us to reduce complexity drastically. Simulations are used to validate the predicted performance of the developed schemes. The performance and computational complexity of the proposed methods as well as that of ZF-SIC are evaluated. The results show that one of the proposed schemes (Method 3) performs like ZF-SIC while its computational complexity is about one-half of that of ZF-SIC. The schemes are extended to OFDM case as OFDM converts a frequency selective channel into a flat fading over each subcarrier [8]. Simulations, conducted for frequency selective channels based on TGn channel models [2] with spatial correlation, show that the performance degrades with increasing correlation. Section II presents signal model for frequency flat and frequency selective fading channels. The decoding schemes for frequency flat fading scenario are introduced in Section III, and diversity orders which they yield are analyzed in Section IV. Implementation complexity of the proposed schemes and simulation results are discussed in Sections V and VI, respectively. Finally, Section VII concludes the paper. Throughout this paper, lower case and upper case letters correspond to time domain and frequency domain variables, respectively and, lower case and upper case bold letters correspond to vectors and matrices, respectively. (.)H denotes conjugate transpose of (.), (.)∗ denotes complex conjugate of (.) and E(.) denotes expectation of (.).
1525-3511/07/$25.00 ©2007 IEEE Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on August 11, 2009 at 16:00 from IEEE Xplore. Restrictions apply.
583
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.
II. S IGNAL M ODEL
Equation (1) can be rewritten as
y1 (1) y1∗ (2) y (1) 2 y2∗ (2)
We first introduce the signal model for frequency flat fading channel case.
In view of the Alamouti scheme [7], we work with the data received in two consecutive symbol intervals. Table I gives the data transmission format according to [1]. s1 and s2 correspond to Alamouti stream while s3 and s4 belong to multiplexed stream. The rows correspond to transmit antennas. As we are sending 4 data symbols in 2 symbol intervals, the code rate is 2. Let yi (l) be the received data at ith receiver in lth symbol interval, hij be the channel tap between ith receiver and j th transmitter, and vi (l) be the complex additive Gaussian noise at the ith receiver during lth symbol interval. Under a frequency flat fading scenario, the received signal for this 3×2 MIMO system over two consecutive symbol intervals is given as
y1 (1) y2 (1) y1 (2) y2 (2)
=
=
h11 h21
h12 h22
h13 h23
h11 h21
h12 h22
h13 h23
s1 −s∗2 + v1 (1) v2 (1) s3 s2 s∗1 + v1 (2) (1) v2 (2) s4
The tap values are assumed to be independent circularly symmetric complex Gaussian random variables with zero mean and unit variance, and fading is assumed to remain constant over two consecutive symbol intervals. We also assume that the transmitted data from all the antennas are independent and equally likely, and also independent of the noise. Without loss of generality, the average total transmitted power is assumed to be unity. The normalization factors are chosen such that E(|s1 |2 ) = E(|s2 |2 ) = 14 and E(|s3 |2 ) = E(|s4 |2 ) = 12 . Note that the power allocation is not specified by [1], but is chosen here to ensure equal power allocation to the Alamouti and multiplexed streams. The additive noise terms are drawn from independent circularly symmetric zero mean complex Gaussian process with variance σ 2 . We assume perfect channel state information at the receiver. SNR is defined as the ratio of average received signal power to noise variance at each receive antenna, which is σ12 in view of the above assumptions. TABLE I DATA TRANSMISSION FORMAT WITH STBC OPTION FOR A 3 X 2 SYSTEM IN EWC DRAFT FOR 802.11 N s1 −s∗2 s3
h11 h∗12 h 21 h∗22
=
y
A. Frequency flat fading channel
−→TIME s2 s∗1 s4
−h12 h13 h∗11 0 −h22 h23 h∗21 0
v1 (1) ∗ v1 (2) v (1) 2 v2∗ (2)
+
H
0 s1 h∗13 s∗2 0 s3 h∗23 s∗4 s
(2)
v
We use the equivalent channel model as depicted by (2) in developing the proposed decoding schemes. Equation (2) can be expressed compactly as y
= h1 s1 + h2 s∗2 + h3 s3 + h4 s∗4 + v
(3)
where hi denotes i column of H. We assume that si is drawn from a constellation A = {a1 , . . . , aγ } of size γ with different normalizations for the Alamouti and multiplexed streams. th
B. Frequency selective fading channel As OFDM converts a frequency selective fading channel into a frequency flat fading channel over each subcarrier, (2) may be modified to include a subcarrier index and rewritten as Y1 (1, k) Y1∗ (2, k) Y (1, k) 2 Y2∗ (2, k)
y(k)
=
H11 (k) ∗ (k) H12 H (k) 21 ∗ (k) H22
−H12 (k) H13 (k) ∗ (k) H11 0 −H22 (k) H23 (k) ∗ (k) H21 0
s1 (k) ∗ s2 (k) s (k) + 3 s∗4 (k)
s(k)
0 ∗ (k) H13 × 0 ∗ H23 (k)
H(k)
V1 (1, k) ∗ V1 (2, k) V2 (1, k) V2∗ (2, k)
(4)
V(k)
where k is the subcarrier index and Hij (k) denotes the tap value in k th bin for the channel between ith receiver and j th transmitter. Yi (l, k) and Vi (l, k) denote the received data and noise, respectively, in k th bin at ith antenna in lth OFDM symbol interval. The tap values are assumed to be circularly symmetric complex Gaussian variables with zero mean and unit variance. The additive noise terms are drawn from independent circularly symmetric zero mean complex Gaussian process with variance σ 2 . III. D ECODING S CHEMES In this section, we present the decoding schemes for frequency flat fading channel. As mentioned in Section II-B, these techniques can be applied in each bin when we use OFDM mode for baseband modulation. The decoding schemes are based on the ZF scheme [6] commonly used for VBLAST detection. In ZF, the desired data point is decoded by projecting the received vector onto a space orthogonal to all other data points, thereby nulling out the other data points. Further, to improve the decoding performance, the previously decoded data can be cancelled out from the received vector so
584 Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on August 11, 2009 at 16:00 from IEEE Xplore. Restrictions apply.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.
that the next data point to be decoded sees less interference. This is called SIC [6]. The schemes are compared on the basis of the diversity orders in Section IV. A. Decoding multiplexed stream first (Method 1) In this case, the Alamouti symbols are nulled and the symbols s3 and s4 are decoded. This is done as follows. sˆ3 sˆ∗4
= =
[0 0 1 0]H−1 y [0 0 0 1]H−1 y
(5)
Projections of h3 and h4 onto the space orthogonal to h1 and h2 are orthogonal to each other (this is shown in Section IV). Hence, there is no interference of s3 on s4 , and therefore, after computing sˆ3 we do not perform SIC before decoding s4 . After decoding the multiplexed stream, they are cancelled from the vector y (i.e., we perform SIC). The modified ˜ is received data vector y ˜ y
= y − h3 sd3 − h4 s∗d 4
=
sˆ∗2
=
˜ hH 1 y H h1 h1 ˜ hH 2 y H h2 h2
(7)
B. Decoding Alamouti stream first (Method 2) In this case, the multiplexed symbols are nulled and the Alamouti stream is decoded as given below. sˆ1 sˆ∗2
= =
[1 0 0 0]H−1 y [0 1 0 0]H−1 y
= y − h1 sd1 − h2 s∗d 2
(8)
=
sˆ∗4
=
¯ hH 3 y hH h 3 3 ¯ hH 4 y hH h 4 4
= y − h3 sd3 − h4 s∗d 4
˘ y
(11)
The Alamouti stream is now decoded as sˆ1
=
sˆ∗2
=
˘ hH 1 y hH h 1 1 ˘ hH 2 y hH h 2 2
(12)
Note that though (11) and (6) are similar, the probability of decoding sˆ3 and sˆ∗4 correctly is not same in Method 1 and Method 3. This fact is brought out in Section IV where it is shown that multiplexed stream gets a diversity order of 2 in Method 2 while it is less than 2 in Method 1. IV. D IVERSITY O RDER A NALYSIS For the techniques described in Section III, we analyze the multiplexed and Alamouti streams separately to obtain the diversity orders of these streams. The diversity orders are obtained from the post-detection SNR as in [6] and [9]. We first show that the projections of h3 and h4 onto the space orthogonal to h1 and h2 are orthogonal to each other. Let P be the projection matrix orthogonal to the space of h1 and h2 . Then, we need to show that (Ph3 )H (Ph4 ) = 0. As h hH h hH h1 and h2 are orthogonal, P is given by P = I− h1H h11 − h2H h22 . 1 2 Then, Ph3
(9)
where sd1 and s∗d are the decoded values of sˆ1 and sˆ∗2 , 2 ¯ contains respectively. Assuming that sd1 and s∗d 2 are correct, y signal part contributed by s3 and s4 only. As h3 and h4 are orthogonal to each other, s3 and s4 are extracted as sˆ3
In this scheme, the decoding scheme of Method 2 is implemented first. i.e., an initial estimate of the Alamouti stream is first obtained as in (8). This initial estimate is used to perform SIC and the multiplexing streams are decoded as in (10). The decoded multiplexed stream symbols are cancelled from the original received vector y giving the modified received data vector as
A. Decoding multiplexed stream first (Method 1)
Projections of h1 and h2 onto the space orthogonal to h3 and h4 are orthogonal to each other (see Section IV). Hence, there is no interference of s1 on s2 , and we do not perform SIC while decoding s2 . After decoding s1 and s2 , they are cancelled from y. The modified received data vector is ¯ y
C. Decoding Alamouti stream twice (Method 3)
(6)
where sd3 and s∗d are the decoded values of sˆ3 and sˆ∗4 , 4 respectively. Assuming that sd3 and s∗d 4 are correct, the signal ˜ contains contributions from s1 and s2 only. Since part of y h1 and h2 are orthogonal to each other (see (2)), s1 , s2 are decoded as sˆ1
using post-detection SNR as the ordering criterion, we used the structure of the channel model as depicted by (2) to decode the data streams. In Method 1, the multiplexed stream is decoded first, and in Method 2, the Alamouti stream is decoded first. We now present an enhancement to Method 2.
Ph4
h1 (hH h2 (hH 1 h3 ) 2 h3 ) − H H h1 h1 h2 h2 h ) (hH h1 (hH h 4 2 1 2 h4 ) = h4 − − Hh hH h h 1 1 2 2 = h3 −
H H We note from (2) that hH 2 h2 = h1 h1 and h2 h1 = 0. H H H Similarly, h3 h3 = h4 h4 and h3 h4 = 0. Combining these relations with (13), we can show that
(Ph3 )H (Ph4 ) = − (10)
While we have not seen any literature explicitly referring to Methods 1 and 2, they are variants of ZF-SIC. Instead of
(13)
H H H (hH 3 h1 )(h1 h4 ) + (h3 h2 )(h2 h4 ) (14) hH 1 h1
Substituting the corresponding vectors from (2) in (14) and simplifying, we obtain (Ph3 )H (Ph4 )
=
0
(15)
585 Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on August 11, 2009 at 16:00 from IEEE Xplore. Restrictions apply.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.
Now, multiplying (3) with P and noting that P projects onto the space orthogonal to h1 and h2 , we have Py
= Ph3 s3 + Ph4 s∗4 + Pv
(16)
B. Decoding Alamouti stream first (Method 2) Following the analysis used in Method 1, we can show that (using the same symbols for P, Q, h for convenience)
Since Ph3 and Ph4 are orthogonal and PH P = P, we get from (16) sˆ3
=
hH 3 Py hH 3 Ph3
=
hH 4 Py hH 4 Ph4
(17)
Similarly, sˆ∗4
(18)
sˆ3 = s3 +
hH 3 Pv hH 3 Ph3
(19)
It is then easy to show that the post-detection SNR of sˆ3 2 σm 2 is given by hH 3 Ph3 σ 2 , where σm is the average power 2 of the multiplexed stream and σ is the variance of the noise. Let Q be a matrix whose columns are an orthonormal basis for P. Since P has rank 2 (because the space orthogonal to h1 and h2 has dimension 2), Q is a 4 × 2 matrix. Then, the post-detection SNR of sˆ3 may be written 2 2 σm H 2 σm H as hH 3 QQ h3 σ 2 = ||h3 || σ 2 , where h3 = Q h3 . We note that h3 is a vector of Gaussian random variables of dimension 2. However, of this vector are not
the elements H H = Q Q = I. As E h h independent, since E h3 hH 3 3 3 the number of independent χ22 random variables corresponds to diversity order ( Ch. 3 in [6]), the above result implies that the diversity order is less than 2. When the elements of h3 are fully correlated, the diversity order falls to unity. Similar analysis holds for sˆ4 too. Thus, the multiplexed stream will get a diversity order between 1 and 2. Once the multiplexed stream is cancelled out, and assuming that the corresponding symbols are decoded correctly, the system will only have Alamouti stream and (6) reduces to ˜ = h1 s1 + h2 s∗2 + v y
(20)
Hence, sˆ1
=
sˆ2
=
˜ hH 1 y hH h 1 1 ˜ hH 2 y H h2 h2
(21)
From (21), we obtain the post-detection SNR of sˆ1 as σ2 ||h1 ||2 σa2 , where σa2 is the average power of each data in the Alamouti stream. Since the elements of vector h1 are independent and identically distributed Gaussian random variables (see Section II), ||h1 ||2 is a sum of 4 independent χ22 random variables. The same analysis holds for sˆ2 . Thus, the Alamouti stream will get a diversity order of 4 in the absence of error propagation. We may point out here that the overall symbol error rate of the method, however, is determined by the data stream with lower diversity order.
=
sˆ∗2
=
hH 1 Py hH 1 Ph1 hH 2 Py hH 2 Ph2
(22)
where P now represents the projection matrix orthogonal to the space of h3 and h4 . Also, we have sˆ1
In view of (15), we can express sˆ3 as
sˆ1
sˆ∗2
hH 1 Pv H h1 Ph1 hH Pv = s∗2 + H2 h2 Ph2 = s1 +
(23) σ2
Note that the post detection SNR of sˆ1 is given by ||h1 ||2 σa2 . Here, h1 = QH h1 , where Q is matrix whose columns are an orthonormal basis for the space orthogonal to h3 and h4 . In the present case,
= QH E h1 hH (24) E h1 hH 1 1 Q=I
because E h1 hH = I. Since Q is a 4 × 2 matrix, the 1 diversity order for sˆ1 is 2. Similar analysis holds for sˆ2 . Thus, the diversity order for the Alamouti stream will be 2. Once the Alamouti stream is cancelled out, the multiplexed stream gets full receive diversity, which is 2, in the absence of error propagation. C. Decoding Alamouti stream twice (Method 3) In this scheme, Method 2 is implemented first. Then, the decoded multiplexed stream is cancelled, and the Alamouti stream is decoded again. In the absence of error propagation, the scheme offers the multiplexed stream a diversity order of 2 (from the discussion in Section IV-B), and the Alamouti stream a diversity order of 4 (see the discussion in Section IV-A). The diversity orders of the Alamouti and multiplexed streams for the three decoding schemes are shown in Table II, where 1 < D < 2. From these results, we predict that the overall Bit Error Rate (BER) will be least for Method 3, followed by Method 2 and then by Method 1. V. C OMPLEXITY While evaluating the complexity of the three decoding schemes, we exploit the fact that h1 and h2 are orthogonal and the same for h3 and h4 , and also note that the vectors h1 and h2 contain same elements, and the same is the case with h3 and h4 . We estimate the computational complexity as the number of complex multiplications. For the purpose of comparison, we compare our schemes with ZF-SIC, which is regarded as a low complexity receiver for V-BLAST.
586 Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on August 11, 2009 at 16:00 from IEEE Xplore. Restrictions apply.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.
TABLE II D IVERSITY ORDERS OF THE DECODING SCHEMES ASSUMING NO ERROR PROPAGATION (1 < D < 2) (MUX STREAM REFERS TO MULTIPLEXED STREAM )
Method 1 2 3
Mux stream D 2 2
Alamouti stream 4 2 4
TABLE III C OMPUTATIONAL C OMPLEXITY OF D IFFERENT SCHEMES Method 1 2 3 ZF-SIC
Complex multiplications 72 72 94 160
1
A. Decoding multiplexing stream first (Method 1) The multiplexed and Alamouti streams are decoded using (17), (18) and (21). Computation of Py is performed as, computing the terms in the braces first, H H H (25) Py = y − h1 hH 1 y/h1 h1 − h2 h2 y/h2 h2 , which requires 22 multiplications including 4 for evaluating H hH = hH 1 h1 . Note that h1 h1 2 h2 . To evaluate (17), we need 27 multiplications where hH 3 Ph3 is evaluated as Ph3 followed by hH (Ph ). Equation (18) requires 5 complex 3 3 multiplications where we exploit the relation that hH 3 (Ph3 ) = ˜ (Ph ). Computation of y requires 8 multiplications while hH 4 4 for evaluating sˆ1 we need 5, and similar number for sˆ2 . Thus, the total number of multiplications required by Method 1 is 72. B. Decoding Alamouti stream first (Method 2) The complexity of this scheme is approximately same as that of Method 1. We, therefore, take the number of multiplications as 72. C. Decoding Alamouti stream twice (Method 3) In addition to Method 2, Method 3 involves computation ˘ , sˆ1 and sˆ2 (see (11) and (12)) which require a total of of y 22 multiplications. Thus, the total number of multiplications required by this method is 94. D. ZF-SIC ZF-SIC can be implemented efficiently using QR decomposition [10] followed by back substitution. However, this assumes a prior ordering of the data based on post-detection SNR. A low complexity method is suggested to obtain postdetection SNR based ordering for the ZF-SIC. The technique is very similar to the one used in [11] for MMSE-SIC. Consider an M × M channel matrix where M data points have to be decoded using M received data. First, obtain a QR decomposition of H where Q is of size M × M with orthonormal columns and R is a square upper triangular matrix of size M × M . The mean square error for ZF is given by PM = (HH H)−1 which can be simplified as PM = (R−1 (R−1 )H ). Thus, R−1 is a square root factor of 1 2 P. Denote R−1 by PM , where the subscript indicates the size 1 2 of the matrix. Thus, minimum norm row of PM corresponds to 1 2 the data layer with maximum post-detection SNR. While PM has to be calculated explicitly using back substitution (R is an
2 upper triangular matrix), PM −1 is calculated recursively as in (see Section 3, Claim 2 in [11]). Note that before obtaining 1 1 2 2 PM −1 , rows of PM should be rearranged so that the chosen 1
2 layer is brought to the last row. Thus, using PM −i for each stage of iteration, we can determine the data ordering on the basis of post-detection SNR. Using this approach, ZFSIC involves roughly 52 M 3 complex multiplications (after neglecting the terms with lower powers of M ). For the underlying problem (M =4), the complexity required is 160 multiplications.
VI. S IMULATION RESULTS A. Frequency flat fading channel The channel and noise assumptions are same as that used in section II-A. 200000 realizations of the channel are used in the simulations. The total transmitted power , the power allocation to the different streams and the SNR definition are same as in section II-A Figure 1 shows the overall Bit Error Rate (BER) performance of the three schemes with QPSK constellation. For the purpose of comparison, we have also evaluated the BER performance of ZF-SIC, as applied to the equivalent received signal model (2), and the results are superimposed in Fig. 1. The plots show that the three schemes, presented in the paper, perform as predicted from the diversity analysis, i.e., Method 1 performs poorer compared to Method 2 and Method 3 performs better than Method 2. Further, Method 3 performs slightly better than ZF-SIC at lower values of SNR while the performance difference between the two vanishes as the SNR increases. Note that the computational complexity of ZF-SIC is significantly higher than that of Method 3 (see Table III). B. Frequency selective channel In practice, channels have multiple taps and undergo frequency selective fading. When we use OFDM, each bin sees a flat fading channel and the techniques of Section III can be readily applied. We have considered TGn channel models [2] in the simulations. The authors of [2] propose six channel models (A-F), of which model A is a single tap channel. The channel models also introduce correlation among the channel taps corresponding to different antennas. This implies that the channel matrix for each bin, defined in (4), will have correlated entries. To understand the effect of correlation on the decoding performance, we have considered TGn channel models B and C [2] corresponding to non-line of sight case, which have different correlation, in our simulations . For the purpose
587 Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on August 11, 2009 at 16:00 from IEEE Xplore. Restrictions apply.
This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.
of a matrix depends on the mutual correlation among the elements, we evaluated the condition number of the channel matrix (4) for each realization and computed the average over 2000 realizations. The resulting values are 3.59, 3.77 and 5.50 for JTC residential B channel, TGn channel models B and C, respectively. The results show that the performance degrades with increasing condition number. Note that the condition number goes up with increasing correlation. VII. C ONCLUSION
Fig. 1.
Performance of different decoding schemes
of comparison, we included results for the JTC residential channel model B [13], which is a frequency selective channel with a delay spread of 350 nano seconds and which is typically used in evaluating the performance of 802.11a WLAN system. For the JTC residential channel model B, no correlation is assumed among the channel taps corresponding to different antennae. We used 2000 different realizations of the channel, with a burst of 50 OFDM symbols for each realization. The time domain channel tap powers are normalized such that each entry in (4) has a unit variance. The noise variance at each receive antenna in each bin is σ 2 . The SNR is varied by changing the noise variance. The BER is evaluated for the decoding scheme of Method 3 for the three channel models and the results are shown in Figure 2. Since the condition number
Fig. 2.
Performance of Method 3 depicting the effect of spatial correlation
We have presented three methods for decoding the combined STBC and V-BLAST option in EWC ([1]) draft for IEEE 802.11n, and analyzed the diversity orders they yield. Simulation results corroborate the performance predicted by the diversity orders of the methods. We evaluated the computational complexity of the methods and compared it with that of ZF-SIC. The BER performance of Method 3 is nearly same as that of ZF-SIC while its complexity is about one-half of that of ZF-SIC. The schemes are applied to the OFDM mode and the TGn channel models, and the results bring out the performance degradation due to spatial correlation. R EFERENCES [1] HT PHY Specification. (2005) Enhanced Wireless Consortium. [Online]. Available: http://www.enhancedwirelessconsortium.org/home/ EWC PHY spec V127.pdf [2] V. Erceg et al., “Tgn channel models,” IEEE 802.11-03/940r4, May 2004. [3] E. Telatar, “Capacity of multi-antenna gaussian channels,” European Transactions on Telecommunications, vol. 10, pp. 585–595, Nov.-Dec. 1999. [4] P. Wolniansky, G. J. Foschini, G. D. Golden, and R. A.Valenzuela, “VBLAST: An architecture for realizing very high data rates over the rich scattering wireless channel,” in Proceedings URSI International Symposium on Signals, Systems and Electronics, New York, USA, 1998, pp. 295–300. [5] G. J. Foschini, G. D. Golden, R. A.Valenzuela, and P. Wolniansky, “Simplified processing for high spectral efficiency wireless communication employing multi-element arrays,” IEEE J. Select. Areas Commun., vol. 17, pp. 1841–1852, Nov. 1999. [6] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. [7] S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [8] R. V. Nee and R. Prasad, OFDM For Wireless Multimedia Communications. Artech House Publishers, 2000. [9] S. Loyka and F. Gagnon, “Performance analysis of the V-BLAST algorithm: An analytical approach,” IEEE Transactions on wireless communications, vol. 3, pp. 1326–1337, No.4, July 2004. [10] G. Golub and C. V. Loan, Matrix Computations. Baltimore, MD: Johns Hopkins University Press, 3rd edition, 1996. [11] B.Hassibi, “An efficient square-root algorithm for BLAST,” in Proc. ICASSP, Istanbul, Turkey, June 2000, pp. 737–740. [12] L. Schumacher. WLAN MIMO channel matlab program. [Online]. Available: http://www.info.fundp.ac.be/∼lsc/Research/IEEE 80211 HTSG CMSC [13] K. Pahlvan and A. Levesque, Wireless Information Networks. John Wiley and Sons Inc., 1995.
588 Authorized licensed use limited to: WASHINGTON UNIVERSITY LIBRARIES. Downloaded on August 11, 2009 at 16:00 from IEEE Xplore. Restrictions apply.
