Diversity Analysis of Bit-Interleaved Coded Multiple Beamforming with Orthogonal Frequency Division Multiplexing

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1 IEEE ICC 03 - Wireless Communiations Symposium Diversity Analysis of Bit-Interleaved Coded Multiple Beamforming with Orthogonal Frequeny Division Multiplexing Boyu Li and Ender Ayanoglu Center for Pervasive Communiations and Computing Department of Eletrial Engineering and Computer Siene The Henry Samueli Shool of Engineering University of California - Irvine Irvine, California boyul@ui.edu, ayanoglu@ui.edu Abstrat For Multiple-Input Multiple-Output (MIMO) systems with frequeny seletive fading hannels, Bit-Interleaved Coded Multiple Beamforming (BICMB) with Orthogonal Frequeny Division Multiplexing (OFDM) an be employed to offer both spatial and multipath diversity, making it an important tehnique. Nevertheless, analyzing its diversity is a hallenging problem. In this paper, the diversity analysis of BICMB-OFDM is arried out. First, the maximum ahievable diversity is derived and a full diversity ondition is proved. Then, the performane degradation due to the subarrier orrelation is investigated. Finally, the subarrier grouping tehnique is applied to ombat the performane degradation and provide multi-user ompatibility. I. INTRODUCTION Fig.. Struture of BICMB-OFDM. In a MIMO system, beamforming with Singular Value Deomposition (SVD) an be employed to improve the data rate or the performane, when the hannel state information is available at both the transmitter and reeiver []. For flat fading MIMO hannels, single beamforming arrying only one symbol at a time ahieves full diversity []. However, without hannel oding, multiple beamforming transmitting multiple streams simultaneously results in full diversity loss. To ombat the performane degradation, BICMB whih interleaves the bit odeword through multiple subhannels, was proposed [3], [4]. BICMB an ahieve full diversity as long as the ode rate R and the number of employed subhannels S satisfy the ondition R S [5], [6]. If the MIMO hannel is in frequeny seletive fading, BICMB an be ombined with OFDM to ombat the inter-symbol interferene aused by multipath propagation and ahieve both spatial and multipath diversity [3]. OFDM is well-suited for broadband data transmission and has been seleted as the air interfae for IEEE 80. WiFi, IEEE 80.6 WiMAX, and 3GPP LTE [7]. Although some more modern standards employ more sophistiated odes than onvolutional odes employed by BICMB, suh as turbo odes and LDPC odes, onvolutional odes are important sine they an be analyzed and there are many legay produts using them. Therefore, BICMB-OFDM is an important tehnique for broadband wireless ommuniation. However, the diversity analysis of BICMB-OFDM is a diffiult hallenge. In this paper, diversity analysis of BICMB-OFDM is arried out. The remainder of this paper is organized as follows: In Setion II, the system model is outlined. In Setion III, the maximum ahievable diversity is derived, and a suffiient and neessary full diversity ondition R SL is proved where S is the number of steams transmitted at eah subarrier and L is the number of hannel taps. Then, Setion IV investigates the performane degradation aused by subarrier orrelation and Setion V disusses the subarrier grouping tehnique [8] to ombat the performane degradation and provide multi-user ompatibility. In Setion VI, simulation results are provided. Finally, a onlusion is drawn in Setion VII. II. SYSTEM MODEL Fig. presents the struture of BICMB-OFDM. First, the onvolutional enoder with rate R generates the bit odeword from the information bits. Then, an interleaved bit sequene is generated by a bit interleaver before being modulated to a symbol sequene. Let N r and N t denote the number of transmit and reeive antennas respetively. Assume that M subarriers are employed, and S min{n t,n r } streams are /3/$ IEEE 50

2 arried on eah subarrier simultaneously. Hene, an S vetor x k (m) is arried on the mth subarrier at the kth time instant with m =,...,M. The length of Cyli Prefix (CP) is L p L where L is the number of hannel taps. The N r N t frequeny seletive fading MIMO hannel with L taps is assumed to be Rayleigh quasi-stati and known by both the transmitter and reeiver, denoted as H(l) with l =,...,L.Let L H(m) = H(l)e i π(m )τ l MT () l= denote the quasi-stati flat fading MIMO hannel observed at the mth subarrier, where T is the sampling period and τ l is the lth tap delay [9]. The beamforming matries at the mth subarrier are derived by SVD of H(m), i.e., H(m) = U(m)Λ(m)V H (m), whereu(m) and V(m) are unitary, and Λ(m) is diagonal whose sth diagonal element, λ s (m), isasingular value of H(m), whih is positive and real, in dereasing order with s =,...,S.ThefirstS olumns of U(m) and V(m), i.e., U S (m) and V S (m), are seleted as beamforming matries at the reeiver and transmitter, respetively. The multipliations with beamforming matries are arried out at eah subarrier before exeuting IFFT and adding CP at the transmitter, and after exeuting FFT and removing CP at the reeiver, respetively. Therefore, the system input-output relation for the mth subarrier at the kth time instant is y s,k (m) =λ s (m)x s,k (m)+n s,k (m), () where y s,k (m) and x s,k (m) are the sth element of the S reeived symbol vetor y k (m) and the transmitted symbol vetor x k (m) respetively, and n s,k (m) is the additive white omplex Gaussian noise with zero mean and variane N 0 = N t /γ, with γ denoting the reeived SNR over all the reeive antennas. Note that the total transmitted power is saled by N t in order to make the reeived SNR γ. The loation of the oded bit k within the transmitted symbol is denoted as k (k, m, s, j), whih means that k is mapped onto the jth bit position on the label of x s,k (m). Let χ denote the signal set of the modulation sheme, and let χ j b denote a subset of χ whose labels have b {0, } at the jth bit position. By using the loation information and the inputoutput relation in (), the reeiver alulates the Maximum Likelihood (ML) bit metris for k = b as Δ[y s,k (m), k ]= min y s,k (m) λ s (m)x. (3) x χ j k Finally, the ML deoder applies the soft-input Viterbi deoding to find a odeword ĉ with the minimum sum weight and its orresponding information bit sequene ˆb as ĉ =argmin Δ[y s,k (m), k ]. (4) k III. MAXIMUM ACHIEVABLE DIVERSITY Aording to (3), the Pairwise Error Probability (PEP) of BICMB-OFDM between the transmitted bit odeword and the deoded bit odeword ĉ is given in [3] as [ ( Pr ( ĉ) E exp d min k,d H λ s (m) )], (5) 4N 0 where d H denotes the Hamming distane between and ĉ, d min is the minimum Eulidean distane in the onstellation, and k,d H stands for the summation of the d H values related to the different bits between the bit odewords. Define an M S matrix A as an α-spetrum, whose element α m,s denotes the number of distint bits transmitting through the sth subhannel of the mth subarrier for an error path, whih implies M S m= s= α m,s = d H.Leta T m denote the mth row of A. Then (5) is rewritten as [ ( Pr ( ĉ) E exp d min m s α m,sλ s (m) 4N 0 ) ]. (6) In this setion, the maximum ahievable diversity is derived for the simplest ase of no subarrier orrelation. Note that subarrier orrelation has a negative effet on the performane, whih will be disussed in Setion IV. Assume that different MIMO delay spread hannels are unorrelated and have equal power, and eah element of eah tap is statistially independent and modeled as a omplex Gaussian random variable with zero mean and variane /L. Further assume that the hannel taps are separated by a onstant sampling time, i.e., τ l =(l )T in (). Then, the orrelation in absolute value between two subarriers is E [ h ρ = u,v (m)h u,v (m ) ] E [ h u,v (m)h u,v(m) ] E [ h u,v (m )h u,v(m ) ] = exp[ i π(m m )L M ] L exp[ i π(m m ) M ]. (7) Note that ρ =0when L = M, implying that all subarriers are unorrelated. Note that h u,v =[h u,v (0),...,h u,v (M)] T is a omplex normal random vetor, where h u,v (m) denotes the (u, v)th element in H(m). Therefore, based on (7), all subarriers are independent when L = M. In the following part of this setion, this speial ase is onsidered. Although it is not pratial, its diversity analysis provides the maximum ahievable diversity for the pratial ase, beause orrelation among subarriers for the pratial ase has a negative effet on performane, whih will be disussed in Setion IV. Sine subarriers are independent when L = M, theλ(m) matries are independent. Hene, (6) is further rewritten as Pr ( ĉ) E m [ exp ( d min s α m,sλ s(m) 4N 0 )]. (8) For eah subarrier, the terms inside the expetation in (8) an be upper bounded by employing a theorem proved in [0]. As a result, an upper bound of PEP is given by Pr ( ĉ) m,a m 0 ( d ζ min α m,min m γ 4N t ) Dm, (9) 50

3 with D m =(N r δ m +)(N t δ m +),whereα m,min is the minimum non-zero element in a m, δ m denotes the index of the first non-zero element in a m,andζ m is a onstant. Therefore, the diversity is D = D m. (0) m,a m 0 The results of (9) and (0) show that the maximum ahievable diversity of BICMB-OFDM diretly depends on the α- spetra beause the PEP with the worst diversity dominates the performane. Note that the α-spetra are related to the bit interleaver and the trellis struture of the onvolutional ode, and an be derived by a similar approah to BICMB of flat fading MIMO hannels presented in [5], or by omputer searh. An example is provided here to show the relation between the α-spetra and the diversity. Consider the parameters N t = N r = S = L = M =. Assume that the R =/ ode with generator polynomial (5, 7) in otal is employed, and the bit interleaver applies simple rotation. In this ase, the dominant α-spetrum is A =[0;], whih implies that δ =and δ =. Hene, D =, D =4. Therefore, the maximum ahievable diversity order is D = D + D =5. Based on (9) and (0), full diversity of N r N t L [] an be ahieved if and only if all elements in the first olumn of A are non-zero, i.e., α m, 0, m, for all error events. To meet the requirement, the ondition R SL needs to be satisfied, and the proof is provided in the following. Proof: To prove the neessity, assume that an information bit sequene b with JR SL bits is transmitted, then a bit sequene m,s with J bits is transmitted at the sth subhannel of the mth subarrier. If R SL >, beause the number of different odewords JRSL is larger than the number of different bit sequenes m,s, J, there always exists at least a pair of odewords with the same m,s. Hene, the pairs of odewords with the same m, ause full diversity loss. To prove the suffiieny, a bit interleaver with simple rotation is onsidered. If R SL, all the SL subhannels ould be assigned to one branh of the trellis for the onvolutional ode. Sine the trellis an be designed suh that the oded bits of the first branh splitting from the zero state are all errored bits, at least one errored bit an be arried on eah subhannel for all error paths, whih guarantees α m, 0, m for all error events. Therefore, full diversity an be ahieved. This onludes the proof. The proof of the neessity above implies that when R SL >, there always exists an error path with no errored bits arried on the first subhannel of a subarrier. Therefore, full diversity annot be ahieved. In this ase, the bit interleaver should be designed suh that onseutive oded bits are transmitted over different subhannels to provide the maximum ahievable diversity, whih depends on the α-spetra. IV. NEGATIVE EFFECT OF SUBCARRIER CORRELATION In pratie, sine M is always muh larger than L, subarrier orrelation exists as shown in (7). Hene, to alulate (6), the joint Probability Density Funtion (PDF) of Λ(m)Λ H (m) for all m satisfying a m 0, whih are eigenvalues of a set of orrelated Wishart matries [], is required. However, this is an extremely diffiult problem. The joint PDF of two orrelated Wishart matries are given in [], [3], whih is already highly ompliated. To the best of our knowledge, the joint PDF of more than two orrelated Wishart matries is not available in the literature. The maximum diversity of an OFDM-MIMO system is known to be N r N t L []. In our ase, however, a performane degradation aused by orrelation is to be expeted. Beause, otherwise, the diversity an exeed the full diversity, whih is a ontradition. In this setion, the negative effet of orrelation on the performane between two subarriers is investigated to provide an intuitive insight. Consider an error path whose d H distint bits between two bit odewords are all transmitted through two orrelated subarriers with orrelation ρ, whih ould be the pratial ase. Define X =max(n t,n r ) and Y = min(n t,n r ).Let Φ =[φ,...,φ Y ] and Φ =[ φ,..., φ Y ] denote the ordered eigenvalues of the two orrelated Wishart matries HH H and H H H, respetively. Note that φ u = λ u.leta =[α,...,α Y ] and ã = [ α,..., α Y ] denote the α-spetra of Φ and Φ respetively. Define p =[p,...,p W ] and p =[ p,..., p W ] whose elements are the indies related to non-zero elements in a and ã, respetively, i.e., α pw 0and α p w 0. Similarly, define q =[q,...,q Y W ] and q =[ q,..., q Y W ] whose elements are the indies related to zero elements in a and ã, respetively, i.e., α qw =0and α q w =0. Then, define Φ p = [φ p,...,φ pw ], Φ p =[ φ p,..., φ p w ], Φ q =[φ q,...,φ qw ], and Φ q =[ φ q,..., φ q w ]. Hene, the PEP in (6) is written as ( ) d Pr ( ĉ) E exp min a T Φ + ã T Φ 4N 0 W W E exp μ φ pw + φ p w w= w= () with μ = ( d min α min) / (4N0 ), where α min indiates the minimum element in a and ã. To solve (), the marginal PDF f(φ p, Φ p ) is needed by alulating f(φ p, Φ p )= f(φ, Φ)dΦ q d Φ q. () D q D q The joint PDF f(φ, Φ) is available in [], [3], as [ f(φ, Φ) =exp ρ Y u= ( φ u + φ u ) ] f (Φ, Φ), (3) with the polynomial f (Φ, Φ) defined as f (Φ, Φ) =[ (φ u φ v )( φ u φ v )] u<v det[(φ u φv ) (X Y )/ I X Y ( ɛφ u φv )], (4) where det[h u,v ] represents the determinant of the matrix with 503

4 the (u, v)th element given by h u,v, I N ( ) is a modified Bessel funtion of order u, and ɛ ρ /( ρ ). Beause the exponent of μ is related to the diversity, the onstant appearing in the literature is ignored in (4) for brevity. Sine the eigenvalues are positive and real, exp( ρ φ u ) and exp( φ ρ u ) in (3). By applying v 0 ut e u du t+ vt+ and u t e u du = t! to Φ 0 q and Φ q,themarginal PDF f(φ p, Φ p ) in () is upper bounded as ( W f(φ p, Φ p w= ) exp φ p w + ) W w= φ p w ρ f (Φ p, Φ p ), (5) where f (Φ p, Φ p ) is a polynomial orresponding to (3). Then () is rewritten as φp φpw φ p φ p W Pr ( ĉ) ( exp μ + ) W W ρ φ pw + φ p w w= w= f (Φ p, Φ p )dφ p d Φ p. (6) Sine f (Φ p, Φ p ) is a polynomial, its multivariate terms an be integrated separately, and the overall performane is dominated by the term with the worst performane. To solve (6), a theorem in [0] an be applied to integrate Φ p and Φ p independently for eah multivariate term, and the term with the smallest degree results in the smallest degree of (μ+ ρ ), whih dominates the overall performane. Note that the smallest degree of f (Φ p, Φ p ) is (X p +)(Y p +)+(X p +)(Y p +) W W, whih is proved in the Appendix. Therefore, (6) is upper bounded by ( d Pr ( ĉ) ζ min α min γ + ) D 4N t ρ, (7) with D =(X p +)(Y p +)+(X p +)(Y p +), where ζ is a onstant. The negative effet of subarrier orrelation ρ is proved by (7). When γ, the diversity is the same as the unorrelated ase. However, on the pratial range, the performane is degraded due to the term ρ, whih is independent of SNR. Speifially, when ρ is small, ρ is also relatively small, and its effet on the performane is not signifiant when SNR is relatively large, and the unorrelated ase ρ = 0 offers the performane upper bound. On the other hand, when ρ is large, ρ is also relatively large ompared to SNR, then signifiant performane loss ould be aused, depending on SNR. When ρ =, whih means all errored bits are arried on one subarrier, no multipath diversity is ahieved, and the diversity equals BICMB of flat fading MIMO hannels [5], [6], whih is the performane lower bound. Note that the analysis in this setion an also be applied to unequal power hannel taps, non-onstant sampling time, and other pratial situations, whih ause different subarrier orrelation. V. SUBCARRIER GROUPING To overome the performane loss aused by subarrier orrelation, the subarrier grouping tehnique [8] is applied. Note that ρ =0in (7) when (m m )L/M is an integer. This means that although orrelation exists among subarriers for L<M, some subarriers ould be unorrelated if M/L is an integer. In this ase, there are G = M/L groups of L unorrelated subarriers. As a result, the subarrier grouping tehnique is applied to transmit multiple streams of bit odewords through these G different groups of unorrelated subarriers, instead of transmitting one stream of the bit odeword through all the orrelated subarriers. As a result, the negative effet of subarrier orrelation is ompletely avoided, and the maximum ahievable diversity is thereby ahieved. For example, onsider the ase of L = and M = 64. Then, the gth and the (g + 3)th subarriers are unorrelated for g =,...,3. The subarrier grouping tehnique an transmit 3 streams of bit odewords simultaneously through the 3 groups of two unorrelated subarriers without performane degradation. Compared to no subarrier grouping, BICMB-OFDM with subarrier grouping ahieves better performane with the same transmission rate and deoding omplexity. Note that the diversity analysis for L = M in Setion III an be applied to BICMB-OFDM with subarrier grouping. Therefore, the full diversity ondition R SL holds for BICMB-OFDM with subarrier grouping. Also note that BICMB-OFDM with subarrier grouping an be onsidered as Orthogonal Frequeny- Division Multiple Aess (OFDMA) [7] version of BICMB- OFDM. OFDMA is a multi-user version of the OFDM and it has been used in the mobility mode of WiMAX as well as the downlink of LTE. In other words, with subarrier grouping, BICMB-OFDM an offer multi-user ompatibility. Note that (7) is derived under the assumption of equal power hannel taps. When they have different power, there are no unorrelated subarriers in general. However, some of them ould have weak orrelation. Therefore, the subarrier grouping an still be applied to ombat the diversity degradation, although it now an no longer fully reover the performane beause of subarrier orrelation. VI. SIMULATION RESULTS To verify the diversity analysis, M =64BICMB- OFDM with L =and L =4using 4-QAM are onsidered for simulations. The number of employed subhannels for eah subarrier is assumed to be the same. The generator polynomials in otal for the onvolutional odes with R = /4 and R =/ are (5, 7, 7, 7), and(5, 7) respetively, and the odes with R =/3 and R =4/5 are puntured from the R =/ ode [4]. Eah OFDM symbol has 4μs duration, of whih 0.8μs is CP and L p =6. Equal and exponential power hannel taps are onsidered. For the exponential hannel model [5], the ratios of non-negligible path power to the first path power are 7dB, the mean exess delays are 30ns for L =and 65ns for L =4, respetively. The bit interleaver 504

5 S=, R =/, A=[; 3] S=, R =/3, A=[0; 5] S=, R =/4, A=[ 3; 3 3] S=, R =/, A=[0 ; ] S=, R =/3, A=[0 ; 0 3] S=, R =4/5, A=[0 0; 0 4] w/o SG, equal w/ SG, equal w/o SG, exponential w/ SG, exponential BER BER SNR in db SNR in db Fig.. BER vs. SNR for L =M =64BICMB-OFDM with subarrier grouping over equal power hannel taps. Fig. 4. BER vs. SNR for L =M =64S =R =/ BICMB- OFDM with and without subarrier grouping over equal and exponential power hannel taps. BER A=[ 0; 0], ρ=0 A=[ 0; 0], ρ=0.4 A=[ 0; 0], ρ=0.7 A=[ 0; 0], ρ=0.9 A=[ 0; 0 0], ρ= A=[ 0; 0 ], ρ=0 A=[ 0; 0 ], ρ=0.4 A=[ 0; 0 ], ρ=0.7 A=[ 0; 0 ], ρ=0.9 A=[ ; 0 0], ρ= Correlation equal exponential SNR in db Fig. 3. BER vs. SNR for examined PEPs of two subarriers with different orrelation for L =M =64S =BICMB-OFDM over equal power hannel taps Subarrier Separation Fig. 5. Correlation vs. subarrier separation for L =M =64 BICMB-OFDM over equal and exponential power hannel taps. employs simple rotation. Note that although simulation results of both L = and L = 4 verify our diversity analysis, only results for L =are shown in this setion beause the diversity values ould be investigated more expliitly through figures, and the results for L =4an be found in [6]. Fig. shows the Bit Error Rate (BER) performane of BICMB-OFDM employing subarrier grouping with different S and R.TheA matries that dominate the performane are provided in the figure. The diversity results of all urves equals the maximum ahievable diversity orders derived from Setion III, whih is diretly deided by the A matries. Speifially, in the ases of S =, R =/4 and R =/3 odes ahieve diversity values of 8 and 4, respetively. For S =, the odes with R =/4, R =/, R =/3, andr =4/5 offer diversity of 8, 5,, and, respetively. Note that full diversity of 8 is ahieved with the ondition R SL. Fig. 3 shows the BER performane of examined PEPs in (6) with S =, where the simplest ase of an error event with d H = is examined for two subarriers with different orrelation oeffiient ρ, whih is derived from the L = M = 64 BICMB-OFDM over equal power hannel taps. The figure shows that when ρ = 0, whih implies the two subarriers are unorrelated, A =[0;0]and A =[0;0]offer diversity of 8 and 5 respetively. On the other hand, when ρ 0, performane degradation is aused by subarrier orrelation, and stronger orrelation results in worse performane loss. When ρ =, whih means d H = distint bits are transmitted through only one subarrier and no multipath diversity is ahieved, both A =[0;00]and A =[;00]provide diversity of 4. The results are onsistent with the analysis provided in Setion IV, and they show the negative effet of subarrier orrelation on performane. Fig. 4 shows the BER performane of S =R =/ BICMB-OFDM with and without subarrier grouping over equal and exponential power hannel taps. In the figure, w/ and w/o denote with and without respetively, while SG denotes subarrier grouping. The results show that the subarrier grouping tehnique an ombat the performane loss aused by subarrier orrelation for both equal and exponential power hannel taps. As disussed in Setion V, the maximum ahievable diversity of 8 is provided by employing subarrier grouping for equal power hannel taps sine there is no subarrier orrelation. As for the ase of exponential power hannel taps, beause subarrier orrelation still exists, subarrier grouping annot fully reover the performane loss. 505

6 Fig. 5 shows the orrelation ρ of two subarriers with different separation for equal and exponential power hannel taps. The figure shows that hannel with exponential power taps ause stronger subarrier orrelation than equal power taps, whih results in worse performane as shown in Fig. 4. VII. CONCLUSIONS In this paper, the diversity analysis of BICMB-OFDM is arried out. As a result, the maximum ahievable diversity is derived and a suffiient and neessary ondition R SL for ahieving full diversity is proved. In addition, the negative effet of subarrier orrelation on the performane in the pratial ase is investigated, and subarrier grouping is employed to overome the performane degradation and provide multi-user ompatibility. Therefore, BICMB-OFDM an be an important tehnique for broadband wireless ommuniation. APPENDIX PROOF OF THE SMALLEST DEGREE OF f (Φ p, Φ p ) The polynomial f (Φ p, Φ p ) in (5) orresponds to (3). Sine v 0 ut e u du t+ vt+ and u t e u du = t!, the 0 smallest degree of f (Φ p, Φ p ) is related to the polynomial f (Φ, Φ), whih is given by (4) and an be rewritten as f (Φ, Φ) =ɛ (X Y )/ [ (φ u φ v )( φ u φ v )] u<v [ (φ u φu ) X Y ]det[ĩx Y (ɛφ u φv )], (8) u= where ĨN (t) = j=0 t j j!(j+n+)!. Note that only the multivariate term of f (Φ, Φ) related to the smallest degree of f (Φ p, Φ p ) needs to be onsidered. Note that the dominant term of f (Φ, Φ) is the one with the smallest degree and the largest eigenvalues, depending on the dominant term of Y u<v (φ u φ v )( φ u φ v ) and the dominant term of det[ĩx Y (ɛφ u φv )]. The dominant term of Y u<v (φ u φ v )( φ u φ v ) is Y u= (φ φ u u ) Y u. On the other hand, the dominant term of det[ĩx Y (ɛφ u φv )] is ζ Y u= (φ φ u u ) Y u with ζ = Y u= ɛy u / [(Y u)!(x u +)!] beause det[ĩx Y (ɛφ u φv )] = Y u= v= k= j k =0 Y ( ) u+v (ɛφ uk φvk ) j k j k!(j k + X Y +)! j k <j k+ (9) where u k =[(u+k ) mod Y ]+ and v k =[(v+k ) mod Y ]+. Hene, ignoring the onstant fator, the dominant term in f (Φ, Φ) is given by f (Φ, Φ) = (φ u φu ) X+Y u. (0) u= Therefore, the degree of f (Φ, Φ) is δ f =Y (X ). () After integration of (), the fator p u u= φx+y u and the fator p X+Y u u= φ u of f (Φ, Φ) vanish due to the fat that u t e u du = t!. Hene, 0 δ vanished =(p )(X + Y p )+( p )(X + Y p ). () Meanwhile, the eigenvalues φ qu with q u >p and φ qu with q u > p result in inreased degree beause v 0 ut e u du t+ vt+. Therefore, δ added =Y W W p p +. (3) As a result, the smallest degree of f (Φ p, Φ p ) is δ = δ f δ vanished + δ added =(X p +)(Y p +) W +(X p +)(Y p +) W. (4) REFERENCES [] H. Jafarkhani, Spae-Time Coding: Theory and Pratie. Cambridge University Press, 005. [] E. Sengul, E. Akay, and E. Ayanoglu, Diversity Analysis of Single and Multiple Beamforming, IEEE Trans. Commun., vol. 54, no. 6, pp , Jun [3] E. Akay, E. Sengul, and E. Ayanoglu, Bit-Interleaved Coded Multiple Beamforming, IEEE Trans. Commun., vol. 55, no. 9, pp. 80 8, Sep [4] E. Akay, H. J. Park, and E. Ayanoglu. (008) On Bit-Interleaved Coded Multiple Beamforming. arxiv: [Online]. Available: [5] H. J. Park and E. Ayanoglu, Diversity Analysis of Bit-Interleaved Coded Multiple Beamforming, in Pro. IEEE ICC 009, Dresden, Germany, Jun [6], Diversity Analysis of Bit-Interleaved Coded Multiple Beamforming, IEEE Trans. Commun., vol. 58, no. 8, pp , Aug. 00. [7] A. Ghosh, J. Zhang, J. G. Andrews, and R. Muhamed, Fundamentals of LTE. Pearson Eduation, In., 0. [8] Z. Liu, Y. Xin, and G. B. Giannakis, Linear Constellation Preoding for OFDM With Maximum Multipath Diversity and Coding Gains, IEEE Trans. Commun., vol. 5, no. 3, pp , Mar [9] I. Lee, A. M. Chan, and C.-E. W. Sundberg, Spae-Time Bit-Interleaved Coded Modulation for OFDM Systems, IEEE Trans. Signal Proess., vol. 5, no. 3, pp , Mar [0] H. J. Park and E. Ayanoglu, An Upper Bound to the Marginal PDF of the Ordered Eigenvalues of Wishart Matries and Its Appliation to MIMO Diversity Analysis, in Pro. IEEE ICC 00, CapeTown, South Afria, May 00. [] A. Edelman, Eigenvalues and Condition Numbers of Random Matries, Ph.D. dissertation, Massahusetts Institute of Tehnology, 989. [] P. J. Smith and L. M. Garth, Distribution and Charateristi Funtions for Correlated Complex Wishart Matries, Journal of Multivariate Analysis, vol. 98, p. 6677, Apr [3] P.-H. Kuo, P. J. Smith, and L. M. Garth, Joint Density for Eigenvalues of Two Correlated Complex Wishart Matries: Charaterization of MIMO Systems, IEEE Trans. Wireless Commun., vol. 6, no., pp , Nov [4] D. Haoun and G. Begin, High-Rate Puntured Convolutional Codes for Viterbi and Sequential Deoding, IEEE Trans. Commun., vol. 37, no., pp. 3 5, Nov [5] Y. S. Cho, J. Kim, W. Y. Yang, and C. G. Kang, MIMO-OFDM Wireless Communiations with MATLAB. Wiley-IEEE Press, 00. [6] B. Li and E. Ayanoglu. (0) Diversity Analysis of Bit-Interleaved Coded Multiple Beamforming with Orthogonal Frequeny Division Multiplexing. arxiv: [Online]. Available: 506