SINGLE-CARRIER Frequency Domain Equalization (SC-

Transcription

1 388 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO 2, NO, JANUARY 203 Single-Carrier Frequency-Domain Equalizer with Multi-Antenna Transmit Diversity Ahmed Hesham Mehana, Student Member, IEEE, and Aria Nosratinia, Fellow, IEEE Abstract Single-carrier SC bloc transmission with cyclic prefix C is a method with several advantages that has been incorporated into standards This paper investigates the performance of multi-antenna SC-FDE under cyclic-delay diversity CDD and Alamouti signaling Our analysis fully characterizes the diversity, showing that it depends not only on the antenna configuration and channel memory, but also on data bloc length and data transmission rate Below a certain rate threshold, full diversity is available to both CDD and Alamouti signaling, while at higher rates their diversity diminishes, albeit not quite in the same way Our analysis shows that at high rates the CDD diversity degenerates to the diversity of the SISO SC-FDE, while Alamouti signaling provides twice the diversity of SISO SC-FDE Index Terms Single-carrier, cyclic prefix, equalization, cyclic delay diversity, Alamouti signaling I INTRODUCTION SINGE-CARRIER Frequency Domain Equalization SC- FDE is an alternative to OFDM that avoids several OFDM drawbacs, including pea-to-average power ratio and the high sensitivity to carrier frequency offset [] SC-FDE has been adopted for the TE uplin [], [2] In this paper, we analyze the performance of SC-FDE in conjunction with either cyclic delay diversity CDD or Alamouti signaling, fully characterizing the diversity as a function of transmission-bloc length, data rate, channel memory, and number of antennas In the process, we obtain a threshold rate as a function of data-bloc length, channel memory, and number of antennas below which the full spatial-temporal diversity is achieved, while at higher rates the diversity of both schemes diminishes, albeit not quite in the same way Our analysis shows that at high rates the CDD diversity degenerates to the diversity of the SISO SC-FDE, while Alamouti signaling provides twice the diversity of SISO SC- FDE We find that beyond a certain rate threshold in either CDD or Alamouti signaling, an increase in transmission rate can reduce the diversity, but this diversity can be recovered by increasing the FFT bloc length Specifically, in this operating regime, the diversity can be maintained if every additional Manuscript received May 22, 202; revised September 4, 202; accepted September 4, 202 The associate editor coordinating the review of this paper and approving it for publication was Deneire The authors are with the Department of Electrical Engineering, The University of Texas at Dallas, Richardson, TX USA ahmedhesham@utdallasedu; aria@utdallasedu The wor in this paper was presented in part at the IEEE International Conference on Communications ICC, June 202 Digital Object Identifier 009/TWC /3$300 c 203 IEEE bit/s/hz of transmission rate is accompanied by a doubling of FFT bloc length Naturally the bloc length cannot exceed the coherence time of the channel, therefore equalizer performance is in practice also limited by the coherence time A brief survey of related literature is as follows It has been nown that SC-FDE in single-antenna SISO systems displays a diversity that is a function of data rate and transmission bloc length hence the FFT size [3] The behavior of SC-FDE has also been analyzed in multi-stream BAST-type MIMO systems, where its diversity multiplexing tradeoff DMT and bounds on its diversity have been obtained [4] Al-Dhahir [5] proposed the Alamouti SC-FDE, but [5] only went so far as to show that the effective channel gain of Alamouti SC-FDE is a sum of two independent components, which only suggests that the diversity is at least two The present wor conclusively settles the question of the diversity of Alamouti SC-FDE Design rules are provided in [6] for achieving maximum diversity gains with linearly precoded OFDM but it requires M decoding Tepedelenlioglu [7] showed that linear equalizers achieve the maximum multipath diversity in linearly precoded OFDM systems The zero-padded SC system with linear equalization was analyzed in [8] where it was shown that the full diversity is achievable by ZF equalizer Muquet et al [9] compared the performance of Z-OFDM and C- OFDM Coded OFDM COFDM schemes were considered in [0], showing that COFDM achieves the maximum channel diversity with M decoding It was shown that the zero-padded and cyclic-prefix single-carrier system are special cases of the COFDM of [0] and thus achieve the maximum diversity with M This paper is organized as follows Section II provides the system model for cyclic-prefix transmission and reviews the recent result for the SC MMSE-FDE receiver diversity Section III provides the performance analysis for the CDD systems Section IV provides the performance analysis for the Alamouti orthogonal-space time coded systems Section V provides simulations that illuminate our results II CYCIC-REFIX TRANSMISSION A System Model We consider a frequency selective quasi-static wireless fading channel The equivalent baseband model for this intersymbol interference ISI channel is given by a multipath model with ν + paths The channel vector is denoted by h [h 0,,h ν ] and the channel coefficients are assumed independent and identically distributed CN0, Weassume a bloc-fading model where the channel is fixed over

2 MEHANA and NOSRATINIA: SINGE-CARRIER FREQUENCY-DOMAIN EQUAIZER WITH MUTI-ANTENNA TRANSMIT DIVERSITY 389 the transmission bloc To remove the inter-bloc interference at the receiver, a cyclic-prefix C with length at least ν is inserted at the beginning of each data-bloc of length The C also transforms linear convolution into circular convolution and thus permits channel diagonalization The input-output system model for a bloc transmission scheme with length-ν C is y ρ Hx + n where x C +ν, y C and n C N denote the transmitted, received and noise vectors respectively The noise vector is assumed white, Gaussian, and zero-mean with covariance matrix σn 2 I and ρ is the transmitted signal power Without loss of generality we assume σn 2 The matrix H C +ν is the convolution channel matrix The linear data extension operation that maps the data bloc, denoted by s, of length to the transmitted vector of length + ν can be expressed by x U cp s where U cp is the C matrix given by [ ] I U cp I ν 0 ν ν Hence the model in is equivalent to: y ρ HU cp s + n ρ H e s + n 2 where H e HU cp is the equivalent circulant channel matrix given by h 0 h h ν h 0 h h ν 0 H e 3 h h 2 h ν 0 h 0 Note that H e has eigen decomposition H e Q H ΛQ where Q is the unitary discrete Fourier transform matrix The diagonal elements of Λ are given by [3] λ i0 2πi h i e for,, 4 In single-carrier frequency domain equalizer SC-FDE, the DFT/IDFT operation is performed at the receiver The DFT/IDFT in the SC-FDE diagonalizes the channel thus a single-tap equalizer can be used, reducing the complexity of equalization The DFT-domain version of Equation 2 is Y Qy ρ ΛS + N 5 where S, Y and N are the DFT of the transmitted, received and noise vectors respectively Assuming perfect channel state information at the receiver CSIR, the linear zero-forcing ZF and MMSE equalizers are given by []: W cρ I + H H H H H 6 where the constant c for MMSE equalizer and c 0for ZF equalizer The matrix H is the channel to be equalized The corresponding unbiased decision-point SINR is γ c I + ρh H H c,, 7 For completeness we mention the definition of the diversity gain log e d lim 8 ρ log ρ and the outage diversity log out d out lim ρ log ρ where e is the pairwise error probability, out is the outage probability given by out Ix; y <R, whereix; y is the mutual information between x and y, andr is the target rate We also denote the exponential equality of two functions log fρ fρ and gρ as fp gp when lim ρ logρ log gρ lim ρ logρ B The Diversity of the MMSE Receiver in SC-FDE System In [3] the linear MMSE receiver is analyzed for SC-FDE, a result that we shall refer to later in this paper and therefore we review it here briefly In [3] the received signal after equalization is given by 9 ỹ Wy ρwh e s + Wn 0 and subject to this model it was shown that MMSE SC-FDE can achieve full diversity for certain values of bloc length and operating rate R b/s/hz The process of developing this result was as follows The analysis performed in [3] consists of two main steps The system outage diversity is first characterized Then lower and upper bounds on the error probability via outage are provided It is shown in [3] that these two bounds are tight and thus the diversity is fully characterized This two-steps approach was first proposed in [2] due to the intractability of the direct pairwise error probability E analysis for many MIMO architectures The diversity of the MMSE SC-FDE is [3] d out ν + 2 R + R log ν R>log ν III CYCIC-DEAY DIVERSITY One common transmit diversity technique used for single carrier and multicarrier systems is antenna delay diversity, which can tae the form of time delay, cyclic delay and phase delay [3], [4] Among them, cyclic delay diversity CDD is more widely adopted for single carrier and multicarrier applications as CDD can be applied to any number of transmit antennas without any rate loss or change in the receiver structure [4] [6] In this section we show that linear MMSE receivers can achieve the maximal spatio-temporal diversity provided that the equalizer and the cyclic delay taps are properly designed

3 390 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO 2, NO, JANUARY 203 s 0 C s 0 Input Data Sequence SC/MC Modulation s M δ s C s Remove C Equalizer Detector δ M- s M- C s M- Fig Single-carrier and multicarrier MISO system with transmitter sided CDD scheme and the proposed MMSE receiver System Model: Consider a MISO system with M transmit antennas and a bloc fading channel model where the channel remains unchanged during the bloc transmission The channel impulse response from the transmit antenna i to the received antenna is given by h i [h i,0,,h i,νi ] with channel memory length denoted by ν i Wealsodefine ν max i ν i We adopt the system model of [4] The model is shown in Figure which displays the front end of a single carrier and multicarrier MISO system with CDD In vector form, the received signal can be written as y M i0 ρ Hi ŝ i + n 2 where H i is an circulant channel matrix whose first row is [h i,0,,h i,νi, 0,,0], ŝ i is the transmitted databloc without the C from transmit antenna i CDD converts the MISO channel into a SISO channel with increased channel selectivity The model can be written as [5] y ρ H cir s + n ρ Q H ΛQs+ n 3 where H cir is circulant matrix, s is the modulated symbols cf Figure, Q is the normalized DFT matrix, and Λ is a diagonal matrix whose diagonal entries are the DFT point of the first row of H cir whicharegivenby [ĥ0,,ĥ ], and ĥl M h i l δ i mod 4 M i0 The selection of the delay samples δ i } and its impact on the data rate, the signal-to-interference-and-noise ratio SINR and maximum achievable diversity is studied in [5], [7] While the C length is independent of δ i,itmustbenoless than the maximum channel delay spread ν [7] Also for the receiver to exploit the full diversity the delays can be chosen as δ i >δ i + ν [5] or simply δ i iδ with δ>ν[6] 2 Diversity Analysis of MMSE Receiver: We first consider the case where ν i ν 0 i ie flat MISO channel and the symbol delays δ i i In this case, the system model is equivalent to a SISO ISI channel under C transmission If the equalizer is designed according to H cir, it is nown that in the SISO ISI C transmission a rate-dependent diversity is observed [3], and due to equivalence of channel models this result can be directly lifted to the flat MISO CDD system This result will be extended to the general case of the multipath MISO channel under CDD For a flat channel ν i 0 i and the delays δ i i, it can be shown that the first row of the circulant channel matrix H cir is [h 0,h,,h M, 0,, 0] where the channel entries ĥl} are given by 4 The equalized signal is ỹ ρ WH cir s + Wn 5 The system model in 5 is equivalent to 0 and we have the following lemma emma : Consider the M MISO flat-fading channel The diversity of the MMSE receiver under uncoded CDD transmission and data-blocs is given by M R log M d 2 R + R>log M 6 Remar : Note that the uncoded C systems can suffer from loss of multipath diversity [8] However if a linear receiver is used and the system parameters rate, transmit antenna delays, and FFT bloc length are appropriately designed, emma indicates that the maximum diversity can be achieved Specifically, the maximum diversity is achieved when 2 R + M or R log M R th 7 For any given values of R and M, the data bloc length can be chosen such that R R th Therefore if we have flexibility in assigning data bloc length, maximum diversity can always be achieved Remar 2: The developments in this paper do not depend on whether the equalization matrix W is multiplied by the received data in the time domain, or is applied in the frequency domain via FFT/IFFT Both approaches lead to the same SINR and outage Therefore, the results of this paper are valid for single-carrier systems regardless of whether the receiver operates in the time domain or frequency domain We now consider the second case: the frequency selective channel, ie, ν i 0 The delay taps are chosen such that δ i n i δ with δ > ν and n i } are distinct integers, and the bloc length is chosen such that the transmitted blocs s i are distinct Notice that the condition on the bloc length guarantees that the channel coefficients seen at the receiver are independent Thus the delay taps and the bloc length must be chosen to satisfy two conditions n i δ 0 mod n j δ 0 mod i j 8 Mν + 9

4 MEHANA and NOSRATINIA: SINGE-CARRIER FREQUENCY-DOMAIN EQUAIZER WITH MUTI-ANTENNA TRANSMIT DIVERSITY 39 We now consider the following case : ν i ν i, δ i iν + It can be shown that this system is equivalent to 5 where the variable M in 5 is replaced by Mν + Thus we obtain the diversity Mν + R log Mν+ d 2 R + R>log Mν+ 20 The maximum diversity is achieved when R log Mν + R th 2 Other choices of δ i } that satisfy 8 exist Each of these choices yields a new H cir whose first row is a permutation of the first row of H cir under δ i iν + We thus have! different choices for the set δ i } Since these circulant matrices have different structures, the diversity analysis of [3] does not directly follow We study the diversity of these systems and show that the! remaining choices δ i } yield the same diversity as shown in 20 The outage probability of the system model in 5 is given by [3] out +ρ λ 2 >2 R where λ are the eigenvalues of H cir Since each eigenvalue is a linear combination DFT points of channel coefficients h [h 0 h h Mν+ ] then the eigenvalues λ } obey a zero-mean complex Gaussian distribution In the special case Mν +the eigenvalues λ } are independent and it can be shown that [3] out Mν+ +ρ λ 2 >2 R ρ d 22 where d is given by 20 Now let > Mν + and δ i iν + The channel matrix is circulant with first-row vector h [h 0 h Mν+ 0 0] and the corresponding eigenvalues λ } are the DFT points of this zero-padded channel vector h In this case, [3, emma 2] applies and we have >m +ρ λ Mν+ +ρ λ 2 >m ρ d 23 where m 2 R and d is given by 20 We now continue with >Mν + but we discard the assumption δ i iν + Instead, the delays δ i } are only required to satisfy 8 In this case the zero-padded channel vector is h 2 [0 h 0 0 h 0 h Mν+ 0] where the locations of zeros depend on the particular choice of the δ i } via Eq 4 We have the following lemma We assume ν i ν for simplicity The general case can be obtained in a straightforward manner emma 2: The DFT of h 2 and the DFT of h satisfy the following equality >m +ρ λ 2 where m is a positive number roof: lease refer to Appendix +ρ λ >m 24 Setting m 2 R, we conclude from 23 and 24 that >m ρ d +ρ λ 2 where d is given by 20 3 Diversity Analysis of Zero-Forcing Receiver: The SINR of the ZF receiver is given by 6 ρ γ [H H e H e ] ρ tr[h H 25 e H e ] ρ 26 λ 2 where 25 follows since H e is circulant thus the diagonal elements of H H e H e are equal, and 26 follows from the equivalence of the trace and the sum of eigenvalues et us first consider the case of Mν + In this case the eigenvalues are independent Gaussian random variables We perform the outage analysis for this case and then show that the result holds for the more general case >Mν + et α be the SNR exponent of the channel eigenvalues, ie α log λ 2 log ρ The outage probability is out l log + γ l <R ρ λ 2 > 2 R > ρ λ 2 > 2 R ρ α > 2 R α > 0 α > ρ 27 Eq 27 follows because λ 2 is exponential random variable and thus we have α > e ρ via arguments similar to [3, emma ] We now obtain an upper bound on outage The outage probability can be bounded as follows out ρ λ 2 > 2 R

5 392 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO 2, NO, JANUARY 203 min ρ λ 2 > 2 R ρ αmax > 2 R α max > } α < [α < ] 28 where 28 follows since Mν + and thus α } are independent Since α > e ρ ρ then α < ρ and thus [α < ] ρ 29 where we have used the fact that ρ i + ρ ni ρ i for any positive integers i and n We then conclude from 27 and 29 that, when Mν +, out ρ λ 2 > 2 R ρ 30 We now consider the case >Mν +Wehavethe following lemma emma 3: Consider the MISO CDD under two transmission scenarios that are similar in all respects except their databloc lengths, where we assume > Mν +The eigenvalues of the channel are denoted λ } when data bloc length is, and are denoted λ } when data bloc length is et m be a positive number We have the following property ρ λ 2 >m ρ λ >m 2 roof: We start with [3, emma 2], which states: +ρ λ 2 >m +ρ λ >m 2 We note the difference with our desired result is only a constant term in the denominators Close inspection reveals that the proof outlined in [3, Appendix A] goes through even without these constant identity terms, therefore the proof of our desired result parallels step-by-step the proof of [3, emma 2] Thus setting m 2 R in emma 3 and using 30 we conclude that the ZF receiver achieves diversity order one irrespective of the bloc length 4 Non-Cyclic Delay Diversity: In this section we consider the more traditional delay diversity without cyclic prefix We establish the equivalence between the delay diversity DD single carrier or multicarrier MISO system 2 and the zero-padding single carrier or multicarrier SISO system We then apply the result of [8] which shows that linear receiver achieves full multipath diversity for the zero-padding SISO 2 For DD SC system, we adopt the model of [3] where there is no C insertion system In order to transform DD MISO channel into SISO ISI channel, the delays are set to δ i iδ with δ>νsymbols For simplicity, let δ ν + the proof is easily extended to other cases It can be shown that the received signal after removing the padding is 3 y ρ H s+ n 3 In the case of single carrier system with no C extension [3], s C is the data bloc and H C is the truncated Toeplitz channel matrix and the model is equivalent to the SISO ISI channel under single carrier zero-padding transmission Using linear equalizers in 3 achieves the full diversity d Mν + [8], [9] In the case of multicarrier system, the channel H C is a circulant channel matrix Hence the result of Section III-2 applies However, the DD multicarrier system incurs a rate loss compared to the CDD system R CDD R DD η N FFT + δ max + ν N FFT + ν where N FFT is the DFT size [7] The diversity gain obtained in Section III-2 is modified to d Mν + R η log Mν+ 2 R + R >ηlog Mν+ 32 IV AAMOUTI SIGNAING The Alamouti method of space-time signaling can also be characterized as a transmit diversity scheme Unlie the CDD system, our analysis shows that Alamouti signaling preserves the transmit diversity and thus provides a larger diversity gain compared with the CDD scheme above a rate threshold R th We consider single-carrier bloc transmission over an additive-noise frequency-selective channel with memory ν, similar to [5] The model supports a 2 system and can be extended to 2 N system Each data-bloc of length is appended with a C of length ν to eliminate interbloc interference x i n denotes the symbol n of the transmitted bloc from antenna i At even time slots, pairs of length-n blocs x n and x 2 n are generated The transmission scheme proposed by [5] is x + n x 2 n N x + 2 n x n N 33 for n 0,,,N and 0, 2, 4, denotes conjugate A length-ν C is added to each transmitted bloc The total transmit power is divided equally among the antennas The transmission scheme is shown in Figure 2 The received blocs at time and + are given by y j ρ H j xj + ρ H j 2 xj 2 + n j for j, + where H j and H j 2 are both circulant, and n j is the noise vector for bloc j A DFT is then applied to y j to diagonalize the channels as follows Y j ρ Λ j Xj + ρ Λ j 2 Xj 2 + Nj for j, + 3 In the DD multicarrier system, the received signal has a C extension added by the multicarrier modulator and a Z extension due to the delays δ i In the DD single carrier system the received signal only has a Z extension

6 MEHANA and NOSRATINIA: SINGE-CARRIER FREQUENCY-DOMAIN EQUAIZER WITH MUTI-ANTENNA TRANSMIT DIVERSITY 393 where Y j, X j and N j are the DFT vectors of y j, x j and n j respectively, and Λ i for i, 2 are diagonal matrices containing the DFT coefficients of the channel impulse responses Using 33 and assuming the channels are fixed over two consecutive blocs indexed by and +, it can be shown that Y ρ Λ Λ Y Y + 2 X Λ 2 Λ N + ρ X N + 2 }} Λ 34 By multiplying both sides of 34 by the orthogonal matrix Λ defined in 34 Λ 0 ρ X Ỹ Λ Y + 0 Λ Ñ 35 ρ X 2 where Ỹ and Ñ are the transformed receive vector Y and noise vector N respectively, and Λ Λ H Λ +Λ H 2 Λ 2 is a N N diagonal matrix whose diagonal element i is Λ i, i 2 + Λ 2 i, i 2 36 A MMSE Receiver We now show that the 2 N Alamouti SC-FDE can achieve full diversity 2Nν + as long as the transmission rate is below a certain threshold, and otherwise full spatio-temporal diversity is not achieved We fully characterize the diversity in all cases We start by considering N receive antenna The received signal for two blocs indexed by and + is given by 34 which can be written as Y ρ ΛX + N The MMSE equalizer is W Λ H Λ+ρ I Λ H In other words, the coefficients of the MMSE FDE are given by Λ, W, Λ, 2 + ρ erforming the equalization process followed by the IDFT operation yields ỹ ρ Q H Λ H Λ+ρ I Λ H ΛQx + ñ ỹ 2 ρ Q H Λ H Λ+ρ I Λ H ΛQx 2 + ñ 2 where ñ and ñ 2 are the filtered noise signals Assuming the transmitted vectors have equal power, the unbiased decisionpoint SINR of the MMSE SC-FDE for detecting the symbol of the vector x and x 2 are denoted by γ, and γ 2, and are given by ρ γ, γ 2, γ [ρ I + Q H ΛQ, ] Observe that all γ are equal because the matrix Q H ΛQ is circulant Thus γ can be written as ρ γ trρ I + Q H ΛQ ρ trρ I + Λ 37 tri + ρ Λ The mutual information is given by I MMSE log + γ log + γ 38 log tri + ρ Λ log [I + ρ Λ ] log +ρ λ the eigenvalues λ are the diagonal elements of Λ, andare given by λ λ, 2 + λ 2, 2 whereλ i, are the eigenvalues of the channel H i for i, 2 cf Equation 36 The outage probability of the MMSE receiver is given by out I MMSE <R +ρ λ >2 R 39 Similar to the SISO analysis presented in [3], we first consider the case when ν + and later extend it to the other cases when >ν+when ν + all the elements of Λ, } and Λ 2, } are iid Gaussian variables and hence the eigenvalues Λ } obey the Gamma distribution with shape parameter M 2 and scale parameter, ie λ ΓM, When>ν+ the elements Λ i, } i,2 are no longer independent and thus analyzing this case requires the unnown distribution of λ } Instead, we indirectly show that the diversity of ν + also holds for >ν+we continue with the case ν + et α log λ log ρ,wehave +ρ λ +ρ α 40 Observe that the term is either zero or one at high +ρ α SNR depending on the value of α [3] lim ρ +ρ λ ρ α ᾱ < ᾱ > 4 therefore to characterize the sum +ρλ in 39 at high SNR, we basically count the ones The outage probability can be written as out + ρ λ >2 R >2 R 42 α >

7 394 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO 2, NO, JANUARY 203 A rigorous proof for 42 follows similarly to [20, Section III-A] We thus need to evaluate α> The probability density function of λ is x f λ ΓM xm e x 43 where ΓM M! is the Gamma function We want to evaluate the probability α > which is the same as λ <ρ Using 43 we have λ <ρ ρ ρ 0 f λ x dx x M e x dx 44 ΓM 0 M ΓM ρ ΓM e ρ 45 ΓM! M e ρ 0 ρ! 0 46 where we have evaluated the integral in 44 according to [2, 336] Thus we have α > λ <ρ M e ρ 0 ρ! M! ρ M 47 ρ M 48 where 47 follows from the Taylor expansion of the exponential function in 46 and the fact that ρ i + ρ ni ρ i for any positive integers i and n From the independence of λ }, and subsequently the independence of α }, we conclude that M α in 42 is binomially distributed with parameter ρ M Hence, similar to [3], we have >2 R M α >2 R +ρ λ i 2 R + i 2 R + ρ M 2 R + M α i ρ i Mi ρ M n i }} which concludes the proof for ν + For the case of > ν +, we will need the following lemma emma 4: Consider the MISO Alamouti signaling given by 33 under two transmission scenarios that are similar in all respects except their data-bloc lengths, 2 where we assume 2 > ν + The eigenvalues of channels H and H 2 are denoted λ, } and λ 2, } respectively when data bloc length is, and are denoted λ, } and λ 2, } when data bloc length is 2 We have the following property + ρ λ, 2 + λ 2, 2 >2 R 2 + ρ λ, 2 + λ 2, 2 >2 R roof: The proof is similar to [22, emma 0] Thus the diversity of 2 Alamouti scheme under MMSE SC-FDE is 2ν + R log d ν 2 2 R + R>log ν 49 where is the data bloc length Remar 3: The result above shows that the Alamouti SC- FDE provides twice the diversity of SISO-SC-FDE At first sight this result seems to be a straight forward product decomposition of a so-called transmit diversity, ie, a factor of two due to Alamouti signaling, and a term due to SC- FDE [3] However, it is important to note that the problem was not fundamentally separable, therefore the results could not be deduced from [3], [5] without a proof, because the distribution of the summation of eigenvalues squared in [5, Eq ] is needed before one can mae a conclusive statement about the overall diversity of the system This subtle point can be further appreciated by noting that diversity in frequencyselective channels cannot in general be decomposed into separate components, eg, due to the transmitter signaling and otherwise For example in the earlier case of CDD, the diversity was not a multiple of the diversity of SISO SC-FDE 4 Thus, the result above was not preordained, even though its form is unsurprising The analysis can easily be generalized for N>receive antennas The outage in the case of N receive antennas will depend on eigenvalues λ 2N i λ i, 2 that have distribution λ Γ2N, Following a similar line of reasoning as earlier, full diversity is achieved when R log ν More broadly, the diversity for all spectral efficiencies is given by the following theorem Theorem : In a 2 N quasi-static frequency-selective channel with channel memory ν, using Alamouti signaling given by 33 the diversity of the MMSE-SC-FDE is given by 2Nν + R log d ν 2N 2 R + R>log ν 50 where is the data bloc length B Zero-Forcing Receiver We now analyze the zero-forcing equalization for Alamouti transmission It can be shown that the outage probability is out > ρ λ 2 R 5 4 There are also other examples showing the non-decomposability of diversity in frequency-selective channels, eg [22]

8 MEHANA and NOSRATINIA: SINGE-CARRIER FREQUENCY-DOMAIN EQUAIZER WITH MUTI-ANTENNA TRANSMIT DIVERSITY 395 -x * 2 -x * 2N- -x * -x * 20 C N- -x * -x * 0 C Ant δ 0 0, δ 2, δ 2 4 δ 0 0, δ, δ 2 2 δ 0 0, δ, δ x * -x * N- -x * 0 C -x * 2N- -x * 2 -x * 20 C Ant 2 Fig 2 Transmission scheme proposed by [5] for communication over frequency-selective fading channels Outage with λ λ, 2 + λ 2, 2,andλ i, is the -th eigenvalue of the channel H i i, 2and λ obey the Gamma distribution Similarly to Section III-3, it is straightforward to show that the diversity of the ZF receiver is only d zf 2 The analysis details are omitted for brevity V SIMUATION RESUTS Figure 3 shows the outage probability out for the equivalent model of the MMSE receiver in the CDD C MISO flat fading channel with 3 transmit antennas, under various choices of the cyclic delay taps The rate is R 2b/s/Hz and 5 In this case, the MMSE diversity is two as predicted from 20 since this rate is greater than R th given by 2 Figure 4 compares CDD-C and DD-without-C systems in a 2 MISO flat fading channel The latter system is equivalent to zero-padding transmission over a SISO ISI channel with three channel coefficients and thus achieves the full diversity for all rates [8] However, the CDD C-system only achieves full diversity for the rates that satisfy 2 Figure 5 compares the performance of zero-forcing and MMSE receivers in 2 Alamouti transmission for bloclength 4 The diversity of the ZF is two for all rates R, whereas the diversity of the MMSE is greater than or equal to two depending on the value of rate R cf Eq 49 Figure 6 compares the performance of zero-forcing receiver in 2 CDD and 2 Alamouti transmission with ν The diversity of the ZF-CDD is one whereas the diversity of the ZF-Alamouti is two The diversities of both systems are independent of R SNR Fig 3 The Outage probability of SC/MC CDD MISO under flat fading with three transmit antennas and R 2b/s/Hz Outage R R35 R0 DD system CDD system SNR Fig 4 erformance of single-carrier delay diversity DD and cyclic delay diversity CDD in 2 MISO under flat fading and R, 35, and 0 b/s/hz R6 ZF MMSE R4 VI CONCUSION This paper analyzes the single-carrier frequency domain equalizer SC-FDE for two common transmit diversity schemes: cyclic delay diversity CDD and Alamouti signaling We characterize the diversity for both schemes at all spectral efficiencies In the process, we obtain a threshold rate as a function of data-bloc length, channel memory, and number of antennas below which the full spatial-temporal diversity is achieved Our analysis shows that at high rates the CDD diversity degenerates to the diversity of the SISO SC- FDE, while Alamouti signaling provides twice the diversity of SISO SC-FDE Outage R SNR Fig 5 The outage probability of zero-forcing and MMSE receiver in 2 Alamouti system, 4, ν and R 2, 4, and 6 b/s/hz

9 396 IEEE TRANSACTIONS ON WIREESS COMMUNICATIONS, VO 2, NO, JANUARY CDD ZF Alamouti ZF Therefore We thus have h CN0, Γ 56 h d h 57 Outage R4 R6 R2 R4 R6 where d denotes equality in distribution 2 Define a vector h that is of length via appropriate number of zero-padding in arbitrary locations of vector h, for example: 0 4 R SNR Fig 6 The outage probability of zero-forcing receiver in 2 single-carrier CDD and Alamouti signaling with ν and R 2, 4, 6 b/s/hz AENDIX Recall that the channel vectors h and h 2 are given by h [h 0 h h Mν+, 0,, 0] h 2 [0 h 0 0 h 0 h Mν+ 0] and the corresponding DFT vectors are respectively given by λ h 2πi i e for,, λ 2 i0 i0 h 2 2πi i e for,, 52 Note that h 0,,h Mν+ } comprise the non-zero elements of both h and h 2 The channel coefficients h i } are assumed independent and identically distributed circular complex normal random variable with zero mean Thus the vector h [h 0 h h Mν+ ] obeys the complex normal distribution CN0, Γ given by fh π Mν+ detγ e hγ h H 53 where the covariance matrix Γ, considering that h is a row vector, is given by ΓE[h H h] We now construct a vector h that has the same distribution as h h [h 0 e jθ0 h e jθ h Mν+ e jθ Mν+ ] hθ 54 where the matrix Θ is a diagonal matrix that has e jθi } on its diagonal, and θ i } are arbitrary real-valued constants Eq 54 shows that h is a linear transform of h Thus h obeys CN0, Θ H ΓΘ However, since the coefficients h i } are independent, the covariance matrix Γ is diagonal and we have Θ H ΓΘ Θ H ΘΓ Γ 55 h 2 [0 h 0e jθ h e jθ 0 0 h Mν+ e jθ Mν+ 0] et λ 2 2 } be the DFT of h λ 2 Note that i0 and therefore i0 h 2 2πi i e for,, 58 h 2 d h 2 λ 2 } d λ 2 } 59 The phases θ i } can be chosen such that for,,, h 2 2πi i e h 2πi i e Using 59 and 60 we get Therefore >m +ρ λ 2 i0 λ 60 λ 2 } d λ 2 } λ } REFERENCES >m +ρ λ [] F ancaldi, G Vitetta, R Kalbasi, N Al-Dhahir, M Uysal, and H Mheidat, Single-carrier frequency domain equalization, IEEE Signal rocess Mag, vol 25, no 5, pp 37 56, Sept 2008 [2] D Falconer, S Ariyavisitaul, A Benyamin-Seeyar, and B Eidson, Frequency domain equalization for single-carrier broadband wireless systems, IEEE Commun Mag, vol 40, no 4, pp 58 66, Apr 2002 [3] A Tajer and A Nosratinia, Diversity order in ISI channels with singlecarrier frequency-domain equalizer, IEEE Trans Wireless Commun, vol 9, no 3, pp , Mar 200 [4] A Hesham Mehana and A Nosratinia, The diversity of MMSE receiver over frequency-selective MIMO channel, in roc 20 IEEE ISIT [5] N Al-Dhahir, Single-carrier frequency-domain equalization for spacetime bloc-coded transmissions over frequency-selective fading channels, IEEE Commun ett, vol 5, no 7, pp , July 200 [6] Z Wang and G Giannais, inearly precoded or coded OFDM against wireless channel fades? in roc 200 IEEE Signal rocess Advances Wireless Commun, pp [7] C Tepedelenlioglu, Maximum multipath diversity with linear equalization in precoded OFDM systems, IEEE Trans Inf Theory, vol 50, pp , Jan 2004 [8] C Tepedelenlioglu and Q Ma, On the performance of linear equalizers for bloc transmission systems, in roc 2005 IEEE GOBECOM, vol 6

10 MEHANA and NOSRATINIA: SINGE-CARRIER FREQUENCY-DOMAIN EQUAIZER WITH MUTI-ANTENNA TRANSMIT DIVERSITY 397 [9] B Muquet, Z Wang, G Giannais, M de Courville, and Duhamel, Cyclic prefixing or zero padding for wireless multicarrier transmissions? IEEE Trans Commun, vol 50, no 2, pp , Dec 2002 [0] Z Wang and G Giannais, Complex-field coding for OFDM over fading wireless channels, IEEE Trans Inf Theory, vol 49, no 3, pp , Mar 2003 [] H Gao, J Smith, and M V Clar, Theoretical reliability of MMSE linear diversity combining in Rayleigh-fading additive interference channels, IEEE Trans Commun, vol 46, no 5, pp , May 998 [2] Zheng and D N C Tse, Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels, IEEE Trans Inf Theory, vol 49, no 5, pp , May 2003 [3] A Wittneben, A new bandwidth efficient transmit antenna modulation diversity scheme for linear digital modulation, in roc 993 IEEE ICC, vol 3, pp [4] A Dammann and S Kaiser, On the equivalence of space-time bloc coding with multipath propagation and/or cyclic delay diversity in OFDM, in 2002 IEEE European Wireless [5] G Bauch and J Mali, Cyclic delay diversity with bit-interleaved coded modulation in orthogonal frequency division multiple access, IEEE Trans Wireless Commun, vol 5, no 8, pp , Aug 2006 [6] U-K Kwon and G-H Im, Cyclic delay diversity with frequency domain turbo equalization for uplin fast fading channels, IEEE Commun ett, vol 3, no 3, pp 84 86, Mar 2009 [7] A Dammann, On antenna diversity techniques for OFDM systems, hd dissertation, Universität Ulm, Germany, June 2005, VDI Verlag Düsseldorf, Series 0, No 766 [8] S Zhou and G Giannais, Single-carrier space-time bloc-coded transmissions over frequency-selective fading channels, IEEE Trans Inf Theory, vol 49, no, pp 64 79, Jan 2003 [9] Groop and D Tse, Diversity-multiplexing tradeoff in ISI channels, IEEE Trans Inf Theory, vol 55, no, pp 09 35, Jan 2009 [20] A Hesham Mehana and A Nosratinia, Diversity of MMSE MIMO receivers, in roc 200 IEEE ISIT [2] I S Gradshteyn and I M Ryzbi, Tables of Integrals, Series, and roducts, 6th edition Academic ress, 2000 [22] A H Mehana and A Nosratinia, Diversity of MMSE MIMO receivers, IEEE Trans Inf Theory, vol 5, no, pp , Nov 202 Ahmed Hesham Mehana S 0 received his BS and MS degrees from Cairo University in 2004 and 2007, respectively, both in Electrical Engineering He is currently pursuing the hd degree in electrical engineering at the University of Texas at Dallas He was an intern at Research in Motion Co td during 200 and research assistant in Texas A&M at Qatar in 2008 His current interests include MIMO precoding, linear receivers, and interference management Aria Nosratinia S 87-M 97-SM 04-F 0 is Eri Jonsson Distinguished rofessor and associate head of the Electrical Engineering Department at the University of Texas at Dallas He received his hd in Electrical and Computer Engineering from the University of Illinois at Urbana-Champaign in 996 He has held visiting appointments at rinceton University, Rice University, and UCA His interests lie in the broad area of information theory and signal processing, with applications in wireless communications He was the secretary for the IEEE Information Theory Society in and was the treasurer for ISIT 200 in Austin, Texas He has served as editor for the IEEE TRANSACTIONS ON INFORMATION THEORY, IEEE TRANSACTIONS ON WIREESS COMMUNI- CATIONS, IEEE SIGNA ROCESSINGETTERS, IEEE TRANSACTIONS ON IMAGE ROCESSING,andIEEE Wireless Communications Magazine Hehas been the recipient of the National Science Foundation career award, and is a fellow of IEEE