MMSE reception of DS-CDMA with transmit diversity

[0001] The present application claims priority from provisional application, Ser. No. 60/234,400, entitled “MMSE RECEPTION OF DS-CDMA WITH TRANSMIT DIVERSITY,” filed Sep. 21, 2000, which is commonly owned and incorporated herein by reference in its entirety.

[0002] The present invention relates generally to Code-Division, Multiple-Access (CDMA) receivers, and in particular, to a method and apparatus for receiving a Direct-Sequence (DS) CDMA signal.

[0003] Transmit diversity is beneficial because it mitigates degradation of the forward link due to signal fading. In its simplest sense, transmit diversity can be thought of as the transmitting a signal to a receiver using multiple antennas. In general diversity can be implemented at either the transmitter, the receiver or at both the transmitter and receiver. Additionally, diversity methods can be combined with feedback from the subscriber to the transmitter. Diversity methods with feedback are referred to as closed-loop methods, while diversity methods without feedback are referred to as open-loop methods. Open-loop transmit diversity methods are desirable because these methods have the least impact on subscriber complexity.

[0004] The three open-loop transmit diversity methods that have been included in the third generation standards are orthogonal transmit diversity (OTD), space-time spreading (STS), and space-time transmit diversity (STTD). The first two methods, OTD and STS, have been included in the current release of the third generation partnership project interim standard 2000 (IS20003GPP2 standard), while STTD is part of the third generation partnership project for wideband CDMA (3GPP standard).

[0005] Current methods for receiving CDMA signals involve the use of a matched-filter receiver, sometimes referred to as a RAKE receiver. More recently, however, it has been proposed to utilize a Minimum Mean Square Error (MMSE) receiver for reception of CDMA signals. MMSE receivers operate to minimize the mean-square error of the symbol estimates. A detailed description of the operation of such receivers for non-transmit diversity applications is given in the paper “Adaptive Interference Suppression for Direct-Sequence CDMA Systems with Long Spreading Codes” (Colin D. Frank and Eugene Visotsky, In Proceedings of the Allerton Conference on Communications, Control and Computing, Monticello, Ill., Sep. 23-25, 1998, pp. 411-420) Although MMSE receivers have been described for receiving CDMA signals in a non-transmit diversity environment, there currently exists no MMSE receiver capable of receiving CDMA signals with transmit diversity. Therefore, a need exists for an MMSE receiver capable of receiving DS-CDMA signals transmitted utilizing transmit diversity.

[0010] To address the need for an MMSE receiver capable of receiving DS-CDMA signals with transmit diversity, several MMSE receivers are provided herein. In particular, three different MMSE receivers—one for OTD, one for STS, and one for STTD are provided.

[0011] In an alternate embodiment of the present invention, a pilot-based adaptation is utilized for the MSME receivers. In particular, a pilot signal enters a first MMSE receiver and a second signal enters a second MMSE receiver. The first MMSE receiver performs a channel estimate for the received pilot signal, determines a mean-square error of the pilot signal, and operates to minimize the mean-square error of the pilot estimate by constantly updating a weighting vector. An estimate of the pilot channel exits the first MMSE receiver. The MMSE weighting vector is also fed into the second MMSE receiver and is applied to the second channel. Symbol estimates (or in other embodiments, chip estimates) exit the second MMSE receiver and enter channel circuitry, where normal channel processing occurs.

[0012] Since the pilot channel data is known, the MMSE receiver for the pilot channel data can be updated using the measured error of the pilot estimate. However, because the MMSE receiver is the same for both the pilot and other channels, the MMSE receiver for the pilot can be used to demodulate the second channel, and this receiver will yield minimum mean-square error estimates of the data channel symbols.

[0013] The present invention encompasses an apparatus comprising a first Minimum Mean Square Error (MMSE) receiver having a signal as an input, wherein the signal was transmitted utilizing a transmit-diversity scheme, and a second MMSE receiver having the signal as an input.

[0014] The present invention additionally encompasses a method for Minimum Mean Square Error (MMSE) reception of a signal. The method comprises the steps of receiving a pilot signal at an MMSE receiver, performing a channel estimate for the received pilot signal, and determining a mean-square error of the pilot signal estimate. A weighting vector is updated for the pilot signal estimate in order to minimize the mean square error of the pilot signal estimate and the weighting vector is applied to a second channel.

[0015] The present invention additionally encompasses a minimum mean square error receiver (MMSE) comprising a pilot channel input, a second channel input, an output comprising an estimate of the pilot channel, wherein the estimate of the pilot channel is determined by applying a weighting vector to the pilot channel, and a second output comprising an estimate of the second channel, wherein the estimate of the second channel is determined by applying the weighting vector to the second channel.

[0016] To address the need for an MMSE receiver capable of receiving DS-CDMA signals with transmit diversity, a MMSE receiver is described herein. In the following discussion, matched filter and MMSE receivers are developed for all three transmit diversity methods. Signal-to-noise ratio expressions for both the matched filter and the MMSE receivers are also developed, and these expressions are used to compare the performance of the different diversity methods. Adaptive implementations of the MMSE receiver for all three forms of transmit diversity are then discussed.

Matched Filter and MMSE Receivers for Orthogonal Transmit Diversity (OTD)

[0017] In two-antenna orthogonal transmit diversity, the coded and interleaved symbols are partitioned into pairs of symbols. Let s₀ and s₁ denote the two symbols in each pair. Several variations of OTD exist. While all forms of OTD can be analyzed using the methods similar to those described herein, we restrict our attention in this discussion to the following particular implementation of OTD. Let w denote the Walsh code assigned to the subscriber of interest, and let N denote the length of this code. For this implementation of OTD, symbols of type s₀ are transmitted over the first antenna using the extended code w⁰=(w,−w), and the symbols of type s₁ are transmitted over the second antenna using the extended code w¹=(w,w). Note that these extended codes are orthogonal to each other and to the set of Walsh codes. Note also that the bandwidth occupied by the subscriber using OTD, measured in terms of the number of Walsh codes allocated to this subscriber, is no greater than for the subscriber not using transmit diversity. Finally, note that the symbol rate of this implementation of OTD is equal to the symbol rate without transmit diversity.

[0018] Let K denote the number of subscribers in the sector, and let P_i denote the total power allocated to the i-th subscriber. Let α_i denote the fraction of the power allocated to each of the two antennas for the i-th subscriber. For subscribers employing orthogonal transmit diversity, α_i is equal to ½, while α_i is equal to 0 or 1 for subscribers not employing transmit diversity.

[0019] The signal received by the subscriber is filtered and converted to baseband prior to sampling. If the received signal is sampled at the chip rate with sampling times {nT_c+τ}, n integer, the performance of the receiver will typically be sensitive to the sampling phase τ. In order to avoid poor performance associated with a particular sampling phase τ₀, the received signal is typically sampled at a multiple of the chip rate. Since the need for oversampling is well understood, this issue will not be addressed further in this paper. Thus, the signal received at the subscriber is filtered, converted to baseband, and sampled at the chip rate to produce the sequence {r_k}, and the demodulator has access only to this chip-spaced sequence {r_k}.

[0020] Let the vectors f₀ and f₁, respectively, denote the channels between the first and second antennas and the subscriber of interest. Since the received signal is sampled at the chip rate, the vector elements consist of chip-spaced samples of the channel impulse responses. In order to simplify the analysis, the channels f₀ and f₁ are assumed to be zero outside of an interval of length M, so that
f_i,j=0 if j<0 or j>M−1.

[0021] The sequence {r_k} is reduced to a set of observation vectors, R₀ and R₁, by despreading this sequence with the complex spreading code, {c_k} of the serving base station, and correlating with the code channel of interest. Let the vectors R₀ and R₁ be defined as $R_{0,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^0=\sum _{i=1}^N{r}_{i+1}{c}_i^{*}{w}_i-\sum _{i=1}^N{r}_{N+i+l}{c}_{N+i}^{*}{w}_i$ $R_{1,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^1=\sum _{i=1}^N{r}_{i+1}{c}_i^{*}{w}_i+\sum _{i=1}^N{r}_{N+i+l}{c}_{N+i}^{*}{w}_i,$
for l in the interval [0, M−1], so that these vectors are of length M. More generally, the vectors R₀ and R₁ may be defined on a set of delays which are possibly non-consecutive and may also lie outside the interval [0, M−1]. This more general definition can be used to evaluate the performance of a Rake architecture with MMSE combining or to determine the performance of the MMSE receiver as a function of the equalizer length. However, for the remainder of the discussion in this paper, the initial definition will be used.

[0022] The objective is to define and evaluate the performance of the matched filter receiver and the MMSE receiver for the symbols s₀ and s₁. Both the matched filter receiver and the MMSE receiver yield estimates of the symbols s₀ and s₁, respectively, of the form
ŝ₀=v₀^HR₀ŝ₁=v₁^HR₁.

[0023] However, before the matched filter and MMSE receivers can be presented, the mean and covariance of the observation vectors R₀ and R₁ must be evaluated. If the Walsh code w corresponds to the first code channel, then the mean vectors are given by
μ_R0(s₀)=E(R₀s*₀)=2N√{square root over (P₁/2)}f₀
μ_R1(s₁)=E(R₁s*₁)=2N√{square root over (P₁/2)}f₁.
The correlation matrix of the vector R₀ is given by ${\Gamma }_{R_0}=E\left(R_0{R}_0^H\right)=2N\left(\sum _{j=0}^{K-1}{P}_i{\alpha }_i\right){\Psi }^{0,0}+2N\left(\sum _{j=0}^{K-1}{P}_i\left(1-{\alpha }_i\right)\right){\Psi }^{1,1}+2N^2{P}_1{f}_0{f}_0^H+2N{I}_{oc}\Phi ,$
where the matrix Ψ^i,j has elements given by ${\Psi }_{l,m}^{i,j}=\sum _{k\ne l}{f}_{i,k}{f}_{j,m-l+k}^{*}=\sum _k{f}_{i,k}{f}_{j,m-l+k}^{*}=f_{i,l}{f}_{j,m}^{*},$
and Φ denotes the covariance matrix for the other-cell interference and noise. The parameter I_oc denotes the power of the other-cell interference and noise, and thus the diagonal elements of Φ are by definition equal to 1. Similarly, the correlation matrix of the vector R₁ is given by ${\Gamma }_{R_1}=E\left(R_1{R}_1^H\right)=2N\left(\sum _{j=0}^{K-1}{P}_i{\alpha }_i\right){\Psi }^{0,0}+2N\left(\sum _{j=0}^{K-1}{P}_i\left(1-{\alpha }_i\right)\right){\Psi }^{1,1}+2N^2{P}_1{f}_1{f}_1^H+2N{I}_{oc}\Phi ,$
Let I_or denote the power per chip received from the serving base station, which is given by $I_{or}=\left(\sum _{j=0}^{K-1}{P}_i{\alpha }_i\right)||f_0{||}^2+\left(\sum _{j=0}^{K-1}{P}_i\left(1-{\alpha }_i\right)\right)||f_1{||}^2.$
Also, let Ω denote the normalized covariance matrix for both R₀ and R₁, which is given by $\begin{array}{c}\Omega =\left({\Gamma }_{R_0}-{\mu }_{R_0}\left(s_0\right){\mu }_{R_0}^H\left(s_0\right)\right)/\left(2N{I}_{or}\right)=\left({\Gamma }_{R_1}-{\mu }_{R_1}\left(s_1\right){\mu }_{R_1}^H\left(s_1\right)\right)/\left(2N{I}_{or}\right)\\ =\left(\sum _{j=0}^{K-1}{P}_i{\alpha }_i\right)I_{or}^{-1}{\Psi }^{0,0}+\left(\sum _{j=0}^{K-1}{P}_i\left(1-{\alpha }_i\right)\right)I_{or}^{-1}{\Psi }^{1,1}+\left(I_{oc}/I_{or}\right)\Phi .\end{array}$
For an arbitrary combining vector, v, the signal-to-noise ratio for symbol s₀, transmitted from the first antenna, is given by $SN{R}_{\mathrm{OTD\_v}{\mathrm{\_s}}_0}=\frac{N\left(P_1/I_{or}\right)||f_0^{H}v{||}^2}{v^H\Omega v}.$
The matched filter receiver for a symbol, s₀, transmitted from the first antenna is f₀, and the signal-to-noise ratio for the matched-filter receiver is given by $SN{R}_{\mathrm{OTD\_MF}{\mathrm{\_s}}_0}=\frac{N\left(P_1/I_{or}\right)||f_0{||}^2}{f_0^H\Omega f_0}.$
Similarly, the matched filter for symbol, s₁, transmitted from the second antenna is f₁, and the signal-to-noise ratio for this receiver is given by $SN{R}_{\mathrm{OTD\_MF}{\mathrm{\_s}}_1}=\frac{N\left(P_1/I_{or}\right)||f_1{||}^2}{f_1^H\Omega f_1}.$
The MMSE receiver for s₀ satisfies the relations $\begin{array}{c}{v}_{\mathrm{OTD\_MMSE}{\mathrm{\_s}}_0}\propto {\Gamma }_{R_0}^{-1}{f}_0\\ \propto {\Omega }^{-1}{f}_0,\end{array}$
where the latter relation is proved using the matrix inverse lemma. The signal-to-noise ratio of the MMSE receiver for s₀ is given by
SNR_{OTD—MMSE—s0}=N(P₁/I_or)f₀^HΩ⁻¹f₀.

[0024] Similarly, the MMSE receiver for a symbol s₁ transmitted from the second antenna satisfies the relation
v_{OTD—MMSE—s1}∝Ω⁻¹f₁,
and the signal-to-noise ratio of the MMSE receiver for s₁ is given by
SNR_{OTD—MMSE—s1}=N(P₁/I_or)f₁^HΩ⁻¹f₁.

[0025] Note that if orthogonal transmit diversity is used for all of the code channels, so that α_i=½ for all i, the intracell interference power I_or becomes $I_{\mathrm{or}}=\left(\sum _{j=0}^{K-1}{P}_i\right)\left({f_0}^2+{f_1}^2\right),$
and the interference covariance matrix Ω becomes $\Omega =\left(I_{\mathrm{or}}^{-1}\sum _{j=0}^{K-1}{P}_i/2\right)\left({\Psi }^{0,0}+{\Psi }^{1,1}\right)+\left(I_{\mathrm{oc}}/I_{\mathrm{or}}\right)\Phi .$

[0026] With appropriate normalization of the powers {P_i}, the sum power of the channels f₀ and f₁ can be taken to be one, so that the P₁/I_or is equal to the fraction of the power allocated by the base station to the first code channel. Thus, with this normalization and with all code channels using orthogonal transmit diversity, we have $P_1/I_{\mathrm{or}}=P_1/\sum _{j=0}^{K-1}{P}_i,\mathrm{and}$ $\Omega =\left({\Psi }^{0,0}+{\Psi }^{1,1}\right)/2+\left(I_{\mathrm{oc}}/I_{\mathrm{or}}\right)\Phi .$
Matched Filter and MMSE Receivers for Space-Time Transmit Diversity and Space Time Spreading

[0027] Space-time transmit diversity (STTD) and space-time spreading (STS) are very closely related forms of transmit diversity. Space-time transmit diversity was first described by Alamouti in “A Simple Transmit Diversity Technique for Wireless Communications” IEEE Journal on Selected Area in Communications, vol. 16, pp 1451-1458, October 1998, and has been introduced into the 3GPP standard. Space-time spreading is a variation of STTD that has been introduced into the 3GPP2 standard. In fact, if implemented with a particular set of extended Walsh codes, STS is related to STTD by a simple mapping of the code symbols.

[0028] Both STTD and STS partition the encoded and interleaved symbols into pairs of symbols, s₀ and s₁. With STTD, the symbol pair is transmitted over two time slots using a single Walsh code. Conversely, with STS, the symbol pair is transmitted over a single time slot using two Walsh codes. The mappings used for STTD and STS are indicated in Table 1.

[0029]

TABLE 1
Modulation Format for STTD and STS
	STTD	STS
	time 1	time 2	code 1	code 2
	Antenna 1	s0	−s1*	s0	−s1*
	Antenna 2	s1	s0*	s1	s0*

[0030] Because STS requires two Walsh codes rather than the single Walsh code used for the STTD, some care is required in comparing the performance of these two diversity methods. In a fair comparison of STS and STTD, the same number of Walsh codes (same bandwidth) must be allocated to each method, and the overall symbol rate of the two methods must also be equal. The equal bandwidth and equal symbol rate requirements can be satisfied by using extended Walsh codes for STS. Thus, if the Walsh code w is assigned to a particular subscriber in the STTD implementation, the extended Walsh codes w⁰=(w,−w) and w¹=(w,w) are assigned to the same subscriber in the STS implementation. With this assignment, STS uses the single Walsh code w, and the two symbols, s₀ and s₁, are transmitted over the period of two of these Walsh symbols. Note also that, since two extended Walsh codes are transmitted simultaneously in the STS implementation, the power allocated to each of these code channels must be equal to one-half of the power allocated to the single code channel in the STTD implementation.

[0031] With the above description of STS and STTD, the observation vectors R₀ and R₁ used in the demodulation of the symbols s₀ and s₁ can now be defined for each of the two diversity methods. For STTD, the vectors R₀ and R₁ result from the correlation of the received sequence {r_k} with the complex spreading code of the serving base station and the code channel of interest in the first and second time intervals, respectively. Specifically, the vectors R₀ and R₁, are defined such that $\begin{array}{c}{R}_{0,l}=\sum _{i=1}^N{r}_{i+l}{c}_i^{*}{w}_i\\ R_{1,l}=\sum _{i=1}^N{r}_{N+i+l}{c}_{N+i}^{*}{w}_i,\end{array}$
for l in the interval [0, M−1]. For STS, the vectors R₀ and R₁ result from the correlation of the received sequence {r_k} with the complex spreading code of the serving base station and the first and second extended Walsh codes, respectively. Thus, if the first and second code channels w⁰=(w,−w) and w¹=(w,w), the vectors R₀ and R₁ are defined such that $\begin{array}{c}{R}_{0,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^0=\sum _{i=1}^N{r}_{i+l}{c}_i^{*}{w}_i-\sum _{i=1}^N{r}_{N+i+l}{c}_{N+i}^{*}{w}_i\\ R_{1,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^1=\sum _{i=1}^N{r}_{i+l}{c}_i^{*}{w}_i+\sum _{i=1}^N{r}_{N+i+l}{c}_{N+i}^{*}{w}_i,\end{array}$
for l in the interval [0, M−1]. For both STTD and STS, the vectors R₀ and R₁ can be defined more generally on a set of delays that are possibly non-consecutive and may also lie outside the interval [0, M−1]. However, for the remainder of this paper, the above definitions will be used.

[0032] The performance of STTD and STS can be shown to be exactly the same for both the matched-filter receiver and the MMSE receiver. Furthermore, with the exception of some very minor differences, the analysis of the matched-filter and MMSE receivers is the same for these two diversity methods. Thus, in the remainder of this section, only STTD will be considered in the discussion of the matched-filter and MMSE receivers and their respective performance.

[0033] Let Q denote the vertical concatenation of the two vectors R₀ and R₁* so that $Q=\left[\begin{array}{c}{R}_0\\ R_1^{*}\end{array}\right]$
The expected value of the product of Q and the conjugates of s₀ and s₁, respectively are given by $\begin{array}{c}{\mu }_Q\left(s_0\right)=E\left({\mathrm{Qs}}_0^{*}\right)=N\sqrt{P_1/2}\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]\\ {\mu }_Q\left(s_1\right)=E\left({\mathrm{Qs}}_1^{*}\right)=N\sqrt{P_1/2}\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]\end{array}$
The correlation matrix Γ_Q is given by ${\Gamma }_Q=E\left({\mathrm{QQ}}^H\right)=E\left\{\left[\begin{array}{cc}{R}_0{R}_0^H& R_0{R}_1^T\\ R_1^{*}{R}_0^H& R_1^{*}{R}_1^T\end{array}\right]\right\}.$
where $\begin{array}{c}E\left(R_0{R}_0^H\right)=N\left(\sum _{j=1}^K{P}_i{\alpha }_i\right){\Psi }^{0,0}+N\left(\sum _{j=1}^K{P}_i\left(1-{\alpha }_i\right)\right){\Psi }^{1,1}+\\ N^2\left(P_1/2\right)\left(f_0{f}_0^H+f_1{f}_1^H\right)+N{I}_{\mathrm{oc}}\Phi ,\\ E\left(R_1{R}_1^H\right)=N\left(\sum _{j=1}^K{P}_i{\alpha }_i\right){\Psi }^{0,0}+N\left(\sum _{j=1}^K{P}_i\left(1-{\alpha }_i\right)\right){\Psi }^{1,1}+\\ N^2\left(P_1/2\right)\left(f_0{f}_0^H+f_1{f}_1^H\right)+N{I}_{\mathrm{oc}}\Phi ,\end{array}$ $\mathrm{and}$ $\begin{array}{c}E\left(R_0{R}_1^T\right)=N^2\left(P_1/2\right)\left(f_0{f}_1^T-f_1{f}_0^T\right),\\ E\left(R_1^{*}{R}_0^H\right)=N^2\left(P_1/2\right)\left(-f_0^{*}{f}_1^H+f_1^{*}{f}_0^H\right).\end{array}$
The matrices Ψ^0,0, Ψ^1,1, and Φ are as defined in Section 2. Let the matrix Ω also be defined as in Section 2 and note that, as before, the power per chip received from the serving base station, I_or, is given by $I_{\mathrm{or}}=\left(\sum _{j=0}^{K-1}{P}_i{\alpha }_i\right){f_0}^2+\left(\sum _{j=0}^{K-1}{P}_i\left(1-{\alpha }_i\right)\right){f_1}^2.$
With these definitions, the normalized correlation matrix can be expressed as $\frac{{\Gamma }_Q}{{\mathrm{NI}}_{\mathrm{or}}}=\left[\begin{array}{cc}\Omega & 0\\ 0& {\Omega }^{*}\end{array}\right]+\frac{{\mathrm{NP}}_1}{2I_{\mathrm{or}}}\left({\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]}^H+{\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]}^H\right).$
The matched-filter receiver for symbol s₀ is given by $v_{\mathrm{STTD\_MF}{\mathrm{\_s}}_0}=\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right].$
The signal-to-noise ratio for the matched filter receiver is given by $\begin{array}{c}{\mathrm{SNR}}_{\mathrm{STTD\_MF}{\mathrm{\_s}}_0}=\frac{\left({\mathrm{NP}}_1/2I_{\mathrm{or}}\right){\left(f_0^H{f}_0+f_1^H{f}_1\right)}^2}{{\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]}^H\frac{\Gamma }{{\mathrm{NI}}_{\mathrm{or}}}\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]-\frac{{\mathrm{NP}}_1}{2I_{\mathrm{or}}}{\left(f_0^H{f}_0+f_1^H{f}_1\right)}^2}\\ =\frac{N\left(P_1/2I_{\mathrm{or}}\right){\left(f_0^H{f}_0+f_1^H{f}_1\right)}^2}{f_0^H\Omega f_0+f_1^H\Omega f_1}.\end{array}$
Similarly, the matched-filter receiver for s₁ is given by $v_{\mathrm{STTD\_MF}{\mathrm{\_s}}_1}=\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right],$
and the signal-to-noise ratio for this matched filter receiver is given by $\begin{array}{c}{\mathrm{SNR}}_{\mathrm{STTD\_MF}{\mathrm{\_s}}_1}=\frac{\left({\mathrm{NP}}_1/2I_{\mathrm{or}}\right){\left(f_1^H{f}_1+f_0^H{f}_0\right)}^2}{{\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]}^H\frac{\Gamma }{{\mathrm{NI}}_{\mathrm{or}}}\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]-\frac{{\mathrm{NP}}_1}{2I_{\mathrm{or}}}{\left(f_0^H{f}_0+f_1^H{f}_1\right)}^2}\\ =\frac{N\left(P_1/2I_{\mathrm{or}}\right){\left(f_1^H{f}_1+f_0^H{f}_0\right)}^2}{f_1^H\Omega f_1+f_0^{H*}{\Omega }^{*}{f}_0^{*}}\\ ={\mathrm{SNR}}_{\mathrm{STTD\_MF}{\mathrm{\_s}}_0}.\end{array}$
Thus, the signal-to-noise ratio expressions for the matched-filter detection of s₀ and s₁ are identical.

[0034] General conditions under which the received signal-to-noise ratios of the symbols s₀ and s₁ are identical can be easily described. Let the receivers for symbols s₀ and s₁ be denoted by $v_{s_0}=\left[\begin{array}{c}{v}_{s_0}^u\\ v_{s_0}^l\end{array}\right],v_{s_1}=\left[\begin{array}{c}{v}_{s_1}^u\\ v_{s_1}^l\end{array}\right].$

[0035] From the expressions given above for the mean and variance of Q, a sufficient condition can be derived for these receivers to yield decision statistics with equal signal-to-noise ratios. The sufficient condition is that
v_s1^u=(v_s0^l)*, v_s1^l=−(v_s0^u)*.
The MMSE receiver for s₀ satisfies the relations $\begin{array}{c}{v}_{\mathrm{STTD\_MMSE}{\mathrm{\_s}}_0}\propto \frac{{\Gamma }^{-1}}{{\mathrm{NI}}_{\mathrm{or}}}\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]\\ \propto \left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right].\end{array}$
where the latter relation is proved with repeated use of the matrix inverse lemma. With this expression, the signal-to-noise ratio for the MMSE receiver can be written as $\begin{array}{c}{\mathrm{SNR}}_{\mathrm{STTD\_MMSE}{\mathrm{\_s}}_0}=\frac{\frac{{\mathrm{NP}}_1}{2I_{\mathrm{or}}}{\left({\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]}^H\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]\right)}^2}{\left\{\begin{array}{c}{\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]}^H\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\frac{\Gamma }{{\mathrm{NI}}_{\mathrm{or}}}\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]-\\ \frac{{\mathrm{NP}}_1}{2I_{\mathrm{or}}}{\left({\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]}^H\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]\right)}^2\end{array}\right\}}\\ =N\left(P_1/2I_{\mathrm{or}}\right)\left(f_0^H{\Omega }^{-1}{f}_0+f_1^H{\Omega }^{-1}{f}_1\right).\end{array}$
Similarly, the MMSE receiver for s₁ satisfy the relations $\begin{array}{c}{v}_{\mathrm{STTD\_MMSE}{\mathrm{\_s}}_1}=\frac{{\Gamma }^{-1}}{{\mathrm{NI}}_{\mathrm{or}}}\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]\\ \propto \left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right].\end{array}$
The signal-to-noise ratio resulting from the use of the MMSE receiver for s₁ is given by $\begin{array}{c}{\mathrm{SNR}}_{\mathrm{STTD\_MMSE}{\mathrm{\_s}}_1}=\frac{\frac{{\mathrm{NP}}_1}{2I_{\mathrm{or}}}{\left({\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]}^H\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]\right)}^2}{\left\{\begin{array}{c}{\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]}^H\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\frac{\Gamma }{{\mathrm{NI}}_{\mathrm{or}}}\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]-\\ N\frac{E_c}{I_{\mathrm{or}}}{\left({\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]}^H\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]\right)}^2\end{array}\right\}}\\ =N\left(P_1/2I_{\mathrm{or}}\right)\left(f_0^H{\Omega }^{-1}{f}_0+f_1^H{\Omega }^{-1}{f}_1\right)\\ ={\mathrm{SNR}}_{\mathrm{STTD\_MMSE}{\mathrm{\_s}}_0},\end{array}$
and thus the signal-to-noise ratios of the MMSE receivers for s₀ and s₁ are equal.

[0036] With trivial modifications of the above analysis, the expressions for the STTD MMSE receivers can be shown to be identical to the expressions given above for the STS MMSE receiver. That is, $\begin{array}{cc}{\hat{s}}_0={\left(v_{\mathrm{STS\_MMSE}{\mathrm{\_s}}_0}\right)}^H\left[\begin{array}{c}{R}_0\\ R_1^{*}\end{array}\right]& {\hat{s}}_1={\left(v_{\mathrm{STS\_MMSE}{\mathrm{\_s}}_1}\right)}^H\left[\begin{array}{c}{R}_0\\ R_1^{*}\end{array}\right],\end{array}$ $\mathrm{where}$ $v_{\mathrm{STS\_MMSE}{\mathrm{\_s}}_0}\propto \left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right],\mathrm{and}$ $v_{\mathrm{STS\_MMSE}{\mathrm{\_s}}_1}\propto \left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right].$
Of course, as noted previously, R₀ ands R₁ are defined slightly differently for STS than for STTD.

[0037] An example illustrating the relative performance of the MMSE and matched-filter receivers for STTD is given in FIG. 1. In this example, the first channel has two non-zero taps and is given by f₀={0, . . . , 0, 0.5, 0.5, 0, . . . 0), while the second channel has a single non-zero tap, and is given by f₁={0, . . . , 0, 0.71, 0, 0, . . . 0). Note that the energy of both channels is equal to ½. The other-cell interference is assumed to have a correlation function with values {. . . , 0, 0.4, 1, 0.4, 0, . . . }.

[0038] In this example, a separate 15-tap MMSE equalizer is allocated to each of the transmit antennas. FIG. 1 indicates the gain in signal-to-noise ratio of the MMSE receiver relative to the matched-filter receiver as a function of the ratio I_oc/I_or. For this example, the MMSE receiver performs 1.5 to 2 dB better than the matched-filter receiver.

[0039] With the signal-to-noise ratio expressions derived in the previous section, several observations can be made regarding the relative performance of OTD and STTD/STS. The signal-to-noise ratio performance of the matched-filter receiver for OTD and STTD/STS can be summarized in the following expressions: $SN{R}_{\mathrm{OTD\_MF}{\mathrm{\_s}}_0}=\frac{N\left(P_1/I_{or}\right){\left(f_0^H{f}_0\right)}^2}{f_0^H\Omega f_0}$ $SN{R}_{\mathrm{OTD\_MF}{\mathrm{\_s}}_1}=\frac{N\left(P_1/I_{or}\right){\left(f_1^H{f}_1\right)}^2}{f_1^H\Omega f_1}$ $SN{R}_{\mathrm{STS\_MF}{\mathrm{\_s}}_0}=SN{R}_{\mathrm{STS\_MF}{\mathrm{\_s}}_1=}\frac{N\left(P_1/2I_{or}\right){\left(f_0^H{f}_0+f_1^H{f}_1\right)}^2}{f_0^H\Omega f_0+f_1^H\Omega f_1}.$

[0040] With the above expressions, it claim can be easily proved that
SNR_{OTD—MF—s0}+SNR_{OTD—MF—s1}≧SNR_{STTD—MF—s0}+SNR_{STTD—MF—s1}=2SNR_{STTD—MF—s0},
with equality if and only if
(f₀^Hf₀)(f₁^HΩf₁)=(f₁^Hf₁)(f₀^HΩf₀).

[0041] The meaning of this inequality is that the sum signal-to-noise ratio for matched filter reception of OTD is always greater than or equal to the sum signal-to-noise ratio for matched-filter reception of STTD/STS. This result is perhaps somewhat surprising, since the generally held belief is that STTD/STS always performs better than OTD. There are two possible interpretations of this result. The first interpretation is that the sum signal-to-noise ratio for OTD is superior to that of STS and STTD because it creates a more favorable interference environment. The second interpretation is that matched filter reception is more nearly optimal for OTD than for STS or STTD. In light of the results given below for the MMSE receiver, it seems that the correct interpretation is that the matched filter receiver is more nearly optimal for OTD than for STTD/STS.

[0042] For uncoded binary modulation, the average bit error rate for STTD/STS will be less than that for OTD if the following inequality is satisfied: $Q\left({\left(2SN{R}_{\mathrm{STS\_MF}{\mathrm{\_s}}_0}\right)}^{\frac{1}{2}}\right)\le \frac{1}{2}Q\left({\left(2SN{R}_{\mathrm{OTD\_MF}{\mathrm{\_s}}_0}\right)}^{\frac{1}{2}}\right)+\frac{1}{2}Q\left({\left(2SN{R}_{\mathrm{OTD\_MF}{\mathrm{\_s}}_1}\right)}^{\frac{1}{2}}\right).$
A necessary (but not sufficient) condition for the average uncoded bit error rate of STTD/STS to be less than the average uncoded error probability of OTD is the following:
min{SNR_{OTD—MF—s0}, SNR_{OTD—MF—s1}}≦SNR_{STS—MF—s0}.

[0043] In general, it appears that not even this weaker inequality will always be satisfied. However, it seems likely that both inequalities will be satisfied for most channel pairs f₀ and f₁.

[0044] The signal-to-noise ratio performance of the MMSE receiver for OTD and STTD/STS can be summarized in the following expressions:
SNR_{OTD—MMSE—s0}=N(P₁/I_or)(f₀^HΩ⁻¹f₀)
SNR_{OTD—MMSE—s1}=N(P₁/I_or)(f₁^HΩ⁻¹f₁).
SNR_{STTD—MMSE—s0}=SNR_{STTD—MMSE—s1}=N(P₁/2I_or)(f₀^HΩ⁻¹f₀+f₁^HΩ⁻¹f₁).
Thus, for the MMSE receiver
SNR_{OTD—MMSE—s0}+SNR_{OTD—MMSE—s1}=2SNR_{STTD—MMSE—s0},
and the sum signal-to-noise ratio for OTD is exactly equal to the sum signal-to-noise ratio for STTD/STS. Since the two STTD/STS sub-channels have the same signal-to-noise ratio, the following inequality holds for all channel pairs f₀ and f₁:
min{SNR_{OTD—MMSE—s0}, SNR_{OTD—MMSE—s1}}≦SNR_{STTD—MMSE—s0}.

[0045] Similarly, for the MMSE receiver, the average uncoded bit error probability (binary modulation) for STTD/STS can be shown to be less than or equal to the average bit error probability for OTD. Thus, $Q\left({\left(2SN{R}_{\mathrm{STTD\_MMSE}{\mathrm{\_s}}_0}\right)}^{\frac{1}{2}}\right)\le \frac{1}{2}Q\left({\left(2SN{R}_{\mathrm{OTD\_MMSE}{\mathrm{\_s}}_0}\right)}^{\frac{1}{2}}\right)+\frac{1}{2}Q\left({\left(2SN{R}_{\mathrm{OTD\_MMSE}{\mathrm{\_s}}_1}\right)}^{\frac{1}{2}}\right).$

[0046] Two basic architectures exist for the implementation of the MMSE receivers for OTD, STS, and STTD. In the first implementation, which we shall refer to as the correlator implementation, the receiver correlates delayed versions of the received signal with the product of the long spreading code and the Walsh code or codes corresponding to the data channel of interest. The correlations are performed at a set of delays that generally correspond to the set of multipath delays introduced by the channel. For each of the data subchannels s₀ and s₁, a vector of correlations (the previously described observation vector for the data subchannel) is passed to the MMSE receiver for that subchannel. In this implementation, the MMSE receiver employs a complex-valued MMSE weighting vector with length equal to the length of the observation vector. The MMSE receiver forms an estimate of the subchannel data symbol by taking the inner product of the MMSE weighting vector and the observation vector. By definition, the MMSE weighting vector is chosen (as derived previously) so as to minimize the mean-square error of the symbol estimate.

[0047] The second implementation of the MMSE receiver will be referred to as the filter implementation. The filter implementation of the MMSE receiver is achieved by interchanging the order of the correlation (or despreading) operation and the application of the MMSE weighting vector. In this implementation, the received signal is filtered with the time-reverse of the MMSE weighting vector. For OTD a single filter is required per data subchannel. For STTD and STS, two filters are required per data subchannel, but the same two filters are required (and thus can be shared) by the two data subchannels. The sequences at the output of these filters are correlated with the product of the long spreading sequence and the code channel (or channels) of interest. For OTD, the two subchannels can be demodulated with a total of two correlators, each of length 2N. For STTD and STS, the two subchannels can be demodulated with a total of four correlators, each of length N (STTD) and 2N (STS), respectively.

[0048] The advantage of the filter implementation of the MMSE receiver is that additional data channels can be demodulated with very little additional complexity. No extra filters are required to demodulate the additional code channels. The filter implementation of the MMSE receiver for OTD permits the demodulation of each additional data channel with two correlators, each of length 2N. The filter implementation of STTD and STS permit the demodulation of each additional data channel with four correlators, each of length N (STTD) and 2N (STS), respectively. In contrast, the complexity of the correlator implementation of the MMSE receiver scales approximately linearly with the number of data channels demodulated.

[0049] In the previous analysis of OTD, the sequence {r_k} was reduced to a pair of vectors, R₀ and R₁, by despreading this sequence with the complex spreading code, {c_k} of the serving base station, and correlating with the code channel of interest. Recall that R₀ and R₁ were defined as $R_{0,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^0$ $R_{1,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^1$
for l in the interval [0, M−1], so that these vectors are of length M. More generally, the vectors R₀ and R₁ may be defined on a set of delays which are possibly non-consecutive and may also lie outside the interval [0, M−1].

[0050] The above definition applies to a particular Walsh code and a particular time interval. More generally, we consider an arbitrary Walsh code and an arbitrary symbol interval. Let R₀(m,n) and R₁(m,n) denote the observation vectors for the m-th Walsh code and the n-th time interval. Let w_ij denote the j-th chip of the i-th Walsh code. The components of the observation vectors R₀(m,n) and R₁(m,n) are then given by $R_{0,l}\left(m,n\right)=\sum _{i=1}^{2N}{r}_{2nN+i+l}{c}_{2nN+i}^{*}{w}_{m,i}^0$ $R_{1,l}\left(m,n\right)=\sum _{i=1}^{2N}{r}_{2nN+i+l}{c}_{2nN+i}^{*}{w}_{m,i}^1.$

[0051] The MMSE estimates of the symbols transmitted using the m-th Walsh code in the n-th time interval are given by
ŝ₀(m,n)=(Ω⁻¹f₀)^HR₀(m,n)ŝ₁(m,n)=(Ω⁻¹f₁)^HR₁(m,n).
The above expressions for ŝ₀(m,n) and ŝ₁(m,n) describe the correlation form of the MMSE receiver for OTD. As indicated in the above definitions, 2M correlations of length 2N are required in order to compute the vectors R₀(m,n) and R₁(m,n). The MMSE estimates of ŝ₀(m,n) and ŝ₁(m,n) are formed by computing the indicated inner products of the MMSE weighting vectors Ω⁻¹f₀ and Ω⁻¹f₁ with the corresponding observation vectors R₀(m,n) and R₁(m,n).

[0052] The development of the filter implementation of the OTD receiver is straightforward. Let g₀=Ω⁻¹f₀, and note that the expression for ŝ₀(m,n) can be written as $\begin{array}{c}{\hat{s}}_0\left(m,n\right)=\sum _{k=0}^{M-1}{g}_{0,l}^{*}{R}_{0,k}\left(m,n\right)\\ =\sum _{l=0}^{M-1}{g}_{0,l}^{*}\left(\sum _{i=1}^{2N}{r}_{2nN+i+l}{c}_{2nN+i}^{*}{w}_{m,i}^0\right)\\ =\sum _{i=1}^{2N}{c}_{2nN+i}^{*}{w}_{m,i}^0\left(\sum _{l=0}^{M-1}{g}_{0,l}^{*}{r}_{2nN+i+l}\right)\\ =\sum _{i=1}^{2N}{c}_{2nN+i}^{*}{w}_{m,i}^0\left({\tilde{g}}_0*r\left(2nN+i\right)\right)\end{array}$
where {tilde over (g)}hd 0,k≡g*_{0, −k} and * denotes convolution. Similarly, let g₁=Ω⁻¹f₁ and note that $\begin{array}{c}{\hat{s}}_1\left(m,n\right)=\sum _{k=0}^{M-1}{g}_{0,l}^{*}{R}_{1,k}\left(m,n\right)\\ =\sum _{l=0}^{M-1}{g}_{0,l}^{*}\left(\sum _{i=1}^{2N}{r}_{2nN+i+l}{c}_{2nN+i}^{*}{w}_{m,i}^1\right)\\ =\sum _{i=1}^{2N}{c}_{2nN+i}^{*}{w}_{m,i}^1\left(\sum _{l=0}^{M-1}{g}_{0,l}^{*}{r}_{2nN+i+l}\right)\\ =\sum _{i=1}^{2N}{c}_{2nN+i}^{*}{w}_{m,i}^1\left({\tilde{g}}_0*r\left(2nN+i\right)\right)\end{array}$
where {tilde over (g)}hd 1,k≡g*_1,−k.

[0053] From these expressions, it is apparent that the estimates ŝ₀(m,n) and ŝ₁(m,n) can be computed by first filtering the received sequence {r_k} with the conjugate of the time reverse of Ω⁻¹f₀ and Ω⁻¹f₁, respectively, and then correlating the filter output with Walsh code channel and subchannel (s₀ or s₁) of interest. Note that the filter implementation is preferred because it greatly reduces the complexity associated with the demodulation of multiple Walsh code channels. If it is necessary to demodulate additional code channels, such as the pilot channel or additional data channels, each additional channel can be demodulated with the use of two additional length-2N correlation operations (one for subchannel s₀ and one for subchannel s₁). Conversely, in the correlation implementation, 2M correlation operations are required for each additional code channel that is demodulated. The filter implementation of the MMSE OTD receiver is illustrated in FIG. 4 and FIG. 5.

[0054] In the previous analysis of STTD, the sequence {r_k} was reduced to a pair of vectors, R₀ and R₁, in the following fashion $R_{0,l}=\sum _{i=1}^N{r}_{i+l}{c}_i^{*}{w}_i$ $R_{1,l}=\sum _{i=1}^N{r}_{N+i+l}{c}_{N+i}^{*}{w}_i,$
for l in the interval [0, M−1], so that these vectors are of length M. More generally, the vectors R₀ and R₁ may be defined on a set of delays which are possibly non-consecutive and may also lie outside the interval [0, M−1]. The above definition is applies to a particular Walsh code and a particular time interval. More generally, we consider an arbitrary Walsh code and an arbitrary symbol interval. Let R₀(m,n) and R₁(m,n) denote the observation vectors for the m-th Walsh code and the n-th time interval. As above, let w_ij denote the j-th chip of the i-th Walsh code. The components of the observation vectors R₀(m,n) and R₁(m,n) are then given by $R_{0,l}\left(m,n\right)=\sum _{i=1}^N{r}_{2\mathrm{nN}+i+l}{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}$ $R_{1,l}\left(m,n\right)=\sum _{i=1}^N{r}_{\left(2n+1\right)N+i+l}{c}_{\left(2n+1\right)N+i}^{*}{w}_{m,i}.$

[0055] The MMSE estimates of the symbols transmitted using the m-th Walsh code in the n-th time interval are given by ${\hat{s}}_0\left(m,n\right)={\left(\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_0\\ f_1^{*}\end{array}\right]\right)}^H\left[\begin{array}{c}{R}_0\left(m,n\right)\\ R_1^{*}\left(m,n\right)\end{array}\right]$ ${\hat{s}}_1\left(m,n\right)={\left(\left[\begin{array}{cc}{\Omega }^{-1}& 0\\ 0& {\left({\Omega }^{-1}\right)}^{*}\end{array}\right]\left[\begin{array}{c}{f}_1\\ -f_0^{*}\end{array}\right]\right)}^H\left[\begin{array}{c}{R}_0\left(m,n\right)\\ R_1^{*}\left(m,n\right)\end{array}\right]$
The above expressions for ŝ₀(m,n) and ŝ₁(m,n) describe the correlation form of the MMSE receiver for STTD. As indicated in the above definitions, 2 M correlations of length N are required in order to compute the vectors R₀(m,n) and R₁(m,n). The MMSE estimates of ŝ₀(m,n) and ŝ₁(m,n) are formed by computing the indicated inner products between the length-2M MMSE weighting vectors and the corresponding length-2M observation vectors.

[0056] The development of the filter implementation of the STTD receiver is straightforward. As above, let g₀=Ω⁻¹f₀ and g₁=Ω⁻¹f₁. Note that ŝ₀(m,n) can be written as $\begin{array}{c}{\hat{s}}_0\left(m,n\right)=g_0^H{R}_0\left(m,n\right)+{\left(g_1^H\right)}^{*}{R}_1^{*}\left(m,n\right)\\ =\sum _{l=0}^{M-1}{g}_{0,l}^{*}\sum _{i=1}^N{r}_{2\mathrm{nN}+i+l}{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}+\\ \sum _{l=0}^{M-1}{g}_{1,l}\sum _{i=1}^N{r}_{\left(2n+1\right)N+i+l}^{*}{c}_{\left(2n+1\right)N+i}{w}_{m,i}\\ =\sum _{i=1}^N{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}\sum _{l=0}^{M-1}{g}_{0,l}^{*}{r}_{2\mathrm{nN}+i+l}+\\ \sum _{i=1}^N{c}_{\left(2n+1\right)N+i}{w}_{m,i}\sum _{l=0}^{M-1}{g}_{1,l}{r}_{\left(2n+1\right)N+i+l}^{*}\\ =\sum _{i=1}^N{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}\left({\tilde{g}}_0*r\left(2\mathrm{nN}+i\right)\right)+\\ {\left(\sum _{i=1}^N{c}_{\left(2n+1\right)N+i}^{*}{w}_{m,i}\left({\tilde{g}}_1*r\left(\left(2n+1\right)N+i\right)\right)\right)}^{*}\end{array}$
where {tilde over (g)}₀ and {tilde over (g)}₁ were defined previously. Similarly, we have $\begin{array}{c}{\hat{s}}_1\left(m,n\right)=g_1^H{R}_0\left(m,n\right)-{\left(g_0^H\right)}^{*}{R}_1^{*}\left(m,n\right)\\ =\sum _{l=0}^{M-1}{g}_{1,l}^{*}\sum _{i=1}^N{r}_{2\mathrm{nN}+i+l}{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}-\\ \sum _{l=0}^{M-1}{g}_{0,l}\sum _{i=1}^N{r}_{\left(2n+1\right)N+i+l}^{*}{c}_{\left(2n+1\right)N+i}{w}_{m,i}\\ =\sum _{i=1}^N{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}\sum _{l=0}^{M-1}{g}_{1,l}^{*}{r}_{2\mathrm{nN}+i+l}-\\ \sum _{i=1}^N{c}_{\left(2n+1\right)N+i}{w}_{m,i}\sum _{l=0}^{M-1}{g}_{0,l}{r}_{\left(2n+1\right)N+i+l}^{*}\\ =\sum _{i=1}^N{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}\left({\tilde{g}}_1*r\left(2\mathrm{nN}+i\right)\right)-\\ {\left(\sum _{i=1}^N{c}_{\left(2n+1\right)N+i}^{*}{w}_{m,i}\left({\tilde{g}}_0*r\left(\left(2n+1\right)N+i\right)\right)\right)}^{*}.\end{array}$

[0057] Note that that the two symbol estimates ŝ₀(m,n) and ŝ₁(m,n) can be written entirely in terms of the convolutions {tilde over (g)}₀*r(·) and {tilde over (g)}₁*r(·). As with OTD, the filter implementation is preferred because it greatly reduces the complexity associated with the demodulation of multiple Walsh code channels. The filter implementation of the MMSE STTD receiver is illustrated in FIG. 6.

[0058] In the previous analysis of STTD, the sequence {r_k} was reduced to a pair of vectors, R₀ and R₁, in the following fashion $R_{0,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^0$ $R_{1,l}=\sum _{i=1}^{2N}{r}_{i+l}{c}_i^{*}{w}_i^1,$
for l in the interval [0, M−1], so that these vectors are of length M. More generally, the vectors R₀ and R₁ may be defined on a set of delays which are possibly non-consecutive and may also lie outside the interval [0, M−1]. The above definition is applies to a particular Walsh code and a particular time interval. More generally, we consider an arbitrary Walsh code and an arbitrary symbol interval. Let R₀(m,n) and R₁(m,n) denote the observation vectors for the m-th Walsh code and the n-th time interval. As above, let w_ij denote the j-th chip of the i-th Walsh code. The components of the observation vectors R₀(m,n) and R₁(m,n) are then given by $R_{0,l}\left(m,n\right)=\sum _{i=1}^{2N}{r}_{2\mathrm{nN}+i+l}{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}^0$ $R_{1,l}\left(m,n\right)=\sum _{i=1}^{2N}{r}_{2\mathrm{nN}+i+l}{c}_{2\mathrm{nN}+i}^{*}{w}_{m,i}^1,$