Channel estimation for OFDM systems with transmitter diversity

[0002] This application is related to an application titled “Diversity Coded OFDM For High Data Pe Commuxication,” which was filed on Dec. 17, 1998 and assigned to the assignee of this application, bearing the Ser. No. 09/213,585.

[0003] This invention relates to OFDM systems and, more particularly, to channel parameter estimation in OFDM systems that employ transmitter diversity.

[0004] In orthogonal frequency division multiplexing (OFDM) the channel is divided into many narrow subbands, which are transmitted in parallel, thereby, increasing the symbol duration and reducing or eliminating the intersymbol interference (ISI) caused by the dispersive Rayleigh fading environment. On the other hand, since the dispersive property of wireless channels causes frequency selective fading, there is higher error probability for those subbands with deep fading. Hence, techniques such as error correction code and diversity have to be used to compensate for the frequency selective fading of wireless channels. In this report, we investigate transmitter diversity using space-time codes for OFDM systems.

[0005] Recently, space-time coding has been developed for high data-rate wireless communications, and such coding has been extended to OFDM systems in a U.S. patent application titled “Orthogonal Designs and Transmit Diversity for Wireless Communications,” filed on Nov. 6, 1998 and bearing the Ser. No. 09/186,908. The space-time code is characterized by high code efficiency and good performance; hence, it is a promising technique to improve the efficiency and performance of OFDM systems. However, decoding of the space-time code requires the channel state information, which is a task that must be performed and, therefore, the efficiency with which the task is carried out is an issue.

[0006] Channel parameter estimation has been successfully used to improve performance of OFDM systems. For systems without co-channel interference, with estimated channel parameters, coherent demodulation is allowed, instead of differential demodulation, to achieve a 3-4 dB signal-to-noise ratio (SNR) gain. Moreover, for systems with receiver diversity, the maximum-ratio diversity combiner (MR-DC), which is equivalent to the minimum-mean-square-error diversity combiner (MMSE-DC) in this case, can be obtained using estimated channel parameters. For systems with co-channel interference, the coefficients for the MMSE-DC can be calculated from estimated channel parameters and instantaneous correlation of the signals from each receiver. However, no teachings can be found on the parameter estimation for OFDM systems with transmitter diversity.

[0007] Enhanced performance is obtained by estimating channel parameters during normal operation, in addition to an initial estimate based on a known training sequence, through use of the detected signals. Once the OFDM-modulated signals that are transmitted by the various transmitting antennas are detected, the signal received at a receiving antenna is employed to compute channel impulse response estimates between the signal received at that receiving antenna and the various transmitting antennas. The process is repeated for each receiving antenna.

[0008] In computing the channel impulse response estimates between the signal received at that receiving antenna and the various transmitting antennas, an nK₀×nK₀matrix of terms (q_xy[l]) is developed. Then, the inverse of the matrix is computed, and the computed matrix inverse is multiplied by a vector of terms (p_i[l]), to obtain a vector of nK₀-sample channel impulse response estimates. Each q_xy[l] term is evaluated by selecting a signal sent by transmitting antenna x (i.e., b_x[k], where k is an index designating a frequency subband of the OFDM transmitter), multiplying it by the conjugated signal of transmitting antenna y, i.e., b_y[k], further multiplying it by $e^{\left(-j\frac{2\pi \mathrm{kl}}{K}\right)},$

[0009] and summing over all of the frequency subbands; all for impulse response sample time l. K is the number of frequency subbands. The p_i[l] term corresponds to $\sum _{k=1}^{K}r\left[k\right]b_i^{*}\left[k\right]e^{\left(-j\frac{2\pi \mathrm{kl}}{K}\right)}$

[0010] where r[k] is the signal of the receiving antenna.

[0011] FIG. 1 presents a block diagram circuitry for estimating channel response in accordance with the principles disclosed herein; and

[0012] FIG. 2 presents block 240 of FIG. 1 in greater detail.

[0013] An OFDM system with transmitter diversity using the spacetime code is shown in FIG. 1. At each time t, a data block is encoded by encoder 110 to develop n blocks of symbols, and these blocks of symbols are applied to OFDM transmitters 120-i, and thence to transmitting antenna 130-i. At the receiver the transmitted signals are detected by antennas 210-j, and applied to OFDM receivers 220-j, where j=1, 2, . . . m. The signal obtained at the output of OFDM receiver 220-j is $\begin{array}{cc}{r}_j\left[k\right]=\sum _{i=1}^n{H}_{\mathrm{ij}}\left[k\right]b_i\left[k\right]+w_j\left[k\right]& \left(1\right)\end{array}$

[0014] where r_j[k] is the signal received at antenna 210-j at frequency subband k, b_i[k] is the is symbol transmitted by antenna 130-i at frequency subband k, H_ij[k] is the channel transfer function between transmitting antenna 130-i and receiving antenna 210-j at frequency subband k, and w_j[k] is the additive complex white Gaussian noise, with zero-mean and variance σ²received at antenna 210-j and is the same for all frequency subbands. The following analysis pertains to signals received at antenna j that were transmitted by transmitting antenna i. To simplify the notations, the subscripts j and i are omitted.

[0015] A channel that includes multi-paths can be described by an impulse time response of the form: $\begin{array}{cc}h\left(t\right)=\sum _p{\gamma }_p{h}_o\left(t-{\tau }_p\right),& \left(7\right)\end{array}$

[0016] where t is time, τ_pis the delay of multi-path p between an originating point and a terminating point, γ_pis the corresponding complex amplitude flinction of time for multi-path p, and h_o(t) is shaping pulse, which is usually a square-root raised-cosine. Hence, the frequency response at time t is $\begin{array}{cc}\begin{array}{c}H\left(f\right)\equiv {\int }_{-\infty }^{+\infty }h\left(t\right)e^{-j2\pi f\tau }dt\\ =H_0\left(f\right)\sum _p{\gamma }_p{e}^{-j2\pi f{\tau }_p},\end{array}& \left(8\right)\end{array}$

[0017] where $H_0\left(f\right)\equiv {\int }_{-\infty }^{+\infty }h\left(t\right)e^{-j2\pi f\tau }dt$

[0018] is basically the channel's frequency response of the one path that has no delay (i.e., the principal, direct, path). It is sometimes called the channel's “shaping filter.” We assume that the γ_p's are wide-sense stationary (WSS), narrow-band complex Gaussian processes, which are independent for different paths.

[0019] The combined channel response, in the time domain, is the sum of the delayed replicas of h_o(t) multiplied by the γ_p's. In an OFDM system with K frequency subbands, one would want to have K samples of the channel's response over the time slot's time interval T_s(recalling that the FFT is a K-to-K transformation). However, when the combined channel response of equation (7) is only rmax long, one needs to consider only ┌τ_max/T_s┐ samples (because the other ones are zero), and any integration (summation) to obtain the frequency response needs to be done only over K₀samples, where K₀=┌τ_max/T_s┐. Hence the frequency response of the channel can be obtained from the following equation $\begin{array}{cc}H\left[k\right]=\sum _{l=0}^{K_o-1}h\left[l\right]\exp \left(-j\frac{2\pi \mathrm{kl}}{K}\right),& \left(9\right)\end{array}$

[0020] where h[l] corresponds to the combined, effective, channel impulse response, l is a time domain index, k is an index designating a subband, and K is the number of subbands employed in the OFDM system. H[k] in equation (9) actually corresponds to H[kΔf], where Δf is the bandwidth of a subband of the OFDM.

[0021] As indicated above, to decode space-time code used for the transmitter diversity of OFDM systems, channel parameters have to be provided. One can provide an initial estimate of the h[l] impulse responses by sending a known training sequence of samples, and evaluating the received signal.

[0022] In accordance with the principles disclosed herein, however, once the initial are obtained and normal operations begin, the impulse responses continue to be estimated based on the decisions made regarding the transmitted signals. Specifically, for a given set of received signals, r_j[k], at the different receiving antennas, j, a decision is made regarding the transmitted signals b_i[k] from each antenna i. Based on the decisions made regarding those transmitted signals, and given the actually received signals, a set of optimized impulse functions is computed and applied to subsequent reception decisions.

[0023] Thus, given that the decisions regarding the signals transmitted by transmitting antenna i are made correctly, the impulse responses can be computed by choosing a set, at each receiving antenna j, a set of impulse responses estimates, {tilde over (h)}_gj[l], that minimize $\begin{array}{cc}C\left(\left\{\tilde{h}\left[l\right]\right\}\right)=\sum _{k=1}^{K_o-1}{\left|r_j\left[k\right]-\sum _{g=1}^n\sum _{l=0}^{K_o-1}{\tilde{h}}_{\mathrm{gj}}\left[l\right]W\left[\mathrm{kl}\right]b_g\left[k\right]\right|}^2,& \left(10\right)\end{array}$

[0024] where $\begin{array}{cc}W\left[\mathrm{kl}\right]\equiv \exp \left(-j\frac{2\pi kl}{K}\right).& \left(11\right)\end{array}$

[0025] The minimum can be identified by taking the derivative of equation (10) with respect to each particular impulse function {tilde over (h)}_ij[l₀] (i.e., of antenna i, at time l₀) and setting it to zero. This minimization is repeated l₀times in connection with a given transmitting antenna, and n times for the n transmitting antennas.

[0026] It can be shown that taking the derivative of equation (10) with respect to {tilde over (h)}_ij[l₀] and setting it to zero corresponds to $\begin{array}{cc}0=\sum _{k=1}^{K}W\left[-{\mathrm{kl}}_o\right]b_i^{*}\left[k\right]·\left(r_j\left[k\right]-\sum _{g=1}^n\sum _{l=0}^{K_o-1}{\tilde{h}}_{\mathrm{gj}}\left[l\right]W\left[\mathrm{kl}\right]b_g\left[k\right]\right),& \left(12\right)\end{array}$

[0027] and as mentioned above, this equation is repeated for i=1, 2, . . . ,n and l₀=0, 1, . . . , K₀−1. Equation (12) can be rewritten as: $\begin{array}{cc}\sum _{g=1}^n\sum _{l=0}^{K_o-1}{q}_{\mathrm{gi}}\left[l_o-l\right]{\tilde{h}}_{\mathrm{gj}}\left[l\right]=p_i\left[l_o\right],& \left(13\right)\end{array}$

[0028] where $\begin{array}{cc}{p}_i\left[l_o\right]=\sum _{k=1}^{K}r\left[k\right]W\left[-{\mathrm{kl}}_o\right]b_i^{*}\left[k\right]\mathrm{and}& \left(14\right)\\ q_{\mathrm{gi}}\left[l_o\right]=\sum _{k=1}^K{b}_g\left[k\right]b_i^{*}\left[k\right]W\left[-{\mathrm{kl}}_o\right].& \left(15\right)\end{array}$

[0029] What equation (13) says is that the elements of {tilde over (h)}_gj[l] that satisfy the equality are the elements that minimize the error signal C({{tilde over (h)}[l]}) of equation (10). Of course, equation (13) has K₀×n unknowns, corresponding to the K₀elements of each {tilde over (h)}_gj[l] in the n different {tilde over (h)}_gj[l]'s (from the n transmitting antennas). However, the differentiation that led to equation (13) is carried out K₀×n times, as indicated above, providing the requisite number of equations to allow solving for the elements of the impulse function estimates.

[0030] Equation (13) can be rewritten as a sum of dot products $\begin{array}{cc}\sum _{g=1}^n{q}_{\mathrm{gi}}\left[l_0\right]{\tilde{h}}_{\mathrm{gj}}=p_i\left[l_0\right]& \left(16\right)\end{array}$

[0031] where

[0032] {tilde over (h)}_gj=(h_gj[0], h_gj[1], . . . h_gj[K₀−1])^Tand  (17)

[0033] q_gi[l₀]=(q_gi[l₀−0], q_gi[l₀−1], . . . q_gi[l₀−K₀+1])  (18)

[0034] or as

[0035] q_i[l₀]{tilde over (h)}_j=p_i[l₀],  (19)

[0036] where

[0037] q_i[l₀]=(q_1i[l₀], q_2i[l₀], . . . q_ni[l₀]), and  (20)

[0038] {tilde over (h)}_j=({tilde over (h)}_1j, {tilde over (h)}_1j, . . . {tilde over (h)}_1j)  (21)

[0039] Expressing equation (19) for each value of l₀and then for each value of i, we get the expression

[0040] Q{tilde over (h)}_j=P  (22)

[0041] where P is a vector of n elements, p_x, where index x=1, 2, . . . n, and

[0042] p_x=(p_x[0], p_x[1], . . . p_x[K₀−1])^T  (23)

[0043] and Q is an n×n matrix of elements Q_xy, each of which is a matrix of the form $\begin{array}{cc}{Q}_{\mathrm{xy}}=\left[\begin{array}{cccc}{q}_{\mathrm{xy}}\left[0\right]& q_{\mathrm{xy}}\left[0-1\right]& \cdots & q_{\mathrm{xy}}\left[0-K_0+1\right]\\ q_{\mathrm{xy}}\left[1-0\right]& q_{\mathrm{xy}}\left[0\right]& \cdots & q_{\mathrm{xy}}\left[-K_0+2\right]\\ ⋮& ⋮& ⋰& ⋮\\ q_{\mathrm{xy}}\left[K_0-1\right]& q_{\mathrm{xy}}\left[K_0-2\right]& \cdots & q_{\mathrm{xy}}\left[0\right]\end{array}\right].& \left(24\right)\end{array}$

[0044] Finally, from equation (22), we can write

[0045] {tilde over (h)}_j=Q⁻¹P.  (25)

[0046] Equation (25) indicates that, in order to compute the channel's impulse response estimates, Q⁻¹must be calculated. While this is a straight forward task, it can be burdensome because the value of K₀×n may be relatively large even for small values of n. Fortunately, however, for wireless channels, the h[l]'s of equation (9) for most of values of l are zero or essentially zero, so these taps can be ignored, leading to a situation where many terms of Q are zero (or small enough to be set to zero). This reduces the volume of computation that is associated with computing Q⁻¹.

[0047] FIG. 1 presents a block diagram of that portion of a receiver that estimates the impulse response of a channel from a transmitter having n antennas, to a receiver having m antennas. Accordingly, FIG. 1 depicts a transmitter having an encoder 110, a plurality of OFDM transmitters 120-i, where i=1, 2, . . . n, and a corresponding plurality of transmitting antennas 130-i. A depicted receiver includes m receiving antennas 210-j that are coupled to n OFDM receivers 220-j. The output signals of receivers 220-i are applied to maximum likelihood detector 230 which detects the signals transmitted by antennas 130-i in accordance with whatever approach is appropriate to the encoding process employed in the transmitter. The aforementioned “Diversity Coded OFDM For High Data Rate Communication,” patent application discloses one approach.

[0048] The detection performed in element 230 employs estimates of the channel impulse responses. Those estimates come from a plurality of m estimation blocks 240, only one of which is shown in FIG. 1. The shown estimation block 240 is responsive to the b_i(k) signals developed by detector 230 and to the signals of OFDM receiver 220-m. The other, not-shown, estimation blocks are each responsive to a different OFDM receiver.

[0049] Within block 240, unit 241 develops the various q_xy[l₀]'s of matrix (24) and equation (15). These values are stored in memory 242 and then applied, as a group, to unit 243 which computes the inverse of matrix Q. Concurrently, unit 244 computes the elements of vector P in accordance with equations (23) and (14), and the computation results of units 244 and 243 are applied to matrix multiplier unit 245. Unit 245 performs the computations of equation (25), yielding the desired channel impulse response estimates that are applied to detector 230.

[0050] FIG. 2 presents a more detailed block diagram of unit 241. Selector 2411 receives the detected signals that represent the signals sent by the different transmitting antennas and selects a particular signal under control of controller 250. That forms the signals b_x[k] of equation (15). Selector 2412 selects of the detected signals, also under control of controller 250, and conjugation unit 2413 conjugates the signal, thus forming signal b_i[k] of equation (15). The outputs of units 2411 and 2413 are multiplied in units 2414, and the product is multiplied by factor W[−kl₀], which is obtained from unit 2415. Unit 2415 generates this factor in response to applied signals k, and l, from controller 250. The product signal developed by multiplication unit 2416 is accumulated in register 2418 with the aid of adder 217. By cycling through the k and l indexes, and cycling through the indexes that effect the selection in units 2412 and 2411, all of the needed q_xy[l₀] terms that form matrix Q are obtained.