Method and apparatus for making a channel estimate

In the method of making a channel estimate, at least first and second confidence levels that a transmitted data symbol has respective first and second values are determined based on a received data symbol corresponding to the transmitted data symbol. A channel estimate is then determined based on the first and second confidence levels.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS In the method of the present invention, estimated individual realizations of the complex-valued fading coefficient, commonly called channel estimates, are generated. According to the inventive method, these realizations are easy to generate and can produce a channel estimate that exhibits a smaller variance in comparison to conventional methods. Using the received signal y i,d , the channel estimate is defined as: â i,d &equals;h(y i,d )y i,d .   (1) where â i,d is the channel estimate, h( ) is any predefined function that can be designed based on a specific constellation and channel being used, i represents the ith estimate, and d represents that the parameter or variable pertains to data symbols (as opposed to a pilot symbols). For the purposes of discussion only, the method of the present invention will described for the bi-phase shift keying (BPSK) constellation and an over-the-air communication channel. In BPSK, the transmitted symbol is either &plus;1 or −1. However, from the following disclosure, it will be understood that application of the method of the present invention is not limited to a particular constellation or channel. In a preferred embodiment for BPSK modulation, h(y i,d ) is defined as: h ( y i,d ) &equals;P ( x i &equals;&plus;1&verbar; y i,d )− P ( x i &equals;−1&verbar; y i,d ),   (2) where P(x i &equals;&plus;1&verbar;y i,d ) is the a posteriori probability that a transmitted data symbol x i &equals;&plus;1 was transmitted conditioned on the observation of the received data symbol y i,d corresponding to the transmitted data symbol x i , and where P(x i &equals;−1&verbar;y i,d ) is the a posteriori probability that a transmitted data symbol x i &equals;−1 was transmitted conditioned on the observation of the received data symbol y i,d corresponding to the transmitted data symbol x i . Here, P(x i &equals;&plus;1&verbar;y i,d ) represents a confidence level, based on the received data symbol y i,d , that the corresponding transmitted data symbol x i had a value of &plus;1. And, P(x i &equals;−1&verbar;y i,d ) represents a confidence level, based on the received data symbol y i,d , that the corresponding transmitted data symbol x i had a value of −1. Using Bayes' rule, equation (2) can be rewritten as equation (3a) below: 1 P &af; ( x i = + 1 &VerticalSeparator; y i , d ) = p &af; ( y i , d &VerticalSeparator; x i = + 1 ) &it; P &af; ( x i = + 1 ) p &af; ( y i , d ) (3a) where p(y i,d ) represents a probability density function of the received statistic evaluated at y i,d . A similar expression exists for P(x i &equals;−1&verbar;y i,d ). 2 P &af; ( x i = - 1 &VerticalSeparator; y i , d ) = p &af; ( y i , d &VerticalSeparator; x i = - 1 ) &it; P &af; ( x i = - 1 ) p &af; ( y i , d ) (3b) Under the assumption that P(x i &equals;&plus;1)&equals;P(x i &equals;−1)&equals;0.5 using the Law of Total Probability, p(y i,d ) is given by equation (4) below: 3 p &af; ( y i , d ) = 1 2 &it; ( p &af; ( y i , d &VerticalSeparator; x i = + 1 ) + p &af; ( y i , d &VerticalSeparator; x i = - 1 ) ) (4) The well-known log-likelihood ratio (LLR) is defined as: 4 λ &af; ( y ) = λ 1 &af; ( y ) - λ - 1 &af; ( y ) = ln &af; ( p &af; ( y &VerticalSeparator; x = + 1 ) p &af; ( y &VerticalSeparator; x = - 1 ) ) (5) where In( ) represents the natural logarithm. It is known that y conditioned on x is a complex Gaussian random variable with mean ax and per-components variance &sgr; 2 , (i.e., noise), where &sgr; 2 is determined according to any well-known technique. Therefore, the LLR is given by equation (6) below: 5 λ &af; ( y ) = λ 1 &af; ( y ) - λ - 1 &af; ( y ) = - ( ( y - a ) 2 2 &it; σ 2 ) - ( ( y + a ) 2 2 &it; σ 2 ) = 2 &it; a * &it; y σ 2 ( 6 ) Combining Equations (1)-(6) results in: 6 a ^ i , d = ( e λ ( y i , d ) - 1 e λ ( y i , d ) + 1 ) &it; y i , d . ( 7 ) Returning to Equation (2), h (y i,d ) is given by: 7 h &af; ( y i , d ) = e λ ( y i , d ) - 1 e λ ( y i , d ) + 1 . ( 8 ) A plot of this function with respect to &lgr; is given in FIG. 1 . The function is odd and is bounded by ±1, and bears a strong resemblance to the sign( ) function. As will be appreciated, h(y i,d ) represents the confidence that the transmitted symbol x i is a particular value in view of the corresponding received symbol y i,d . Stated another way, h(y i,d ) indicates the strength or degree of confidence that the transmitted symbol x i is a particular value in view of the corresponding received symbol y i,d . Next, the overall data-based estimate is found by averaging the individual realizations of the channel estimate over a weighted time window: 8 a ^ d = 1 2 &it; N d + 1 &it; &Sum; j = i - N d i + N d &it; K j , d &it; a ^ j , d = 1 2 &it; N d + 1 &it; &Sum; j = i - N d i + N d &it; K j , d &af; ( e λ &af; ( y i , d ) - 1 e λ &af; ( y i , d ) + 1 ) &it; y i , d ( 9 ) where K i,d is a weighting constant, 2N d &plus;1 is the window over which the estimate is averaged, and N d is a number of samples. Unlike conventional data-aided channel estimate methods, the method according to the present invention does not require making an explicit calculation (called a hard decision) of an estimate of the transmitted symbol. Instead the present invention offers a soft decision alternative that does not require performing any maximizations. A simple evaluation of the LLR, a well-known receiver calculation already made in most receivers, is used. Consequently, the methodology of the present invention offers an easy means of determining a data- aided channel estimate that exhibits a smaller variance in comparison to conventional methods. Once the appropriate pilot-aided (PA) and data-aided (DA) channel estimates and their variances are obtained, they can be combined in an optimal manner. In the present invention, optimality is defined as minimum variance in the final estimate. The PA channel estimate â (p) can be determined according to any well-known technique, and therefore will not be described. The variances &sgr; p 2 and &sgr; d 2 of the PA and DA channel estimates â (p) and â (d) can be determined according to any well-known statistical technique; and therefore will not be described. The final channel estimate, â is a linear combination of the PA and DA channel estimates, 9 a ^ = w p &it; a ^ ( p ) + w d &it; a ^ ( d ) , (10) where w p and w d are non-negative constants. Assuming that E&lsqb;â (p) &rsqb;&equals;E&lsqb;â (d) &rsqb;&equals;a, where E&lsqb;&rsqb; represents the average value, the added constraint that w p &plus;w d &equals;1   (11) ensures that E&lsqb;â&rsqb;&equals;a. Under the assumption that the PA and DA channel estimates are independent, the variance of the overall estimate is Var( â ) &plus;w p 2 &sgr; p 2 &plus;w d 2 &sgr; d 2 .   (12) To minimize this variance subject to the constraint in equation (11) and w p ,w d being non-negative, w d &equals;1−w p is substituted in Equation (12). Then, equation (12) is differentiated with respect to w p , set equal to zero, and solved for w p . The result is 10 w p = σ d 2 σ p 2 + σ d 2 , &NewLine; &it; and ( 13 ) w d = σ d 2 σ p 2 + σ d 2 (14) A check of the second derivative confirms that the solution is indeed a minimum. Accordingly, by substituting equations (13) and (14) into equation (10), the channel estimate can be calculated using the PA channel estimate, the variance of the PA channel estimate, the DA channel estimate and the variance of the DA channel estimate. An apparatus for implementing the above-described embodiment of the present invention will now be described with reference to FIG. 2 . As will be appreciated from the forgoing, the apparatus of FIG. 2 forms a part of a receiver. Because the other components of the receiver are well-known, applicants have not illustrated and will not describe these other components for the sake of brevity. As shown in FIG. 2, a shift register 10 inputs the received symbols y i from a demodulator (not shown). As alluded to above, also not shown are the well-known components for determining the per-component noise or variance &sgr; 2 , the pilot-aided channel estimate â p , the variance &sgr; p 2 of the pilot-aided channel estimate and the variance &sgr; d 2 of the data aided channel estimate. An LLR calculator 12 receives the received symbol y i from the shift register 10 and the square of the standard deviation, and calculates the LLR of the received symbol y i according to equation (6). As will be appreciated, equation (6) requires, during a first iteration, an initial channel estimate as an input variable. In a preferred embodiment, the channel estimate based on the pilot symbols is used as the initial channel estimate, and then each subsequent iteration uses the channel estimate determined based on the combined data aided and pilot-aided channel estimates as shown in FIG. 2 . Next, a confidence factor generator 14 generates a confidence factor h(y) according to equation (8) using the output of the LLR calculator 12 . A multiplier 16 multiplies the confidence factor with the received symbol y i on a per component basis to obtain an individual realization of the channel estimate based on the received data according to equation (1). A weighted time window averager 18 stores the output of the multiplier 16 and calculates a weighted average of the data-aided channel estimate according to equation (9). A channel estimate combiner 20 receives the output of the weighted time window averager 18 and the pilot symbol based channel estimate, calculates the variances of the DA channel estimate and the PA channel estimate, and generates the channel estimate according to equations (10), (13) and (14). The channel estimate is then used in the conventional manner to determine the transmitted symbols x i , and is also feedback to the LLR calculator 12 to be used in the LLR calculation. The invention being thus described, it will be obvious that the same may be varied in many ways. Such variations are not to be regarded as a departure from the spirit and scope of the invention, and all such modifications are intended to be included within the scope of the following claims.