A class of balanced binary sequences with optimal autocorrelation properties

From (15) and (II), it follows that because N(0) = IV(O). The resulting estimate of X(w) is given by m bee (2211 c J(Wi-w/&(w) = J(w-Wk), k = 1,. .. ,m. (16) i=l With Aik the elements of the inverse of a matrix whose elements X(w) = X(w) + &) [N(w)-iv(O) +$q equal d(wi-wk), the system (16) yields and the mean-square error by [see (21)] m @i(w) = C A;J(w-wk). k=l (17) From (16), it follows that Without the correction, the mean-square value of the error is @'.i(Wk) = 1, k=i o > k # i. (18) Inserting (17) into (9), we conclude that the optimum estimator N(w) is given by m N(W) = kgl BkJ(w-Wk) (19) where the random variables & are linearly dependent on the data Wi. Setting w = wi in (19), we obtain the system fJ BkJ(wi-Wk) = A(Ui) = Wi, k=l i = 1, fee ,m. (20) The last equality follows from (9) and (18). Thus N(w) is a linear combination of the functions J(w-Wk) and it interpolates N(W) at the zero-crossings wi of Y(w). The coefficients & can be found by solving the system (20). Clearly [see (9) and (ll)], the difference N(w)-R(w) is orthogonal to N(w), hence the mean-square error e is given by e = E(I N(w)-AVOW) = E([iV(w)-N(w)]iV*(w)]. The above yields e = J(0)-f %pi(W)J(Wi-W). i=l (21) The error is zero at w = wi [see (18)]. With N(w) so corrected, the best estimate of X(w) takes the form S(w) = j& [W(w)-mJd1 = X(w) + & [iv(w)-AYW,]. The correction has the following time-domain interpretation: Since the inverse of J(w) equals Z(t), it follows from (19) that the inverse a(t) of the estimator R(w) is given by m fi(t) = z(t) c &ej " 'kt. k=l (2% Thus, to improve the estimation of x (t) with the above method, we deconvolve not the data w(t), but the difference w(t)-a(t). Example: We wish to deconvolve the function w(t) = x(t) * y(t) + n(t), where y(t) is a known function whose transform Y(w) has a single zero at w = 0. We assume that w(t) vanishes for ] t ] > T and that n(t) is stationary white noise with R(t1,tz) = LWl-t'd, It11 < T, ltzl < 7'. Clearly, Z(t) equals Z, for (t ] < T and is 0 otherwise. Hence, J(w) = Z,s_TT e-jut & = 2z0 'r wT. …