Detection and diagnosis of abrupt changes in modal characteristics of nonstationary digital signals

where E, = l?',]f(t)]' dt is the signal energy. Note that the where E, = l?',]f(t)]' dt is the signal energy. Note that the coefficients a;,, coefficients a;,, in (9) are obtainable in explicit form as the in (9) are obtainable in explicit form as the coefficients in the binomial expansion of coefficients in the binomial expansion of Clearly, the technique used here can be extended using three parameters to introduce another degree of freedom: 3n (1-au;z-'y;z2) " = 1-; a;,zk k=l where cz; = 2-fi and Q; = 1-0. III. RANDOM CASE In this section x(t) is assumed to be a complex-valued WSS process with power spectral density S,(w) = 0 for ]o] > 8, and R,(T) = (1/2?r)~T!,Sx(w)eJWTdw for all r, where R,(T) 6 E{x(t + 7)x*(t)}. With Z,,(w) L 1-C$=lukneK'WT for o E [-rr, n] it is an easy exercise to verify that E x(t)-E a,,x(t-kT) k=l From the preceding section we have (1-(~ie~'~r-a2e-J20T) ' = 1-k$1 a&JkWT = I,,,(0) thus 1 I,,,(0) I2 = (I1-alePior-a2e-J207.12) " .(w)]2 5 p " where p = 4(3-2fi) = 0.6863, since ]I-cyfe-lwT-a$e-j20T]2 =< p for-r/2 6 wT s 7r/2. Theorem 2: For x(t) a WSS random process with power spectral density having support on [-rr, V] and 0 < T 5 l/2 the coefficients { a;,, } determined from the identity [ 1-(2-0) e-jwT-(1 _ a) e-j2wT] " = 1-g a;,,e-jkwT k=l are such that E x(t)-F &x(t-kT) Sr(w) da k-l n where /3 = 0.6863. Again we observe that the mean-square convergence is uniform in t for t E (co , co) and the coefficients { ai,,} are independent of both T and the spectral properties of the particular random process being estimated. IV. REMARKS We have determined a set of numerical coefficients which yields a one-step prediction of either a deterministic or random band-limited process with error that decreases geometrically as the number of past samples used becomes infinite. While the deterministic signal was assumed to have finite energy, the assumption that the signal spectrum is absolutely integrable works equally well fort -(t)-?a,,,f(f-W 1 where ]Z2,? (o)12 j /? " as in the previous section. 11-ale-/~T-a2e-i20T-a3e-13wT~r~ = 1-C ak,,e-lkWT k=l Using this approach with normalized band-width we have found explicit coefficients { akn } such that the prediction error goes to zero at a geometric rate for intersample spacings T satisfying 0 < T 5 2/3; that is, …