Estimation and detection of signals in multiplicative noise (Corresp.)

Several recent articles [l]-[3] have discussed the use ofLagrange polynomials for band-limited signal estimation. We find errorbounds when using Lagrange polynomials for interpolation and extra-polation of finite-power band-limited signals if a finite number of regularlyspaced noisy samples are used. I. INTR~DUC~ONSuppose f(t) has a (2N + 1)st derivative and the samplesf(tk) are given, for 0 5 k 5 2N. The Lagrange polynomialand error term that pass through the f(tk) points and used toestimatef(t) for points other than tk are given by [4] f(t) = 2 f(tk) w0 w(t)+ (2N + l)! f (2N+1)(U) (1)k=OW ’(tk)(ttk) where w(t) = (t to)(t tl) . l . (t t2N) and u is somepoint in the smallest interval containing t and [t0,t2N]. If eachsample f(tk) contains an error &k;, then the system error e(t) inreconstructing f(t) from the Lagrange polynomial is 2Ne(t) = c ek40w(t) f ‘(2N+ 1’(u). (2)k=OWl(tk)(ttk) + (2N + l)! The first term .of (2) is called the channel errorterm of (2) is called the truncation error.We shall consider as a deterministic signalbounded by M, which can be represented bywhile the second