In many fields of research, there naturally arise many problems such as whether echoes exist or not in time series and how is their timing, and there exist many problems easily convertible to the forrn mentioned above. The use of correlation technique er ・Cepstrum analysis has proved powerful to solve these problems. However, it is diMcult to detect an echo by those techniques under such conditions as the band width of the time series is not wide enough and the delay time of the echo is small.

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... his paper 'presents a new method which is useful even under these conditions. The new method is based on the principle that an echo shows up as a delta function ln the impulse response function of the'optimum predictor for the signal containing the echo. In the simplest echo, values of a time series g(t) are multiplied by a constant a(lal<1), delayed by a time difference o, and added to the original series to give a new series as given by Eq, (1). The transfer function of the optimum predictor EIC);to) for the signal is given by Eq. (11), where a is the prediction time. Its inverse Fourier transform or the impulse response function h(r) is given by Eq. (14) and also shown schema-'tically in Fig. 1 (b) . A train of delta functions is contained in h(r) and the biggest delta function gives us the information about the echo. Namely, its position and height /correspond to the timing (a-a) seconds and the intensity a respectively. Then this pre-・dietion rnethod converts the detection problem of the echo inte that of the delta function in the impulse responce function of the predictor. This transformation brings forth an ・enormous advantage that the delta function showing the existence of the eeho is too sharp to be covered up by other noisy terms in the impulse response function and thencan be ・scarcely overlooked even if the signal band width is narrow andlor if the time delay of the echo is small. Simulation results (Fig. 2) , experiments of sound propagation in an anechoic room - (Fig. 4) and those of flexural wave propagation in a steel strip (Fig. 6) show the effectiveness of this prediction method.