#### Abstract

The problem, of estimating the impulse response of a linear system arises is adaptive, control problems and elsewhere. Often it is necessary to make the system identification in the presence of external noise disturbances. This work considers the problem of determining the time that is necessary to estimate the impulse response of a linear system with a specified variance. It is assumed that' essentially no a priori knowledge about the unknown system is available, and that the output signal of the system is corrupted by an additive stationary noise signal. An ideal identifier is defined as a device that yields, for a given identification time, minimum variance estimates of samples of the unknown impulse response function. Statistical parameter estimation techniques are used to determine the identification time required by an ideal identifier. The results show that when the external disturbance is Gaussian and white, and the output signal energy is large compared to the power spectral density of the noise, the identification time is. proportional to the power spectral density of the noise and inversely proportional to the variance of the estimate and the mean square value of the input test signal. The identification time is independent of the impulse reopens® being estimated. . The identification times required by several practical identification schemes are calculated and compared to the identification time of the ideal identifier. It is established that, when the input test signal is optimized and the noise is white, the methods of cross correlation sampling input-output data, and matched filter identification are all equivalent to the ideal identifier. Depending upon the size of the variance in the impulse response estimate that is required it is concluded that, in the absence of a priori knowledge about the system, and when the rms response of the system to the input test signal is of the same order of magnitude as the variance of the external noise, the time required to identify an unknown system is an order of magnitude or more greater than the significant length of the impulse response. It is also concluded that, when the noise is white and the test signal is optimized, no measurement technique will yield a smaller identification time than that of the ideal identifier. It is pointed out that further reduction in identification time could probably be achieved by identification schemes making maximum use of all available a priori knowledge about the system.

#### Date of this Version

2-1-1961

## Comments

This report is under USAF Contract No. AF 33(6l6)-6890s Project No 8225, Task No. -