Effect of input cross spectra in estimation of frequency response and prediction in multiple‐input, single‐output systems

It is shown that under certain conditions cross spectra between correlated inputs may be neglected when estimating frequency response of multiple‐input, single‐output systems. When the input time histories are similar except for specified time lags, the above conditions exist, providing appropriate frequency smoothing is used. An additional requirement is that the magnitude of the spectra of the inputs should be constant, or vary periodically over the smoothing bandwidth. When these conditions are fulfilled, the frequency‐smoothed input cross spectra will be considerably less than the input auto spectra and can be neglected in the spectral matrix. Neglecting input cross spectra can save experimental and computer time, since for n inputs, n(n − 1) cross spectra need to be measured. The disadvantage is in a loss of frequency resolution due to the frequency smoothing which forms the basis of the technique. However, in most estimation procedures, some frequency smoothing is needed to reduce the random error. Two examples are presented to illustrate the effect for a two‐input and a six‐input system. This work has been directed at measuring the frequency responses and predicting the noise of a six‐cylinder diesel engine.