applications exist today where these results can identify acoustics and 

 vibration sources, and can predict their separate and combined effects at 

 any output point in a general evnironment. In particular, differences 

 can be detected due to system changes between excitation points and response 

 points indicating system failures. These techniques can also be used to 

 quantify overall nonlinear system features that may be present at different 

 frequencies. 



When input noise sources are not related or when mechanical systems 

 are not structurally connected, then ordinary coherence functions and 

 associated coherent output spectra can provide many useful answers as 

 discussed in 5j. However, when multiple noise sources are measured on 

 structurally connected systems, the ordinary coherence function will not 

 separate out the effects of the various sources or distinguish between the 

 possible transmission paths. Use of the ordinary coherence function by itself 

 in these situations will give erroneous results and lead to wrong interpre- 

 tations. For these physical cases, correct results can be obtained only by 

 using partial coherence functions and multiple coherence functions as employed 

 in [6]. 



Material to follow discusses multiple input/output models for given 

 arbitrary input records and for derived ordered sets of conditioned input 

 records. Iterative computational formulas are explained to compute conditioned 

 spectral density functions, partial and multiple coherence functions. Results 

 are then illustrated for a general three input/output model. 



2. Multiple Input/Output Models for Arbitrary Inputs 



As shown in Figure 1, the input records are assumed to pass through 

 physically realizable constant parameter linear systems described by frequency 



111 



