Smith and Cummins 



function or the impulse response; these being Fourier transforms of each other. 

 The testing procedure then consists of driving the system with some function 

 and measuring both the input and output and performing relevant calculations. 

 The minimum requirement of a good test function is that it should be rich in the 

 frequencies of interest . 



It will usually be the case that the calculations will involve some sort of 

 division. If the numerator has an error component which is fairly uniform over 

 the entire frequency range, then the ratio will be of lower quality for those val- 

 ues at which the denominator is smallest. Since the denominator will consist of 

 some characteristic of the input function, it is best that the denominator be 

 fairly constant. With these characteristics in mind, let us now look at three 

 possible classes of input functions: (1) sine waves; (2) deterministic functions 

 like the ramp, or pulse, or step, and (3) stationary random input. The sine wave 

 is of course the worst as far as frequency content is concerned since it has only 

 one frequency. As a compensatory feature, the transfer function at this fre- 

 quency is estimated by a very simple operation on the output and it can be esti- 

 mated quite accurately since all we have to extract from the continuous record 

 is just the amplitude and the phase angle. The effect of noise in the measuring 

 system should be quite small. We also have a measure of linearity of the sys- 

 tem from visual inspection of the output. 



This is a very expensive way to proceed since it requires a very large 

 number of sine wave tests to complete the analysis of the entire frequency band. 



The pulse test function is a very convenient one because all frequencies are 

 present in an equal amount. The calculations to be performed on the output are 

 quite simple. To estimate the transfer function, one merely takes the Fourier 

 transform of the response.* The drawback to this test function is that it may be 

 difficult to generate as the Davis, etc., paper indicates. 



Now no system is exactly linear, but this observation should bother no one 

 since it is sufficient for dealing with problems of nature that the systems are 

 enough so for the purposes of its use. Therefore, one should try to test under 

 conditions which are representative of the conditions of use. One would not test 

 in very high waves. Similarly the fast rise time of a good pulse may activate 

 nonlinear modes of response. 



Finally, the stationary random test function is in some way a mixture of the 

 previous two. There is some sort of repetition, albeit, an average one, and a 

 "pulsiness." We may make the function broadband in frequencies (in an average 

 sense), and its spectrum as flat as our generating methods will allow. As 

 pointed out by Davis, the system will have to be brought into a steady state be- 

 fore the relevant arithmetic may be performed on the output, and in case of 



''If I seem casual about this operation, I am just reflecting the speakers. Let 

 me assure you though that this is a numerical operation fraught with error es- 

 pecially at the higher frequencies as we will then be taking differences of a 

 large number of numbers of approximately equal magnitude. Since experimen- 

 tal data usually has low significance (numerically speaking) this is a real 

 problenn. 



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