In this chapter, the role of Laplace transforms in the analysis of dynamic systems has been outlined. They are used in solving differential equations and in formulating transfer functions. The frequency response may be obtained simply by substituting jw for s in the transfer function. The frequency response is also experimentally observable, giving a convenient link between theory and experiment.

In Section 2.4, we began a discussion of mathematical analyses that will be used in frequency response test data analysis. The key operation is Fourier analysis. The Fourier coefficients are related to the power spectrum of the signal by Parseval’s relation. The power spectrum is important for assess­ing signal strength and as an intermediate results in some data-analysis methods. Correlation functions are related to the Fourier coefficients of a signal by Wiener’s theorem.

Analysis procedures for obtaining frequency response results from non­sinusoidal signals were described in Section 2.9. We found that the frequency response is given by the Fourier transform of the output signal divided by the Fourier transform of the input signal. This is the most important result of Chapter 2.

Coherence functions are used to assess the influence of background noise on the test results. We examined the process of Fourier analysis and found that it is equivalent to a band-pass filtering of the signal. The filter has a shape given by the sampling function (sin x)/x.

We considered the bandwidth of a binary pulse chain so that the range of harmonics with suitably large amplitudes could be estimated. We con­sidered nonlinear contamination and found that antisymmetric periodic signals discriminate against nonlinear effects.

A number of topics that may appear somewhat unrelated appear in this chapter. However, the reader will find that they all enter into questions of test-signal characterization and selection, data analysis, and data interpreta­tion that arise in later chapters.