TABLE OF CONTENTS In Section 10.2, we discussed how some of the properties of a random waveform can be expressed in terms of its amplitude probability distribution. However, if we wish to predict or measure the response of systems to random inputs, we must know how the power is spread over the frequency range, and this information is conveniently represented by the power spectrum. Strictly, the rigorous derivation [10.1] of the power spectrum is as the Fourier transform of the autocorrelation function, which we shall discuss later, and it is double-sided, i.e. it extends to negative frequencies. We shall look at this more mathematical approach later, but for most practical work, the following derivation from the Fourier series, leading to a single-sided definition of the power spectrum, is adequate.
Although the main application of the power spectrum is in dealing with random vibration, it can also be used with periodic waveforms. Let us look at this simpler application first. From Eq. (9.3), we know that a periodic time history, x( t), say, can be expressed as a Fourier series:
| (10.23) |  |
where a 0 is the mean level, a n and b n are the amplitudes of the cosine and sine components that together represent the waveform and ? 0 is the fundamental frequency in rad./s. From Eq. (9.2), ? 0 = 2 ?/ T, where T is the period of the...
