HARMONIC ANALYSIS OF v(t)

The deviation v(t) may be represented over the interval — T to T (fx having been set equal to zero) by the real Fourier series (2.3,12), or by its complex counterpart (2.3,2). Since v(t) has a zero mean, then from (2.3,12c) A0 — 0. Since (2.3,12cZ) shows that B0 also is zero, it follows from (2.3,126) that G0 = 0 too. The Fourier series representation consists of replacing the actual function over the specified interval by the sum of an infinite set of sine and cosine waves—i. e. we have a spectral representation of x(t). The amplitudes and frequencies of the individual components can be portrayed by a line spectrum, as in Fig. 2.6. The lines are uniformly spaced at the interval a)0 = 7т/T, the fundamental frequency corresponding to the interval 2T.

The function described by the Fourier series is periodic, with period 2 T, while the random function we wish to represent is not periodic. Nevertheless, a good approximation to it is obtained by taking a very large interval 2 T. This makes the interval co0 very small, and the spectrum lines become more densely packed.

If this procedure is carried to the limit T —* oo, the coefficients An, Bn, Gn all tend to zero, and this method of spectral representation of x(t) fails. This limiting process is just that which leads to the Fourier integral (see

2.3,4 to 2.3,6) with the limiting value of Gn leading to C(o>) as shown by (2.3,13). A random variable over the range — oo < t < oo does not satisfy the condition for G(w) to exist as a point function of со. Nevertheless, over any infinitesimal dw there is a well-defined average value, which allows a proper representation in the form of the Fourier-Stieltjes integral

v{t) = Г eiatdc (2.6,4)

J (0*=—00

It may be regarded simply as the limit of the sum (2.3,2) with исо0 —* со and Gn —>■ dc. Equation (2.6,4) states that we may conceive of the function

v(t) as being made up of an infinite sum of elementary spectral components, each of bandwidth dm, of the form eiwt, i. e. sinusoidal and of amplitude dc. If the derivative dcjdm existed, it would be the G(m) of (2.3,4).

Leave a reply

You may use these HTML tags and attributes: <a href="" title=""> <abbr title=""> <acronym title=""> <b> <blockquote cite=""> <cite> <code> <del datetime=""> <em> <i> <q cite=""> <s> <strike> <strong>