By Braja M. Das
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Extra info for Fundamentals of Soil Dynamics
This is valid even though the solution itself may not be of the exponential form (as in the Bessel solution, for instance). Thus it will be sufficient to study superpositions of this form. Phase Speed At a typical frequency, the wavenumber can be written in terms of its real and imaginary parts as giving the wave in the form u(x, t) = E Fn e-kIZe-·(kRz-wt) This is recognized as having three parts: an amplitude spectrum Fn , a spatially decaying term e- kI "', and the propagating sinusoids e-·(kR",-wt).
200. 5: Transform of a modulated pulse with q = 50 kHz and a = 50ILS. It is apparent what is happening. 4). The product of the two in the time domain is a convolution in the frequency domain. The convolution tends to distribute the effect of the pulse. ) Otherwise, it will appear broad-banded. 8 Sampled Waveforms By necessity, real-signal records are finite in length, and this can add other problems to the scheme of estimating its spectral content. This section reviews the major effects that should be considered.
In that case, the computations are on the order of NS. The purpose of the FFT is to take advantage of the special form of the exponential terms to reduce the number of computations to even less than N2. Repeated Patterns The key to understanding the FFT algorithm lies in seeing the repeated forms of numbers. This will be motivated by considering the special cases of N being 4 and 8. First consider the matrix of the exponents -i21f(~) -i21f N 0 0 0 0 0 0 1 2 3 2 4 6 3 6 9 0 0 (N -1) 2(N - 1) 3(N -1) 0 (N -1) 2(N - 1) 3(N - 1) (N -1)(N -1) It is apparent that for an arbitrary value of N, 21f will be multiplied by non-integer numbers (in general).