By D. Gibbons
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Additional info for Time-Series Analysis and Inverse Theory for Geophysicists
In practice we have a discretised series of finite length, which introduces two fundamental limií tations on the derived spectrum. The finite length limits the frequency spacing í ❭✐⑤❻❵ ❧✼⑧ , and the sampling interval ❭❪❨ limits the maximum meaningful frequency ⑤✼➡ . In the previous section we found that adding zeroes to a time series reduces the frequency spacing of the spectrum without changing its content. Adding zeroes therefore improves the spectrum’s appearance without introducing any spurious new information; it is also a convenient device for lenghthening the time series to the next power of two when using the FFT.
1): any component of this sine wave in the original continuous function will not appear in the digitised sequence. 1. A useful illustration of aliasing is the appearance of spokes on a wheel, each an angle ✤ apart, captured on film. 025 s ( ❭❪❨ ) appears to us to rotate with angular velocity ❡ because the spoke moves through an angle ❢ ❵❣❡✧❭❪❨ between successive frames. As the wheel speeds up it reaches a point where ❢ exceeds half the angle between the spokes, and if the spokes are identical we cannot tell from the digitised film whether the wheel is rotating forwards at a rate ❢✲⑧✵❭❪❨ or backwards at a rate ❳❚✤ ❯❤❢❡❬④⑧✵❭✃❨ .
Think of the short sequence as a “train” running along the “rails” of the long sequence as the subscript ➾ is increased. If multiplication by a ❚ -transform corresponds to convolution, division by a ❚ -transform corresponds to the opposite procedure, deconvolution since ➭➯❳➦❚❡❬Õ❵ ➥ ➵ ❳➦❚❡❬④⑧ ❳➦❚❡❬ . 13) and so on. 14) This procedure is called recursion because the result of each equation is used in all subsequent equations. 14) impossible when ❲ ③ ❵Ý❥ . Since ➭❾❳➦❚❙❬➨❵ must be equivalent to division by the ❚ -transform.
Time-Series Analysis and Inverse Theory for Geophysicists by D. Gibbons