The Sampling Process Geophysical processes are continuous processes in time, but are not sampled continuously and never for infinitely long time spans. Processes are almost always (1) sampled at discrete points in time (usually equally spaced) for (2) a finite length of time, and (3) recorded in digital form. Digital recording has the advantage of high resolution and ease in the subsequent analysis by the use of a computer. Understanding how to sample the environment properly so the statistics of the process can be accurately estimated involves knowledge of: (1) The discrete sampling process. This includes the effects of quantization in time or the sample interval, δt, and the quantization of the parameter being measured, δx. (2) The response of our instrument to the environment. (3) Gating (sampling for only a finite length of time). The relationship between the discrete, digitized sample set (what we have to work with) and the original continuous function is covered by certain sampling theorems. The Sampler - Define the sampler, III, (Shah) as ∞ III(t) = ∑ δ(t-n) Eq 47 n=-∞ which is an infinite set of unit spaced impulses or delta functions. By suitable limiting process to take care of sharp peaks and infinite length, Bracewell shows that 1. III(t) ⊃ III(f) Eq 48 Then some properties of the sampler, (given ‘n’ as an integer and ‘a’ as a scalar) are 2. III(t+n) = III(t) 3. III(t-1/2) = III(t+1/2) 4. III(-t) = III(t) ∞ 5. III(at) = 1/|a| ∑ δ(t-n/a) Eq 50 n=-∞ ⌠ n+½ 6. ⎮ III(t)dt = 1 ⌡ n-½ ∞ 7. III(t) * x(t) = ∑ x(t-n) Eq 51 n=-∞ This says that the convolution of x(t) with the sampler produces an infinite sum of shifted versions of x(t). The general results of this is a complex mess. If, however, x(t) is gated so that it 24is time limited to |t| < 1/2, and x(t)=0 otherwise, the convolution produces an infinite replication of x(t) in each interval t. ∞ 8. III(t)·x(t) = ∑ x(n) δ(t-n) Eq 52 n=-∞ This is the sampling operation. The continuous function x(t) is changed into a set of pulses. The product function is determined by its values at t=n, and can be represented by a series of numbers, xt, as the discrete representation of x(t). xt = x(t)·III(t) Here we have made assumptions that we have equally spaced samples. Note that time series of monthly means are not equally spaced samples and may cause some difficulties in analysis. If xt = x(t)·III(t) and we define X(f) as the Fourier transform of our observed series, x(t) ⊃ X(f), and X’(f) as the transform of our sampled series, xt ⊃ X’(f) 25and we had III(t) ⊃ III(f) So if we have, xt = x(t)·III(t) then by the convolution theorem X’(f) = X(f) * III(f) where X’(f) is the estimate of the true transform, X(f), resulting from our sampling of the continuous geophysical process. By property number 7 above, it is obvious that this could be a mess. However, if X(f) were band limited to |f| < 1⁄2, (that is X(f) = 0 for |f| > 1⁄2) then the transform of the sampled function, X’(f) is a replicated version of the true transform X(f). These replications are called aliases. Where the solid line is the true transform, X(f), and the dashed curves are the replicated versions of X(f). Now since the true transform is bandlimited to frequencies of magnitude less than 1⁄2, we can reconstruct the original, continuous function by applying an ideal low pass filter, the gate function, Π. X(f) = X’(f)·Π(f) Eq 53 Note that the function Π is also sometimes called the “boxcar filter” because applying it as a filter in the time domain, we just average all the points within the boxcar as the filtered value. We can then reconstruct the original continuous series, x(t), by transforming Equation 53 back to the time domain, 26x(t) = xt * sinc(t). Thus, the sinc function is the interpolator that enables one to reconstruct the continuous series from the discrete time series. It is obvious that if X(f) is not band limited to |f| < 1⁄2, we get energy from the alias peaks falling in the interval |f| < 1⁄2, and we can not design an ideal filter to reconstruct the original series from the estimated transform. Our sampled transform is the sum of the solid curves and is shown dashed. There is high frequency energy which is not eliminated properly by the gate filter. This means that we have not sampled the function properly. Expressing it another way, we can interpolate between the sampled points in our discrete time series, xt, to reconstruct the original time series, x(t), if we have at least two samples per cycle of the highest frequency present in x(t), and this sinc function is the ideal interpolation function. Scaling - The sampling theorem for equally spaced data is given for δt = 1. However, if δt = a, then ∑ δ(t-an) = ∑ δ{a(t/a - n)} n n = 1/|a| ∑ δ(t/a - n) n = 1/|a| III(t/a) and 271/|a| III(t/a) ⊃ 1/|a| III(t/a) ⊃ 1/|a| a III(af) ⊃ III(af) Now our sampling is xt = 1/|a| x(t)·III(t/a) Eq 54 and transforming X’(f) = X(f) * III(af) Eq 55 where III(af) = 1/|a| δ(f - n/a). So we see that X(f) is replicated and multiplied by 1/a. If x(t) is bandlimited to |f| < 1/(2 a), we can recover X(t) from xt by the low pass filter aΠ(f) X(f) = a Π(af)·X’(f) Eq 56 and transforming x(t) = Sinc(t/a) * xt Eq 57 This convolution will give x(t) for any t from the sampled function, xt, so the sinc is an interpolation function. SAMPLING THEOREM - If x(t) is a band limited function, the discrete sample function xt can be used to reconstruct the original series x(t) if we have sampled at twice the highest frequency present. The sampling theorem works the other way also. If X(f) is sampled at equally spaced frequency points, then the function can be reconstructed from the sample frequency points provided x(t) is time limited. This says that a time