Data Transforms

Define the underlying continuous data function to be dcont(t).  The measured data is spaced Δt apart.  Fourier sums are made using spacing Δ∞.

Figure 1 Pictorial representation of dcont, dsamp, Δ∞, and Δt   

Define

                               (1.1)

                                (1.2)

                    (1.3)

      (1.4)

        (1.5)

               (1.6)

            The continuous function is that at the values iΔ∞.  These become continuous in the limit that M ∞.  The M in (1.6) is needed to make integrals of dcont(t)s(t) equal to trap rule integrals of dsamp(t).  Note that the sum in (1.6) is over the N points in dsamp, not over the N∞ points representing the continuous data. 

The Fourier transform of s

The Fourier transform of s is defined according to the definitions in Convolution2.htm for

(1.7)

The fm in (1.7) is needed between -N∞/2 and N∞/2 for the back transform for the Discrete Fourier Transform definitions.  The last sum in this range is zero except for those cases where m=kN.  Note that there are M of these in the range of fm.  That is m={-M/2,-M/2+1,…,0,1,…,M/2-1}.

                                                       (1.8)

The Fourier transform of d

The measured data values are

                                                                    (1.9)

 

Comment  - This is a strange function, all spikes at the data points.  Its transform should be expected to smaller than that of xc by a factor of M to reflect that fact that it is zero for all but one of the M values of xc in the range of Δt

 

The convolution theorem (Convolution2.htm ) states that if dsamp is a product as in (1.9) then its transform is

                (1.10)

Using (1.8) this becomes

(1.11)

Thus the Fourier transform of the measured spectral points is equal to a sum of shifted transforms of the original. 

(1.12)

This is twice the maximum frequency in an N point transform of the data, xs, at the Δt points.  Note that the natural periodicity with respect to f = k/Δ∞ has been shrunk by these periodic repeats to a periodicity with respect to f=k/Δt.  The back transform of (1.11), Xs(f), is exactly (1.9) complete with the zeroes at the intermediate points.  These zeroes are due to the sums of the repeated transforms.

Back transform

            Equation (1.11) says that the transform of the sample multiplied by s(t) is the transform of the underlying data periodically repeated.  If it were not periodically repeated, the back transform of (1.11) would be the underlying data.  Define

(1.13)

A symmetrical definition of H as in (1.13) is necessary to make h as defined in (1.14) real.

(1.14)

The sum in (1.14) includes both a point at m=-N/2 and at m=N/2.  The DFT sum includes only the point at N/2.  The  in the definition of H in (1.13) compensates for this by making the value at the first and last point equal to  the average so that the sum in (1.14) becomes an end point trapezoidal rule approximation to the integral of a constant H times the exponential factor.

(1.15)

If there is no overlap and if Dcont(fm) is zero for |f| > 1/(2Δt), that is if

(1.16)

Then owing to the presence of H, the sum in (1.15) reduces to a single term so that

(1.17)

Back transform (1.17) as

(1.18)

Note that (1.18) includes the point at m=N/2.  The D is the same as at m=-N/2, but not exp unless t=iΔt.

 

Using the convolution theorem (1.18) can also be written as

(1.19)

The N/2 -1 in (1.19) comes from the delta function defining dsamp.  It is correctly to N/2 - 1.  The definition of H and h use the N∞ data and hence in limiting the range for H, the trap rule correction or half values at the end points are needed.

Note that in the limit that M ∞, that (1.19) is valid for all t.  The Nyquist theorem is the statement that for band limited data as in (1.16), the complete underlying function is given by  (1.18) or (1.19).