Time extends from –T/2 to T/2 in N steps of Dt = T/N. Frequency extends from –N/(2Dt) to N/(2Dt) in steps of Df = 1/ NDt = 1/Time.
The DFT is fundamental, the integral relationships are induced.
(1.1)
The data consists of N points spaced Dt apart so that T = NDt. The conceptually infinite data set is N¥ = M ´ N data points spaced D¥ apart so that T = N¥D¥. The frequencies are for all values of m. Note that Time in this limit is not changed.
The forward transforms in (1.3) are
Both the top and the bottom lines of (1.5) are defined for frequencies given by values of m from -¥ to ¥. Replacing the D’s in the exponential by their values as Time/N and Time/N¥, gives
This show that DN[m+N] = DN[m], while D¥[m+N¥] = D¥[m]. In other words there are N¥ distinct values of the frequency in the bottom lines of (1.5) and (1.6) and only N distinct values in the top lines. As M à ¥, the points in the bottom lines of (1.5) and (1.6) are spaced zero distance apart making the sums become integrals.
The back transforms in (1.4) are
The back transforms in (1.7) are defined for all values of i. For i' = i + N in the upper line or i + N¥ in the lower line we see that both d’s have the property that . Thus these transforms have periodically extended data originally defined from –Time/2 to Time/2 to all times. The frequency spacing in the top and bottom line in (1.7) is same, but the upper and lower limits in the bottom line are M times those in the top line -- the bottom line extends to much higher values of |fm|.
The DFT becomes an end point trap rule by correcting for the end points. To do this subtract ½ the value of the first term and add ½ the value of the last term. Note that d[-N/2] = d[N/2] so that for any frequency f
The second term in (2.1) is a trap rule correction. It is zero for f=m/T. For other f values, the term d[-N/2] is assumed equal to d[N/2] by the periodicity of the system. The d(N/2) value is then added to the sum multiplied by ½ and ½ of the term d(-N/2) is subtracted from the sum. This gives the usual trap rule equation with end points multiplied by ½. Trap rule.doc Trap rule2.doc