booklet format - inaf iasf bologna

Temporal Data Analysis A.A. 2011/2012
correct
wrong
Figure 3.1: Correctly wrapped-around (top); incorrectly wrapped-around (bottom)
truncate at the wrong time (meaning: not after an integer number of periods. See Figure 2.7).
To make life easier, we’ll also take for granted that N is a power of 2. We’ll have to assume the
latter anyway for the Fast Fourier Transformation (FFT) which we’ll cover in Section 3.5.
3.1.1 Even and Odd Series and Wrap-around
A series is called even if the following is true for all f k :
A series is called odd if the following is true for all f k :
f −k = f k (3.1)
f −k = −f k (3.2)
Please note that f 0 is compulsory! Any series can be broken up into an even and an odd series.
But what about negative indices? We’ll extend the series periodically:
f −k = f N−k (3.3)
This allows us, by adding N, to shift the negative indices to the right end of the interval, or
using another word, “wrap them around”, as shown in Figure 3.1. Please make sure f 0 doesn’t
get wrapped, something that often is done by mistake. The periodicity with period N, which we
always assume as given for the discrete Fourier transformation, requires f N = f 0 . In the second
example – the one with the mistake – we would get f 0 twice next to each other (and apart from
that, we would have overwritten f 4 ).
3.1.2 The Kronecker Symbol or the Discrete δ-Function
Before we get into the definition of the discrete Fourier transformation (forward and inverse
transformation), a few preliminary remarks are in order. From the continuous Fourier transfor-
44 M.Orlandini