Physics and Engineering of Radiation Detection

564 Chapter 9. Essential Statistics for Data Analysis

data, which generally can not be represented in a functional form, it is not very
useful. For such a situation the above equation can also be written in discrete form,
that is

Fn=

N∑− 1

j=0

gje−i^2 πnj/N, (9.10.2)

whereNis the number of data points.
Discrete Fourier transformation is an extensively used technique in data analy-
sis. There are different algorithms exist to solve the above equations but the most
common one is the so calledfast Fourier transformor FFT. As the name suggests,
this algorithm performs the transformation faster than any other technique, which
is actually due to the lesser number of computations required by it. FFT performs
2 Nlog 2 N(log 2 is the base 2 log) computations as compared to the usual algorithm
that requires 2N^2 computations.
Fig.9.10.1 shows an example of the utility of Fourier transformation. The figure
shows a sinusoid function and its power spectrum obtained by taking the square
of the amplitude of the Fourier transformed data. Suppose the sinusoid represents
the output of an electronics chain in response to a perfect sinusoidal input. The
broadness of the peak in the power spectrum tells us that the system is behaving as
a damped harmonic oscillator. But we know that the damping is electronic systems
is characterized by charge injection. Hence, if the peak is too broad we can say that
there is significant charge injection in the circuitry and if it was used to read out a
detector output the signal to noise ratio may not be acceptable.

Amplitude

Power

Frequency

Time

Figure 9.10.1: Fourier trans-

formation of an imperfect sinu-

soidal function and its power

spectrum. The power is ob-

tained by taking the square of

the amplitude of the Fourier

transform. The broadness of the

peak determines thequalityof

the sinusoid.