Python Programming for Biology: Bioinformatics and Beyond

Figure 19.3.  The results of a Fast Fourier transform. Illustrated are real and

imaginary parts of the Fourier transform (for the time series shown in

Figure 19.2

) and its

power spectrum. The original time signal has been converted into a frequency spectrum

where the intensity axis shows the amount of each frequency present in the signal.

Peaks

Once we have a Fourier transformed signal, which in certain contexts is called a spectrum,

the  next  thing  to  do  is  to  analyse  the  frequency  peaks,  or  at  least  the  significant  ones.

These correspond to the underlying frequency components of the signal. We would like to

determine  the  parameters  for  each  peak,  namely  the  amplitude,  frequency  and  decay.  If

there are many components then the peaks can overlap and this job becomes difficult. To

simplify our introduction we will assume here that there is no overlap between the peaks.

The  frequency  is  the  point  in  the  spectrum  where  the  given  peak  value  is  at  its

maximum, but there are a few subtleties to this. The spectrum is specified on a grid, so at

equally  spaced  frequencies.  Hence,  it’s  quite  likely  that  the  actual  underlying  frequency

does not lie exactly on the point of the sampled grid, but in between two such points. This

means  that  the  maximum  points  need  to  be  interpolated  somehow  to  find  the  peak

frequency positions.

There is another subtle issue to do with frequency, which comes about because of the

discrete  time  sampling  of  the  signal.  If  the  signal  is  sampled  at  time  intervals  Δt  then  a

pure signal at frequency ω and another one at frequency ω + 1/Δt  give  the  same  Fourier

transform, since

. Indeed, in general we get the same Fourier transform

for  frequency  ω  +  n/Δt  for  any  integer  n.  In  effect,  a  signal  at  one  of  these  frequencies

cannot be distinguished from a signal at any of the other frequencies. The frequency with

an  absolute  value  less  than  1/(2Δt)  is  called  the  fundamental  frequency  and  the  other

frequencies are said to be aliased to this one.

8

The height or intensity of a peak is the value at its maximum. This does not determine

the  underlying  amplitude  by  itself,  because  the  observed  height  is  also  affected  by  the

decay parameter, and as with the frequency there is also the issue that the peak values are

only defined on a grid. The amplitude is proportional to the volume (integral) of the peak,

which is the summation of the frequency values around the peak, but there is the question

of exactly how that is done, such as how far away from the maximum position to include

in  the  sum.  The  decay  parameter  is  determined  by  the  linewidth  of  the  peak,  which  is

---

roughly  speaking  how  wide  the  peak  is.  A  common  way  to  measure  this  is  the  width  at

half the peak height. These parameters can be determined by fitting the observed data to a

theoretical description of a peak.