Python Programming for Biology: Bioinformatics and Beyond

Figure 19.2.  An example signal time series generated using Python. A plot of the

one-dimensional signal, or time series created using the synthetic signal-generation

function. The trace shows two superimposed frequencies with different maximum

intensities and decay rates. The jagged nature of the line illustrates how only few points

are sampled relative to the period of the signals, which is typical in many real-world

situations.

The Fourier transform can also be used in other circumstances, unrelated to time. For

example,  a  crystal  has  a  periodic  three-dimensional  atomic  structure  and  as  a  result  the

Fourier transform can be used to determine the structure of a molecule in a crystal from X-

ray diffraction patterns. Also, the Fourier transform can be used to detect patterns in DNA

and protein sequences, e.g. to look at amino acid properties that occur with a periodicity

corresponding to alpha-helical structures.

Once  a  signal  has  been  Fourier  transformed,  we  generally  find  that  only  certain

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frequencies  have  significant  representation,  and  so  various  algorithms  have  been

developed  to  determine  what  those  frequencies  are.  Because  of  the  imprecision  of  the

signal  measurement,  and  perhaps  also  because  of  a  spread  in  the  phenomenon  being

observed,  normally  when  there  are  discrete  frequencies  the  transformed  distribution  at

those  points  is  not  perfectly  sharp.  Instead,  for  each  there  is  an  observed  range  of

frequencies,  called  a  peak,  with  the  maximum  value  occurring  close  to  the  actual

frequency. In general we want to determine the centre of each peak and other parameters

associated  with  the  peak,  such  as  the  intensity  (its  maximum  height)  and  also  the

integration of the peak over its frequencies (often called its volume).  Integration  is  often

used because it allows us to relate the signal strength to the underlying amount of causal

phenomenon.