SIGNAL CONDITIONING, DATA ACQUISITION, AND SPECTRAL ANALYSIS - Design and Development of Medical Electronic Instrumentation - page 235

In most instances, the powerful Fourier techniques used in EEG systems and CT or ultra-
sound scanners are hidden from the user, who does not have to worry about their mathe-
matical implications. In other cases, however, human interpreters must make diagnostic
decisions based on frequency-domain representations of data processed through Fourier
transforms. For example, many digital storage oscilloscopes o er the user the option to con-

vert time-domain signals into the frequency domain through the use of an FFT running on
an embedded DSP that displays the results directly on screen. It is also common for scien-
tists and engineers to write short FFT-based routines to display a spectral representation of
experimental data acquired into a personal computer. It is in these cases where the unwary
may fall into one of the many traps that FFTs conceal.
FFT users often forget that real-world signals are seldom periodic, free of noise, and
distortion, and that signal and noise statistics play an important role in their analysis. Because
of this, FFTs and other methods can only provide estimates of the actual spectrum of signals,
which require competent interpretation by the user for their correct analysis. Moreover, the
FFT has certain inherent problems that make it unsuitable for high-resolution applications.

ff

FFTs and the Power Spectral Density

Using a typical data acquisition setup, a signal is sampled at a fixed rate of f S (samples/s), which

yields discrete data samples x 0 , x 1 , ... , x N 1 . These N samples are then spaced equally by the
discrete sampling period

fi

1/ f S . The discrete Fourier transform (DFT) represents the
time-domain data with N spaced samples in the frequency domain X 0 , X 1 , ... , X N 1 through

∆ t (s)

N 1

X ( f ) ∆ t x n e 2π jf n ∆ t

n 0

where the frequency f f(Hz) is de fi ned over the interval ∆ t /2 f ∆ t /2. The FFT will

e ciently evaluate this expression at a discrete set of N frequencies spaced equally by

∆ t.

In its simplest form, the energy spectral density estimate of the time-domain data is
given by the squared modulus of the FFT of these data, and the power spectral density
(PSD) estimate P ( f ) (or simply, the spectrum) at every discrete frequency f is obtained by
dividing the latter by the time interval N

f (Hz) 1/ N ∆

∆ t :

X m
∆

2
t

N P ( f m ) m 0,1,..., N 1

f . When real data are used (usually, the case when sampling from real-
world signals), the PSD for negative frequencies will be symmetrical to the PSD for posi-
tive frequencies, making only half of the PSD useful. However, at times it may be
necessary to compute the PSD for complex data, and relevant results will be obtained for
both positive and negative frequencies.
Although obtaining the PSD seems to be as simple as computing the FFT and obtain-
ing the square modulus of the results, it must be noted that because the data set employed
to obtain the Fourier transform is only a limited record of the actual data series, the PSD
obtained is only an estimate of the true PSD. Moreover, as will be seen later, meaningless
spectral estimates may be obtained by using the estimate of P ( f m ) without performing
some type of statistical averaging of the PSD.

where f m (Hz) m ∆

Pitfalls of the FFT

When sampling a continuous signal, information may be lost because no data are avail-
able between the sample points. As the sampling rate is increased, a larger portion of the