Download Frequency Domain Representation of Biomedical Signals

Frequency Domain
Representation of
Biomedical Signals
Fourier’s findings provide a general
theory for approximating complex
waveforms with simpler functions
that has numerous applications in
mathematics, physics, and
engineering.
This section summarizes the Fourier
transform and variants of this
technique that play an important
conceptual role in the analysis and
interpretation of biological signals
Periodic Signal representation: The
Trigonometric Fourier Series
In 1807, Fourier showed that an
arbitrary periodic signal of
period T can be represented
mathematically as a sum of
trigonometric functions
this is achieved by summing or mixing
sinusoids while simultaneously
adjusting their amplitudes and
frequency as illustrated for a square
wave function
in the Figure
If the amplitudes and frequencies are
chosen appropriately,
the trigonometric signals add
constructively, thus recreating an
arbitrary periodic
signal. This is akin to combining prime
colors in precise ratios to recreate an
arbitrary color and shade.
Because…..
RGB are the building blocks for more
elaborate colors
much as sinusoids of different
frequencies serve as the building
blocks for more
complex signals.
For the example, a first-order approximation
of the square wave is achieved by
fitting the square wave to a single sinusoid
of appropriate frequency and amplitude.
Successive improvements in the
approximation are obtained by adding
higher-frequency
sinusoid components, or harmonics, to the
first-order approximation. If this
procedure is repeated indefinitely, it is
possible to approximate the square wave
signal
with infinite accuracy.
The Fourier series summarizes this
result
Compact Fourier Series
The most widely used counterparts for
approximating
and modeling biological signals are the
exponential and compact Fourier
series.
The compact Fourier series is a close
cousin of the standard Fourier series.
This
version of the Fourier series is
obtained by noting that the sum of
sinusoids and cosines can be
rewritten by a single cosine term
with the addition of a phase constant
Which lead to the compact form of the series:
Exponential Fourier Series
An alternative and somewhat more
convenient form of this result is
obtained
by noting that complex exponential
functions are directly related to
sinusoids and
cosines through Euler’s identities
an arbitrary periodic signal can be expressed as a
sum of complex exponential functions:
Where:
Fourier Transform
The Fourier integral, also referred to as the Fourier
transform, is used to decompose a continuous a
periodic signal into its constituent frequency
components
Properties of the Fourier Transform
Linearity
 Time Shifting/Delay
 Frequency Shifting
 Convolution Theorem

Discrete Fourier Transform
The DFT is essentially the digital version of the Fourier transform.
The index m represents the digital frequency index, x(k) is the sampled
approximation of x(t), k is the discrete time variable, N is an even
number that represents the number of samples for x(k), and
X(m) is the DFTof x(k)
The Z Transform
The z transform provides an
alternative tool for analyzing discrete
signals in the
frequency domain. This transform is
essentially a variant of the DFT,
which converts
a discrete sequence into its z domain
representation
The z transform plays a similar role for
digital signals as
the Laplace transform does for the
analysis of continuous signals
If a discrete sequence x(k) is
represented by xk , the (one-sided) z
transform of the
discrete sequence is expressed by
Properties of the Z transform
Linearity
 Delay
 Convolution
