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Fourier Series for Biological Signal Processing - Coggle Diagram
Fourier Series for Biological Signal Processing
Applications
Images
High Pass Filtering
Low Pass Filtering
noise minimization
signals
harvest forecast
biological signal processing
assist in the transformation of biomedical signals into electrical signals
electrical engineering
noise cancellation
electrical signal analysis
vibration analysis
quantum physics
econometrics
Voice
noise cancellation
Converting acoustic signals into electrical
Medicine
electrocardiogram
History
Origin
The concept of adding an infinite number of real or complex numbers goes back to the original ideas of Archimedes. He was probably the first to devise a method (which he called exhaustion) according to which it is possible to assign numerical meaning (convergence) to these sums. By this method he obtained a very accurate approximation of the number π, among other notable feats
The idea of representing functions by means of series emerged in India around the turn of the century. XIV, period in which the precursor techniques were conceived to deal with what today is is known as Power Series. Particular examples of this type of series are the Taylor and Maclaurin series, taught in regular Calculus courses. These arise as a limit of polynomial series, and allow to represent a collection reasonably large number of nested functions on an interval (called a convergence interval)
Other types of series exist and are more suitable for representing periodic functions defined on the line in terms of sines and cosines. In honor of Jean-Baptiste Joseph Fourier (1768-1830), who was the first to systematically study such series, we now call them Fourier Series. Other researchers, such as Euler, D'Alembert and Bernoulli had already come across this concept but had not developed it to the same degree of depth and breadth obtained by Fourier.
Equations
Is when utilized ?
used to represent infinite and complex periodic functions of physical processes, in the form of simple trigonometric functions of sines and cosines. That is, simplifying the visualization and manipulation of complex functions.
Fourier Series vs Taylor Series
Fourier series are analogous to Taylor series in the sense that both series provide a form
of representing relatively complicated functions in terms of elementary and familiar functions.
If the Fourier series converges then it represents a function f(x) and we can represent this relationship
this way:
We know that for a function to be representable by a series of powers, the conditions are the following for a real x:
The function must be infinitely differentiable.
The remainder of the Taylor formula must tend to zero
properties
A function is even if its domain contains the point −x whenever it contains the point x and satisfies the
relationship:
f(x) = f(-x)
A function is odd if its domain contains −x whenever it contains x and the relation satisfies:
f(-x) = -f(x)
But most functions are neither even nor odd. From (3.28) we conclude that if x = 0 does part of the domain of f(x) then f(0) = 0. Also the identically null function is the only one that is at the same even and odd time.
