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The Fundamental Concepts Behind Signal Processing

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A signal is a time dependent, numerical representation of events in the physical world. In typical applications, the signal is in the form of a current or a voltage. For the signal to be useful, it must be modeled. Signal processing takes time dependent data, and manipulates it to create a mathematical model useful to practical problem solvers. Many techniques for signal processing exist, including Fourier Transforms, moving averages, filtering, and spectral analysis. Spectral analysis uses sampled data to reconstruct a given signal. Though conceptually simple, sampling is typically impractical for most applications due to the large quantity of data involved in the calculations. However, the fundamental concepts behind signal processing …show more content…

Fourier Series require an infinite number of frequencies, but the sampling frequency is subdivided into a finite number of frequency ranges to reduce calculations. One cosine and one sine function is needed to represent the signal for each subdivision of the sampling frequency. If the sampled data is represented by a vector, it can be written as a linear combination of two vectors, each composed of the appropriate sinusoidal entries. Let b m contain the cosine entries at frequency subdivision m, and c contain the sine entries at m subdivision m. Then, B=[b  b] , C=[c  c] , D=[B C] , and the columns 0m0m of D form a basis for the vector space V. It can be shown that the columns of D are, in fact, an orthogonal basis because the dot product of any two vectors in D is zero.
The signal s∈V , and s=BuCv . This can be rewritten as s=Dw where w=[u] . Given that the columns of D form an orthogonal basis, the weights can be
v
[s⋅b ] [s⋅c ] calculated using the following relation: u = m , v = m . This discussion forms the foundation for the calculations in the following example. m  m c⋅c bm⋅bm m m

EXAMPLE: SAMPLING AT 60 HZ
Take the following signal: s={1, 5, 9, 1, 2, 1} where s is sampled at at a rate of 60 Hz. Subdividing into 6 equal frequency ranges yields the following sinusoidal vectors:
10
1033 b = [ ] c = [ ] b = c = cos sin 33
10
1 0 cos2 sin2
0,0,1 ,1 , 1 0 cos sin
[][] cos5

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