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Signal Analysis |
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Signals are single valued functions
of time (t) and are of complex nature. A signal wave
form, however complex it may be, comprises one or more
sine and/or cosine function. Assume that we have a square
wave given by the expression: |
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This signal
is represented by the below given illustration. Let us try to see as
to how a sine function of the same time period can be
used to represent this square wave function. |
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(a) |
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(b) |
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(c) |
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(d) |
Square wave function approximated by sine wave function |
If we choose
a sine function marked X, with its peak magnitude same
as that of the square wave, its magnitude equals the
square wave magnitude only at the peak point and it
is a very poor approximation of the square wave. If
the magnitude of the sine function is increased, as
shown by the curve Y, its magnitude becomes equal to
the square wave magnitude at two points. This provides
an approximation slightly better than the first curve,
but it is still a poor approximation.
In the above illustration,another sine wave component is added to improve the
approximation. This componet has frequence thrice the first component and as can be seen, this provides a better approximation. The approxmated wave approaches more closly to the square wave when more sine wave components are added.
The graphical method of approximating one function to
another gives a clear understanding, but is difficult
to use in practice. We now discuss an analytical method
of approximating two functions.
Consider two signals f(t) and g(t). Assume that f(t)
is to be approximated in terms of g(t) over the interval
(t1 - t2). This approximation may be written as |
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Where C is a
constant and has a value such that error between the
actual function and approximated function is minimum
over the time-interval considered. If the error function
is denoted as F e(t), then
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One possible
method of minimizing the error F e(t) over this time
interval is to minimize the average value of the error
F e(t)
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In other words an
average error should be kept minimum. However, there
may occur large positive and negative errors in the
approximation. These errors cancel each other in the
average, giving false indication that the error is minimum.
This situation may be improved if average or mean of
the squares of the error denoted by E is minimized,
instead of the error itself. In other words, |
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Minimum value of E can be obtained for a value of
C which makes  |
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or |
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Interchanging
the order of integration and differentiation gives the value
of C for obtaining the best approximation. |
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