Biomedical Engineering Reference
In-Depth Information
For any linear operator there is a class of functions, called
eigenfunctions
,thatare
not distorted when the operator is applied to them. The only result of application
of the operator to its eigenfunction is multiplication by a number (in general it can
be a complex number). Complex exponentials are the eigenfunctions for the LTI
system. For an LTI system with signals in the real domain
4
such eigenfunctions are
the sinusoids. It follows from the linearity of LTI system and the Euler formulas:
2
e
ix
e
−
ix
1
cos
x
=
+
(1.21)
2
i
e
ix
e
−
ix
1
sin
x
=
−
Please note, that when expressing a real signal in the language of complex ex-
ponentials we must introduce a negative valued frequency since any oscillation in
a real signal described by
f
cycles per second is represented by a pair of complex
exponentials with frequencies
f
and
f
. Since complex exponentials are eigenfunc-
tions of LTI system, from equation (1.21) it follows that when a sinusoid is passed
through the LTI system it can change the amplitude and phase but cannot change
the frequency. This property is very useful when dealing with the LTI systems, since
sinusoids form a basis in the space of real functions. Hence we can express exactly
any real function as a (possibly infinite) sum (or integral) of sinusoids with specific
amplitudes, frequencies, and phases. This property is the reason Fourier analysis is
used so extensively in signal processing.
−
1.4 Duality of time and frequency domain
In the previous section we noticed that a tool which allows to translate input signal
to a sum or integral of sinusoids would be very handy when dealing with LTI systems.
Such a tool is the Fourier transform. In fact, there is always a pair of transforms: one,
from time domain to the frequency domain, we shall denote as
F
{
x
(
t
)
}
and the
inverse transform, from frequency to time domain, is denoted as
F
−
1
{
X
(
ω
)
}
.The
operation of the transformation is illustrated in the scheme below:
F
X
x
(
t
)
(
ω
)
F
−
1
as a series of values at certain moments
in time; in the frequency domain the same signal
X
In time domain we think of a signal
x
(
t
)
is thought of as a specific
set of frequencies, and each frequency has its own amplitude and phase. Those two
(
ω
)
4
It means that the signal values are real numbers.
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