IEEE Solid-State Circuits Magazine - Spring 2017 - 12
y^ t h = 1
2r
#-+33 H^ j~hX^ j~he j~t d~. (6)
The reader can verify that H ^ j~h is also
the Fourier transform of the impulse
response of the system [2].
Finally, although we try our best
to operate our circuits in so-called
linear regions, there is always some
level of nonlinearity present. Interestingly enough, to characterize the nonlinearity of a circuit, we often apply
a sinusoid with a single frequency
(known as a single tone) and observe
the output. If the system is completely
linear, we would expect a sinusoid of
the same frequency at the output. Any
nonlinearity, i.e., any deviation from
a linear relationship, results in additional sinusoids at the output whose
frequencies are multiples of the input
frequency. These additional sinusoids
are known as harmonics. By measuring the power of the harmonics relative to the power of the main tone, we
quantify the amount of nonlinearity in
our circuit.
In summary, complex sinusoids
are simply multiplied by a complex
number when they go through an LTI
system, and because they span the
signal space, they can combine linearly to form any signal of interest. An
arbitrary input signal can be written
as a linear combination of complex
sinusoids as shown in (4), which, when
fed to an LTI system, maintains the
shapes of its sinusoids and only multiplies them by H ^ j~h, whose value is
only a function of ~. For this reason,
an LTI circuit is fully characterized by
H ^ j~h, which is also the Fourier transform of the LTI system response to
an impulse.
References
[1] D. Norman and D. Wolczuk, Introduction
to Linear Algebra for Science and Engineering, 2nd ed. Canada: Pearson, 2011.
[2] A. V. Oppenheim, A. S. Willsky, and S.
H. Nawab, Signals and Systems, 2nd ed.
Canada: Pearson, 1997.
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