IEEE Solid-State Circuits Magazine - Spring 2017 - 11

models the transistor as an LTI circuit for small signals.
Now let us consider a general LTI
circuit as shown in Figure 2, where
the input and output variables could
be either voltage or current waveforms. A very important property of
LTI circuits is that a sinusoid input
results in a sinusoid output. In other
words, if we assume the input variable is of the form x ^ t h = cos ^~t h,
then the output variable is of the
form y ^ t h = A cos ^~t + {h, where A
and { are constants (i.e., not functions of time) and the output frequency is the same as the input
frequency. In other words, the LTI system maintains the general shape of
a sinusoid except for a scaling in
amplitude and a shift in phase (constant delay in time) where both parameters are only functions of ~.
More interestingly, if we consider
the input to be a complex sinusoid
x ^ t h = e j~t , the output will be the
same as the input except for a scaling factor. The scaling factor is constant, i.e., it is not a function of time,
but it is a function of ~. If we represent this scaling factor as H ^ j~h,
implying its dependence on ~, we
can write the following expression
for the output: y ^ t h = H ^ j~h e j~t. This
equation states that the complex
exponentials simply go through an
LTI system unchanged except for a
constant multiplication factor.
Do you know any other function
that maintains its shape as it goes
through any LTI system? Certainly
none of the rectangular or triangular
functions possesses this property.
Consider, for example, a rectangular
voltage waveform applied to a series
RC circuit and observe the voltage
across the capacitor as the output of
this circuit. In this case, the output
has little resemblance to the input,
especially when the RC time constant
of the circuit is much larger than the
period of the input waveform.
A function that goes through an
LTI system without change in shape
(except for a scaling factor) is called
an eigenfunction of the LTI system.
This is quite similar to the concept
of eigenvectors in linear algebra [1],

X=

x1
x2
x3

A=

a b c
d e f
g h i
(a)

y1
Y = AX = y2
y3

Y = AV = λV
Y = AX

X

V
(b)

(c)

Figure 3: (a) In linear algebra, a system may be characterized by a matrix A (3×3 in this
example) where the input vector X is multiplied by A to produce the output vector Y. (b) An
input vector A is both scaled and rotated by the system. (c) An input eigenvector V results in
an output vector that is aligned with the input vector.

sinusoids knowing that each comwhere a linear system receives, for
plex sinusoid is simply multiplied
example, a three-dimensional vector
X as its input and multiby a constant H ^ j~h
plies it by a fixed 3 × 3
to produce the corresmatrix A to produce a
ponding output sinuDo you know any
three-dimensional vecsoid (see Figure 4). The
other function
tor Y at the output [see
process of representthat maintains its
F i g ure 3(a)]. That is,
ing a signal as a linshape as it goes
Y3 # 1 = A 3 # 3 X 3 # 1. M o s t
ear combination of
through any LTI
input vectors will scale
complex sinusoids is
system?
and rotate as they go
done through a Fouthrough this syst e m
rier transform of the
[see Figure 3(b)], except
input signal. The Foufor three sets of input vectors that
rier transform simply takes the input
x ^ t h and represents it as a linear
are called eigenvectors. The eigencontinuous sum (integral) of comvectors are defined as input vectors
plex sinusoids:
that are scaled as they go through the
system but maintain their direc+3
x^ t h = 1 #
X ^ j~h e j~t d~
(4)
tion [see Figure 3(c)]. For an eigen2r -3
vector, the matrix multiplication
where X ^ j~h is found by correlatreduces to a simple scalar multipliing the input signal with each of the
cation because the input and output
eigenfunctions:
vectors are aligned. In other words,
we can write: AV = mV, where V is
+3
(5)
X ^ j~ h = #
x ^ t h e -j~t dt.
an eigenvector and m is its corre-3
sponding eigenvalue.
Once we find three independent
The process of finding the outnormalized eigenvectors and their
put waveform for an arbitrary input
corresponding eigenvalues, we can
is now simple. The output is the
represent any input vector as a linear
continuous sum (integral) of all the
combination of these eigenvectors.
input sinusoids multiplied by their
In doing so, we can replace lengthy
corresponding eigenvalues H ^ j~h. In
other words,
matrix multiplications by three scalar multiplications, one corresponding to each of the eigenvectors in the
linear representation.
We now return to the fact that
H ( jω)e jωt
e jωt
complex sinusoids are the eigenfuncH ( jω)
tions of the LTI systems. We note that,
in this case, we have infinitely many
eigenfunctions corresponding to all
Figure 4: Complex sinusoids (e j~t ) are
values of ~ (from zero to 3). We can
the eigenfunctions of any LTI system as
represent any arbitrary input signal
they preserve their shapes when they pass
as a linear combination of complex
through the system, only scaled by H ( j~) .

IEEE SOLID-STATE CIRCUITS MAGAZINE

s p r i n g 2 0 17

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