As an example of
statespace analysis, we will use it to determine the
frequency of oscillation of the system of Fig.
G.3 [
90].
Note the assignments of unitdelay outputs to
state variables and
.
From the diagram, we see that
and
In
matrix form, the state timeupdate can be written
or, in vector notation,
We have two natural choices of output,
and
:
A basic fact from
linear algebra is that the
determinant of a
matrix is equal to the
product of its eigenvalues. As a quick
check, we find that the determinant of
is
Since an undriven
sinusoidal oscillator must not lose energy, and
since every lossless statespace system has unitmodulus eigenvalues
(consider the modal representation), we expect
.
Note that
. If we
diagonalize this system to
obtain
, where
diag
,
and
is the matrix of
eigenvectors of
,
then we have
where
denotes the state vector in these
new ``modal coordinates''. Since
is diagonal, the modes are
decoupled, and we can write
If this system is to generate a real sampled
sinusoid at radian frequency
, the eigenvalues
and
must be of the form
(in either order) where
is real, and
denotes the
sampling
interval in seconds.
Thus, we can determine the frequency of oscillation
(and
verify that the system actually oscillates) by determining the
eigenvalues
of
. Note that, as a prerequisite, it will
also be necessary to find two
linearly independent eigenvectors of
(columns of
).
Starting with the defining equation for an
eigenvector
and its
corresponding
eigenvalue ,
we get

(G.23) 
We normalized the first element of
to 1 since
is an
eigenvector whenever
is. (If there is a missing solution
because its first element happens to be zero, we can repeat the
analysis normalizing the second element to 1 instead.)
Equation (
G.23) gives us two equations in two unknowns:
Substituting the first into the second to eliminate
, we get
Thus, we have found both eigenvectors
They are
linearly independent provided
and finite provided
.
We can now use Eq.
(
G.24) to find the eigenvalues:
Assuming
, the eigenvalues are

(G.26) 
and so this is the range of
corresponding to
sinusoidal
oscillation. For
, the eigenvalues are real, corresponding
to
exponential growth and decay. The values
yield a repeated
root (
dc or
oscillation).
Let us henceforth assume
. In this range
is real, and we have
,
. Thus, the eigenvalues can be expressed as follows:
Equating
to
, we obtain
, or
, where
denotes the
sampling rate. Thus the
relationship between the coefficient
in the digital
waveguide
oscillator and the frequency of sinusoidal oscillation
is
expressed succinctly as
We see that the coefficient range (1,1) corresponds to frequencies in
the range
, and that's the complete set of available
digital frequencies.
We have now shown that the system of Fig.
G.3 oscillates
sinusoidally at any desired digital frequency
rad/sec by simply
setting
, where
denotes the
sampling interval.
Recalling that
, the output
signal from any diagonal
statespace model is a
linear combination of the modal signals. The
two immediate outputs
and
in Fig.
G.3 are given
in terms of the modal signals
and
as
The output signal from the first
state variable is
The
initial condition
corresponds to modal initial
state
For this initialization, the output
from the first state
variable
is simply
A similar derivation can be carried out to show that the output
is proportional to
,
i.e., it is
in
phase quadrature with respect to
).
Phase
quadrature outputs are often useful in practice,
e.g., for
generating complex
sinusoids.
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