State-Space Analysis
We will now use state-space analysisC.15[449] to determine Equations (C.133-C.136).
![$\displaystyle x_1(n+1) = c[g x_1(n) + x_2(n)] - x_2(n) = c\,g x_1(n) + (c-1) x_2(n)
$](http://www.dsprelated.com/josimages_new/pasp/img4191.png)
![$\displaystyle x_2(n+1) = g x_1(n) + c[g x_1(n) + x_2(n)] = (1+c) g x_1(n) + c\,x_2(n)
$](http://www.dsprelated.com/josimages_new/pasp/img4192.png)
or, in vector notation,
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(C.137) |
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(C.138) |
where we have introduced an input signal









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
When the eigenvalues








Note that
. If we diagonalize this system to
obtain
, where
diag
, and
is the matrix of eigenvectors
of
, then we have
![$\displaystyle \tilde{\underline{x}}(n) = \tilde{A}^n\,\tilde{\underline{x}}(0) ...
...eft[\begin{array}{c} \tilde{x}_1(0) \\ [2pt] \tilde{x}_2(0) \end{array}\right]
$](http://www.dsprelated.com/josimages_new/pasp/img4209.png)



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
).
Next Section:
Eigenstructure
Previous Section:
Digital Waveguide Resonator