Modal Representation
When the state transition matrix is diagonal, we have the
so-called modal representation. In the single-input,
single-output (SISO) case, the general diagonal system looks like
Since the state transition matrix is diagonal, the modes are decoupled, and we can write each mode's time-update independently:

Thus, the diagonalized state-space system consists of
parallel one-pole systems. See §9.2.2
and §6.8.7 regarding the conversion of direct-form filter
transfer functions to parallel (complex) one-pole form.
Diagonalizing a State-Space Model
To obtain the modal representation, we may diagonalize
any state-space representation. This is accomplished by means of a
particular similarity transformation specified by the
eigenvectors of the state transition matrix . An eigenvector
of the square matrix
is any vector
for which




![$\displaystyle E= \left[ \underline{e}_1 \; \cdots \; \underline{e}_N \right],
$](http://www.dsprelated.com/josimages_new/filters/img2189.png)
A system can be diagonalized whenever the eigenvectors of are
linearly independent. This always holds when the system
poles are distinct. It may or may not hold when poles are
repeated.
To see how this works, suppose we are able to find linearly
independent eigenvectors of
, denoted
,
.
Then we can form an
matrix
having these eigenvectors
as columns. Since the eigenvectors are linearly independent,
is
full rank and can be used as a one-to-one linear transformation, or
change-of-coordinates matrix. From Eq.
(G.19), we have that
the transformed state transition matrix is given by







![$\displaystyle \Lambda \isdef \left[\begin{array}{ccc}
\lambda_1 & & 0\\ [2pt]
& \ddots & \\ [2pt]
0 & & \lambda_N
\end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img2193.png)



The transfer function is now, from Eq.(G.5), in the SISO case,
We have incidentally shown that the eigenvalues of the state-transition matrix




Notice that the diagonalized state-space form is essentially
equivalent to a partial-fraction expansion form (§6.8).
In particular, the residue of the th pole is given by
. When complex-conjugate poles are combined to form real,
second-order blocks (in which case
is block-diagonal with
blocks along the diagonal), this is
corresponds to a partial-fraction expansion into real, second-order,
parallel filter sections.
Finding the Eigenvalues of A in Practice
Small problems may be solved by hand by solving the system of equations


Example of State-Space Diagonalization
For the example of Eq.(G.7), we obtain the following results:
>> % Initial state space filter from example above: >> A = [-1/2, -1/3; 1, 0]; % state transition matrix >> B = [1; 0]; >> C = [2-1/2, 3-1/3]; >> D = 1; >> >> eig(A) % find eigenvalues of state transition matrix A ans = -0.2500 + 0.5204i -0.2500 - 0.5204i >> roots(den) % find poles of transfer function H(z) ans = -0.2500 + 0.5204i -0.2500 - 0.5204i >> abs(roots(den)) % check stability while we're here ans = 0.5774 0.5774 % The system is stable since each pole has magnitude < 1.
Our second-order example is already in real form,
because it is only second order. However, to illustrate the
computations, let's obtain the eigenvectors and compute the
complex modal representation:
>> [E,L] = eig(A) % [Evects,Evals] = eig(A) E = -0.4507 - 0.2165i -0.4507 + 0.2165i 0 + 0.8660i 0 - 0.8660i L = -0.2500 + 0.5204i 0 0 -0.2500 - 0.5204i >> A * E - E * L % should be zero (A * evect = eval * evect) ans = 1.0e-016 * 0 + 0.2776i 0 - 0.2776i 0 0 % Now form the complete diagonalized state-space model (complex): >> Ei = inv(E); % matrix inverse >> Ab = Ei*A*E % new state transition matrix (diagonal) Ab = -0.2500 + 0.5204i 0.0000 + 0.0000i -0.0000 -0.2500 - 0.5204i >> Bb = Ei*B % vector routing input signal to internal modes Bb = -1.1094 -1.1094 >> Cb = C*E % vector taking mode linear combination to output Cb = -0.6760 + 1.9846i -0.6760 - 1.9846i >> Db = D % feed-through term unchanged Db = 1 % Verify that we still have the same transfer function: >> [numb,denb] = ss2tf(Ab,Bb,Cb,Db) numb = 1.0000 2.0000 + 0.0000i 3.0000 + 0.0000i denb = 1.0000 0.5000 - 0.0000i 0.3333 >> num = [1, 2, 3]; % original numerator >> norm(num-numb) ans = 1.5543e-015 >> den = [1, 1/2, 1/3]; % original denominator >> norm(den-denb) ans = 1.3597e-016
Properties of the Modal Representation
The vector
in a modal representation (Eq.
(G.21)) specifies how
the modes are driven by the input. That is, the
th mode
receives the input signal
weighted by
. In a computational
model of a drum, for example,
may be changed corresponding to
different striking locations on the drumhead.
The vector
in a modal representation (Eq.
(G.21)) specifies how
the modes are to be mixed into the output. In other words,
specifies how the output signal is to be created as a
linear combination of the mode states:


The modal representation is not unique since
and
may be scaled in compensating ways to produce the same transfer
function. (The diagonal elements of
may also be permuted along
with
and
.) Each element of the state vector
holds the state of a single first-order mode of the system.
For oscillatory systems, the diagonalized state transition matrix must
contain complex elements. In particular, if mode is both
oscillatory and undamped (lossless), then an excited
state-variable
will oscillate sinusoidally,
after the input becomes zero, at some frequency
, where







In practice, we often prefer to combine complex-conjugate pole-pairs
to form a real, ``block-diagonal'' system; in this case, the
transition matrix is block-diagonal with two-by-two real matrices
along its diagonal of the form
![$\displaystyle \mathbf{A}_i = \left[\begin{array}{cc} 2R_iC_i & -R_i^2 \\ [2pt] 1 & 0 \end{array}\right]
$](http://www.dsprelated.com/josimages_new/filters/img2214.png)



Next Section:
Repeated Poles
Previous Section:
Similarity Transformations