### State-Space Analysis

We will now use state-space analysisC.15[449] to determine Equations (C.133-C.136).

From Equations (C.128-C.132),

and

In matrix form, the state time-update can be written

 (C.136)

or, in vector notation,
 (C.137) (C.138)

where we have introduced an input signal , which sums into the state vector via the (or ) vector . The output signal is defined as the vector times the state vector . Multiple outputs may be defined by choosing to be an matrix.

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

 (C.139)

When the eigenvalues of (system poles) are complex, then they must form a complex conjugate pair (since is real), and we have . Therefore, the system is stable if and only if . When making a digital sinusoidal oscillator from the system impulse response, we have , and the system can be said to be marginally stable''. Since an undriven sinusoidal oscillator must not lose energy, and since every lossless state-space system has unit-modulus eigenvalues (consider the modal representation), we expect , which occurs for .

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 ).

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