### State-Space Analysis

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

From Equations (C.128-C.132),

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

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

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Eigenstructure

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Digital Waveguide Resonator