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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),

$\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) $

and

$\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) $

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

$\displaystyle \left[\begin{array}{c} x_1(n+1) \\ [2pt] x_2(n+1) \end{array}\rig... ...bf{A} \left[\begin{array}{c} x_1(n) \\ [2pt] x_2(n) \end{array}\right] \protect$ (C.136)

or, in vector notation,
$\displaystyle \underline{x}(n+1)$ $\displaystyle =$ $\displaystyle \mathbf{A}\underline{x}(n) + \mathbf{B}u(n)$ (C.137)
$\displaystyle y(n)$ $\displaystyle =$ $\displaystyle \mathbf{C}\underline{x}(n)$ (C.138)

where we have introduced an input signal $ u(n)$, which sums into the state vector via the $ 2\times 1$ (or $ 2\times N_u$) vector $ \mathbf{B}$. The output signal is defined as the $ 1\times 2$ vector $ \mathbf{C}$ times the state vector $ \underline{x}(n)$. Multiple outputs may be defined by choosing $ \mathbf{C}$ to be an $ N_y \times 2$ 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 $ A$ is

$\displaystyle \det{\mathbf{A}} = c^2g - g(c+1)(c-1) = c^2g - g(c^2-1) = c^2g - gc^2+g = g. \protect$ (C.139)

When the eigenvalues $ {\lambda_i}$ of $ \mathbf{A}$ (system poles) are complex, then they must form a complex conjugate pair (since $ \mathbf{A}$ is real), and we have $ \det{\mathbf{A}} = \vert{\lambda_i}\vert^2 = g$. Therefore, the system is stable if and only if $ \vert g\vert<1$. When making a digital sinusoidal oscillator from the system impulse response, we have $ \vert g\vert=1$, 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 $ \left\vert\det{A}\right\vert=1$, which occurs for $ g=1$.

Note that $ \underline{x}(n) = \mathbf{A}^n\underline{x}(0)$. If we diagonalize this system to obtain $ \tilde{\mathbf{A}}= \mathbf{E}^{-1}\mathbf{A}\mathbf{E}$, where $ \tilde{\mathbf{A}}=$   diag$ [\lambda_1,\lambda_2]$, and $ \mathbf{E}$ is the matrix of eigenvectors of $ \mathbf{A}$, 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] $

where $ \tilde{\underline{x}}(n) \isdef \mathbf{E}^{-1}\underline{x}(n)$ denotes the state vector in these new ``modal coordinates''. Since $ \tilde{A}$ is diagonal, the modes are decoupled, and we can write

\begin{eqnarray*} \tilde{x}_1(n) &=& \lambda_1^n\,\tilde{x}_1(0)\\ \tilde{x}_2(n) &=& \lambda_2^n\,\tilde{x}_2(0) \end{eqnarray*}

If this system is to generate a real sampled sinusoid at radian frequency $ \omega $, the eigenvalues $ \lambda_1$ and $ \lambda_2$ must be of the form

\begin{eqnarray*} \lambda_1 &=& e^{j\omega T}\\ \lambda_2 &=& e^{-j\omega T}, \end{eqnarray*}

(in either order) where $ \omega $ is real, and $ T$ denotes the sampling interval in seconds.

Thus, we can determine the frequency of oscillation $ \omega $ (and verify that the system actually oscillates) by determining the eigenvalues $ \lambda_i$ of $ A$. Note that, as a prerequisite, it will also be necessary to find two linearly independent eigenvectors of $ A$ (columns of $ \mathbf{E}$).

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