Linear State Space Models

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Physical Audio Signal Processing

Physical Models

State Definition

Impulse Response of State Space Models

Matlab is a high-level scripting language for scientific computing — Click for https://ccrma.stanford.edu/~jos/matlab/

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

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According to Hooke's law, the force F exerted by a spring stretched a distance x from its rest position is F=-kx. — Click for http://en.wikipedia.org/wiki/Spring_%28device%29

The volume level of an audio device corresponds to the gain applied to the signal right before it reaches the output. — Click for http://ccrma.stanford.edu/~jos/pasp/Amplifier_Feedback.html

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Any function that is not linear is nonlinear. There are no general methods for solving nonlinear equations or nonlinear differential equations. This is why it is usually easier, although generally less accurate, to approximate a nonlinear function by a linear function if possible. — Click for http://en.wikipedia.org/wiki/Nonlinear

The sampling interval (or sampling period) of a discrete-time signal is defined as the reciprocal of the sampling rate. Since the sampling rate is normally in units of samples per second (Hz), the sampling interval is normally in units of seconds. See also: sampling rate, sampling theorem — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html

The difference equation is a recipe for computing samples of the output signal of a digital filter based on samples of the input signal and the filter coefficients. In an Infinite-Impulse-Response (IIR) digital filter, there are typically both feedforward and feedback coefficients in the difference equation. — Click for https://ccrma.stanford.edu/~jos/filters/Difference_Equation_I.html

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

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A force is required to change the momentum of an object. In the absence of external forces, momentum is conserved. For a mass m in flight, the momentum is m v, where v denotes the velocity of the mass. Newton's second law, F = m a, says that force equals mass times acceleration, i.e., force equals the time derivative of momentum. — Click for https://scienceworld.wolfram.com/physics/Force.html

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The topic of linear systems theory is primarily about linear, time-invariant (LTI) filters. A linear filter is characterized by the property that its output-signal amplitude is linearly proportional to its input-signal amplitude. A time-invariant filter, or 'constant-coefficient' filter, performs the same filtering operation at all times. Only LTI filters can be characeterized by their impulse response, or their frequency response, or their poles and zeros plus a gain, or the like. Also, only LTI filters have the superposition property for audio signals, in which the filtering of an audio mix can be obtained alternatively by filtering each channel separately and adding the filtered channels together. Equivalently, only LTI filters exhibit no intermodulation distortion for audio signals. — Click for https://ccrma.stanford.edu/~jos/filters/Linear_Time_Invariant_Digital_Filters.html

In the context of linear time-invariant filter analysis, a zero is a point in the complex s or z plane at which the transfer function goes to zero. A pole is a point in the s or z plane at which the transfer function goes to infinity. The s plane is used to analyze analog filters while the z plane is used for digital filters. — Click for https://ccrma.stanford.edu/~jos/filters/Pole_Zero_Analysis_I.html

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Index: Physical Audio Signal Processing

Physical Audio Signal Processing

Physical Models

State Definition

Impulse Response of State Space Models

Linear State Space Models

As introduced in Book II [452, Appendix G], in the linear, time-invariant case, a discrete-time state-space model looks like a vector first-order finite-difference model:

$\displaystyle \underline{y}(n)$	$\displaystyle =$	$\displaystyle C \underline{x}(n) + D\underline{u}(n)$
$\displaystyle \underline{x}(n+1)$	$\displaystyle =$	$\displaystyle A \underline{x}(n) + B \underline{u}(n) \protect$	(2.8)

where $\underline{x}(n)$ is the length

state vector at discrete time

, $\underline{u}(n)$ is in general a $q\times 1$ vector of inputs, and $\underline{y}(n)$ the $p\times 1$ output vector.

is the $N\times N$ state transition matrix, and it determines the dynamics of the system (its poles, or modal resonant frequencies and damping).

The state-space representation is especially powerful for multi-input, multi-output (MIMO) linear systems, and also for time-varying linear systems (in which case any or all of the matrices in Eq.(1.8) may have time subscripts ) [221].

To cast the previous force-driven mass example in state-space form, we may first observe that the state of the mass is specified by its velocity and position $x(t)=\int v(t)\, dt$ , or $x[(n+1)T] = x(nT) + T\,v(nT)$ .^2.9Thus, to Eq.(1.5) we may add the explicit difference equation

$\displaystyle x[(n+1)T] \eqsp x(nT) + T\,v(nT) \eqsp x(nT) + T\,v[(n-1)T] + \frac{T^2}{m} f[(n-1)T]$

which, in canonical state-space form, becomes (letting $x_n\isdef x(nT)$ , etc., for notational simplicity)

$\displaystyle \left[\begin{array}{c} x_{n+1} \\ [2pt] v_{n+1} \end{array}\right] \eqsp \left[\begin{array}{cc} 1 & T \\ [2pt] 0 & 1 \end{array}\right]\left[\begin{array}{c} x_n \\ [2pt] v_n \end{array}\right] + \left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right] f_n, \quad n=0,1,2,\ldots\,, \protect$

(2.9)

with

being a typical initial state.

General features of this example are that the entire physical state of the system is collected together into a single vector, and the elements of the matrices include physical parameters (and the sampling interval, in the discrete-time case). The parameters may also vary with time (time-varying systems), or be functions of the state (nonlinear systems).

The general procedure for building a state-space model is to label all the state variables and collect them into a vector $\underline {x}$ , and then work out the state-transition matrix , input gains , output gains , and any direct coefficient . A state variable $\underline{x}_i(n)$ is needed for each lumped energy-storage element (mass, spring, capacitor, inductor), and one for each sample of delay in sampled distributed systems. After that, various equivalent (but numerically preferable) forms can be generated by means of similarity transformations [452, pp. 360-374]. We will make sparing use of state-space models in this book, because they can be linear-algebra intensive, and therefore rarely used in practical real-time signal processing systems for music and audio effects. However, the state-space framework is an important general-purpose tool that should be kept in mind [221], and there is extensive support for state-space models in the matlab (``matrix laboratory'') language and its libraries. We will use it mainly as an analytical tool from time to time.