Transfer Functions

A digital filter can be visualized as a 'black box' that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. — Click for https://ccrma.stanford.edu/~jos/filters/

The BiQuad is a ubiquitous filter in audio signal processing, so named because its transfer function H(z)=B(z)/A(z) has a quadratic numerator polynomial B(z) and a quadratic denominator polynomial A(z). — Click for https://ccrma.stanford.edu/~jos/filters/BiQuad_Section.html

A delay-free loop is a feedback loop containing no delays. It therefore cannot be implemented in that form. Typically a unit delay is inserted, or system is rederived in some other way that avoids the delay-free loop. — Click for https://ccrma.stanford.edu/~jos/DigitizingNewton/Delay_Free_Loops.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.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

A transfer function matrix is simply a matrix of transfer functions. That is, each element of the matrix is a scalar transfer function. A matrix transfer function accommodates multiple inputs and multiple outputs (MIMO) in a linear, time-invariant (LTI) system, while each entry of the matrix relates a single-input to a single-output (SISO). — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_State_Space.html

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

The impulse signal is the shortest pulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). The impulse is typically denoted by the Greek delta symbol δ and is often called the 'Dirac delta function', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.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

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

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

In science and technology, a physical model allows us to simulate or visualize something about the thing it represents. — Click for http://en.wikipedia.org/wiki/Model_(physical)

The impulse response of a system is its output signal in response to the impulse signal. For discrete time (digital) systems, the impulse is a 1 followed by zeros. In continuous time, the impulse is a narrow, unit-area pulse (ideally infinitely narrow). — Click for https://ccrma.stanford.edu/~jos/filters/Impulse_Response_Representation.html

The transfer function is defined for LTI filters as the z transform of the filter output signal, divided by the z transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Transfer_Function_Analysis.html

Transfer Functions

As developed in Book II [452], a discrete-time transfer function is the z transform of the impulse response of a linear, time-invariant (LTI) system. In a physical modeling context, we must specify the input and output signals we mean for each transfer function to be associated with the LTI model. For example, if the system is a simple mass sliding on a surface, the input signal could be an external applied force, and the output could be the velocity of the mass in the direction of the applied force. In systems containing many masses and other elements, there are many possible different input and output signals. It is worth emphasizing that a system can be reduced to a set of transfer functions only in the LTI case, or when the physical system is at least nearly linear and only slowly time-varying (compared with its impulse-response duration).

As we saw in the previous section, the state-space formulation nicely organizes all possible input and output signals in a linear system. Specifically, for inputs, each input signal is multiplied by a ``

vector'' (the corresponding column of the

matrix) and added to the state vector; that is, each input signal may be arbitrarily scaled and added to any state variable. Similarly, each state variable may be arbitrarily scaled and added to each output signal via the row of the

matrix corresponding to that output signal.

Using the closed-form sum of a matrix geometric series (again as detailed in Book II), we may easily calculate the transfer function of the state-space model of Eq.(1.8) above as the z transform of the impulse response given in Eq.(1.10) above:

In the force-driven-mass example of the previous section, defining the input signal as the driving force

and the output signal as the mass velocity

, we have

, and

, so that the force-to-velocity transfer function is given by

$\begin{eqnarray*} H(z) &=& D + C \left(zI - A\right)^{-1}B\\ [5pt] &=&\begin{array}{r}\left[\begin{array}{cc} 0 & 1 \end{array}\right]\\ [2pt]{}\end{array}\left[\begin{array}{cc} z-1 & -T \\ [2pt] 0 & z-1 \end{array}\right]^{-1}\left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right]\\ [5pt] &=&\frac{1}{(z-1)^2}\begin{array}{r}\left[\begin{array}{cc} 0 & 1 \end{array}\right]\\ [2pt]{}\end{array}\left[\begin{array}{cc} z-1 & T \\ [2pt] 0 & z-1 \end{array}\right]\left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right]\\ [5pt] &=&\frac{T}{m}\frac{z-1}{(z-1)^2} \eqsp \zbox {\frac{T}{m}\frac{z^{-1}}{1-z^{-1}}.} \end{eqnarray*}$

Thus, the force-to-velocity transfer function is a one-pole filter with its pole at

(an integrator). The unit-sample delay in the numerator guards against delay-free loops when this element (a mass) is combined with other elements to build larger filter structures.

$\begin{eqnarray*} H(z) &=&\frac{1}{(z-1)^2}\begin{array}{r}\left[\begin{array}{cc} 1 & 0 \end{array}\right]\\ [2pt]{}\end{array}\left[\begin{array}{cc} z-1 & T \\ [2pt] 0 & z-1 \end{array}\right]\left[\begin{array}{c} 0 \\ [2pt] T/m \end{array}\right]\\ [5pt] &=&\frac{T^2/m}{(z-1)^2} \eqsp \frac{T^2}{m}\,\frac{z^{-2}}{1-2z^{-1}+z^{-2}} \end{eqnarray*}$

Now we have two poles on the unit circle at

, and the impulse response of this filter is a ramp, as already discovered from the previous impulse-response calculation.

Once we have transfer-function coefficients, we can realize any of a large number of digital filter types, as detailed in Book II [452, Chapter 9].

Transfer Functions

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University