Inverse Filters

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Matrix Filter Representations

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State Space Realization

An Infinite Impulse Response (IIR) digital filter has an impulse response that lasts forever (in principle) due (in practice) to the use of feedback within the filter. Digital filters with feedback are also called recursive. The transfer function of a practical (finite-order) IIR filter is a ratio of polynomials. — Click for https://ccrma.stanford.edu/~jos/filters/Difference_Equation.html

In digital signal processing, FIR is an acronym for Finite Impulse Response — Click for https://ccrma.stanford.edu/~jos/filters/FIR_Digital_Filters.html

A causal filter is any filter whose impulse response is zero prior to time zero. — Click for https://ccrma.stanford.edu/~jos/filters/Causal_Recursive_Filters.html

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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

An Finite Impulse Response (FIR) digital filter has an impulse response that reaches zero in a finite number of samples. Such filters cannot have any feedback loops. FIR filters are also called nonrecursive. The transfer function of an FIR filter is a polynomial. — Click for https://ccrma.stanford.edu/~jos/filters/FIR_Digital_Filters.html

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

An exponential is any function of the form Ae^-t/τ, where t is the independent variable (often denoting time in seconds), A is an arbitrary scale factor, τ is called the time constant, and e is Euler's number (2.718281828459...). Many systems in nature exhibit exponential decay over time, because an exponential decay is produced whenever the rate of decay is proportional to how much is left. — Click for https://ccrma.stanford.edu/~jos/mdft/Exponentials.html

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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

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JOS Home Page

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Matrix Filter Representations

Cyclic Convolution Matrix

State Space Realization

Inverse Filters

Note that the filter matrix $\mathbf{h}$ is often invertible [58]. In that case, we can effectively run the filter backwards:

$\displaystyle {\underline{x}}= \mathbf{h}^{-1} \underline{y}$

However, an invertible filter matrix does not necessarily correspond to a stable inverse-filter when the lengths of the input and output vectors are allowed to grow larger. For example, the inverted filter matrix may contain truncated growing exponentials, as illustrated in the following matlab example:

> h = toeplitz([1,2,0,0,0],[1,0,0,0,0])
h =
  1  0  0  0  0
  2  1  0  0  0
  0  2  1  0  0
  0  0  2  1  0
  0  0  0  2  1
> inv(h)
ans =
    1    0    0    0    0
   -2    1    0    0    0
    4   -2    1    0    0
   -8    4   -2    1    0
   16   -8    4   -2    1

The inverse of the FIR filter

is in fact unstable, having impulse response

, $n=0,1,2,\ldots\,$ , which grows to $\infty$ with

Another point to notice is that the inverse of a banded Toeplitz matrix is not banded (although the inverse of lower-triangular [causal] matrix remains lower triangular). This corresponds to the fact that the inverse of an FIR filter is an IIR filter.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Inverse Filters

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University