Length L FIR Frame Filters

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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

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

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

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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Spectral Audio Signal Processing

Time Varying OLA Modifications

Time-Varying STFT Modifications

Weighted Overlap Add

Length L FIR Frame Filters

To avoid time aliasing, we restrict the filter length to a maximum of samples. Since $H_m(\omega_k)$ is an arbitrary multiplicative weighting of the th spectral frame, the frame filter need not be causal. For odd , the filter impulse response indices may run from to , where

$\displaystyle L_h \isdef \frac{L-1}{2}$

(9.43)

This gives

$\begin{eqnarray*} y(n) &=& \sum_{r=-L_h}^{L_h} x(n-r) {\hat h}_{n-r}(r) \\ &=& x(n) {\hat h}_n(0) \\ & & + x(n-1) {\hat h}_{n-1}(1) + \cdots + x(n-L_h) {\hat h}_{n-L_h}(L_h) \\ & & + x(n+1) {\hat h}_{n+1}(-1) + \cdots + x(n+L_h) {\hat h}_{n+L_h}(-L_h) \end{eqnarray*}$

This is the general length time-varying FIR filter convolution sum for time , when is odd.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Length L FIR Frame Filters

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1. Copyright © 2022-02-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University