Minimum-Phase and Causal Cepstra

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Spectral Audio Signal Processing

FIR Digital Filter Design

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The subject of Digital Signal Processing (DSP) is concerned with the computational processing of digital signals. An example digital signal is music from a compact disk (CD). An example DSP task is modifying the tone of a digital audio stream, such as boosting its bass or treble. Another DSP example is compressing digital audio into MP3 format, or decoding an MP3 stream. — Click for https://ccrma.stanford.edu/~jos/filters/Book_Series_Overview.html

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Spectrum analysis of sound is analogous to decomposing white light into its component colors by means of a prism — Click for https://ccrma.stanford.edu/~jos/mdft/Example_Applications_DFT.html

Fast Fourier Transforms (FFT) are fast algorithms for computing the Discrete Fourier Transform (DFT) — Click for https://ccrma.stanford.edu/~jos/mdft/Fast_Fourier_Transform_FFT.html

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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

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

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The frequency response is defined for LTI filters as the Fourier transform of the filter output signal divided by the Fourier transform of the filter input signal — Click for https://ccrma.stanford.edu/~jos/filters/Frequency_Response_I.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

Minimum-phase polynomials have all their zeros inside the unit circle of the complex plane. — Click for https://ccrma.stanford.edu/~jos/filters/Minimum_Phase_Polynomials.html

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FIR Digital Filter Design

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Minimum-Phase and Causal Cepstra

To show that a frequency response is minimum phase if and only if the corresponding cepstrum is causal, we may take the log of the corresponding transfer function, obtaining a sum of terms of the form $\ln(1-\xi_iz^{-1})$ for the zeros and $-\ln(1-\rho _iz^{-1})$ for the poles. Since all poles and zeros of a minimum-phase system must be inside the unit circle of the plane, the Laurent expansion of all such terms (the cepstrum) must be causal. In practice, as discussed in [263], we may compute an approximate cepstrum as an inverse FFT of the log spectrum, and make it causal by ``flipping'' the negative-time cepstral coefficients around to positive time (adding them to the positive-time coefficients). That is $c(n) \leftarrow c(n) + c(-n)$ , for $n=1,2,\ldots\,,$ and $c(n)\leftarrow 0$ for . This effectively inverts all unstable poles and all non-minimum-phase zeros with respect to the unit circle. In other terms, $p_i \leftarrow 1/p_i$ (if unstable), and $\xi_i\leftarrow 1/\xi_i$ (if non-minimum phase).

The Laurent expansion of a differentiable function of a complex variable can be thought of as a two-sided Taylor expansion, i.e., it includes both positive and negative powers of , e.g.,

$\displaystyle H(z) = \cdots + h(-2)z^2 + h(-1)z + h(0) + h(1)z^{-1}+ h(2)z^{-2}+ \cdots\,.$

(5.26)

In digital signal processing, a Laurent series is typically expanded about points on the unit circle in the plane, because the unit circle--our frequency axis--must lie within the annulus of convergence of the series expansion in most applications. The power-of- terms are the noncausal terms, while the power-of- $z^{-1}$ terms are considered causal. The term in the above general example is associated with time 0, and is included with the causal terms.

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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

Minimum-Phase and Causal Cepstra

``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