Resolution versus Stability

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Welch's Method

Welch Autocorrelation Estimate

Welch's Method with Windows

Spectral resolution refers to the resolving power of a spectrum analysis. In particular, taking the Fourier transform of a longer data segment yields higher spectral resolution. To resolve two sinusoids which are closely spaced in frequency, a spectrum analysis needs to use on the order of two cycles (or more) of the difference frequency. — Click for https://ccrma.stanford.edu/~jos/sasp/Resolving_Sinusoids.html

Welch's method for estimating power spectral densities may be defined as the average of periodograms of successive signal blocks over time. — Click for https://ccrma.stanford.edu/~jos/sasp/Welch_s_Method.html

A filter is said to be stable if its impulse response decays to zero as time goes to infinity — Click for https://ccrma.stanford.edu/~jos/filters/Stability_Revisited.html

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Welch's Method

Welch Autocorrelation Estimate

Welch's Method with Windows

Resolution versus Stability

A fundamental trade-off exists in Welch's method between spectral resolution and statistical stability. As discussed in §5.4.1, we wish to maximize the block size in order to maximize spectral resolution. On the other hand, more blocks (larger ) gives more averaging and hence greater spectral stability. A typical default choice is $M\approx K\approx \sqrt{N_x}$ , where denotes the number of available data samples.

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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

Resolution versus Stability

``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