What is Noise?

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Introduction to Noise

Why Analyze Noise?

Spectral Characteristics of Noise

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

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

White noise is a random signal in which all frequencies are equally present over time. It is analogous to white light which, as Isaac Newton first realized, consists of the superposition of all visible light frequencies. — Click for https://ccrma.stanford.edu/~jos/sasp/White_Noise.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

Turbulent flow is the opposite of laminar flow. — Click for http://en.wikipedia.org/wiki/Turbulence

In signal processing, noise is a random signal. Broadband noise contains energy over a wide range of frequencies. — Click for http://en.wikipedia.org/wiki/White_noise

The Gaussian probability density function is, in Matlab syntax, exp(-(x-mu)^2/(2*sigma^2))/sqrt(2*pi*sigma^2), where mu is the mean and sigma is the standard deviation. — Click for http://en.wikipedia.org/wiki/Normal_distribution

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

Click for https://scienceworld.wolfram.com/physics/

The theory of quantum mechanics describes fundamental particles and their interactions at the smallest observable scales. — Click for https://quantummechanics.ucsd.edu/ph130a/130_notes/130_notes.html

A measure of the average thermal energy of a substance, typically expressed in degrees Celsius, or degrees Kelvin. — Click for http://en.wikipedia.org/wiki/Temperature

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

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JOS Home Page

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Index: Spectral Audio Signal Processing

Spectral Audio Signal Processing

Introduction to Noise

Why Analyze Noise?

Spectral Characteristics of Noise

What is Noise?

Consider the spectrum analysis of the following sequence:

x = [-1.55, -1.35, -0.33, -0.93, 0.39, 0.45, -0.45, -1.98]

In the absence of any other information, this is just a list of numbers. It could be temperature fluctuations in some location from one day to the next, or it could be some normalization of successive samples from a music CD. There is no way to know if the numbers are ``random'' or just ``complicated''.^7.2 More than a century ago, before the dawn of quantum mechanics in physics, it was thought that there was no such thing as true randomness--given the positions and momenta of all particles, the future could be predicted exactly; now, ``probability'' is a fundamental component of all elementary particle interactions in the Standard Model of physics [59].

It so happens that, in the example above, the numbers were generated by the randn function in matlab, thereby simulating normally distributed random variables with unit variance. However, this cannot be definitively inferred from a finite list of numbers. The best we can do is estimate the likelihood that these numbers were generated according to some normal distribution. The point here is that any such analysis of noise imposes the assumption that the noise data were generated by some ``random'' process. This turns out to be a very effective model for many kinds of physical processes such as thermal motions or sounds from turbulent flow. However, we should always keep in mind that any analysis we perform is carried out in terms of some underlying signal model which represents assumptions we are making regarding the nature of the data. Ultimately, we are fitting models to data.

We will consider only one type of noise: the stationary stochastic process (defined in Appendix C). All such noises can be created by passing white noise through a linear time-invariant (stable) filter [263]. Thus, for purposes of this book, the term noise always means ``filtered white noise''.

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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

What is Noise?

``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