Introduction to Noise
Why Analyze Noise?
An example application of noise spectral analysis is denoising, in which noise is to be removed from some recording. On magnetic tape, for example, ``tape hiss'' is well modeled mathematically as a noise process. If we know the noise level in each frequency band (its power level), we can construct time-varying band gains to suppress the noise when it is audible. That is, the gain in each band is close to 1 when the music is louder than the noise, and close to 0 when the noise is louder than the music. Since tape hiss is well modeled as stationary (constant in nature over time), we can estimate the noise level during periods of ``silence'' on the tape.
Another application of noise spectral analysis is spectral modeling synthesis (the subject of §10.4). In this sound modeling technique, sinusoidal peaks are measured and removed from each frame of a short-time Fourier transform (sequence of FFTs over time). The remaining signal energy, whatever it may be, is defined as ``noise'' and resynthesized using white noise through a filter determined by the upper spectral envelope of the ``noise floor''.
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 Characteristics of Noise
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Optimal Peak-Finding in the Spectrum
