Band-limited noise

Spirou G.A. and Young E.D. (1991). Organization of dorsal cochlear nucleus type IV unit response maps and their relationship to activation by band-limited noise. J. Neurophysiol. 66 1750-1768. 

A stochastic process is also characterized by its spectral density, the Fourier transform of its autocorrelation function. The autocorrelation function of a (stationary stochastic process) measures the correlation of the process at different time intervals while the spectral density measures the amplitudes of the component waves of different frequencies. A white noise process has a constant spectral density (i.e., the same amplitude for all frequencies) and the band-limited noise has a frequency band over which the spectral density is nearly constant. 

Since the disturbance power spectrum S w) in this example is limited to a narrow frequency band and it can be shown that the spectrum does not significantly overlap any of the weighting functions Wt( u ) for z = 1,..., 5, it is expected from the variance analysis for the band-limited noise case (Equation (2.95)) that the accuracy of the estimated coefficients should not not be significantly compromised by the disturbance. This is confirmed by the results given in Table 2.1 and Figures 2.10 and 2.12 where it can be seen that the estimated Laguerre model gives a very accurate representation of the true process. 

Each sound grain produced by Chaosynth is composed of several spectral components. Each component is a waveform produced by a digital oscillator which needs two parameters to function frequency (Hz) and amplitude (dB). In Chaosynth, the oscillators can produce various types of waveforms such as sinusoid, square, sawtooth, and band-limited noise. The CA control the frequency and duration values of each grain, but the amplitude values are set up by the user beforehand via Chaosynth s Oscillators panel. The system can, however, interpolate between two different amplitude settings in order to render sound morphing effects. 

The described approach is suitable for the reconstruction of complicated dielectric profiles of high contrast and demonstrates good stability with respect to the noise in the input data. However, the convergence and the stability of the solution deteriorate if the low-frequency information is lacking. Thus, the method needs to be modified before using in praetiee with real microwave and millimeter wave sourees and antennas, whieh are usually essentially band-limited elements. 

In practice, since x(t) is a frequency band-limited signal, equation (11) shows that H(u) is known only on the finite interval wherein X(u) 0. There are also problems when the input signal is small, reduced to noise. 

As regards the noise spectrum, the different situations can be analyzed ap proximately with NC (noise criterion) and NR (noise rating) curves (Fig. 9.6.3). NC and NR curves define the octave band limits of an acceptable back ground noise each of them is characterized by a number representing the sound pressure level at 1000 Hz. 

Continuous Memoryless Channels.—The coding theorem of the last section will be extended here to the following three types of channel models channels with discrete input and continuous output channels with continuous input and continuous output and channels with band limited time functions for input and output. Although these models are still somewhat crude approximations to most physical communication channels, they still provide considerable insight into the effects of the noise and the relative merits of various transmission and detection schemes. 

Figure 9.7. Noise content of a fiberoptic oxygen sensor signal (a) in the time and (b) in the frequency domains. Time domain signals require broad frequency bandwidths. Frequency domain signals require very limited-frequency bandwidths. Noise is reduced by band limiting the signal, an advantage of frequency domain methods.

At this point, we note that there is no mechanism presently built into the relaxation methods to prevent undesirable high-frequency noise from growing with each iteration. Any spurious solution 6(x) satisfies Eq. (1) (see also Chapter 1, Sections V.A and V.B) for co beyond the band limit. If we know that the object 6 is truly band limited, with frequency cutoff co = 2, we can band-limit both data i and first object estimate d(1). The relaxation methods cannot then propagate noise having frequencies greater than Q into an estimate o(k). (One possible exception involves computer roundoff error. Sufficient precision is usually available to avoid this problem.)... 

Typically, t(co) is small for co large. A spectrometer suppresses high frequencies. If the data i(x) have appreciable noise content at those frequencies, it is certain that the restored object will show the noise in a more-pronounced way. It is clearly not possible to restore frequencies beyond the band limit Q by this method when such a limit exists. (Optical spectrometers having sine or sine-squared response-function components do indeed band-limit the data.) Furthermore, where the frequencies are strongly suppressed, the signal-to-noise ratio is poor, and T(cu) will amplify mainly the noise, thus producing a noisy and unusable object estimate. 

It is thus possible to convolve both spread function and data i(x) with s( — x). We may then use the relaxation methods as before. This time, however, we replace i(x) with s( — x) (g) i(x) and s(x) with s( — x) (x) s(x). Not only are we assured convergence, but we have also succeeded in band-limiting the data i(x) in such a way as to guarantee that all noise is removed from i(x) at frequencies where i(x) contains no information about o(x). Furthermore, Ichioka and Nakajima (1981) have shown that reblurring reduces noise in the sense of minimum mean-square error. 

The answer lies in the relatively poor performance of linear methods, especially with band-limited data. Frequently a linear restoration reveals little true structure that could not have been seen in the original data. Even worse, noise-based artifacts often call the result into question. One might even say that the linear methods have helped to give deconvolution a bad reputation in spectroscopy. 

To be sure, linear methods have value where fast computation is necessary. They perform reasonably well when the experimental data are not band limited, and in trials with computer-generated data devoid of noise. Spectroscopic data are often band limited, however, and computation time is becoming less of a problem with advances in computer hardware. The quantity of data required in spectroscopy is far less than that in image processing, for example, another field that has given much attention to deconvolution. Image processing problems are two and sometimes three-dimensional, whereas spectral problems are usually one dimensional. 

B. Deconvolving Spectra with Band-Limited White Noise Added 196... 

Spectral analysis techniques to study the behavior of pol3rmers subjected to dynamic mechanical loads and/or deformation is called Fourier Transform Mechanical Analysis (FTMA). FTMA measures the complex moduli over a range of frequencies in one test by exciting the sample by a random signal (band limited white noise) (13.14). FTMA overcomes or circumvents problems inherent in other test methods because it measures dynamic mechanical properties over a wide range of frequency with minimal temperature and moisture changes within the sample. 

A signal generator feeds band limited white noise into a power amplifier which drives an electro-mechanical shaker. A piezoelectric impedance head is mounted between the shaker and the... 

Digitisation of data during a 2D experiment is subject to the same thermal noise arising from the probe head and preamplifier as in a ID experiment, and this contributes to the noise baseplane observed in the 2D spectrum. There also exists a particularly objectionable artefact associated with 2D experiments, referred to as noise (note the here refers to the evolution period and should not be confused with the relaxation time constant). This appears as bands of noise parallel to the/i axis where an NMR resonance exists, and it is sometimes this that limits the observation of peaks in the spectrum rather than the true thermal noise. Indeed, it appears that the very earliest work on 2D NMR was unpublished due to excessive ti noise present in the spectra. Generally speaking, this is caused by... 

Here, the stochastic process x represents the response of a single-degree-of-freedom (SDOF) system or the response of a particular degree of freedom of a multi-degree-of-lfeedom (MDOF) system. The prediction error is modeled as a zero-mean discrete (band-limited) white noise process e with variance and spectral intensity ... 

Although the above formulation was presented for displacement time history, it can be easily modified for velocity or acceleration measurements. In this case the right hand side of Equations (3.4) or (3.17) can be modified with the corresponding expressions for velocity or acceleration. Of course, the case of relative acceleration with white noise excitation is not realistic since the response variance is infinity. However, the absolute acceleration measurements can be considered for ground excitation. Another choice is to utilize a band-limited excitation model. 

Band-limited white noise H Chrip signal (3 Clock Constant... 

Often it is necessary to represent the amplimde of noise (whether RMS or average-peak) in a constant relative bandwidth RBW over the whole frequency range. This means that the bandwidth B around each frequency Pq is a fixed fraction or multiple of pq. The lower and upper band limits, /i and/u, are chosen so that/o is the geometric mean of the two (i.e., Vq appears in the middle between Vi and Vu on a logarithmic frequency scale). When the relative bandwidth is n octaves or m decades (n and m normally being fractions, not integers), then the following relationships hold ... 

A well-known PSD of the ground motion available in literature is the so-called Kanai-Tajimi spectrum. It is based on the hypothesis that the ground acceleration during earthquakes may be considered as a filtered band-limited white-noise process expressed by the following function ... 

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