Noise-free signal

Figure 3.11. Smoothing a noisy signal. The synthetic, noise-free signal is given at the top. After the addition of noise by means of the Monte Carlo technique, the panels in the second row are obtained (little noise, left, five times as much noise, right). A seven-point Savitzky-Golay filter of order 2 (third row) and a seven-point moving average (bottom row) filter are...

FIGURE 10.9 Smoothed data from application of the trapezoidal smoother shown in Figure 10.8. The true, noise-free signal is shown as a dotted line. 

In order to remove noise by Fourier-transform filtering we can look at the transform, as in Fig. 7.2-6. However, it is often more convenient to inspect the power spectrum, which is a (usually semi-logarithmic) plot of the magnitude (i.e., of the square root of the sum of squares of the real and imaginary components) of the Fourier transform. Such a power spectrum is shown in Fig. 7.2-9, both for a noise-free signal, and for the same signal with noise. The power spectrum is symmetrical, i.e., the information at negative and... 

Fig. 7.6-2 The power spectrum 14 log (Re2 + Im2) vs. f of the original spectrum (left) and that of the same with added Gaussian noise, na = 0.001 (middle panel) or na = 0.01 (right). The bo ttom panels show the same with an enlarged vertical scale. Large solid circles the power spectrum small open circles the power spectmm of the noise-free signal. Color is used to indicate those data points that are mostly signal , while black signifies mostly noise . As indicated by a fewpoints in the middle panels, that distinction is somewhat ambiguous.

Equation (29.5) yields the discrete transfer function of a first-order digital filter. The noise-free signal (the output of the filter) is given by... 

Signal with deterministic changes. In this example, the noise-free signal is deterministic with some sudden changes in the mean. The variables are contaminated by iid Gaussian error of standard deviation 0.5, and the results are summarized in Fig. 11. Wavelet thresholding of the result of maximum... 

In previous sections, we examined several physically important noise-free signals. [We did briefly consider the effect of noise as a radiation source, but did not consider noise contributions to the observed response to an excitation.] In the absence of noise, a signal of any shape can be analyzed to determine its parameters (e.g., spectral line position, width, area, etc.). However, noise superimposed on a signal can obscure its information content, and it may therefore become desirable to sacrifice one kind of information (e.g., resolution) in order to improve the quality of other information (e.g., signal-to-noise ratio). When an already acquired signal is modified before Fourier transformation, the modification is called apodization (literally, "removal of feet", named after early efforts to smooth FT/IR line shapes—see de Haseth Chapter). 

The necessary power for a clear and noise-free signal can be determined from the noise of the electrical conductivities and the sensitivity of the sensor. The inverse ratio gives the detectivity of the sensor. The time constant of the sensor, which determines the response time to flow changes or the start-up time of the sensor, is a function of the thermal diffusivity of the heater, thermocouple(s), and the substrate, as well as the thermal conductance of the entire device. 

Lithium niobate modulators have been used for a number of years in high capacity fiber optic transmission systems. The combination of an external modulator and a CW laser produces a more noise-free signal than a diode laser modulated by current drive. In spite of this advantage, large-scale use of modulators was not realized until recently because of earlier stability problems (Ko-rotky and Veselka, 1996). 

Noise in the signal, combined with oversampling, is therefore helping to increase the dynamic range. However, with a completely constant, noise-free signal, oversampling would not be able to increase dynamic range. Normally,... 

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