Fourier signal filtering

Steady-state extrapolation Fourier analysis Gating and signal filtering Transient signals Speeding up of response Spatial and temporal information Enhancement of selectivity Increasing order of measurement... 

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... 

Most modern infrared and NMR spectrometers collect data in the time domain with interferometers, and then the data are transformed to the familiar frequency domain by the Fourier transform. Filtering and signal enhancement in the Fourier domain before transformation is often an attractive approach to signal processing. 

A.3. Assume that you have been given sampled data of the signal / (r) of problem 9. A. 1. You identify that it has a component at Fourier techniques, filter out this componeut and plot the remaining signal. Compare your result to... 

In fig. 2 an ideal profile across a pipe is simulated. The unsharpness of the exposure rounds the edges. To detect these edges normally a differentiation is used. Edges are extrema in the second derivative. But a twofold numerical differentiation reduces the signal to noise ratio (SNR) of experimental data considerably. To avoid this a special filter procedure is used as known from Computerised Tomography (CT) /4/. This filter based on Fast Fourier transforms (1 dimensional FFT s) calculates a function like a second derivative based on the first derivative of the profile P (r) ... 

Figure 8.40 The k ySk) extended X-ray absorption fine structure (EXAFS) signal, Fourier transformed and then retransformed after application of the filter window indicated, in (a) osmium metal and (b) a 1% osmium catalyst supported on silica. (Reproduced, with permission, Ifom Winnick, FI. and Doniach, S. (Eds), Synchrotron Radiation Research, p. 413, Plenum, New York, 1980)...

Assuming that the Fourier transformed spectra 5(v) and N v) contribute at specific frequencies, the true signal, s t), can be recovered from M(v) after elimination of N y). This is called filtering (see further Section 40.5.3)... 

Many other filter functions can be designed, e.g. an exponential or a trapezoidal function, or a band pass filter. As a rule exponential and trapezoidal filters perform better than cut-off filters, because an abrupt truncation of the Fourier coefficients may introduce artifacts, such as the annoying appearance of periodicities on the signal. The problem of choosing filter shapes is discussed in more detail by Lam and Isenhour [11] with references to a more thorough mathematical treatment of the subject. The expression for a band-pass filter is H v) = 1 for v j < v < else... 

Filtering and smoothing are related and are in fact complementary. Filtering is more complicated because it involves a forward and a backward Fourier transform. However, in the frequency domain the noise and signal frequencies are distinguished, allowing the design of a filter that is tailor-made for these frequency characteristics. 

The linearity property of the Fourier transform does not hold for phase delay values. Let a, be the intrinsic, single measurement, time domain standard deviation of the filtered signal (in units of amplitude). Also let o>(n) be the standard deviation of the phase of the nth harmonic averaged j times. The phase noise as a function of harmonic (frequency) is given by... 

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