Fast Fourier transform Filtering

In Figure 13, a typical high resolution contact mode SFM scan of uniax-ially oriented POM is shown together with the corresponding 2-D fast Fourier transform filtered image (139). Since the polymer chain direction is in this particular case known a priori, the observed periodicities can be related to the well-established hexagonal crystal structure of POM in a straightforward manner. Uniaxially oriented or epitaxially crystallized specimens thus help in the analysis of the data, as has been discussed in recent review articles (140,141). 

Bowen, W. R., and Doneva, T. A. (2000). Artefacts in AFM studies of membranes Correcting pore images using fast fourier transform filtering. J. Membr. Sci. 171, 141. 

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) ... 

If further resolution is necessary one-third octave filters can be used but the number of required measurements is most unwieldy. It may be necessary to record the noise onto tape loops for the repeated re-analysis that is necessary. One-third octave filters are commonly used for building acoustics, and narrow-band real-time analysis can be employed. This is the fastest of the methods and is the most suitable for transient noises. Narrow-band analysis uses a VDU to show the graphical results of the fast Fourier transform and can also display octave or one-third octave bar graphs. 

A spectrum is the distribution of physical characteristics in a system. In this sense, the Power Spectrum Density (PSD) provides information about fundamental frequencies (and their harmonics) in dynamical systems with oscillatory behavior. PSD can be used to study periodic-quasiperiodic-chaotic routes [27]. The filtered temperature measurements y t) were obtained as discrete-time functions, then PSD s were computed from Fast Fourier Transform (FFT) in order to compute the fundamental frequencies. 

The actual limit of the summation is the extent of the weighting filter. Zero padding is used to ensure that the discretized matrices have sizes which are a power of two so that the computation can be done in the frequency domain using fast fourier transform (FFT) techniques. The effective discretized density, pin, Wj), is then given by... 

When using the fast-Fourier-transform algorithm to calculate the DFT, inverse filtering can be very fast indeed. By keeping the most noise-free inverse-filtered spectral components, and adding to these an additional band of restored spectral components, it is usually found that only a small number of components are needed to produce a result that closely approximates the original function. This is an additional reason for the efficiency of the method developed in this research. 

The Phase Vocoder. The Phase Vocoder [Flanagan and Golden, 1966][Gordon and Strawn, 1985] is a common analysis technique because it provides an extremely flexible method of spectral modification. The phase vocoder models the signal as a bank of equally spaced bandpass filters with magnitude and phase outputs from each band. Portnoff s implementation of the Short Time Fourier Transform (STFT) provides a time-efficient implementation of the Phase Vocoder. The STFT requires a fast implementation of the Fast Fourier Transform (FFT), which typically involves bit addressed arithmetic. 

FIGURE 4.8. Schematic representation of the tetrarosette assembly. (A) Unfiltered high resolution tapping mode AFM phase image of the nanorod domain stmcture (inset) fast Fourier transform. (B) Fourier filtered section of raw data shown in A and unit cell of the lattice stmcture. (Reproduced with permission from Proc. Natl. Acad. Sci. USA 2002, 99, 5024-5027. Copyright 2002 National Academy of Sciences, U.S.A.). 

In general, there are two types of compression (1) individual spectra can be compressed and filtered and (2) the entire dataset can be compressed and filtered by representing each of the individual spectra as a linear combination of some smaller set of data, which is referred to as a basis set. In this section, we will address the processing of individual spectra by applying the fast fourier transform (FFT) algorithm and followed this discussion with one on processing sets of spectra with principal component analysis (PCA). 

First, either a Lorentzian or Gaussian filter is applied to the FID to reduce the amount of noise. The choice of lineshape will depend on the shape of the frequency domain spectrum, the lineshape is related to how the fluorine spins interact with their environment. The filter linewidth is generally similar to or slightly less than the T2 value (T2 can be estimated from the spectral linewidth). After application of the time domain filter, a fast Fourier transform (FFT) is performed. The resultant frequency domain spectrum will then need to undergo phase adjustment to obtain a pure absorption spectrum. The amount of receiver dead time (time lost between the end of the excitation pulse and the first useful detection time point) will determine the presence and extent of baseline artifact present as well as how difficult phase adjustment will be to accomplish. 

One solution to the problem is to increase the ionization probability. This can be done by choosing primary ions with heavy mass, for example, Bi+ or even Ccarbon atoms. The noise level can also be reduced by techniques of digital image processing. For example, a fast Fourier transform technique has been used to remove noise from the image. This technique transforms an image from a space domain to a reciprocal domain by sine and cosine functions. Noise can be readily filtered out in such domain. After a reverse Fourier transform, filtered data produces an image with much less noise. 

One of the most important limitations of derivative spectroscopy is the background noise associated with any experimental measurement. This random noise usually has a higher frequency than the signal to be measured. This means that it will give weak but narrow peaks. The use of derivatives will strengthen this kind of peaks in front of the broader and stronger ones of the compounds, which are the most important in the usual spectrum. A solution to this problem is to use the Fourier transform (or indeed better, the fast Fourier transform, FFT), which allows the use of filters in order to remove high-frequency... 

Figure 4. Multipexed detection of interleukins, (a) Spectra for resonators labeled 1-5 that correspond to control, streptavidin-functionalized control, anti-interleukin 6, anti-interleukin 4, and anti-interleukin 8, respectively. The trace in blue shows the initial baseline spectrum. The red trace corresponds to the test spectrum after introducing 10 pg/ml of interleukin 6 along with 1 pg/ml of interleukin 8, followed by the sequential association of secondary antibodies corresponding to each of these interleukins. We clearly see shifts corresponding to the resonators functionalized with anti-interleukin 6 and 8 (Resonance 3 and 5, respectively) while the other resonances do not shift appreciably thus indicating the lack of non-specific binding. Fabry-Perot resonances were filtered out in both spectra by performing a fast Fourier transform, (b) Reaction stages at each of five resonators.

The data are also filtered before the Fast Fourier transform, with a suitable analytic window (I>(f), as explained in Section 4. 

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