Frequency aliased

The choice of sampling rate is sometimes governed by a tradeoff between frequency aliasing (low rate) and noise sensitivity (high rate). Quick and Bolgiano (1976) have employed the Poisson transformation to avert this complication. Consult Piovoso and Bolgiano (1970) for additional background on the Poisson transform. 

Fig. 40.11. Aliasing or folding, (a) Sine of 8 Hz sampled at 16 Hz (Nyquist frequency), (b) Sine of 11 Hz sampled at 16 Hz (under-sampled), (c) A sine of 5 Hz fitted through the data points of signal (b).

Aliased signals Signals that fall outside the spectral window (i.e., those that fail to meet the Nyquist condition). Such signals still appear in the spectrum but at the wrong frequency because they become folded back into the spectrum and are characterised by being out of phase with respect to the other signals. 

Van Munster, E. B. and Gadella, T. W. J. (2004). phi FLIM A new method to avoid aliasing in frequency-domain fluorescence lifetime imaging microscopy. J. Microsc. 213, 29-38. 

If there were no noise in the world, these factors would control the required sampling rate. However, because noise exists and noise can be aliased as well as data, the maximum noise frequency passed by the electronics system actually determines the required sampling rate. 

The truth of this is readily seen in the following hypothetical example. Assume that a spectrum of Gaussian lines is to be scanned at a rate such that the maximum Fourier component is 100 Hz. We might then establish the electronic bandpass such that a 100-Hz component is attenuated less than 3 dB. Without noise we would sample at 200 Hz. However, significant noise signals exist out to at least six times the passband frequency of 100 Hz, which means that we must sample at 1200 Hz to avoid aliasing the noise as we deconvolve. This is an extremely conservative approach, and one might well sample less frequently without difficulty. 

Time domain aliasing cancellation based filter banks. The Modified Discrete Cosine Transform (MDCT) was first proposed in [Princen et al., 1987] as a sub-band/transform coding scheme using Time Domain Aliasing Cancellation (TDAC). It can be viewed as a dual to the QMF-approach doing frequency domain aliasing cancellation. The window is constructed in a way that satisfies the perfect reconstruction condition ... 

Today (in 1997), most commercial sampling playback implementations use only two point linear interpolation. This is described as two point because only two input samples are involved in the interpolation calculation. While this method works reasonably well, implementors have to be careful not to use waveforms with significant high frequency content (energy above ), or aliasing distortion will be noticeable. 

In [Edler, 1992] a solution to this problem has been proposed. It is based on the fact that every frequency component of the input signal influences two subbands of the cascaded filter bank, one as a signal component and the other as an aliasing component. Since this influence is symmetric, a compensation can be achieved using a butterfly... 

The sampling time At must be sufficiently short to avoid aliasing from signal intensity at frequencies above the Nyquist frequency (0Ny = n I At, which are folded back into the frequency interval -0)Ny < (0 < 0)Ny As a rule, At < Tm( /10 should be fulfilled, where Tmin is the shortest diffusion time constant. 

Figure 5,20 Portion of a 3D X-filtered NOESY spectrum of uniformly 13C/15N-labeled, stability-enhanced kinaseX in complex with kinaseX inhibitor 2. The protein and inhibitor concentrations used were 300 pM. The F3 (inhibitor1H) plane is at 7.83 ppm. Peaks with protein resonance assignments are labeled. (Note Val-A and Val-B refer to the y-i and y methyl, respectively of the same valine residue.) The spectrum was recorded at 35 °C, 600 MHz 1H frequency using a NOESY mixing time of 100 ms on a Varian Inova spectrometer equipped with a Cold Probe. The spectrum is aliased in the 13C (F2) dimension.

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