Nyquist frequency aliasing

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).

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 3.6 (Continued) (b) Example of foldover of frequencies above the Nyquist frequency with two phase-sensitive detectors. The lower spectrum (spectral width 5000 Hz) faithfully reproduces the true spectrum, but the upper spectrum (spectral width 3600 Hz) shows that peaks near + 2000 Hz now display an aliased frequency near — 1800 Hz and appear near the right-hand end of the spectrum. Sample D-glucorono-6,3-lactone-l,2-acetonide in DMSO-<4 at 500 MHz. Spectra courtesy of Joseph J. Barchi (National Institutes of Health). 

Aliasing, which is caused by overlap of intensities in the spectrum and its mirror image automatically generated by the DFT. Care must be taken to avoid overlap of intensities with the mirror symmetrical replicate (alias) above a certain frequency (Nyquist frequency). As a result, only N/2 points in the calculated spectrum can be used. FT routines are usually dealiased . 

As the wheel picks up speed, the spokes first appear to rotate faster up to a certain rate, then slow down, then rotate in the opposite direction, and so on. The maximum displayable frequency is the Nyquist frequency. Higher frequencies are "folded over" or "aliased", to a frequency obtained by reflection about the Nyquist frequency as shown in Figure 17. 

Figure 17. Demonstrate of foldover aliasing, (a) Hypothetical spectrum, with peaks located at their true frequencies, (b) Discrete cosine Fourier transform of the time-domain signal corresponding to (a), with sampling and Nyquist frequencies as shown. The peaks in (b) have correct relative intensities, but are folded-back to lower apparent displayed frequencies.

Because computerized Fourier Transforms are performed on discrete, digital data arrays of finite length, two well-known problems arise. The discreteness of the data array leads to a phenomenon referred to as "aliasing" in which frequencies which are higher than one-half the data point acquisition frequency (the Nyquist frequency) appear at values which are lower than the true frequency. This effect is illustrated in Figure 6 for the case of a sine wave. 

Sampling theory states that the signal must be sampled at twice the highest frequency present in the signal. If this criterion is not met then frequencies above the Nyquist frequency (half the digitalsampling rate) will be aliased that is, they appear at spurious lower frequencies. To avoid this, very high sampling rates (10 MHz) must be employed. 

In summary, the sampled waveform should contain a whole number of periods to avoid leakage and the sampling should be with at least the Nyquist frequency or faster to avoid aliasing. In EIS practice, a waveform containing a predetermined number of frequencies and whole number of periods of waveforms is used and sampling is synchronized (Chap. 3.7). 

Figure 18.29 shows the spectral content of the input signal after sampling. Frequencies below 50 Hz, the Nyquist frequency (/s/2), appear correctly. However, frequencies above the Nyquist appear as aliases below the Nyquist frequency. For example, FI appears correctly however, both F2, F3, and F4 have aliases at 30, 40, and 10 Hz, respectively The resulting frequency of aliased signals can be calculated with the following formula ... 

Failure to obey the Nyquist sampling theorem results in aliasing, that is, the effect of frequencies greater than the Nyquist frequency are reflected into the band of frequencies between DC and one-half the sampling rate. This situation is demonstrated in Fig. 20.70, where the samples of the high-frequency sinusoid are indistinguishable from a uniform sampling of the low-frequency sinusoid. 

The ADC needs at least two samples per waveform cycle in order to represent the frequency of the sound if not, then the frequency information is lost. Digital recording systems place a low-pass filter before the ADC in order to ensure that only signals below the Nyquist frequency enter the converter. Otherwise, the conversion process creates foldback frequencies, thus causing a phenomenon known as aliasing distortion (Figure 1.3). 

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. 

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