Aliasing folded signals

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. 

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. 

Answer The spectral width is too narrow to allow the Nyquist limit ( 1.3.1) to be satisfied for all the frequencies in the spectrum and the methyl signals are folded into the window. On spectrometers that use a different version of the Fourier transformation, the aliased data may appear at the other end of the frequency spectrum but will still be out of phase with the rest of the signals. Clearly, the spectral width needs to be increased. 

In Section 3.2.3 it was shown that a resonance falling outside the spectral window (because it violates the Nyquist condition) will still be detected but will appear at an incorrect frequency and is said to be aliased or folded back into the spectrum. This can be confusing if one is unable to tell whether the resonance exhibits the correct chemical shift or not. The precise location of the aliased signal in the spectrum depends on the quadrature detection scheme in use and on how far outside the window it truly resonates. With the simultaneous (complex FT) scheme, signals appear to be wrapped around the spectral window and appear at the opposite end of the spectrum (Fig. 3.24b), whereas with the sequential (real FT) scheme signals are folded back at the same end of the spectrum (Fig. 3.24c). If you are interested to know why this difference occurs see reference [7]. 

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.

Aliasing can sometimes be used to advantage. For example, if a spectrum has no signal between 0 and Vmax/2, then every second point can be deleted ( decimation ) because this undersampling by a factor of 2 will fold the signal from v J2 to v ,ax backwards into the empty region from 0 to... 

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