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So far we have considered the baseband sampling case where all signals of interest lie within the first Nyquist zone. The figure shows another case where the sampled signal band is limited to the first Nyquist zone and the original band image appears in every other Nyquist zone.
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The case shown is an example where the sampled signal band lies entirely within the second Nyquist zone. The process of sampling a signal outside the first Nyquist zone is often referred to as "undersampling" or "harmonic sampling". Note that the image in the first Nyquist zone contains all the information in the original signal, except its original location (the order of the frequency components in the spectrum is reversed, but this can easily be corrected by reordering the FFT output correct).
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The sampled signal restricted to the third Nyquist zone is shown. Note that the image in the first Nyquist zone is not spectrally inverted. In fact, the sampled signal frequency may lie in any unique Nyquist zone, and the image in the first Nyquist zone is still an exact representation (except for the spectral inversion that occurs when the signal is in an even-numbered Nyquist zone ). At this point, we can revisit the Nyquist criterion, as it applies to wideband signals: A signal with a bandwidth of BW must be sampled at a rate equal to or greater than twice its bandwidth (2BW) in order to preserve all the information in the signal . Note that there is no reference to the absolute position of the frequency band of the sampled signal in the frequency spectrum relative to the sampled frequency. The only restriction is that the sampled signal band must be confined to a single Nyquist zone, i.e. the signals must not overlap arbitrarily many fs/2 (in fact, this is the main function of the anti-aliasing filter).
Sampling signals above the first Nyquist zone is equivalent to analog demodulation in communication applications and is therefore gaining popularity. It has become increasingly common to directly sample an IF signal and then use digital techniques to process that signal, eliminating the need for IF demodulators and filters. Obviously, however, the higher the IF frequency, the more stringent the dynamic performance requirements for the ADC. The ADC input bandwidth and distortion performance must be sufficient to handle the IF frequency, not just the baseband. This presents a challenge for most ADCs that are only designed to handle signals in the first Nyquist zone—ADCs suitable for undersampling applications must maintain constant dynamic performance in higher order Nyquist zones.
Shows the signal in the second Nyquist zone centered on the carrier frequency fc, where the lower and upper frequency limits are f1 and f2, respectively. The antialiasing filter is a bandpass filter. The desired dynamic range is DR, which defines the filter stopband attenuation. The upper limit of the transition zone is f2 to 2fs–f2, while the lower limit is f1 to fs–f1. For baseband sampling, the anti-aliasing filter requirements can be reduced by scaling up the sampling frequency, but fc must also be changed so that it is always in the center of the second Nyquist zone.
In general, larger NZ is better, allowing handling of high IF frequencies. Regardless of the choice of NZ, the Nyquist criterion requires fs>2Δf. If NZ is chosen to be odd, then fc and its signal will lie in the odd-numbered Nyquist zone, and the image frequency in the first Nyquist zone will not be inverted. For example, assume a signal is centered on a carrier frequency of 71 MHz and is 4 MHz wide. Therefore, the minimum sampling frequency requirement is 8MSPS. Solving for NZ by substituting fc=71MHz and fs=8MSPS into Equation 6 yields NZ=18.25. However, NZ must be a whole number, so we round 18.25 to the nearest whole number, which is 18. Solving for fs by Equation 6 again yields fs=8.1143MSPS. Therefore, the final values are fs=8.1143MSPS, fc=71MHz, NZ=18.
Now suppose we need more headroom in the anti-aliasing filter, so fs is chosen to be 10MSPS. Solving for NZ by substituting fc=71MHz and fs=10MSPS into Equation 6 yields NZ=14.7. We round 14.7 to the nearest integer to get NZ=14. Solving for fs by Equation 6 again gives fs=10.519MSPS. Therefore, the final values are fs=10.519MSPS, fc=71MHz, NZ=14. The iterative process described above can also produce an integer for NZ by starting from fs and adjusting the carrier frequency.
summary
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This article covered the basics of the Nyquist criterion and the effects of aliasing in the time and frequency domains. It also shows how to properly specify anti-aliasing filters using working knowledge of this criterion. Examples of oversampling and undersampling relevant to modern communication system applications are presented.