Nyquist “on the edge”

Summary

[-1, 1, -1, 1, ... ] is the “evil twin brother” of [1, 1, 1, 1, ...]
Unlike all other frequencies, they carry information only in the amplitude, but not in the phase.

What happens, if I use exactly two samples per period?

Answer: “Nothing, as long as the sampling phase is fixed.
In digital signal processing, it is perfectly reasonable to use a stream of samples such as

a=[-1, 1, -1, 1, -1, 1, -1, 1, ...]

There are exactly two samples per period, and everything is fine.

In some cases (for example random numbers), there is a valid signal component at the Nyquist limit.
If it is removed, information is lost and the signal cannot be reconstructed without error.

However, something is special: it has no phase.
Unlike any other nonzero frequency, delaying the signal does not convey any information:
Time-shifting the signal will reduce its amplitude. At a delay of half a sample, it disappears completely.

Relevance

In many real-life examples, the question is completely irrelevant:
One cannot approach the Nyquist limit, simply because an implementable lowpass filter requires some frequency margin.

In some situations, it needs to be considered, for example interpolation by FFT.

Real-world example: a communications link

Often, the sampling clock timing between a transmitter and a receiver has an unknown phase offset. If so, one needs to use a rate higher than “one sample per symbol” at the receiver! Otherwise, information is lost during sampling.
Since analog-to-digital converter cannot operate at the Nyquist limit, the problem is likely to appear, when trying to downsample later in the chain.
It is possible to go back to one sample per symbol, once the timing has been restored. For example, a polyphase filter as variable delay can be used to shift the signal back on the sample grid.

Link level simulators

Typically, link level simulators may interpolate and downsample the signal at will, but the sample timing is always fixed.
Introducing an element with fractional-sample delay will break the simulation. An example for such a model is an IIR model of an analog baseband filter.

FFT based interpolation and the center bin

One application where special attention is required is FFT-based interpolation:
The center bin corresponds to the term $e^{ik\pi}$ , which is the sum of two rotating phasors $\frac{1}{2}e^{-ik\pi}$ and $\frac{1}{2}e^{ik\pi}$ .
When changing to a wider FFT, it splits into a positive frequency and a negative frequency term.
No error is introduced, and no information is lost.

© Markus Nentwig 2007-2008
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Please send me a mail! mnentwig@elisanet.fi