How to construct an unknown signal from sparse periodic samples?
Nonuniform Sampling

Assuming you know what the actual sampling rate is, and exactly which samples are missed, it is possible to reconstruct the signal over a time interval T as long as the bandwidth W of the signal still satisfies the Nyquist criterion: W &lt; N/T/2, where N is the number of samples you do have.

Visualize it this way:
A signal of bandwidth W over a time interval T can be decomposed into a
linear combination of N sinc functions (sinc is shorthand for sin x/x),
where N&nbsp;=&nbsp;2&times;W&times;T.
The sinc functions are equally spaced over the interval in question and
their widths are all the same.
At the instant where each sinc pulse reaches its peak, all of the other
sinc functions have a value of zero.



If you sample the signal at the instants corresponding to the peaks of
the sinc functions (tA0, tA1 and tA2),
each sample directly yields the amplitude of the corresponding sinc pulse
(A0, A1 and A2).
The original signal can be reconstructed by creating new sinc pulses of the
correct amplitudes and overlaying them, usually by using a digital or
analog filter with the correct brick-wall frequency response.
The reconstructed signal will be described by the equation:

Signal(t) = A0 sinc (t &#150; tA0)
     + A1 sinc (t &#150; tA1)
     + A2 sinc (t &#150; tA2)

If you take N samples of the signal at other instants, each sample will be
a linear combination of all of the sinc functions.
Each sinc pulse will contribute an amount to the sample based on the
time interval between the sample and the center of the sinc pulse.
It is still possible to calculate the amplitudes of the sinc pulses, but
now it is necessary to solve a solve a system of linear equations in order
to do so.



If you have samples B1, B2 and B3 taken at times tB1,
tB2 and tB3 as shown above, you can write the
following equations:

B1 = A0 sinc (tB1 &#150; tA0)
     + A1 sinc (tB1 &#150; tA1)
     + A2 sinc (tB1 &#150; tA2)

B2 = A0 sinc (tB2 &#150; tA0)
     + A1 sinc (tB2 &#150; tA1)
     + A2 sinc (tB2 &#150; tA2)

B3 = A0 sinc (tB3 &#150; tA0)
     + A1 sinc (tB3 &#150; tA1)
     + A2 sinc (tB3 &#150; tA2)

Given that you know B1, B2 and B3, and all of the time values, it's a
straightforward matter to solve this system for A0, A1 and A2.
Once you have these coefficients, you can plug them into the reconstruction
equation given previously to reconstruct the original signal.

Note that in the above explanation, I have implicitly assumed that the
uniform sampling interval (tA1 &#150; tA0) equals
&pi;, in order to simplify the diagrams and
equations.
The same argument holds for other sampling intervals (and bandwidths) as
well.

Averaging Multiple Periods

Using multiple periods of the signal to improve the reconstruction requires that you can precisely correlate those periods in the time tomain. Once you have done the reconstruction described above, it should be possible to precisely locate features such as peaks and zero-crossings, even if these features fall between samples. Use this information to time-align multiple periods so that they can be averaged together to reduce errors such as noise, quantization error or timing jitter.