🚀 go-pugleaf

In article <1991Mar2.230829.21497@eedsp.gatech.edu> dar@euler.eedsp.gatech.edu (Doug Reynolds) writes:
>In a pervious post, someone brought up the theoretical problem that you
>sample a sine wave with frequency f at exactly 2*f, but your first sample
>is perfectly lined up at t=0. Thus it would seem that your sample stream would
>then be all zeros makeing it impossible to reconstruct the original sine wave.
>
>I have tried to reconcile this, but to no ava as of yet. Does anyone else
>have any ideas on this? 

It's just a matter of "greater than" versus "greater than or equal to." 
Nyquist's theorem claims that the sampling frequency should be strictly 
greater than twice the frequency of the highest component of your signal.
Usually this distiction isn't a big deal, but there is indeed aliasing if you
sample at exactly the Nyquist rate. In the sine wave example of the previous
posting, you'll get impulses at +/- f with magnitudes +/- 1/2j. When you 
periodically replicate the sine spectrum, the positive impulses cancel the 
negative impulses and you get a DTFT of zero, as predicted. In the case of a
cosine you get lucky and the impulses add, so the aliasing isn't destructive
and you can reconstruct your signal (although you will be off by a constant
factor).

Casimir Wierzynski

The Nyquist rate is the rate needed to sample a given bandwidth from zero
up to that frequency but not including that frequency since the frequency
of the sampling introduces harmonics which will be lost, including the
top of the band, and any multiple of the Nyquist rate. This is not as bad
as the frequencies outside of the band, which will be aliased. This is the
whole basis of the Nyquist sampling theorem, to be able to determine what 
sampling rate to use for a given bandwidth. Realistically, for any decent
resolution of the wave, you should sample at four times the maximum, but
the absolute minimum (the limit, which will obviously not work at all)
is the Nyquist rate, or twice the maximum frequency.

In article <13687@life.ai.mit.edu>, campbell@churchy.ai.mit.edu (Paul Campbell) writes:
> The Nyquist rate is the rate needed to sample a given bandwidth from zero
> up to that frequency but not including that frequency ...

Correct so far.

> This is not as bad
> as the frequencies outside of the band, which will be aliased.

A frequency that is an exact multiple of the sampling rate gets aliased
as a DC signal. The amplitude of this aliased signal depends on the phase
where sampling occurs.

> 
>  ... Realistically, for any decent
> resolution of the wave, you should sample at four times the maximum, but
> the absolute minimum (the limit, which will obviously not work at all)
> is the Nyquist rate, or twice the maximum frequency.

When you are sampling at slightly more than twice the minimum frequency
you can completely reconstruct the original signal *provided your sampling
sequence is infinitely long*. Of course, it never is. You can
oversample to increase frequency resolution, but you can also accomplish
this by increasing your sample length.

One point that may be confusing to some people is that
an FFT allows you to determine the spectrum from a finite sample
series.  However, it assumes that the infinite series from which this
finite series is taken keeps repeating the same sequence of samples out
to infinity. To the degree this assumption is incorrect, the FFT does
not give the exact spectrum of the infinite sequence. Thus
a longer sampling length or increased sample rate can provide better
resolution in an FFT when this assumption is false. 

Paul Budnik

>
>In a pervious post, someone brought up the theoretical problem that you
>sample a sine wave with frequency f at exactly 2*f, but your first sample
>is perfectly lined up at t=0. Thus it would seem that your sample stream would
>then be all zeros makeing it impossible to reconstruct the original sine wave.
>
>I have tried to reconcile this, but to no ava as of yet. Does anyone else
>have any ideas on this? 
>

   Since this is a theoretical discussion, I'll go out on a limb and 
 claim that although a sine wave with frequency Fs/2 could not be
 sampled without ambiguity, a complex exponential [e^j(.5Fst+@)] could
 be, for any phase value @.  This is only possible, though, if there
 is no component at [e^-j(.5Fst+@)].

   In other words, you could accurately sample a "tone" at .5 Hz
 with a 1 Hz A/D converter, provided there was no "tone" at -.5 Hz.
 Of course, you'd need a complex (Re+jIm) A/D converter, since the
 signal to be sampled is not purely real anymore.

   As I recall the "nyquist sampling theorem", the sampling rate 
 dictates the maximum bandwidth that can be sampled unambiguously
 with a given sampling rate, and this bandwidth interval is closed
 on one end and open on the other, (-.5,.5] in the above example.

   It is interesting to note that the nyquist bandwidth in this
 example violates the 2X criteria, with almost a 1 Hz bandwidth
 for a 1 Hz sample rate.

   It's been a long time since my college DSP class, so feel free
 to correct me if I got something wrong!

   Bryan Hoog