"Jerry Avins" <
[email protected]> wrote in message
news:
[email protected]...
> Vladimir Vassilevsky wrote:
> >
> > Jerry Avins wrote:
> >
> >>Samples are valid only when the sampling rate exceeds twice the
> >>highest frequency in the signal.
> >
> >
> > Common misconception. Use "bandwidth" instead of "frequency".
> >
> > Vladimir Vassilevsky
> >
> > DSP and Mixed Signal Design Consultant
> >
> > http://www.abvolt.com
>
> Start slowly. He wanted to know how to interpret the 1.,0 and 1.1 KHz
> sampled at 2 Khz. Sub band sampling is a little too deep right now.
>
I agree with Jerry that we should keep it simple. And, I agree with
Vladimir. My reasoning is below. This came up a few months ago when one of
our stalwarts seemed to insist that a particular waveform was bandlimited
(it wasn't).
Start with your favorite sinusoid: f(t) = sin(2*pi*f1*t); -inf <= t <= +inf
Here, there is a single frequency and the waveform is bandlimited.
Now, time limit f(t): multiply by g(t)=1.0, -0.5 <= t <= +0.5; g(t)=0,
|t|>0.5
Resulting in:
f'(t)=sin(2*pi*f1*t), -0.5 <= t <= +0.5; f'(t)=0, |t|>0.5
Because of the notation above for f'(t), it may appear that "there is only
one frequency" and it may be OK to say that there is only one frequency if
the comment is suitably qualified. However, the bandwidth is infinite. So,
some band limitation is necessary if suitable sampling / reconstruction is
going to be possible. The band limitation that's suitable will depend on
the width of g(t), the temporal epoch or length of the sequence of samples.
In order to accomplish this, you are going to have to lowpass filter the
gated signal - which will modify the temporal samples at least at the edges.
I can imagine that it's sometimes OK to do this filtering after sampling and
sometimes not. I'll leave that discussion up to others.
I realize that gilbert asked about what *could* be continuous sampling - in
which case this all might be moot. However, he also asked about
"calculating the periodic" [sic] and that implies a temporal epoch - the
time over which this measurement is going to happen - or the integration
time or...... So, a part of the answer to gilbert is:
**Make sure that the length of the time sample is long compared to
1/((fs/2) - f1). Otherwise, there may be more frequency aliasing (and
error) than you can tolerate.**
Interesting isn't it? The closer the frequency of the sinusoid is to fs/2,
the longer one has to sample to avoid aliasing. I haven't thought much
further about it. It's simply saying that the sinc or Dirichlet width
around the frequency spike (due to finite temporal epoch) should not be such
that it is very large at fs/2. The closer f1 is to fs/2, the narrower the
sinc needs to be. The same applies for overlap around f=0, so fs/4
conceptually allows the shortest epoch it appears - for the same degree of
error. Of course, the error varies up and down as the frequency is changed
and the tails of the sinc add or subtract from one another. So the actual
frequency and the length of the epoch (period of the sinc) interact in
generating the error.
Fred