today's puzzle

Suppose there is a continuous-time signal x(t) which is sampled at integer values of t to produce a sequence 

..... , x[-2], x[-1], x[0], x[1], x[2], .......

I have this sequence available to me and I notice
that it has certain properties that allow me to
reconstruct it at will with a short computer program.
In particular, I find that x[n] = 1-n. So I see
that I don't need to store the sequence x[n] at
all.  So I begin to wonder, What if I plugged
in a non-integer value, say 1.5. Isn't x(1.5) just
1-1.5? The reason for even thinking that this might 
work is given **any** sequence of numbers y[n], 
I can always reconstruct the original signal y(t) 
from which these samples y[n] were drawn.
I mean, that's the whole crux of DSP, right? A sample 
sequence corresponds to a unique signal that can be
reconstructed from the samples. So the original
signal must be 1-t, right?  Let me ignore the
other guy who analyzed the same sequence but came
up with a more complicated solution 

       x[n] = cos(2*pi*n) - n

which matches the given sample sequence but leads
to x(t) = cos(2*pi*t) - t.  That cannot be right
because the samples x[n] uniquely specify x(t),
or equivalently, one and only one signal x(t)
can give rise to any specific sequence of sample values.
If multiple signals can give the same sequence of
sample, the whole of DSP falls apart.

dvsarwate wrote:
> 
> Suppose there is a continuous-time signal x(t) which is sampled at integer values of t to produce a sequence
> 
> ..... , x[-2], x[-1], x[0], x[1], x[2], .......
> 
> I have this sequence available to me and I notice
> that it has certain properties that allow me to
> reconstruct it at will with a short computer program.
> In particular, I find that x[n] = 1-n. So I see
> that I don't need to store the sequence x[n] at
> all.  So I begin to wonder, What if I plugged
> in a non-integer value, say 1.5. Isn't x(1.5) just
> 1-1.5? The reason for even thinking that this might
> work is given **any** sequence of numbers y[n],
> I can always reconstruct the original signal y(t)
> from which these samples y[n] were drawn.
> I mean, that's the whole crux of DSP, right? A sample
> sequence corresponds to a unique signal that can be
> reconstructed from the samples. 

No. That only applies if you have some prior 
assumptions in place. Things like being linear and
unchanging for all time and bandlimited. 
Without the right conditions your statement is pretty 
much guaranteed to be not hold true.


> So the original
> signal must be 1-t, right?  Let me ignore the
> other guy who analyzed the same sequence but came
> up with a more complicated solution
> 
>        x[n] = cos(2*pi*n) - n

The other guy gave that as a counter example 
to the implied claim that only one function can 
produce those samples. 


> 
> which matches the given sample sequence but leads
> to x(t) = cos(2*pi*t) - t.  That cannot be right
> because the samples x[n] uniquely specify x(t),
> or equivalently, one and only one signal x(t)
> can give rise to any specific sequence of sample values.
> If multiple signals can give the same sequence of
> sample, the whole of DSP falls apart.

If all that were true then the whole of DSP does 
fall apart. But its not true. DSP recognizes that more 
than one function will produce the same set of samples. 
It calls that concept aliasing

On Thursday, November 1, 2012 7:21:04 AM UTC-5, jim wrote:
> > response deleted to save space....

Do tell! I never knew that. Thanks for
setting me straight.