FPGA Central

Hi all,

Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest sampling
rate for x(t) is Fs=2B...

Does this matter when x(t) is real or complex-valued?

Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no
matter when x(t) is real or complex-valued. Am I right?

More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we can
still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I right?
Does this result hold if x(t) is complex-valued?

Thanks a lot!

-L

lucy wrote:
>
> Hi all,
>
> Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest sampling
> rate for x(t) is Fs=2B...
>
> Does this matter when x(t) is real or complex-valued?
>
> Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no
> matter when x(t) is real or complex-valued. Am I right?
>
> More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we can
> still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I right?
> Does this result hold if x(t) is complex-valued?
>
> Thanks a lot!
>
> -L

In the thread "Nyquist rate for sampling complex-valued data?" starting at 12 November this item was
discussed already. You may find some answers there.

Jeroen

Hi,

If x(t) is real, then the spectrum is symetric. So for example, you can
multiply x(t) by a complex exponential with frequency B/2 and then low
pass filter with bandwidth B/2. Then sampling at Fs=B is enough.

If it is complex you need to sample at Fs=2B

I don't understand why you think x(t)^3 can be sampled at Fs=2B instead of
Fs=6B? Unless x(t) as special properties...

Séb



Hi all,
>
>Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest
sampling
>rate for x(t) is Fs=2B...
>
>Does this matter when x(t) is real or complex-valued?
>
>Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no
>matter when x(t) is real or complex-valued. Am I right?
>
>More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we can

>still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I
right?
>Does this result hold if x(t) is complex-valued?
>
>Thanks a lot!
>
>-L
>
>
>



This message was sent from http://www.DSPRelated.com

Sorry, just to correct as I must have confused myself.

if x(t) is real, then the spectrum is symetric (in power, but the phase is
the conjugate). But it still need to be sampled to Fs=2B.

If x(t) is complex, then it needs to be sampled at Fs=2B too. Then it
spectrum is not symetric, and the samples are complex (which takes twice
as much room to store than id x(t) was real).

Séb


Hi,
>
>If x(t) is real, then the spectrum is symetric. So for example, you can
>multiply x(t) by a complex exponential with frequency B/2 and then low
>pass filter with bandwidth B/2. Then sampling at Fs=B is enough.
>
>If it is complex you need to sample at Fs=2B
>
>I don't understand why you think x(t)^3 can be sampled at Fs=2B instead
of
>Fs=6B? Unless x(t) as special properties...
>
>Séb
>
>
>
>Hi all,
>>
>>Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest
>sampling
>>rate for x(t) is Fs=2B...
>>
>>Does this matter when x(t) is real or complex-valued?
>>
>>Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no

>>matter when x(t) is real or complex-valued. Am I right?
>>
>>More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we
can
>
>>still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I
>right?
>>Does this result hold if x(t) is complex-valued?
>>
>>Thanks a lot!
>>
>>-L
>>
>>
>>
>
>
>
> This message was sent from http://www.DSPRelated.com
>



This message was sent from http://www.DSPRelated.com

Lucy,

lucy wrote:
> Hi all,
>
> Suppose x(t) has bandwith bandlimited in [-B, B]... so the lowest sampling
> rate for x(t) is Fs=2B...
>
> Does this matter when x(t) is real or complex-valued?

Correct, Fs=2B whether x(t) is real or complex. The resulting samples
would correspondingly be real and complex.

>
> Moreover, for (x(t))^2, the bandwidth is [-2B, 2B], the lowest Fs=4B no
> matter when x(t) is real or complex-valued. Am I right?

Correct.

>
> More interestingly, for (x(t))^3, the bandwidth is [-3B, 3B], but we can
> still use Fs=2B(the same as x(t))... if x(t) is real-valued... Am I right?
> Does this result hold if x(t) is complex-valued?

Nope. Fs = 6B i.e. 2* 3B.

What usually confuses people is the definition of Bandwidth in real and
complex sampling. In your first example ,with a real signal, the
bandwidth is sometimes said to be B (not 2B) (No pun intended). The
thing to remember is that the sample rate = the full bandwidth , by full
I mean including the negative frequency range.

Cheers,
David