When does sampled autocorrelation = autocorrelation of sampled sequence?

Hello,

It feels like this will be a standard DSP result that I should be able 
to dig up, but I'm not sure where to look, so here goes!

If we have a continuous-time function, h(t) (assuming real and 
"sufficiently well-behaved" for now), then define the autocorrelation as:

r(t) = int { h(tau) h(tau - t) d tau }

Let's also define the discrete-time sampled sequence, h[n] = h(nT), 
where T is the sampling period, and the discrete-time autocorrelation:

r[n] = sum { h[k] h[k-n] }

My question is, what are the conditions under which r[n] = r(nT) for all 
n?  It's trivially true when h(t) is a Nyquist filter.  I believe it's 
also true if h(t) is bandlimited appropriately for the sampling rate. 
However, is there a more general condition?


Note: this is not a "homework" question!  For context, I'm considering 
the whitened-matched-filter (WMF) receiver model in e.g. "Digital 
Communications" by Proakis, and trying to understand when (if ever) the 
WMF and a ZF equaliser cancel.


Regards,

-- 
Oli

Oli Charlesworth  <catch@olifilth.co.uk> wrote:

>If we have a continuous-time function, h(t) (assuming real and 
>"sufficiently well-behaved" for now), then define the autocorrelation as:
>
>r(t) = int { h(tau) h(tau - t) d tau }
>
>Let's also define the discrete-time sampled sequence, h[n] = h(nT), 
>where T is the sampling period, and the discrete-time autocorrelation:
>
>r[n] = sum { h[k] h[k-n] }
>
>My question is, what are the conditions under which r[n] = r(nT) for all 
>n?  It's trivially true when h(t) is a Nyquist filter.  I believe it's 
>also true if h(t) is bandlimited appropriately for the sampling rate. 
>However, is there a more general condition?

I don't think so.  A loose argument would be that if the
signal does not satisfy the Nyquist criterion, after sampling
higher-frequency components will have aliased into the baseband
and the spectrum will change, hence the autocorrelation function
will change.  But there could be exceptions to this argument if 
the signal is sufficiently exotic.

Steve

Steve Pope wrote:
> Oli Charlesworth  <catch@olifilth.co.uk> wrote:
> 
>> If we have a continuous-time function, h(t) (assuming real and 
>> "sufficiently well-behaved" for now), then define the autocorrelation as:
>>
>> r(t) = int { h(tau) h(tau - t) d tau }
>>
>> Let's also define the discrete-time sampled sequence, h[n] = h(nT), 
>> where T is the sampling period, and the discrete-time autocorrelation:
>>
>> r[n] = sum { h[k] h[k-n] }
>>
>> My question is, what are the conditions under which r[n] = r(nT) for all 
>> n?  It's trivially true when h(t) is a Nyquist filter.  I believe it's 
>> also true if h(t) is bandlimited appropriately for the sampling rate. 
>> However, is there a more general condition?
> 
> I don't think so.  A loose argument would be that if the
> signal does not satisfy the Nyquist criterion, after sampling
> higher-frequency components will have aliased into the baseband
> and the spectrum will change, hence the autocorrelation function
> will change.  But there could be exceptions to this argument if 
> the signal is sufficiently exotic.

I think that a Nyquist filter (e.g. raised-cosine) is an exception.  In 
general, a Nyquist filter doesn't have to be bandlimited, it just has to 
satisfy the ISI criterion (i.e. h(nT) = 0, n =/= 0).


-- 
Oli

>Hello,
>
>It feels like this will be a standard DSP result that I should be able 
>to dig up, but I'm not sure where to look, so here goes!
>
>If we have a continuous-time function, h(t) (assuming real and 
>"sufficiently well-behaved" for now), then define the autocorrelation
as:
>
>r(t) = int { h(tau) h(tau - t) d tau }
>
>Let's also define the discrete-time sampled sequence, h[n] = h(nT), 
>where T is the sampling period, and the discrete-time autocorrelation:
>
>r[n] = sum { h[k] h[k-n] }
>
>My question is, what are the conditions under which r[n] = r(nT) for all

>n?  It's trivially true when h(t) is a Nyquist filter.  I believe it's 
>also true if h(t) is bandlimited appropriately for the sampling rate. 
>However, is there a more general condition?

Oli,

I believe the condition is that the signal be sampled at twice the Nyquist
rate.  I'll let you have the fun of justifying this yourself, though.  :-)

Emre

On 1 Mrz., 04:56, "emre" <egu...@ece.neu.edu> wrote:
> >Hello,
>
> >It feels like this will be a standard DSP result that I should be able
> >to dig up, but I'm not sure where to look, so here goes!
>
> >If we have a continuous-time function, h(t) (assuming real and
> >"sufficiently well-behaved" for now), then define the autocorrelation
> as:
>
> >r(t) = int { h(tau) h(tau - t) d tau }
>
> >Let's also define the discrete-time sampled sequence, h[n] = h(nT),
> >where T is the sampling period, and the discrete-time autocorrelation:
>
> >r[n] = sum { h[k] h[k-n] }
>
> >My question is, what are the conditions under which r[n] = r(nT) for all
> >n? &#4294967295;It's trivially true when h(t) is a Nyquist filter. &#4294967295;I believe it's
> >also true if h(t) is bandlimited appropriately for the sampling rate.
> >However, is there a more general condition?
>
> Oli,
>
> I believe the condition is that the signal be sampled at twice the Nyquist
> rate. &#4294967295;I'll let you have the fun of justifying this yourself, though. &#4294967295;:-)

The condition you suggest is sufficient but not necessary. To see
this, try the following DSP Riddle:

Find a function h(t) defined on continuous-time that is not
bandlimited but such that Oli's property holds.

:-)

Regards,
Andor

Andor wrote:
> On 1 Mrz., 04:56, "emre" <egu...@ece.neu.edu> wrote:
>>> Hello,
>>> It feels like this will be a standard DSP result that I should be able
>>> to dig up, but I'm not sure where to look, so here goes!
>>> If we have a continuous-time function, h(t) (assuming real and
>>> "sufficiently well-behaved" for now), then define the autocorrelation
>> as:
>>
>>> r(t) = int { h(tau) h(tau - t) d tau }
>>> Let's also define the discrete-time sampled sequence, h[n] = h(nT),
>>> where T is the sampling period, and the discrete-time autocorrelation:
>>> r[n] = sum { h[k] h[k-n] }
>>> My question is, what are the conditions under which r[n] = r(nT) for all
>>> n?  It's trivially true when h(t) is a Nyquist filter.  I believe it's
>>> also true if h(t) is bandlimited appropriately for the sampling rate.
>>> However, is there a more general condition?
>> Oli,
>>
>> I believe the condition is that the signal be sampled at twice the Nyquist
>> rate.  I'll let you have the fun of justifying this yourself, though.  :-)
> 
> The condition you suggest is sufficient but not necessary. To see
> this, try the following DSP Riddle:
> 
> Find a function h(t) defined on continuous-time that is not
> bandlimited but such that Oli's property holds.

I believe h(t) = rect(t/T) is such a function!

So, is it possible to find a *necessary* condition?


-- 
Oli

>> Oli,
>>
>> I believe the condition is that the signal be sampled at twice the
Nyquis=
>t
>> rate. =A0I'll let you have the fun of justifying this yourself, though.
=
>=A0:-)
>
>The condition you suggest is sufficient but not necessary. To see
>this, try the following DSP Riddle:
>
>Find a function h(t) defined on continuous-time that is not
>bandlimited but such that Oli's property holds.
>
>:-)
>
>Regards,
>Andor

Thanks, Andor.  Indeed, I meant "the sufficient condition".  However, I
should point that this is what you need to satisfy for unknown
deterministic signals.  I think in that case it becomes "the necessary and
coefficient condition".  Please correct me if I am wrong.

Emre

>Thanks, Andor.  Indeed, I meant "the sufficient condition".  However, I
>should point that this is what you need to satisfy for unknown
>deterministic signals.  I think in that case it becomes "the necessary
and
>coefficient condition".  Please correct me if I am wrong.
>
>Emre

I meant necessary and *sufficient* condition here, not coefficient
condition.  :-)

Recapping my conjecture:  
    For unknown (deterministic) bandlimited signals, the necessary and
sufficient condition for Oli's property to hold is that the signal be
sampled at twice the Nyquist rate.

Emre

> Recapping my conjecture: �
> � � For unknown (deterministic) bandlimited signals, the necessary and
> sufficient condition for Oli's property to hold is that the signal be
> sampled at twice the Nyquist rate.


I dont think that is true. If the signal is periodic for instance, we
can devise a scheme that samples *below* the Nyquist based on the
signal's periodicity.

> Recapping my conjecture:
>     For unknown (deterministic) bandlimited signals, the necessary and
> sufficient condition for Oli's property to hold is that the signal be
> sampled at twice the Nyquist rate.


I dont think thats 100% true. If the signal is narrowband and periodic
for instance, we
can devise a scheme that samples *below* the Nyquist based on the
signal's periodicity.