Analog and Digital Convolution

Richard Owlett wrote:
> robert bristow-johnson wrote:
>> On 3/12/15 12:48 PM, Acomp26 wrote:
>>>
>>> I have two digital signals that have to be convolved. Lets say
>>> we do that
>>> and get an output. If  we then pass both these signals through
>>> Digital to
>>> Analog converters (DAC) and convert both signals to analog and
>>> then perform
>>> convolution in continuous time will the output be the same?
>>>
>>> In other words what is the relation between a discrete time and
>>> continuous
>>> time convolution of two signals?
>>
>> how do you *do* the convolution in the continuous-time domain?
>> you likely will not just directly implement the convolution
>> integral.
>> [massive snip]
>
> More basic question. In the physically realizable world, does the
> convolution of two real time dependent "values" have any physical meaning?
>


For most? many? all? filters, there is an equivalent convolution, so yes.

-- 
Les Cargill

On 03/13/2015 04:08 AM, Tim Wescott wrote:
> There are analog systems that convolve some signal with a prototype.  Most
> don't actually do that explicitly -- most run the input signal through
> some lumped-constant filter, and get an answer.  There are some (rather
> esoteric) radar systems that actually run an input signal through a delay
> line, tap the delay line with taps of varying gains, and add up the
> result.  This gives you a filter that can do a more-or-less arbitrary
> convolution, but only by a signal that is a collection of discrete
> impulses -- as far as I know, you can't just draw some arbitrary,
> continuous-time signal on a white board and then build a filter for it.
> Again, as far as I know, you _really_ can't build a system that somehow
> lets you pump in an arbitrary impulse response, capture it, and then
> convolve by that impulse response.

Try looking up SAW convolvers. You put signals into either end, and out 
comes their convolution - roughly. They convolve over the physical 
length of the device, and not to infinity, so they are an approximation. 
They can be a bit noisy, too, limiting their dynamic range. They used to 
see practical application before digital methods totally displaced them.

Regards,
Steve

"Acomp26" <104113@dsprelated> writes:

> Hello,
>
> I have two digital signals that have to be convolved. Lets say we do that
> and get an output. If  we then pass both these signals through  Digital to
> Analog converters (DAC) and convert both signals to analog and then perform
> convolution in continuous time will the output be the same?
>
> In other words what is the relation between a discrete time and continuous
> time convolution of two signals?
>
> Could someone please explain this? Thanks in advance. 

With the possible exception of a difference in delay through each
system, yes, they should be the same, if you're speaking analytically
and not practically. That is, if you mean ideal D/A conversion and ideal
analog convolution.

Linear system theory dictates that they are the same apart from a delay.
The digital system will almost certain perform with more delay than the
analog system since the analog system operates with theoretically
infinite bandwidth (even if the input and the impulse response are
band-limited).
-- 
Randy Yates
Digital Signal Labs
http://www.digitalsignallabs.com

On Friday, March 13, 2015 at 5:49:02 AM UTC+13, Acomp26 wrote:
> Hello,
> 
> I have two digital signals that have to be convolved. Lets say we do that
> and get an output. If  we then pass both these signals through  Digital to
> Analog converters (DAC) and convert both signals to analog and then perform
> convolution in continuous time will the output be the same?
> 
> In other words what is the relation between a discrete time and continuous
> time convolution of two signals?
> 
> Could someone please explain this? Thanks in advance. 
> 
> 	 
> 
> _____________________________		
> Posted through www.DSPRelated.com

Yes of course, in an ideal world. Your D/As would need filtering at the output and that's not ideal of course. (they have a zero-order hold included). The digital theory of linear TI systems is a good approximation to the analogue case and it gets better as the sampling frequency increases.