In another thread, when comparing analog and digital filters, Jerry 
wrote "The behaviors of these analog elements are governed by 
continuous-time differential equations, while the behaviors of sampled 
elements are governed by discrete-time difference equations."

I don't recall seeing digital filters presented as a set of "difference 
equations". Or, have I seen it and not recognized what I was looking at?

On Mon, 26 Nov 2007 14:26:41 -0600, Richard Owlett wrote:

> In another thread, when comparing analog and digital filters, Jerry 
> wrote "The behaviors of these analog elements are governed by 
> continuous-time differential equations, while the behaviors of sampled 
> elements are governed by discrete-time difference equations."
> 
> I don't recall seeing digital filters presented as a set of "difference 
> equations". Or, have I seen it and not recognized what I was looking at?

That's how I present them in my book, and in this article:
http://www.wescottdesign.com/articles/zTransform/z-transforms.html.  It's
a pretty common way of showing them -- the reality of a filter, after all,
is it's difference equation.  A transfer function is just a convenient
mathematical model one can use to analyze a filter's behavior if it
happens to be linear.

-- 
Tim Wescott
Control systems and communications consulting
http://www.wescottdesign.com

Need to learn how to apply control theory in your embedded system?
"Applied Control Theory for Embedded Systems" by Tim Wescott
Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html

Hi Richard, 
This nomenclature is probably intended to shows how the two fields ar
analogous, but your are right, it is confusing.  Consider this simpl
example of a true difference equation: 

y(t) = k0 ( x[t] - x[t-1] )   

This can be rewritten in the more standard form: 

y(t) = a0 x[t] + a1 x[t-1]  

where k0, a0, a1 are coefficients


That point is, you could write everything in terms of true differenc
equations, it is just more straightforward to place a coefficient in fron
of each sample.  Hope this helps.
Regards,
Steve 
Steve.Smith1@SpectrumSDI.com

SteveSmith wrote:
> Hi Richard, 
> This nomenclature is probably intended to shows how the two fields are
> analogous, but your are right, it is confusing.  Consider this simple
> example of a true difference equation: 
> 
> y(t) = k0 ( x[t] - x[t-1] )   
> 
> This can be rewritten in the more standard form: 
> 
> y(t) = a0 x[t] + a1 x[t-1]  
> 
> where k0, a0, a1 are coefficients
> 
> 
> That point is, you could write everything in terms of true difference
> equations, it is just more straightforward to place a coefficient in front
> of each sample.  Hope this helps.
> Regards,
> Steve 
> Steve.Smith1@SpectrumSDI.com

And as I noted in another thread it's been over 40 years since I had the 
courses in college and never used them after I left school ;)

SteveSmith wrote:

> This nomenclature is probably intended to shows how the two fields are
> analogous, but your are right, it is confusing.  Consider this simple
> example of a true difference equation: 

> y(t) = k0 ( x[t] - x[t-1] )   

> This can be rewritten in the more standard form: 

> y(t) = a0 x[t] + a1 x[t-1]  

> where k0, a0, a1 are coefficients

> That point is, you could write everything in terms of true difference
> equations, it is just more straightforward to place a coefficient in front
> of each sample.  Hope this helps.

In the form given above, a true difference equation would have the 
coefficients add to zero.  a0 + a1 = 0.   But like differential 
equations, difference equations can have terms that are not differences.

-- glen