I'm working on an instructional video, and while I don't mind tossing out
some hand-waving, and even some minor inaccuracies, I don't want to get
things ENTIRELY wrong.
First question:
Quite a while ago, quite by accident, I discovered that one could solve
linear shift-invariant difference equations via "the hard way" in much
the same way that one solved linear differential equations.
In other words, given
y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k)
then one could solve the thing by first finding the homogeneous solution
(arrived at by setting f(k) = 0), and then finding the homogeneous
solution (assuming that f(k) was, by chance or design, equal to something
"easy", i.e. some sum of A * k^n * d^m).
All of this was done by direct analogy to how it's done with differential
equations.
Does anyone have any references to this? It can't be entirely my own
invention. Even a name that coughs up results in a Google search would
help.
Second question:
When you come up with the solution to a non-homogeneous solution (f(k) !=
0, above), the the range of functions that f(k) can be to keep the
problem "easy" is constrained to functions that have "easy" z transforms
(i.e., some limited sum of A * k^n * d^m). Is there a theorem associated
with that? Again, references or a name I can Google (really, a name that
I can set my viewers to Googling) would be cool.
If I don't get any answers in a day or a few I'm just going to hand-wave,
and leave people puzzled. But I'd much rather be able to point them in a
fruitful direction.
--
Tim Wescott
Wescott Design Services
http://www.wescottdesign.com
I'm looking for work -- see my website!
Two questions about differential and difference equations
Started by ●October 3, 2016
Reply by ●October 3, 20162016-10-03
"Tim Wescott" wrote in message
news:xvadnRfVx_UMc2_KnZ2dnUU7-RmdnZ2d@giganews.com...
I'm working on an instructional video, and while I don't mind tossing out
some hand-waving, and even some minor inaccuracies, I don't want to get
things ENTIRELY wrong.
First question:
Quite a while ago, quite by accident, I discovered that one could solve
linear shift-invariant difference equations via "the hard way" in much
the same way that one solved linear differential equations.
In other words, given
y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k)
then one could solve the thing by first finding the homogeneous solution
(arrived at by setting f(k) = 0), and then finding the homogeneous
solution (assuming that f(k) was, by chance or design, equal to something
"easy", i.e. some sum of A * k^n * d^m).
All of this was done by direct analogy to how it's done with differential
equations.
Does anyone have any references to this? It can't be entirely my own
invention. Even a name that coughs up results in a Google search would
help.
Second question:
When you come up with the solution to a non-homogeneous solution (f(k) !=
0, above), the the range of functions that f(k) can be to keep the
problem "easy" is constrained to functions that have "easy" z transforms
(i.e., some limited sum of A * k^n * d^m). Is there a theorem associated
with that? Again, references or a name I can Google (really, a name that
I can set my viewers to Googling) would be cool.
If I don't get any answers in a day or a few I'm just going to hand-wave,
and leave people puzzled. But I'd much rather be able to point them in a
fruitful direction.
================================================================
Way, way back (35 yrs, sigh) I used the book "Continuous and Discrete Signal
and System Analysis" by McGillem and Cooper in my signals and systems
course. In the chapter on z transforms there is a little section on solving
difference equations with z transforms that definitely covers your first
question. Sorry, but it's been long enough since I looked at this that I'm
not going to think hard enough to make sure they cover your second question,
but a casual glance looks promising :-). So, I'll bet you can find it in
any standard sig and sys text.
-----
Regards,
Carl Ijames
Reply by ●October 3, 20162016-10-03
On 10/3/2016 7:49 PM, Tim Wescott wrote:> I'm working on an instructional video, and while I don't mind tossing out > some hand-waving, and even some minor inaccuracies, I don't want to get > things ENTIRELY wrong. > > First question: > > Quite a while ago, quite by accident, I discovered that one could solve > linear shift-invariant difference equations via "the hard way" in much > the same way that one solved linear differential equations. > > In other words, given > > y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k) > > then one could solve the thing by first finding the homogeneous solution > (arrived at by setting f(k) = 0), and then finding the homogeneous > solution (assuming that f(k) was, by chance or design, equal to something > "easy", i.e. some sum of A * k^n * d^m). > > All of this was done by direct analogy to how it's done with differential > equations. > > Does anyone have any references to this? It can't be entirely my own > invention. Even a name that coughs up results in a Google search would > help.Here are two: 1. "Digital Signal Processing" by Proakis and Manolakis 2. "Numerical Methods for Scientists and Engineers" by Richard Hamming> > Second question: > > When you come up with the solution to a non-homogeneous solution (f(k) != > 0, above), the the range of functions that f(k) can be to keep the > problem "easy" is constrained to functions that have "easy" z transforms > (i.e., some limited sum of A * k^n * d^m). Is there a theorem associated > with that? Again, references or a name I can Google (really, a name that > I can set my viewers to Googling) would be cool.I would say all this falls out of a z-transform analysis. The system function H(z) = Y(z)/F(z) is the solution of the homogeneous problem. The solution of the non-homogeneous problem is Y(z) = H(z)F(z) in the z-domain and y() therefore is a convolution of h(k) and f(k). This is why the solution has a factor that resembles the right-hand side.> > If I don't get any answers in a day or a few I'm just going to hand-wave, > and leave people puzzled. But I'd much rather be able to point them in a > fruitful direction. >
Reply by ●October 3, 20162016-10-03
On Mon, 03 Oct 2016 20:30:23 -0400, Michael Soyka wrote:> On 10/3/2016 7:49 PM, Tim Wescott wrote: >> I'm working on an instructional video, and while I don't mind tossing >> out some hand-waving, and even some minor inaccuracies, I don't want to >> get things ENTIRELY wrong. >> >> First question: >> >> Quite a while ago, quite by accident, I discovered that one could solve >> linear shift-invariant difference equations via "the hard way" in much >> the same way that one solved linear differential equations. >> >> In other words, given >> >> y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k) >> >> then one could solve the thing by first finding the homogeneous >> solution (arrived at by setting f(k) = 0), and then finding the >> homogeneous solution (assuming that f(k) was, by chance or design, >> equal to something "easy", i.e. some sum of A * k^n * d^m). >> >> All of this was done by direct analogy to how it's done with >> differential equations. >> >> Does anyone have any references to this? It can't be entirely my own >> invention. Even a name that coughs up results in a Google search would >> help. > > Here are two: > 1. "Digital Signal Processing" by Proakis and Manolakis 2. "Numerical > Methods for Scientists and Engineers" by Richard Hamming > > >> Second question: >> >> When you come up with the solution to a non-homogeneous solution (f(k) >> != >> 0, above), the the range of functions that f(k) can be to keep the >> problem "easy" is constrained to functions that have "easy" z >> transforms (i.e., some limited sum of A * k^n * d^m). Is there a >> theorem associated with that? Again, references or a name I can Google >> (really, a name that I can set my viewers to Googling) would be cool. > > I would say all this falls out of a z-transform analysis. The system > function H(z) = Y(z)/F(z) is the solution of the homogeneous problem. > The solution of the non-homogeneous problem is Y(z) = H(z)F(z) in the > z-domain and y() therefore is a convolution of h(k) and f(k). This is > why the solution has a factor that resembles the right-hand side. > > >> If I don't get any answers in a day or a few I'm just going to >> hand-wave, >> and leave people puzzled. But I'd much rather be able to point them in >> a fruitful direction. >>I'm trying to do this _without_ introducing the z transform. It's a 15- minute talk; not 15 hours. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com I'm looking for work -- see my website!
Reply by ●October 4, 20162016-10-04
[...snip...]> >Does anyone have any references to this? It can't be entirely my own >invention. Even a name that coughs up results in a Google search would >help. >[...snip...] In the paragraph following your first equations, I think when you wrote "and then finding the homogeneous" you should have written "and then finding a particular". If you type "homogenous linear differ" into the Google(tm). you will get autocompletion options for both "differential" and "difference". Suppose "h" is your homogenous solution, and "p", is your particular solution, then any constant times "h" plus your particular solution "p" will also be a solution. This is due to the linearity of the differential operator and that differentials are also linear. Keep in mind that "h" can be a linear combination of solutions. Let the general solution be "g". g = c*h + p For the differential case: g' = c*h' + p' g" = c*h" + p" etc. Plug this into your original equation: Original Equation in g = RHS (right hand side) Separate the solutions: c*(Original Equation in h) + (Original Equation in p) = RHS c*0 + RHS = RHS For difference equations: g_{k} = c*h_{k} + p_{k} g_{k-1} = c*h_{k-1} + p_{k-1} g_{k-2} = c*h_{k-2} + p_{k-2} etc. Therefore, when you plug them in they separate just like the differential case. I don't think this concept has a special name. Hope this helps. Ced --------------------------------------- Posted through http://www.DSPRelated.com
Reply by ●October 4, 20162016-10-04
On Tuesday, October 4, 2016 at 12:49:44 PM UTC+13, Tim Wescott wrote:> I'm working on an instructional video, and while I don't mind tossing out > some hand-waving, and even some minor inaccuracies, I don't want to get > things ENTIRELY wrong. > > First question: > > Quite a while ago, quite by accident, I discovered that one could solve > linear shift-invariant difference equations via "the hard way" in much > the same way that one solved linear differential equations. > > In other words, given > > y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k) > > then one could solve the thing by first finding the homogeneous solution > (arrived at by setting f(k) = 0), and then finding the homogeneous > solution (assuming that f(k) was, by chance or design, equal to something > "easy", i.e. some sum of A * k^n * d^m). > > All of this was done by direct analogy to how it's done with differential > equations. > > Does anyone have any references to this? It can't be entirely my own > invention. Even a name that coughs up results in a Google search would > help. > > Second question: > > When you come up with the solution to a non-homogeneous solution (f(k) != > 0, above), the the range of functions that f(k) can be to keep the > problem "easy" is constrained to functions that have "easy" z transforms > (i.e., some limited sum of A * k^n * d^m). Is there a theorem associated > with that? Again, references or a name I can Google (really, a name that > I can set my viewers to Googling) would be cool. > > If I don't get any answers in a day or a few I'm just going to hand-wave, > and leave people puzzled. But I'd much rather be able to point them in a > fruitful direction. > > -- > > Tim Wescott > Wescott Design Services > http://www.wescottdesign.com > > I'm looking for work -- see my website!Probably the best book on difference equations is Introduction to difference equations Elaydi
Reply by ●October 4, 20162016-10-04
On Tue, 04 Oct 2016 12:49:10 -0700, gyansorova wrote:> On Tuesday, October 4, 2016 at 12:49:44 PM UTC+13, Tim Wescott wrote: >> I'm working on an instructional video, and while I don't mind tossing >> out some hand-waving, and even some minor inaccuracies, I don't want to >> get things ENTIRELY wrong. >> >> First question: >> >> Quite a while ago, quite by accident, I discovered that one could solve >> linear shift-invariant difference equations via "the hard way" in much >> the same way that one solved linear differential equations. >> >> In other words, given >> >> y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k) >> >> then one could solve the thing by first finding the homogeneous >> solution (arrived at by setting f(k) = 0), and then finding the >> homogeneous solution (assuming that f(k) was, by chance or design, >> equal to something "easy", i.e. some sum of A * k^n * d^m). >> >> All of this was done by direct analogy to how it's done with >> differential equations. >> >> Does anyone have any references to this? It can't be entirely my own >> invention. Even a name that coughs up results in a Google search would >> help. >> >> Second question: >> >> When you come up with the solution to a non-homogeneous solution (f(k) >> != >> 0, above), the the range of functions that f(k) can be to keep the >> problem "easy" is constrained to functions that have "easy" z >> transforms (i.e., some limited sum of A * k^n * d^m). Is there a >> theorem associated with that? Again, references or a name I can Google >> (really, a name that I can set my viewers to Googling) would be cool. >> >> If I don't get any answers in a day or a few I'm just going to >> hand-wave, >> and leave people puzzled. But I'd much rather be able to point them in >> a fruitful direction. >> >> -- >> >> Tim Wescott Wescott Design Services http://www.wescottdesign.com >> >> I'm looking for work -- see my website! > > Probably the best book on difference equations is > > Introduction to difference equations ElaydiHmm. Thanks. Amazon reviews are positive; it's going on my wish list at least. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com I'm looking for work -- see my website!
Reply by ●October 4, 20162016-10-04
On Wednesday, October 5, 2016 at 11:06:31 AM UTC+13, Tim Wescott wrote:> On Tue, 04 Oct 2016 12:49:10 -0700, gyansorova wrote: > > > On Tuesday, October 4, 2016 at 12:49:44 PM UTC+13, Tim Wescott wrote: > >> I'm working on an instructional video, and while I don't mind tossing > >> out some hand-waving, and even some minor inaccuracies, I don't want to > >> get things ENTIRELY wrong. > >> > >> First question: > >> > >> Quite a while ago, quite by accident, I discovered that one could solve > >> linear shift-invariant difference equations via "the hard way" in much > >> the same way that one solved linear differential equations. > >> > >> In other words, given > >> > >> y_k + a_1 * y_{k-1} + a_0 * y_{k-2} = f(k) > >> > >> then one could solve the thing by first finding the homogeneous > >> solution (arrived at by setting f(k) = 0), and then finding the > >> homogeneous solution (assuming that f(k) was, by chance or design, > >> equal to something "easy", i.e. some sum of A * k^n * d^m). > >> > >> All of this was done by direct analogy to how it's done with > >> differential equations. > >> > >> Does anyone have any references to this? It can't be entirely my own > >> invention. Even a name that coughs up results in a Google search would > >> help. > >> > >> Second question: > >> > >> When you come up with the solution to a non-homogeneous solution (f(k) > >> != > >> 0, above), the the range of functions that f(k) can be to keep the > >> problem "easy" is constrained to functions that have "easy" z > >> transforms (i.e., some limited sum of A * k^n * d^m). Is there a > >> theorem associated with that? Again, references or a name I can Google > >> (really, a name that I can set my viewers to Googling) would be cool. > >> > >> If I don't get any answers in a day or a few I'm just going to > >> hand-wave, > >> and leave people puzzled. But I'd much rather be able to point them in > >> a fruitful direction. > >> > >> -- > >> > >> Tim Wescott Wescott Design Services http://www.wescottdesign.com > >> > >> I'm looking for work -- see my website! > > > > Probably the best book on difference equations is > > > > Introduction to difference equations Elaydi > > Hmm. Thanks. Amazon reviews are positive; it's going on my wish list at > least. > > -- > > Tim Wescott > Wescott Design Services > http://www.wescottdesign.com > > I'm looking for work -- see my website!a quick google and the whole PDf is there. I never said that though.
