Time Reversal and Frequency Response - paper

Hello All,

I recently had a reason to delve into the details of time reversal
filters, and then we recently had a post asking about just that. So I
put together a paper this morning detailing it. (I'm in the mood to
write I guess). Since some details of this are what students should be
learning, I put in baby steps so it will be easy to follow.

So here is a link to the paper: http://www.claysturner.com/dsp/timereversal.pdf

Any comments, suggestions, and criticisms accepted. I'll even take
money, but I won't hold my breath ;-)

Thanks,

Clay

On Thu, 23 Oct 2008 10:24:21 -0700 (PDT), clay@claysturner.com wrote:

>Hello All,
>
>I recently had a reason to delve into the details of time reversal
>filters, and then we recently had a post asking about just that. So I
>put together a paper this morning detailing it. (I'm in the mood to
>write I guess). Since some details of this are what students should be
>learning, I put in baby steps so it will be easy to follow.
>
>So here is a link to the paper: http://www.claysturner.com/dsp/timereversal.pdf
>
>Any comments, suggestions, and criticisms accepted. I'll even take
>money, but I won't hold my breath ;-)
>
>Thanks,
>
>Clay

Nice treatment, and I like that it's spelled out clearly and
completely. 

An aside comment is that that can be used to help better understand
the difference between convolution and correlation, since they only
really differ in the sense of the reference function being reversed or
not.

Eric Jacobsen
Minister of Algorithms
Abineau Communications
http://www.ericjacobsen.org

Blog: http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php

On Oct 23, 3:24&#4294967295;pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:
> On Thu, 23 Oct 2008 10:24:21 -0700 (PDT), c...@claysturner.com wrote:
> >Hello All,
>
> >I recently had a reason to delve into the details of time reversal
> >filters, and then we recently had a post asking about just that. So I
> >put together a paper this morning detailing it. (I'm in the mood to
> >write I guess). Since some details of this are what students should be
> >learning, I put in baby steps so it will be easy to follow.
>
> >So here is a link to the paper:http://www.claysturner.com/dsp/timereversal.pdf
>
> >Any comments, suggestions, and criticisms accepted. I'll even take
> >money, but I won't hold my breath ;-)
>
> >Thanks,
>
> >Clay
>
> Nice treatment, and I like that it's spelled out clearly and
> completely.
>
> An aside comment is that that can be used to help better understand
> the difference between convolution and correlation, since they only
> really differ in the sense of the reference function being reversed or
> not.
>
> Eric Jacobsen
> Minister of Algorithms
> Abineau Communicationshttp://www.ericjacobsen.org
>
> Blog:http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php- Hide quoted text -
>
> - Show quoted text -

Hello Eric,

Thanks for the comments. I could add sections on convolution and
correlation. I'll give it some thought.

Clay

On Oct 24, 8:16�am, c...@claysturner.com wrote:
> On Oct 23, 3:24&#4294967295;pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:
>
>
>
>
>
> > On Thu, 23 Oct 2008 10:24:21 -0700 (PDT), c...@claysturner.com wrote:
> > >Hello All,
>
> > >I recently had a reason to delve into the details of time reversal
> > >filters, and then we recently had a post asking about just that. So I
> > >put together a paper this morning detailing it. (I'm in the mood to
> > >write I guess). Since some details of this are what students should be
> > >learning, I put in baby steps so it will be easy to follow.
>
> > >So here is a link to the paper:http://www.claysturner.com/dsp/timereversal.pdf
>
> > >Any comments, suggestions, and criticisms accepted. I'll even take
> > >money, but I won't hold my breath ;-)
>
> > >Thanks,
>
> > >Clay
>
> > Nice treatment, and I like that it's spelled out clearly and
> > completely.
>
> > An aside comment is that that can be used to help better understand
> > the difference between convolution and correlation, since they only
> > really differ in the sense of the reference function being reversed or
> > not.
>
> > Eric Jacobsen
> > Minister of Algorithms
> > Abineau Communicationshttp://www.ericjacobsen.org
>
> > Blog:http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php-Hide quoted text -
>
> > - Show quoted text -
>
> Hello Eric,
>
> Thanks for the comments. I could add sections on convolution and
> correlation. I'll give it some thought.
>
> Clay- Hide quoted text -
>
> - Show quoted text -

A suggestion on more standard notation.  On the DFTs, move the 'i' in
the exponent of 'e' so that either 'i' or '-i' (if the '-' is present)
is the first term in the exponent of 'e'.

Dirk

>Hello All,
>
>I recently had a reason to delve into the details of time reversal
>filters, and then we recently had a post asking about just that. So I
>put together a paper this morning detailing it. (I'm in the mood to
>write I guess). Since some details of this are what students should be
>learning, I put in baby steps so it will be easy to follow.
>
>So here is a link to the paper:
http://www.claysturner.com/dsp/timereversal.pdf
>
>Any comments, suggestions, and criticisms accepted. I'll even take
>money, but I won't hold my breath ;-)
>
>Thanks,
>
>Clay
>
>

Clay - that is a well-written paper.  Having taken only one signals class,
it was right around my level of comprehension.  You didn't explicitly
mention some things, which forced me to reason about what is happening with
the math.  For instance, eq. 16-17 required me to notice that the exponent
of e is negated when you do the conjugate.  If I had the time, I would
probably look into the math more, as I'm not exactly sure what's going on
with the math.

Even though you go step by step, you use very short descriptions between
the steps.  This could be good (force student to do more research on their
own) or bad (lazy students won't get the most out of it).  For instance, I
also haven't seen arg() before, but a quick google shows that it's just
shorthand for something I had to do a few times many years ago...

Cheers,
Silash

On Oct 24, 10:36&#4294967295;am, "Silash" <deadca...@gmail.com> wrote:
> >Hello All,
>
> >I recently had a reason to delve into the details of time reversal
> >filters, and then we recently had a post asking about just that. So I
> >put together a paper this morning detailing it. (I'm in the mood to
> >write I guess). Since some details of this are what students should be
> >learning, I put in baby steps so it will be easy to follow.
>
> >So here is a link to the paper:
>
> http://www.claysturner.com/dsp/timereversal.pdf
>
>
>
> >Any comments, suggestions, and criticisms accepted. I'll even take
> >money, but I won't hold my breath ;-)
>
> >Thanks,
>
> >Clay
>
> Clay - that is a well-written paper. &#4294967295;Having taken only one signals class,
> it was right around my level of comprehension. &#4294967295;You didn't explicitly
> mention some things, which forced me to reason about what is happening with
> the math. &#4294967295;For instance, eq. 16-17 required me to notice that the exponent
> of e is negated when you do the conjugate. &#4294967295;If I had the time, I would
> probably look into the math more, as I'm not exactly sure what's going on
> with the math.
>
> Even though you go step by step, you use very short descriptions between
> the steps. &#4294967295;This could be good (force student to do more research on their
> own) or bad (lazy students won't get the most out of it). &#4294967295;For instance, I
> also haven't seen arg() before, but a quick google shows that it's just
> shorthand for something I had to do a few times many years ago...
>
> Cheers,
> Silash

Thanks Silash,

I was considering putting in a description of arg() so then I could be
explicit about my using its being an odd function - odd as in -f(x) =
f(-x). Since this was my 1st pass, I can add a lot more to it. Also
the question about what level of mathematics to expect a student to
have. Certainly for a signals class, one expects you have had complex
number theory. But I do get too terse, and I should be more explicit
with the text from step to step. Plus just because one has had a
class, doesn't mean he has absorbed all of what was taught.

Clay

On Oct 24, 10:31&#4294967295;am, dbell <bellda2...@cox.net> wrote:
> On Oct 24, 8:16&#4294967295;am, c...@claysturner.com wrote:
>
>
>
>
>
> > On Oct 23, 3:24&#4294967295;pm, Eric Jacobsen <eric.jacob...@ieee.org> wrote:
>
> > > On Thu, 23 Oct 2008 10:24:21 -0700 (PDT), c...@claysturner.com wrote:
> > > >Hello All,
>
> > > >I recently had a reason to delve into the details of time reversal
> > > >filters, and then we recently had a post asking about just that. So I
> > > >put together a paper this morning detailing it. (I'm in the mood to
> > > >write I guess). Since some details of this are what students should be
> > > >learning, I put in baby steps so it will be easy to follow.
>
> > > >So here is a link to the paper:http://www.claysturner.com/dsp/timereversal.pdf
>
> > > >Any comments, suggestions, and criticisms accepted. I'll even take
> > > >money, but I won't hold my breath ;-)
>
> > > >Thanks,
>
> > > >Clay
>
> > > Nice treatment, and I like that it's spelled out clearly and
> > > completely.
>
> > > An aside comment is that that can be used to help better understand
> > > the difference between convolution and correlation, since they only
> > > really differ in the sense of the reference function being reversed or
> > > not.
>
> > > Eric Jacobsen
> > > Minister of Algorithms
> > > Abineau Communicationshttp://www.ericjacobsen.org
>
> > > Blog:http://www.dsprelated.com/blogs-1/hf/Eric_Jacobsen.php-Hidequoted text -
>
> > > - Show quoted text -
>
> > Hello Eric,
>
> > Thanks for the comments. I could add sections on convolution and
> > correlation. I'll give it some thought.
>
> > Clay- Hide quoted text -
>
> > - Show quoted text -
>
> A suggestion on more standard notation. &#4294967295;On the DFTs, move the 'i' in
> the exponent of 'e' so that either 'i' or '-i' (if the '-' is present)
> is the first term in the exponent of 'e'.
>
> Dirk- Hide quoted text -
>
> - Show quoted text -

Thanks Dirk,

I have often seen 2*pi*i but I have also seen the exponent lead with
an "i" or "-i". I think the issue here is the typesetting that
squeezes the "i" inbetween the other symbols making it hard to see. So
leading with the "i" or "-i" should help. I should get an updated Tex
and LaTex program so I can set the equations properly instead of using
Microsoft's equation editor.

Thanks,

Clay

clay@claysturner.com wrote:

   ...

> I have often seen 2*pi*i but I have also seen the exponent lead with
> an "i" or "-i". I think the issue here is the typesetting that
> squeezes the "i" inbetween the other symbols making it hard to see. So
> leading with the "i" or "-i" should help. I should get an updated Tex
> and LaTex program so I can set the equations properly instead of using
> Microsoft's equation editor.

The dot over the 'i' (aka tittle) is supposed to make the letter stand 
out better. A hook underneath it -- 'j' -- is more effective yet. :-)

I think the best reason to put the i (or j) first is to get it next to 
the sign. After all -sin(wt) can indicate a subtraction as - sin(wt), 
negative frequency as sin(-wt), negative time as sin(w*-t), or an 
inverted waveform. It makes no difference to the math.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

clay@claysturner.com wrote:
> Hello All,
>
> I recently had a reason to delve into the details of time reversal
> filters, and then we recently had a post asking about just that. So I
> put together a paper this morning detailing it. (I'm in the mood to
> write I guess). Since some details of this are what students should be
> learning, I put in baby steps so it will be easy to follow.
>
> So here is a link to the paper:
> http://www.claysturner.com/dsp/timereversal.pdf
>
> Any comments, suggestions, and criticisms accepted. I'll even take
> money, but I won't hold my breath ;-)

!pictures!  would help.

Fred 

On Thu, 23 Oct 2008 10:24:21 -0700 (PDT), clay@claysturner.com wrote:

>Hello All,
>
>I recently had a reason to delve into the details of time reversal
>filters, and then we recently had a post asking about just that. So I
>put together a paper this morning detailing it. (I'm in the mood to
>write I guess). Since some details of this are what students should be
>learning, I put in baby steps so it will be easy to follow.
>
>So here is a link to the paper: http://www.claysturner.com/dsp/timereversal.pdf
>
>Any comments, suggestions, and criticisms accepted. I'll even take
>money, but I won't hold my breath ;-)
>
>Thanks,
>
>Clay

Hi Clay,
  I like papers like the one you wrote.  They 
help reinforce my understanding of DSP.

Your time reversal, as far as I can tell, is a 
straightforward "flipping" of the order of a 
time sequence.  In the 3rd edition of the 
Proakis & Manolakis DSP book, they discuss 
(pages 411 & 425) what they call a "new definition" 
of "time reversal" that they claim is necessary 
when dealing with discrete sequences.  

For example, if an original time sequence is:


    x(0),x(1),x(2),x(3),x(4),x(5),x(6),x(7)

your time reversal is:

    x(7),x(6),x(5),x(4),x(3),x(2),x(1),x(0).

However Proakis & Manolakis' time reversal is: 

    x(0),x(7),x(6),x(5),x(4),x(3),x(2),x(1).

Notice that P&M's first sample is the same in both 
the original and their time-reversed sequences.
The DFT of P&M's time-reversed sequence is the 
conjugate of the DFT of the original sequence.

Now if you write the original sequence repetitively 
on a piece of paper, you'll see that your 
time-reversed sequence is merely a shifted-by-one-sample
version of P&M's time-reversed sequence.  By the 
DFT Shifting Theorem, that should make your 
time-reversed sequence's DFT differ from P&M's 
time-reversed sequence's DFT by a phase factor 
of exp(2*pi*k/N).  Which is your Equation (19).
So what I'm mumblin' about here is that I think 
your results are correct.  :-)

Nice Job Clay.
[-Rick-]