On Oct 30, 8:29�am, Rick Lyons <R.Lyons@_BOGUS_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 > > 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-]- Hide quoted text - > > - Show quoted text -Interesting! I don't have that Proakis book. I'm glad you found another way to verify my result. It seems to work both in terms of symbolic math and numerically. The periodicity of the DFT shows up in a lot of places! Thanks for reading. Clay
Time Reversal and Frequency Response - paper
Started by ●October 23, 2008
Reply by ●October 30, 20082008-10-30
Reply by ●October 30, 20082008-10-30
On Oct 24, 7:43�pm, "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote:> 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 ;-) > > !pictures! �would help. > > FredHello Fred, I agree some drawings would be helpful. I'll work on some. Thanks for reading. Clay
Reply by ●October 30, 20082008-10-30
On Oct 24, 11:36�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. �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, > SilashSilash, I hope you figured out why the conjugation equates to negation in a complex exponential. From Euler we have: exp(i*theta) = cos(theta) + i*sin(theta) and exp(-i*theta) = cos(theta) - i*sin(theta) The 2nd follows from the 1st via the parity relations for the trig functions recall sin() is odd -> sin(-x) = -sin(x) and cos() is even -> cos(-x) = cos(x) So now looking at the 1st two expressions you should see conj(exp(i*theta)) = exp(-i*theta) IHTH, Clay
Reply by ●October 30, 20082008-10-30
On Oct 30, 7:29�am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote:> 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. �:-)Perhaps another way of looking at it is that Proakis and Manolakis are doing something that is analogous to the continuous-time version: y(t) = x(-t). In terms of samples, this becomes y[n] = x[-n] which leaves x[0] in place. If the sample sequence is periodic with period 8, then x[-1] (which is going to become y[1]) is the same as x[7], x[-2] (which is going to become y[2]) is the same as x[6], etc. leading to the observation that the the time-reversed sequence of original time sequence> > x(0),x(1),x(2),x(3),x(4),x(5),x(6),x(7)is Proakis & Manolakis' time reversal:> > x(0),x(7),x(6),x(5),x(4),x(3),x(2),x(1).Put another way, why choose> x(0),x(1),x(2),x(3),x(4),x(5),x(6),x(7)as the *only* possible period to "flip over" to get "Clay's flip"> x(7),x(6),x(5),x(4),x(3),x(2),x(1),x(0)??It is equally legitimate to "flip over" the period x(1),x(2),x(3),x(4),x(5),x(6),x(7),x(8) (remember that x(8) = x(0)) to get "P&M's flip" x(8),x(7),x(6),x(5),x(4),x(3),x(2),x(1) Ah, where is MATLAB when I need it? Just to add to the confusion..... --Dilip Sarwate
Reply by ●October 30, 20082008-10-30
On Thu, 30 Oct 2008 09:44:35 -0700 (PDT), "dvsarwate@yahoo.com" <dvsarwate@gmail.com> wrote:>On Oct 30, 7:29�am, Rick Lyons <R.Lyons@_BOGUS_ieee.org> wrote: > >> 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. �:-) > > >Perhaps another way of looking at it is that Proakis >and Manolakis are doing something that is analogous >to the continuous-time version: y(t) = x(-t). In terms >of samples, this becomes y[n] = x[-n] which leaves >x[0] in place. If the sample sequence is periodic with >period 8, then x[-1] (which is going to become y[1]) >is the same as x[7], x[-2] (which is going to become >y[2]) is the same as x[6], etc. leading to the >observation that the the time-reversed sequence >of original time sequence >> >> x(0),x(1),x(2),x(3),x(4),x(5),x(6),x(7) > >is Proakis & Manolakis' time reversal: > >> >> x(0),x(7),x(6),x(5),x(4),x(3),x(2),x(1). > >Put another way, why choose > >> x(0),x(1),x(2),x(3),x(4),x(5),x(6),x(7) > >as the *only* possible period to "flip over" to get >"Clay's flip" > >> x(7),x(6),x(5),x(4),x(3),x(2),x(1),x(0)?? > >It is equally legitimate to "flip over" the period > >x(1),x(2),x(3),x(4),x(5),x(6),x(7),x(8) > >(remember that x(8) = x(0)) to get "P&M's flip" > >x(8),x(7),x(6),x(5),x(4),x(3),x(2),x(1) > >Ah, where is MATLAB when I need it? > >Just to add to the confusion..... > >--Dilip SarwateHi Dilip, Ah ha. Interesting. . This entire subject of two possible (and different) definitions of discrete-signal time reversal touches on the very "slippery" concept of the circular nature of the duality between time and frequency in the world of discrete systems. A subject that is excruciatingly difficult to describe. Regards, [-Rick-]
