Powered by Invision Power Board


Forum Rules Forum Rules (Please read before posting)
  Reply to this topicStart new topicStart Poll

> F{sinc(t-a) + Sinc(t+a)}, Inconsistency
apalopohapa
Posted: July 15, 2009 12:34 am
Reply to this postQuote Post


Newbie
*

Group: Members+
Posts: 9
Member No.: 5,481
Joined: June 24, 2006




Hello. F{sinc(t)} = rect(w).

Since F{f(t-a)} = F(w)*exp(-j*w*a),
if follows that F{sinc(t-a)} = rect(w)*exp(-j*w*a)
which looks in the magnitude plot exacly before, but the phase plot is a positively inclined straight line (phase increases linearly with w).

In the same way, F{sinc(t+a)} = rect(w)*exp(j*w*a)
which looks in the magnitude plot exacly before, but the phase plot is a negatively inclined straight line (phase decreases linearly with w).

Here's my dilemma,
If I define f(t) = sinc(t-a) + sinc(t+a),
then F(w) would be 2*rect(w), and the positively inclined phase cancels the negatively inclined phase plot. But the inverse of 2*rect(w) is simply 2*sinc(t), which is not the f(t) that I started with.

Where is the mistake?

Thanks.

PMEmail Poster
Top
Sch3mat1c
Posted: July 15, 2009 03:25 am
Reply to this postQuote Post


Forum Addict ++
Group Icon

Group: Moderators
Posts: 20,541
Member No.: 73
Joined: July 24, 2002




Have you just proven, in an exceedingly roundabout way, that sinc(t - a) + sinc(t + a) = 2 * sinc(t)? That is, is there an identity that you've missed?

Tim


--------------------
Answering questions is a tricky subject to practice. Not due to the difficulty of formulating or locating answers, but due to the human inability of asking the right questions; a skill that, were one to possess, would put them in the "answering" category.
PMEmail PosterUsers Website
Top
cdstahl
Posted: July 17, 2009 04:30 am
Reply to this postQuote Post


Forum Addict ++
*******

Group: Trusted Members
Posts: 1,009
Member No.: 11,808
Joined: November 04, 2007




e^jw = cosw +jsinw

edit -- also:
e^-jw = cosw -jsinw
PMEmail Poster
Top
apalopohapa
Posted: October 03, 2012 11:42 am
Reply to this postQuote Post


Newbie
*

Group: Members+
Posts: 9
Member No.: 5,481
Joined: June 24, 2006




To answer my own question, the phase parts don't just 'cancel' each other out. That is like saying that a vector of angle a added with another of same magnitud but angle -a produces a vector with angle zero but twice the magnitude. The phase is in fact zero, but the magnitude will end up being a cosine.

This post has been edited by apalopohapa on October 03, 2012 07:03 pm
PMEmail Poster
Top
Sch3mat1c
Posted: October 03, 2012 04:29 pm
Reply to this postQuote Post


Forum Addict ++
Group Icon

Group: Moderators
Posts: 20,541
Member No.: 73
Joined: July 24, 2002




On closer inspection, just do the algebra: Fourier of the sum of the offset functions is rect(w) * (e^jwa + e^-jwa). That phase factor is (cos wa + j sin wa + cos wa - j sin wa), the sines cancel and the cosines add, yielding (2 cos wa), which is not just 2. The phase parts don't cancel (the phase slopes would cancel on multiplication, but not addition), they interfere. So the phase ends up real while the amplitude oscillates priodically with w.

Geez, this was three years ago already wacko.gif smile.gif

Tim


--------------------
Answering questions is a tricky subject to practice. Not due to the difficulty of formulating or locating answers, but due to the human inability of asking the right questions; a skill that, were one to possess, would put them in the "answering" category.
PMEmail PosterUsers Website
Top
0 User(s) are reading this topic (0 Guests and 0 Anonymous Users)
0 Members:

Topic Options Reply to this topicStart new topicStart Poll

 


:: support us ::




ElectronicsSkin by DutchDork & The-Force