Convergence properties of FT and LT [Archive]

Mehmet Baran

Sep19-04, 06:56 AM

robert bristow-johnson

Sep20-04, 03:38 AM

<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>\nin article 85774fb8.0409171255.6e01aebd@posting.google.com, Mehmet Baran at\nbaranm@kablonet.com.tr wrote on 09/19/2004 07:56:\n\n>\n> Hi,\n>\n> FT is not a special case of LT.\n\nof course it is. at least the FT is a special case of the bilateral LT.\nwhatever you can say about the FT you can also say about the bilateral LT\nwhen sigma is zero because the two are identical (s = sigma + i*omega).\n\nthe single-sided LT can be compared to the FT with the restriction that the\nargument function is what we electrical engineers call a "causal function"\n(which is a signal that is zero for all t<0, causal impulse responses might\nbe realizable in a real linear system - non-causal impulse responses of\ncourse have some ability to see into the future and are not realizable).\n\n> The two transformations have quite\n> different convergence properties\n\ndifferentiation -> s\nintegration -> 1/s\ndelay -> exp(-s*T)\n\n> and applications. Very roughly:\n>\n> 1)If a function f(x) is bounded above and below by a constant it has\n> FT.\n>\n> 2)If a function f(x) grows exponentially towards t->+inf and decays\n> exponentially for t->-inf it has LT.\n\nbut for different values of sigma.\n\n>\n> Hence, 2+sin(x), x in (-inf,inf) will have a FT (in terms of\n> distributions) but no LT (it doesnt decay as x->-inf). e^x will have a\n> LT but no FT (it is not bounded).\n>\n> I believe this point is very badly treated in all the available\n> textbooks.\n\ni might agree that textbooks often do not do a good job of explaining these\nfundamental concepts in a linear systems course (physikers and math guys\nmight call it a "linear algebra" course), but i still maintain that the\none-sided LT and the bilateral FT (the normal FT) are both degenerate cases\nof the general bilateral LT. i think it\'s pretty obvious (but i\'ve gotten\ninto debates here about if you can say alpha=e^2/(h_bar*c) at the same time\nas saying epsilon0=1, which i think is obvious that you cannot do).\n\nr b-j\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form">  View this Usenet post in original ASCII form </a></div><P></jabberwocky>in article 85774fb8.0409171255.6e01aebd@posting.google.com, Mehmet Baran at
baranm@kablonet.com.tr wrote on 09/19/2004 07:56:

>
> Hi,
>
> FT is not a special case of LT.

of course it is. at least the FT is a special case of the bilateral LT.
whatever you can say about the FT you can also say about the bilateral LT
when \sigma is zero because the two are identical (s = \sigma + i*\omega).

the single-sided LT can be compared to the FT with the restriction that the
argument function is what we electrical engineers call a "causal function"
(which is a signal that is zero for all t<0, causal impulse responses might
be realizable in a real linear system - non-causal impulse responses of
course have some ability to see into the future and are not realizable).

> The two transformations have quite
> different convergence properties

differentiation -> s
integration -> 1/s
delay -> \exp(-s*T)

> and applications. Very roughly:
>
> 1)If a function f(x) is bounded above and below by a constant it has
> FT.
>
> 2)If a function f(x) grows exponentially towards t->+inf and decays
> exponentially for t->-inf it has LT.

but for different values of \sigma.

>
> Hence, 2+sin(x), x in (-inf,inf) will have a FT (in terms of
> distributions) but no LT (it doesnt decay as x->-inf). e^x will have a
> LT but no FT (it is not bounded).
>
> I believe this point is very badly treated in all the available
> textbooks.

i might agree that textbooks often do not do a good job of explaining these
fundamental concepts in a linear systems course (physikers and math guys
might call it a "linear algebra" course), but i still maintain that the
one-sided LT and the bilateral FT (the normal FT) are both degenerate cases
of the general bilateral LT. i think it's pretty obvious (but i've gotten
into debates here about if you can say \alpha=e^2/(h_{bar}*c) at the same time
as saying epsilon0=1, which i think is obvious that you cannot do).

r b-j