Hello All,
When we were deriving ideal HPF equation we came across
interesting definition of impulse,
delta[n] = sin(pi*n)/(pi*n)
Maybe I am getting to understand the sampling theorem....
Regards
Bharat Pathak
Arithos Designs
www.Arithos.com
interesting insight into impulse definition
Started by ●March 11, 2008
Reply by ●March 11, 20082008-03-11
bharat pathak wrote:> Hello All, > > When we were deriving ideal HPF equation we came across > interesting definition of impulse, > > delta[n] = sin(pi*n)/(pi*n) > > Maybe I am getting to understand the sampling theorem.... > > Regards > Bharat Pathak > > Arithos Designs > www.Arithos.com >That expression is indeterminate. The expression delta[n] = lim sin(pi * n + e) / (pi * n + e) e -> 0 works. Or, if you use sinc(x) = lim sin(x0) / x0 x0 -> x then delta[n] = sinc(pi * n) works also. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●March 11, 20082008-03-11
On Mar 11, 1:40 pm, Tim Wescott <t...@seemywebsite.com> wrote:> bharat pathak wrote: > > Hello All, > > > When we were deriving ideal HPF equation we came across > > interesting definition of impulse, > > > delta[n] = sin(pi*n)/(pi*n)of course, but with the "removable" singularity explicitly defined as delta[0]=1.> > That expression is indeterminate.not if you fix the removable singularity. it says that delta[0]=1 and for all other integer n, then delta[n]=0.> The expression > > delta[n] = lim sin(pi * n + e) / (pi * n + e) > e -> 0 > > works.okay, Tim, we're on the same page now. (another case where i don't read carefully through the post before starting a response.)> Or, if you use > > sinc(x) = lim sin(x0) / x0 > x0 -> x > > then delta[n] = sinc(pi * n) > > works also.that's probably the cleanest way to say it. r b-j
Reply by ●March 11, 20082008-03-11
On Mar 11, 1:20�pm, "bharat pathak" <bha...@arithos.com> wrote:> � � � delta[n] = sin(pi*n)/(pi*n)Maybe I'm reading this too quickly, but I recall all definitions of delta stating that the amplitude is infinite, with unit area under the curve. Doesn't L'Hospital's Rule give an amplitude of one in the expression above? Greg
Reply by ●March 11, 20082008-03-11
On Mar 11, 10:20 am, "bharat pathak" <bha...@arithos.com> wrote:> Hello All,> When we were deriving ideal HPF equation we came across > interesting definition of impulse, > > delta[n] = sin(pi*n)/(pi*n) > > Maybe I am getting to understand the sampling theorem.... > > Regards > Bharat Pathak > > Arithos Designswww.Arithos.comThat equation is not a definition of the delta functional. It is a relationship that is true of the delta functional acting on the set of integers as described in common engineering language. In engineering it is common practice to substitute the values of the delta functional acting on the real number line for the delta functional itself. See: http://mathworld.wolfram.com/DeltaFunction.html There is further confusion added by the use of the expression 'sampling function' for the sync function: http://mathworld.wolfram.com/SincFunction.html Dale B. Dalrymple http://dbdimages.com
Reply by ●March 11, 20082008-03-11
Greg Berchin wrote:> On Mar 11, 1:20 pm, "bharat pathak" <bha...@arithos.com> wrote: > >> delta[n] = sin(pi*n)/(pi*n) > > Maybe I'm reading this too quickly, but I recall all definitions of > delta stating that the amplitude is infinite, with unit area under the > curve. Doesn't L'Hospital's Rule give an amplitude of one in the > expression above? > > GregThe Dirac delta* has an infinite amplitude and an area of one. It's very handy for continuous-time analysis. The Kroneker (SP?) delta is only defined for integer 'times', it has an amplitude of one for sample zero, and is zero everywhere else. I certainly hope the OP meant the Kroneker delta, because the above function (with all due respect for the problem as n approaches zero) is just a sinc for continuous n. * An engineer will call it a "function" and scold you for being too picky if you call it a "functional". A mathematician will call it a "functional" and scold you for being too lax if you call it a function. I'll just avoid either name... -- Tim Wescott Wescott Design Services http://www.wescottdesign.com Do you need to implement control loops in software? "Applied Control Theory for Embedded Systems" gives you just what it says. See details at http://www.wescottdesign.com/actfes/actfes.html
Reply by ●March 11, 20082008-03-11
On Tue, 11 Mar 2008 14:41:18 -0800, Tim Wescott <tim@seemywebsite.com> wrote:>I certainly hope the OP meant the Kroneker delta, because the above >function (with all due respect for the problem as n approaches zero) is >just a sinc for continuous n.Exactly the point that I was trying to make. I, however, interpreted "delta" to mean "Dirac delta"; hence my confusion.>* An engineer will call it a "function" and scold you for being too >picky if you call it a "functional". A mathematician will call it a >"functional" and scold you for being too lax if you call it a function. > I'll just avoid either name...Yeah, no sense grabbing that tiger by the tail again. Greg
Reply by ●March 11, 20082008-03-11
On Mar 11, 8:15 pm, Greg Berchin <gberc...@comicast.net> wrote:> On Tue, 11 Mar 2008 14:41:18 -0800, Tim Wescott <t...@seemywebsite.com> > wrote: > > >I certainly hope the OP meant the Kroneker delta, because the above > >function (with all due respect for the problem as n approaches zero) is > >just a sinc for continuous n. > > Exactly the point that I was trying to make. I, however, interpreted > "delta" to mean "Dirac delta"; hence my confusion. > > >* An engineer will call it a "function" and scold you for being too > >picky if you call it a "functional". A mathematician will call it a > >"functional" and scold you for being too lax if you call it a function. > > I'll just avoid either name... > > Yeah, no sense grabbing that tiger by the tail again.why not? tigers are cute and fuzzy and warm and cuddly. especially when they lay waste to everything they taught up in our continuous- time signals and systems course. maybe i'm a glutton for punishment. r b-j
Reply by ●March 12, 20082008-03-12
On 11 Mar, 18:40, Tim Wescott <t...@seemywebsite.com> wrote:> bharat pathak wrote: > > Hello All, > > > � � � When we were deriving ideal HPF equation we came across > > � � � interesting definition of impulse, > > > � � � delta[n] = sin(pi*n)/(pi*n) > > > � � � Maybe I am getting to understand the sampling theorem.... > > > Regards > > Bharat Pathak > > > Arithos Designs > >www.Arithos.com > > That expression is indeterminate. �The expression > > delta[n] = �lim � sin(pi * n + e) / (pi * n + e) > � � � � � � e -> 0 > > works. > > Or, if you use > > sinc(x) = � lim � sin(x0) / x0 > � � � � � �x0 -> x > > then delta[n] = sinc(pi * n) > > works also.Or you could use the 'box' function (view with fixed/width font) | 1/T -T/2 <= x <= T/2 Delta(x) = lim | T->0| 0 elsewhere As you can see, there onr takes a limit in all these expressions for Delta(x). This limit is the key, once you grasp the significance of that you gain some insight. To give you the big picure, there is a mathematical term 'function space.' There are lots of such spaces, one of them is 'the collection of functions f(x) which has a finite integral.' Usually, these definitions specfiy the functions that belong in a bit more detail, but leave that for now. The box function is easily seen to have a finite integral, T/2 T/2 integral 1/T dx = [x/T] = 1 -T/2 -T/2 so it belongs to this space of integrable functions. Then there is the distinction between closed and open spaces. In a 'closed' linear space, all Cauchy points also belong to the space. A Cauchy point is the limit of a converging series, e.g. Delta(x) is the Cauchy points of all the limits above. Now, this space I am talking about, L2, is closed. That can be proved without using the Delta(x). Since all these sequences which belong in L2 converge to Delta(x) and L2 is a closed space, it follows that Delta(x) belongs to L2 and therefore can be used as any other function in L2. That's the maths part of it. The technical details are involved, but the big picture is nice and tidy. It seems that lots of problem occur when people try to assign 'physical' meaning to this (I prefer the term 'intuitive' rather than 'physical'). In that sense you are approaching a minefield, so handle with care! Rune
Reply by ●March 12, 20082008-03-12
robert bristow-johnson wrote:> On Mar 11, 8:15 pm, Greg Berchin <gberc...@comicast.net> wrote: >> On Tue, 11 Mar 2008 14:41:18 -0800, Tim Wescott <t...@seemywebsite.com> >> wrote: >> >>> I certainly hope the OP meant the Kroneker delta, because the above >>> function (with all due respect for the problem as n approaches zero) is >>> just a sinc for continuous n. >> Exactly the point that I was trying to make. I, however, interpreted >> "delta" to mean "Dirac delta"; hence my confusion. >> >>> * An engineer will call it a "function" and scold you for being too >>> picky if you call it a "functional". A mathematician will call it a >>> "functional" and scold you for being too lax if you call it a function. >>> I'll just avoid either name... >> Yeah, no sense grabbing that tiger by the tail again. > > why not? tigers are cute and fuzzy and warm and cuddly. especially > when they lay waste to everything they taught up in our continuous- > time signals and systems course. > > maybe i'm a glutton for punishment.I puzzles me why people who have unpopular views about sampled systems can't be a little bit more discrete about them. :-\ Steve
