Folks, just an informal question: do you think that the filter with impulse response h(t) = u(t) / sqrt(t) (u(t) is unit step function) is a physically realizable system? Regards, Andor
Poll
Started by ●September 4, 2007
Reply by ●September 4, 20072007-09-04
Andor wrote:> Folks, > > just an informal question: do you think that the filter with impulse > response > > h(t) = u(t) / sqrt(t) > > (u(t) is unit step function) > > is a physically realizable system? > > Regards, > Andor >Presuming that you're dealing with a linear system I suspect you could approximate it as closely as you'd ever like, given enough states, but that you couldn't realize it as a finite-sized system. But I can't prove the above assertion. -- 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 ●September 4, 20072007-09-04
Andor <andor.bariska@gmail.com> writes:> Folks, > > just an informal question: do you think that the filter with impulse > response > > h(t) = u(t) / sqrt(t) > > (u(t) is unit step function) > > is a physically realizable system?Of course not. The output of the system can be made arbitrarily large since u(t) / sqrt(t) approaches infinity as t approaches 0. For example, what is the output with input x(t) = u(-t) - u(-t - 1)? -- % Randy Yates % "And all that I can do %% Fuquay-Varina, NC % is say I'm sorry, %%% 919-577-9882 % that's the way it goes..." %%%% <yates@ieee.org> % Getting To The Point', *Balance of Power*, ELO http://home.earthlink.net/~yatescr
Reply by ●September 4, 20072007-09-04
Andor wrote:> Folks, > > just an informal question: do you think that the filter with impulse > response > > h(t) = u(t) / sqrt(t) > > (u(t) is unit step function) > > is a physically realizable system?No. The square of the Fourier transform is infinite. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●September 4, 20072007-09-04
Vladimir Vassilevsky wrote:> Andor wrote: > >> Folks, >> >> just an informal question: do you think that the filter with impulse >> response >> >> h(t) = u(t) / sqrt(t) >> >> (u(t) is unit step function) >> >> is a physically realizable system? > > > No. The square of the Fourier transform is infinite.I meant the integral of the [F(w)]^2, of course. It diverges. Vladimir Vassilevsky DSP and Mixed Signal Design Consultant http://www.abvolt.com
Reply by ●September 5, 20072007-09-05
On Tue, 04 Sep 2007 14:11:39 -0700, Tim Wescott wrote:> Andor wrote: >> Folks, >> >> just an informal question: do you think that the filter with impulse >> response >> >> h(t) = u(t) / sqrt(t) >> >> (u(t) is unit step function) >> >> is a physically realizable system? >> >> Regards, >> Andor >> > Presuming that you're dealing with a linear system I suspect you could > approximate it as closely as you'd ever like, given enough states, but > that you couldn't realize it as a finite-sized system. > > But I can't prove the above assertion. >After reading Randy's and Vladimir's responses I realized that I missed the u(t) / sqrt(t) -- I thought it was u(t) * sqrt(t). That infinitely big spike at the front end will be hard to realize in practice. You could easily approximate the response with dynamite, however. -- Tim Wescott Control systems and communications consulting http://www.wescottdesign.com Need to learn how to apply control theory in your embedded system? "Applied Control Theory for Embedded Systems" by Tim Wescott Elsevier/Newnes, http://www.wescottdesign.com/actfes/actfes.html
Reply by ●September 5, 20072007-09-05
On 5 Sep., 00:10, Randy Yates <ya...@ieee.org> wrote:> Andor <andor.bari...@gmail.com> writes: > > Folks, > > > just an informal question: do you think that the filter with impulse > > response > > > h(t) = u(t) / sqrt(t) > > > (u(t) is unit step function) > > > is a physically realizable system? > > Of course not. The output of the system can be made arbitrarily > large since u(t) / sqrt(t) approaches infinity as t approaches 0. > For example, what is the output with input x(t) = u(-t) - u(-t - 1)?Ok, you want to know the response of my system to a rectangular pulse of duration 1. I'll shift the pulse to the interval [0,1] which makes the integrals look nicer (and only shifts the output function by the same amount, as the system is LTI). The convolution is y(t) = x(t) * h(t) = integral_{s=-infinity}^infinity x(s) h(t-s) ds = integral_{s=0}^infinity x(s) h(t-s) ds (lower limit is 0, because the pulse h(t) is causal) = / integral_{s=0}^t 1/sqrt(t-s) ds, 0 <= t <= 1 \ integral_{s=t-1}^t 1/sqrt(t-s) ds, t > 1 = / 2 sqrt(t), 0 <= t <= 1 \ 2 sqrt(t) - 2 sqrt(t-1), t > 1. (The impulse has finite integral over the interval [0,T] for any T > 0) As you see, the system integrates the rectangular pulse, then decays slowly back to 0. The system acts like a "weak" or "leaky" integrator. The "leak" rate is however not fast enough for DC. The frequency response for positive frequencies of the system is H(f) = (1-j)/2 1/sqrt(f). It clearly shows that the system cannot handle DC, but has otherwise finite magnitude and phase response. The impulse response is peculiar, in that the integrals of neither |h(t)| (DC response) or h(t)^2 (energy) are finite. However, is it necessary for the impulse response of a system have to have finite energy to represent a "physical" system? Regards, Andor
Reply by ●September 5, 20072007-09-05
"Andor" <andor.bariska@gmail.com> wrote in message news:1188977909.056618.96680@y42g2000hsy.googlegroups.com...> > > > > just an informal question: do you think that the filter with impulse > > > response > > > > > h(t) = u(t) / sqrt(t) > > > > > (u(t) is unit step function) > > > > > is a physically realizable system?It depends :-) What is your definition of u(0) ? Is it 0 or 1 ?> The "leak" rate is however not fast enough for DC. The frequency > response for positive frequencies of the system is > > H(f) = (1-j)/2 1/sqrt(f).Here we go. Since the Fourier is not L2 integrable from 0 to inf, the system is not physically realizable.> It clearly shows that the system cannot handle DC, but has otherwise > finite magnitude and phase response. The impulse response is peculiar, > in that the integrals of neither |h(t)| (DC response) or h(t)^2 > (energy) are finite. However, is it necessary for the impulse response > of a system have to have finite energy to represent a "physical" > system?No, it is not. Integrators or differentiators are physically realizable. The unstable systems are physically realizable, too. Vladimir Vassilevsky DSP and Mixed Signal Consultant www.abvolt.com
Reply by ●September 5, 20072007-09-05
On 5 Sep., 09:55, "Vladimir Vassilevsky" <antispam_bo...@hotmail.com> wrote:> "Andor" <andor.bari...@gmail.com> wrote in message > > news:1188977909.056618.96680@y42g2000hsy.googlegroups.com... > > > > > > > just an informal question: do you think that the filter with impulse > > > > response > > > > > h(t) = u(t) / sqrt(t) > > > > > (u(t) is unit step function) > > > > > is a physically realizable system? > > It depends :-) > What is your definition of u(0) ? Is it 0 or 1 ?I guess you really mean "what is the definition of h(0)?". Well, I say it doesn't matter. The response of the system is given as an integral transform of the input with the impulse response. For integrals, point- wise definition of functions is irrelevant. So, you can set h(0) = 0 or h(0) = inifinity as you please.> > > The "leak" rate is however not fast enough for DC. The frequency > > response for positive frequencies of the system is > > > H(f) = (1-j)/2 1/sqrt(f). > > Here we go. Since the Fourier is not L2 integrable from 0 to inf, the system > is not physically realizable. > > > It clearly shows that the system cannot handle DC, but has otherwise > > finite magnitude and phase response. The impulse response is peculiar, > > in that the integrals of neither |h(t)| (DC response) or h(t)^2 > > (energy) are finite. However, is it necessary for the impulse response > > of a system have to have finite energy to represent a "physical" > > system? > > No, it is not. Integrators or differentiators are physically realizable. The > unstable systems are physically realizable, too.So if integrators are physically realizable, what about two integrators in series? Is that realizable? Regards, Andor
Reply by ●September 5, 20072007-09-05
On 5 Sep., 06:54, Tim Wescott <t...@seemywebsite.com> wrote:> On Tue, 04 Sep 2007 14:11:39 -0700, Tim Wescott wrote: > > Andor wrote: > >> Folks, > > >> just an informal question: do you think that the filter with impulse > >> response > > >> h(t) = u(t) / sqrt(t) > > >> (u(t) is unit step function) > > >> is a physically realizable system? > > >> Regards, > >> Andor > > > Presuming that you're dealing with a linear system I suspect you could > > approximate it as closely as you'd ever like, given enough states, but > > that you couldn't realize it as a finite-sized system. > > > But I can't prove the above assertion. > > After reading Randy's and Vladimir's responses I realized that I missed > the u(t) / sqrt(t) -- I thought it was u(t) * sqrt(t). > > That infinitely big spike at the front end will be hard to realize in > practice. You could easily approximate the response with dynamite, > however.Well, you only get an infinity spike if you put an infinity spike (dirac) into the system. Is the infinity spike input a "physical" input?... For many other input functions, the output function is well defined (see my response to Randy). Regards, Andor
