You know the problems you get at college - convolving a pulse with another pulse or a pulse with a triangular pulse etc. Loads and loads of such examples, many of them start with negative time (ie pulses go from -1 to +1 !). But although convolution as a classical method can be used for solving ODE's what practical use is it? I am not talking about digital or discrete-time convolution - fine with that FFTs and FIR filters, but with continuous time you don't get a system with a finite-time impulse response (let alone one starting at time -1 !). Hardy
Piecewise Convolution
Started by ●July 30, 2014
Reply by ●July 31, 20142014-07-31
On Wed, 30 Jul 2014 19:35:13 -0700, gyansorova wrote:> You know the problems you get at college - convolving a pulse with > another pulse or a pulse with a triangular pulse etc. Loads and loads of > such examples, many of them start with negative time (ie pulses go from > -1 to +1 !). But although convolution as a classical method can be used > for solving ODE's what practical use is it? I am not talking about > digital or discrete-time convolution - fine with that FFTs and FIR > filters, but with continuous time you don't get a system with a > finite-time impulse response (let alone one starting at time -1 !).I don't know how much it's done these days, but back when digital communications systems were done with analog circuitry, an integrate-and- dump circuit came pretty close to convolution. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●July 31, 20142014-07-31
On Thursday, July 31, 2014 10:19:59 AM UTC+5:30, Tim Wescott wrote:> On Wed, 30 Jul 2014 19:35:13 -0700, gyansorova wrote: > > > > > You know the problems you get at college - convolving a pulse with > > > another pulse or a pulse with a triangular pulse etc. Loads and loads of > > > such examples, many of them start with negative time (ie pulses go from > > > -1 to +1 !). But although convolution as a classical method can be used > > > for solving ODE's what practical use is it? I am not talking about > > > digital or discrete-time convolution - fine with that FFTs and FIR > > > filters, but with continuous time you don't get a system with a > > > finite-time impulse response (let alone one starting at time -1 !). > > > > I don't know how much it's done these days, but back when digital > > communications systems were done with analog circuitry, an integrate-and- > > dump circuit came pretty close to convolution. > > > > -- > > > > Tim Wescott > > Wescott Design Services > > http://www.wescottdesign.comErrr, no. An integrate-and-dump circuit (more generally, a correlator, as a circuit) does NOT give the results of a convolution; it tells you the value of a specific convolution integral at ONE point in time, and, in general, does not give the value of the convolution integral at any OTHER time. The convolution operation gives the value of the convolution integral for all possible times. In other words, the result of a convolution is a FUNCTION; the output of an Integrate-and-Dump circuit is a NUMBER. For an illustration of the output of an integrate-and-dump circuit as compared to a filter (LTI system), see the end of an answer on dsp.SE http://dsp.stackexchange.com/a/9389/235
Reply by ●July 31, 20142014-07-31
On Wednesday, July 30, 2014 10:35:13 PM UTC-4, gyans...@gmail.com wrote:> You know the problems you get at college - convolving a pulse with another pulse or a pulse with a triangular pulse etc. Loads and loads of such examples, many of them start with negative time (ie pulses go from -1 to +1 !). But although convolution as a classical method can be used for solving ODE's what practical use is it? I am not talking about digital or discrete-time convolution - fine with that FFTs and FIR filters, but with continuous time you don't get a system with a finite-time impulse response (let alone one starting at time -1 !). > > > > > > HardyHardy, Continuous time convolution is useful in some cases for predicting linear system response to specific inputs. Like for analog linear filters. Can also be used for computing correlations. You don't really care about a impulse response not being finite time when doing computations because 1) you limit the functions convolved to what you can computationally deal with, 2)you may assume a zero input before a certain time, 3) you may assume an input that exists for all time (like finding the response for a sinusoidal input - frequency response). You may care about computing a linear system input step response, or a burst of a sinusoidal signal... go back and look at the examples in your courses. As mentioned here (and disagreed with here) convolution is used in the modeling of a DAC as outputting impulses into a zero- or first-order output hold. The analysis that you do with continuous time convolution can help you understand or describe a system. You should not be bothered by pulses that go from t=-1 to t=1. It makes the math easier and you can adjust the result if you would rather have the pulse start at t=0 or elsewhere. Your teachers and texts undoubtedly showed you how to adjust for a time shift; it might not have registered at the time. It's in your books and should be in your notes. Also it can be used to help explain parts of DSP. Check your DSP books. Dirk
Reply by ●July 31, 20142014-07-31
On Thursday, July 31, 2014 12:25:29 PM UTC-4, belld...@gmail.com wrote:> On Wednesday, July 30, 2014 10:35:13 PM UTC-4, gyans...@gmail.com wrote: > > > You know the problems you get at college - convolving a pulse with another pulse or a pulse with a triangular pulse etc. Loads and loads of such examples, many of them start with negative time (ie pulses go from -1 to +1 !). But although convolution as a classical method can be used for solving ODE's what practical use is it? I am not talking about digital or discrete-time convolution - fine with that FFTs and FIR filters, but with continuous time you don't get a system with a finite-time impulse response (let alone one starting at time -1 !). > > > > > > > > > > > > > > > > > > Hardy > > > > Hardy, > > > > Continuous time convolution is useful in some cases for predicting linear system response to specific inputs. Like for analog linear filters. Can also be used for computing correlations. > > > > You don't really care about a impulse response not being finite time when doing computations because 1) you limit the functions convolved to what you can computationally deal with, 2)you may assume a zero input before a certain time, 3) you may assume an input that exists for all time (like finding the response for a sinusoidal input - frequency response). > > > > You may care about computing a linear system input step response, or a burst of a sinusoidal signal... go back and look at the examples in your courses. As mentioned here (and disagreed with here) convolution is used in the modeling of a DAC as outputting impulses into a zero- or first-order output hold. > > > > The analysis that you do with continuous time convolution can help you understand or describe a system. > > > > You should not be bothered by pulses that go from t=-1 to t=1. It makes the math easier and you can adjust the result if you would rather have the pulse start at t=0 or elsewhere. Your teachers and texts undoubtedly showed you how to adjust for a time shift; it might not have registered at the time. It's in your books and should be in your notes. > > > > Also it can be used to help explain parts of DSP. Check your DSP books. > > > > DirkI want to correct myself. Integrate and dump was discussed, not DACs. Dirk
Reply by ●July 31, 20142014-07-31
On Thu, 31 Jul 2014 04:17:08 -0700, dvsarwate wrote:> On Thursday, July 31, 2014 10:19:59 AM UTC+5:30, Tim Wescott wrote: >> On Wed, 30 Jul 2014 19:35:13 -0700, gyansorova wrote: >> >> >> >> > You know the problems you get at college - convolving a pulse with >> >> > another pulse or a pulse with a triangular pulse etc. Loads and loads >> > of >> >> > such examples, many of them start with negative time (ie pulses go >> > from >> >> > -1 to +1 !). But although convolution as a classical method can be >> > used >> >> > for solving ODE's what practical use is it? I am not talking about >> >> > digital or discrete-time convolution - fine with that FFTs and FIR >> >> > filters, but with continuous time you don't get a system with a >> >> > finite-time impulse response (let alone one starting at time -1 !). >> >> >> >> I don't know how much it's done these days, but back when digital >> >> communications systems were done with analog circuitry, an >> integrate-and- >> >> dump circuit came pretty close to convolution. >> >> >> >> -- >> >> >> >> Tim Wescott >> >> Wescott Design Services >> >> http://www.wescottdesign.com > > > Errr, no. An integrate-and-dump circuit (more generally, a correlator, > as a circuit) does NOT give the results of a convolution; it tells you > the value of a specific convolution integral at ONE point in time, and, > in general, does not give the value of the convolution integral at any > OTHER time. The convolution operation gives the value of the > convolution integral for all possible times. In other words, the result > of a convolution is a FUNCTION; the output of an Integrate-and-Dump > circuit is a NUMBER. > > For an illustration of the output of an integrate-and-dump circuit as > compared to a filter (LTI system), see the end of an answer on dsp.SE > > http://dsp.stackexchange.com/a/9389/235Well, yes. But if your bit and phase synchronization is correct, integrate-and-dump gives you the result of convolution where it _matters_. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●July 31, 20142014-07-31
On Thu, 31 Jul 2014 12:27:07 -0500, Tim Wescott <tim@seemywebsite.really> wrote:>On Thu, 31 Jul 2014 04:17:08 -0700, dvsarwate wrote: > >> On Thursday, July 31, 2014 10:19:59 AM UTC+5:30, Tim Wescott wrote: >>> On Wed, 30 Jul 2014 19:35:13 -0700, gyansorova wrote: >>> >>> >>> >>> > You know the problems you get at college - convolving a pulse with >>> >>> > another pulse or a pulse with a triangular pulse etc. Loads and loads >>> > of >>> >>> > such examples, many of them start with negative time (ie pulses go >>> > from >>> >>> > -1 to +1 !). But although convolution as a classical method can be >>> > used >>> >>> > for solving ODE's what practical use is it? I am not talking about >>> >>> > digital or discrete-time convolution - fine with that FFTs and FIR >>> >>> > filters, but with continuous time you don't get a system with a >>> >>> > finite-time impulse response (let alone one starting at time -1 !). >>> >>> >>> >>> I don't know how much it's done these days, but back when digital >>> >>> communications systems were done with analog circuitry, an >>> integrate-and- >>> >>> dump circuit came pretty close to convolution. >>> >>> >>> >>> -- >>> >>> >>> >>> Tim Wescott >>> >>> Wescott Design Services >>> >>> http://www.wescottdesign.com >> >> >> Errr, no. An integrate-and-dump circuit (more generally, a correlator, >> as a circuit) does NOT give the results of a convolution; it tells you >> the value of a specific convolution integral at ONE point in time, and, >> in general, does not give the value of the convolution integral at any >> OTHER time. The convolution operation gives the value of the >> convolution integral for all possible times. In other words, the result >> of a convolution is a FUNCTION; the output of an Integrate-and-Dump >> circuit is a NUMBER. >> >> For an illustration of the output of an integrate-and-dump circuit as >> compared to a filter (LTI system), see the end of an answer on dsp.SE >> >> http://dsp.stackexchange.com/a/9389/235 > >Well, yes. But if your bit and phase synchronization is correct, >integrate-and-dump gives you the result of convolution where it _matters_.An I&D is just a sampler with a rectangular impulse response. Really nothing more or less. That winds up being a matched Rx filter to rectangular transmit pulses, so it makes for a practical way to implement a simple (albeit not very spectrally efficient) system. There are many cases where it does wind up being very useful.>-- > >Tim Wescott >Wescott Design Services >http://www.wescottdesign.com >Eric Jacobsen Anchor Hill Communications http://www.anchorhill.com
Reply by ●July 31, 20142014-07-31
On Thu, 31 Jul 2014 23:58:27 +0000, Eric Jacobsen wrote:> On Thu, 31 Jul 2014 12:27:07 -0500, Tim Wescott > <tim@seemywebsite.really> wrote: > >>On Thu, 31 Jul 2014 04:17:08 -0700, dvsarwate wrote: >> >>> On Thursday, July 31, 2014 10:19:59 AM UTC+5:30, Tim Wescott wrote: >>>> On Wed, 30 Jul 2014 19:35:13 -0700, gyansorova wrote: >>>> >>>> >>>> >>>> > You know the problems you get at college - convolving a pulse with >>>> >>>> > another pulse or a pulse with a triangular pulse etc. Loads and >>>> > loads of >>>> >>>> > such examples, many of them start with negative time (ie pulses go >>>> > from >>>> >>>> > -1 to +1 !). But although convolution as a classical method can be >>>> > used >>>> >>>> > for solving ODE's what practical use is it? I am not talking about >>>> >>>> > digital or discrete-time convolution - fine with that FFTs and FIR >>>> >>>> > filters, but with continuous time you don't get a system with a >>>> >>>> > finite-time impulse response (let alone one starting at time -1 !). >>>> >>>> >>>> >>>> I don't know how much it's done these days, but back when digital >>>> >>>> communications systems were done with analog circuitry, an >>>> integrate-and- >>>> >>>> dump circuit came pretty close to convolution. >>>> >>>> >>>> >>>> -- >>>> >>>> >>>> >>>> Tim Wescott >>>> >>>> Wescott Design Services >>>> >>>> http://www.wescottdesign.com >>> >>> >>> Errr, no. An integrate-and-dump circuit (more generally, a >>> correlator, >>> as a circuit) does NOT give the results of a convolution; it tells you >>> the value of a specific convolution integral at ONE point in time, >>> and, >>> in general, does not give the value of the convolution integral at any >>> OTHER time. The convolution operation gives the value of the >>> convolution integral for all possible times. In other words, the >>> result of a convolution is a FUNCTION; the output of an >>> Integrate-and-Dump circuit is a NUMBER. >>> >>> For an illustration of the output of an integrate-and-dump circuit as >>> compared to a filter (LTI system), see the end of an answer on dsp.SE >>> >>> http://dsp.stackexchange.com/a/9389/235 >> >>Well, yes. But if your bit and phase synchronization is correct, >>integrate-and-dump gives you the result of convolution where it >>_matters_. > > An I&D is just a sampler with a rectangular impulse response. Really > nothing more or less. That winds up being a matched Rx filter to > rectangular transmit pulses, so it makes for a practical way to > implement a simple (albeit not very spectrally efficient) system. > > There are many cases where it does wind up being very useful.The one and only time I implemented one it was preceded by a multiplier so that the overall effect was to match the half-cycle cosine pulses of MSK. -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
