LTI systems
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From: achuttys on 27 Oct 2009 12:08 Hi all Please help... I have some doubts on LTI systems. Recently I read the second chapter of Digital Signal Processing by John G Proakis. In the second chapter, it gives a method to charecterize LTI system by LCCDE(Linear Constant Coefficient Differential Equation). The differential equation is solved to find the system parameters. Once the system parameters are found, one can find the response of an LTI system for any arbitary signal. The response of an LTI system can also be found by knowing the impulse response of the system through Convolution method. Why does one need LCCDE method when Convolution method is there to find the response of an LTI system? Is it because Convolution cannot be applied to recursive systems? On computation efficiency, is LCCDE better than Convolution? Thanks in Advance
From: Jerry Avins on 27 Oct 2009 13:06 achuttys wrote: > Hi all > Please help... > I have some doubts on LTI systems. Recently I read the second chapter of > Digital Signal Processing by John G Proakis. In the second chapter, it > gives a method to charecterize LTI system by LCCDE(Linear Constant > Coefficient Differential Equation). The differential equation is solved to > find the system parameters. Once the system parameters are found, one can > find the response of an LTI system for any arbitary signal. > > The response of an LTI system can also be found by knowing the impulse > response of the system through Convolution method. > > Why does one need LCCDE method when Convolution method is there to find > the response of an LTI system? Is it because Convolution cannot be applied > to recursive systems? > > On computation efficiency, is LCCDE better than Convolution? > Thanks in Advance LCCDE systems can also be solved with Laplace transformations. Personally, I would rather, when working by hand, solve the differential equation than work out a convolution integral. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
From: Rune Allnor on 27 Oct 2009 13:46 On 27 Okt, 17:08, "achuttys" <subhish_ku...(a)yahoo.co.in> wrote: > Hi all > Please help... > I have some doubts on LTI systems. Recently I read the second chapter of > Digital Signal Processing by John G Proakis. In the second chapter, it > gives a method to charecterize LTI system by LCCDE(Linear Constant > Coefficient Differential Equation). The differential equation is solved to > find the system parameters. Once the system parameters are found, one can > find the response of an LTI system for any arbitary signal. > > The response of an LTI system can also be found by knowing the impulse > response of the system through Convolution method. > > Why does one need LCCDE method when Convolution method is there to find > the response of an LTI system? Is it because Convolution cannot be applied > to recursive systems? > > On computation efficiency, is LCCDE better than Convolution? There are a number of different but equivalent (formally, although not necessarily practically) representations of the LTI system. Which one is 'best' depends on what information or assumptions you start out from, and what you want to achieve. If you have the LCCDE, then work from that POV. If you don't, but instead have the impulse response, then work from that POV. I have noticed that entry-level DSP classes often start with the case of the known difference equations, and the question is to work out frequency responses, impulse responses, and so on. On medium to advanced levels, the problem is often to work out the diffence equation from measured impulse responses or transfer functions. Rune
From: Tim Wescott on 27 Oct 2009 14:43 On Tue, 27 Oct 2009 10:46:04 -0700, Rune Allnor wrote: > On 27 Okt, 17:08, "achuttys" <subhish_ku...(a)yahoo.co.in> wrote: >> Hi all >> Please help... >> I have some doubts on LTI systems. Recently I read the second chapter >> of Digital Signal Processing by John G Proakis. In the second chapter, >> it gives a method to charecterize LTI system by LCCDE(Linear Constant >> Coefficient Differential Equation). The differential equation is solved >> to find the system parameters. Once the system parameters are found, >> one can find the response of an LTI system for any arbitary signal. >> >> The response of an LTI system can also be found by knowing the impulse >> response of the system through Convolution method. >> >> Why does one need LCCDE method when Convolution method is there to find >> the response of an LTI system? Is it because Convolution cannot be >> applied to recursive systems? >> >> On computation efficiency, is LCCDE better than Convolution? > > There are a number of different but equivalent (formally, although not > necessarily practically) representations of the LTI system. Which one is > 'best' depends on what information or assumptions you start out from, > and what you want to achieve. If you have the LCCDE, then work from that > POV. If you don't, but instead have the impulse response, then work from > that POV. > > I have noticed that entry-level DSP classes often start with the case of > the known difference equations, and the question is to work out > frequency responses, impulse responses, and so on. On medium to advanced > levels, the problem is often to work out the diffence equation from > measured impulse responses or transfer functions. > > Rune And if you're designing control systems you must do both -- first you predict your plant behavior with differential and difference equations, then you design a controller with differential equations, then once the plant is built you verify your model by measurement (and usually adjust it), then you tweak the controller's difference equations yet again. Toss in the need to quantify the system's behavior in the frequency domain and you'll be doing plenty of different kinds of analysis. -- www.wescottdesign.com
From: Tim Wescott on 27 Oct 2009 16:54 On Tue, 27 Oct 2009 11:08:20 -0500, achuttys wrote: > Hi all > Please help... > I have some doubts on LTI systems. Recently I read the second chapter of > Digital Signal Processing by John G Proakis. In the second chapter, it > gives a method to charecterize LTI system by LCCDE(Linear Constant > Coefficient Differential Equation). The differential equation is solved > to find the system parameters. Once the system parameters are found, one > can find the response of an LTI system for any arbitary signal. > > The response of an LTI system can also be found by knowing the impulse > response of the system through Convolution method. > > Why does one need LCCDE method when Convolution method is there to find > the response of an LTI system? Is it because Convolution cannot be > applied to recursive systems? > > On computation efficiency, is LCCDE better than Convolution? Thanks in > Advance I'll more or less rephrase what Rune said: There are a number of different ways to do this, each one completely valid, each one fitting a given problem differently. Knowing a variety of methods lets you choose the method that fits your problem well, and makes your work lighter. -- www.wescottdesign.com
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