Hi,
Let
X(n)-----> Input Discreate Time Sequnece.
H(n)-----> Impulse response of the system.
Y(n)-----> Output Discreate time Sequnece.
X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece and
H(Z) is the transfer function of the system.
Then the mathamatical relation ship holds
Y(n) = X(n)(+)H(n) // Convolution Operation.
Y(Z) = X(Z)*Y(Z) // Multipilcation Operation.
The Above relationships are only valid of LTI system ?? or they also valid
for Non-Linear, Non-Time Invaraint System ??
Thanks in advance.
LTI system and Non LTI system
Started by ●February 15, 2005
Reply by ●February 15, 20052005-02-15
The impulse response, h[n], and the convolution operation are DEFINED based on a system being LTI. Therefore they are in a strict sense only applicable to linear, time-invariant systems. However, you will still probably see them used even for systems that aren't LTI. In this case there is some "fudge factor" involved. For example in a LTV system you could treat it as LTI for short periods of time. In a nonlinear, time-invariant system you might treat it as LTI but then throw in some kind of correction factor to account for non linearities. Brad "Abdul" <abdul_araheem@yahoo.com> wrote in message news:cuslvf$52c$1@rzsun03.rrz.uni-hamburg.de...> Hi, > > Let > > X(n)-----> Input Discreate Time Sequnece. > H(n)-----> Impulse response of the system. > Y(n)-----> Output Discreate time Sequnece. > > X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece > and > H(Z) is the transfer function of the system. > > Then the mathamatical relation ship holds > > Y(n) = X(n)(+)H(n) // Convolution Operation. > Y(Z) = X(Z)*Y(Z) // Multipilcation Operation. > > The Above relationships are only valid of LTI system ?? or they also valid > for Non-Linear, Non-Time Invaraint System ?? > > Thanks in advance. > > > >
Reply by ●February 15, 20052005-02-15
"Brad Griffis" <bradgriffis@hotmail.com> wrote in message news:AUmQd.5744$NO3.572@newssvr31.news.prodigy.com...> The impulse response, h[n], and the convolution operation are DEFINEDbased> on a system being LTI. Therefore they are in a strict sense onlyapplicable> to linear, time-invariant systems. However, you will still probably see > them used even for systems that aren't LTI. In this case there is some > "fudge factor" involved. For example in a LTV system you could treat itas> LTI for short periods of time. In a nonlinear, time-invariant system you > might treat it as LTI but then throw in some kind of correction factor to > account for non linearities. > > Brad > > "Abdul" <abdul_araheem@yahoo.com> wrote in message > news:cuslvf$52c$1@rzsun03.rrz.uni-hamburg.de... > > Hi, > > > > Let > > > > X(n)-----> Input Discreate Time Sequnece. > > H(n)-----> Impulse response of the system. > > Y(n)-----> Output Discreate time Sequnece. > > > > X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequnece > > and > > H(Z) is the transfer function of the system. > > > > Then the mathamatical relation ship holds > > > > Y(n) = X(n)(+)H(n) // Convolution Operation. > > Y(Z) = X(Z)*Y(Z) // Multipilcation Operation. > > > > The Above relationships are only valid of LTI system ?? or they alsovalid> > for Non-Linear, Non-Time Invaraint System ?? > > > > Thanks in advance.Actually, I believe that impulse responses and convolutions are applicable to all linear systems, whether or not they are time invariant or not. It's just that the formulas are a little more complicated since h[n, k] is not necessarily expressable as h[n-k]. Transfer functions and other frequency-domain relationships, however, only apply to linear, time-invariant systems.
Reply by ●February 15, 20052005-02-15
Brad Griffis wrote:> The impulse response, h[n], and the convolution operation are DEFINED based > on a system being LTI. Therefore they are in a strict sense only applicable > to linear, time-invariant systems. However, you will still probably see > them used even for systems that aren't LTI. In this case there is some > "fudge factor" involved. For example in a LTV system you could treat it as > LTI for short periods of time. In a nonlinear, time-invariant system you > might treat it as LTI but then throw in some kind of correction factor to > account for non linearities. > > Brad > > "Abdul" <abdul_araheem@yahoo.com> wrote in message > news:cuslvf$52c$1@rzsun03.rrz.uni-hamburg.de...... And Abdul, keep your notation straight. X(a), X[a], x(a), and x[a] have different conventional meanings. ('a' doesn't stand for the same symbol in all instances.) Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 15, 20052005-02-15
in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at bradgriffis@hotmail.com wrote on 02/15/2005 08:35:> The impulse response, h[n], and the convolution operation are DEFINED based > on a system being LTI.Brad, i have to disagree with this statement. any system, whether it's LTI or not, can be banged with a unit impulse and be allowed to respond (or "ring") and that response, whether it's an LTI system or not, would the be "impulse response". if the system is not LTI, i would think that the impulse response has less meaning than it does for an LTI system where the impulse response completely characterizes the system from an input-output perspective. you could not say as much form the impulse response of a system that is not LTI. the convolution operation is *not* defined but is a *consequence* of the fact that the LTI system is linear and time-invariant. you can derive it from the LTI properties and, particularly for a discrete-time system, it's pretty easy. the only DEFINED properties of an LTI system that i can think of is that it is linear and that it is time-invariant. there are many other properties, but they can be derived. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●February 15, 20052005-02-15
robert bristow-johnson wrote:> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at > bradgriffis@hotmail.com wrote on 02/15/2005 08:35: > > >>The impulse response, h[n], and the convolution operation are DEFINED based >>on a system being LTI. > > > Brad, i have to disagree with this statement. any system, whether it's LTI > or not, can be banged with a unit impulse and be allowed to respond (or > "ring") and that response, whether it's an LTI system or not, would the be > "impulse response". if the system is not LTI, i would think that the > impulse response has less meaning than it does for an LTI system where the > impulse response completely characterizes the system from an input-output > perspective. you could not say as much form the impulse response of a > system that is not LTI. > > the convolution operation is *not* defined but is a *consequence* of the > fact that the LTI system is linear and time-invariant. you can derive it > from the LTI properties and, particularly for a discrete-time system, it's > pretty easy. the only DEFINED properties of an LTI system that i can think > of is that it is linear and that it is time-invariant. there are many other > properties, but they can be derived. >And I'll have to disagree with you both (whee!). Robert: While any system can be banged with an impulse and it's response checked this is meaningless in general with nonlinear systems -- what if the system is a block that limits any input to +/- 10V followed by a purely linear system? In that case the impulse will disappear entirely, and there'll be no response. Brad: This has already been pointed out, but a time-varying linear system can be completely defined by its impulse response, the only caveat being that the impulse response has two time parameters: the time of the output, and the time the impulse happened. So instead of talking about h(n) and assuming that the impulse happened at n=0 with no loss of generality you have to talk about h(n_0, n), where the impulse happened at n=n_0. Neither a nonlinear system or a time-varying one can be reduced exactly to a transfer function. Some time-varying systems can be expressed as a combination of multiplies and convolutions in the frequency domain -- this is particularly useful in analyzing heterodyne radio receivers (either superhets or direct-conversion). The idea of using the time-varying impulse response and doing a convolution on it is _very_ useful in analyzing communications systems. Some nonlinear systems can be expressed with a "describing function" where a real input signal is chosen, the system is excited with it, and the frequency response of the system is extracted. This technique is very useful for control system design. It can be done directly in the case of a swept-sine measurement or indirectly in the case of random noise or step excitation, and you will often find yourself taking measurements at different amplitudes or waveshapes and seeing a range of "frequency responses". -- Tim Wescott Wescott Design Services http://www.wescottdesign.com
Reply by ●February 15, 20052005-02-15
Tim Wescott wrote:> robert bristow-johnson wrote: > >> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at >> bradgriffis@hotmail.com wrote on 02/15/2005 08:35: >> >> >>> The impulse response, h[n], and the convolution operation are DEFINED >>> based >>> on a system being LTI. >> >> >> >> Brad, i have to disagree with this statement. any system, whether >> it's LTI >> or not, can be banged with a unit impulse and be allowed to respond (or >> "ring") and that response, whether it's an LTI system or not, would >> the be >> "impulse response". if the system is not LTI, i would think that the >> impulse response has less meaning than it does for an LTI system where >> the >> impulse response completely characterizes the system from an input-output >> perspective. you could not say as much form the impulse response of a >> system that is not LTI. >> >> the convolution operation is *not* defined but is a *consequence* of the >> fact that the LTI system is linear and time-invariant. you can derive it >> from the LTI properties and, particularly for a discrete-time system, >> it's >> pretty easy. the only DEFINED properties of an LTI system that i can >> think >> of is that it is linear and that it is time-invariant. there are many >> other >> properties, but they can be derived. >> > And I'll have to disagree with you both (whee!).Well Tim, I disagree with you. (Double whee!)> Robert: While any system can be banged with an impulse and its > response checked, this is meaningless in general with nonlinear systems > -- what if the system is a block that limits any input to +/- 10V > followed by a purely linear system? In that case the impulse will > disappear entirely, and there'll be no response.That is true only if you insist that an impulse has zero width and infinite height. In that case, no impulse can get through to any real system. It will always be clipped, maybe to a megavolt, but what's left still disappears entirely. A practical impulse has a width so narrow that it doesn't matter and a height that's "reasonable". One of those _will_ be seen on the other side of your limiter. Try it. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 15, 20052005-02-15
in article 1114g3h2hle7s9d@corp.supernews.com, Tim Wescott at tim@wescottnospamdesign.com wrote on 02/15/2005 13:33:> robert bristow-johnson wrote: >> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at >> bradgriffis@hotmail.com wrote on 02/15/2005 08:35: >> >> >>> The impulse response, h[n], and the convolution operation are DEFINED based >>> on a system being LTI. >> >> >> Brad, i have to disagree with this statement. any system, whether it's LTI >> or not, can be banged with a unit impulse and be allowed to respond (or >> "ring") and that response, whether it's an LTI system or not, would the be >> "impulse response". if the system is not LTI, i would think that the >> impulse response has less meaning than it does for an LTI system where the >> impulse response completely characterizes the system from an input-output >> perspective. you could not say as much form the impulse response of a >> system that is not LTI. >> >> the convolution operation is *not* defined but is a *consequence* of the >> fact that the LTI system is linear and time-invariant. you can derive it >> from the LTI properties and, particularly for a discrete-time system, it's >> pretty easy. the only DEFINED properties of an LTI system that i can think >> of is that it is linear and that it is time-invariant. there are many other >> properties, but they can be derived. >> > And I'll have to disagree with you both (whee!). > > Robert: While any system can be banged with an impulse and it's > response checked this is meaningless in general with nonlinear systems > -- what if the system is a block that limits any input to +/- 10V > followed by a purely linear system? In that case the impulse will > disappear entirely, and there'll be no response.so where is the disagreement? doesn't "meaningless" have less meaning than "completely characterizes the system from an input-output perspective"? i don't see any disagreement. :-\> Brad: This has already been pointed out, but a time-varying linear > system can be completely defined by its impulse response, the only > caveat being that the impulse response has two time parameters: the time > of the output, and the time the impulse happened. So instead of talking > about h(n) and assuming that the impulse happened at n=0 with no loss of > generality you have to talk about h(n_0, n), where the impulse happened > at n=n_0.not a substantive disagreement, but a semantic one: i wouldn't say a time-varying linear system can be completely defined (i would say "described") by its impulse response (emphasis _singular_ tense). i would say it's completely defined (or described) by *all* possible impulse responses of unit impulses applied at all possible times. for the LTI, h[n] which is h[n0, n] where n0 is the special case of being 0, is completely sufficient, whereas it is not sufficient for LTV. you're saying this, but i wouldn't use the singular tense. -- r b-j rbj@audioimagination.com "Imagination is more important than knowledge."
Reply by ●February 15, 20052005-02-15
"Abdul" <abdul_araheem@yahoo.com> wrote in message news:cuslvf$52c$1@rzsun03.rrz.uni-hamburg.de...> Hi, > > Let > > X(n)-----> Input Discreate Time Sequnece. > H(n)-----> Impulse response of the system. > Y(n)-----> Output Discreate time Sequnece. > > X(Z), Y(Z) are the z trasfroms of Input & output Discreate time sequneceand> H(Z) is the transfer function of the system. > > Then the mathamatical relation ship holds > > Y(n) = X(n)(+)H(n) // Convolution Operation. > Y(Z) = X(Z)*Y(Z) // Multipilcation Operation. > > The Above relationships are only valid of LTI system ?? or they also valid > for Non-Linear, Non-Time Invaraint System ?? > > Thanks in advance.To reiterate after the discussion... Linear and Time invariance are two distinct properties, neither one implies the other. Linearity is all that is required for a system to be characterized by an impulse response FUNCTION. Normally this is a function of two variables, the "current" time and the time of the impulse. In the case of a TIME INVARIANT linear system, the response to an impulse depends only on the difference between the current time and the time of the impulse so only a single parameter is needed. I misspoke earlier when I suggested that convolution can be used for a linear system that is not time invariant. Both linearity and time invariance are required for convolution or transfer functions to be usable to calculate the response to an arbitrary input. It should be noted that even for time-varying linear systems, the response to an arbitrary input can be calculated by an integral/summation. It is just a matter of "decomposing" the arbitrary input into a sum of impulses of different amplitudes at different times and then, by linearity, summing their responses. HTH
Reply by ●February 16, 20052005-02-16
"robert bristow-johnson" <rbj@audioimagination.com> wrote in message news:BE37A0C2.482A%rbj@audioimagination.com...> in article AUmQd.5744$NO3.572@newssvr31.news.prodigy.com, Brad Griffis at > bradgriffis@hotmail.com wrote on 02/15/2005 08:35: > >> The impulse response, h[n], and the convolution operation are DEFINED >> based >> on a system being LTI. > > Brad, i have to disagree with this statement. any system, whether it's > LTI > or not, can be banged with a unit impulse and be allowed to respond (or > "ring") and that response, whether it's an LTI system or not, would the be > "impulse response". if the system is not LTI, i would think that the > impulse response has less meaning than it does for an LTI system where the > impulse response completely characterizes the system from an input-output > perspective. you could not say as much form the impulse response of a > system that is not LTI.True enough. Point taken.> the convolution operation is *not* defined but is a *consequence* of the > fact that the LTI system is linear and time-invariant. you can derive it > from the LTI properties and, particularly for a discrete-time system, it's > pretty easy. the only DEFINED properties of an LTI system that i can > think > of is that it is linear and that it is time-invariant. there are many > other > properties, but they can be derived.Convolution itself is a definition. (See the triangle over the equal sign at web page below.) http://ccrma.stanford.edu/~jos/filters/Convolution_Representation.html Calculating the output of an LTI filter using convolution is a consequence of linearity and time-invariance. When convolution was originally defined it was not the whim of a madman. It was created methodically based on calculating the output of an LTI filter. Hence my statement that its DEFINITION was based on LTI filters. It's the physical correspondence that is a consequence. Brad
