Forums

Linear System Properties?

Started by Till Crueger March 26, 2004
Randy Yates wrote:

> Jerry Avins <jya@ieee.org> writes: > > >>Randy Yates wrote: >> >> >>>Hey Till, >>>You're right - it isn't just sine waves that a linear system will not >>>produce new frequencies of - ANY waveform will be unaltered in >>>frequency by a linear system. >> >>Randy, > > > Jerry, > > I hear you (somewhat) but I'm going to play the opposite side here to > the hilt. > > >>This is true in a sense, > > > This is true absolutely. No "sense" or interpretation required. > > >>but misleading. > > > How can the truth be misleading? > > >>For example, you can't expect >>the output of a linear system to be a square wave just because the input >>is excited by one. > > > I do not expect that. What I do expect is that the output > will not contain any frequencies that weren't in the > input. > > >>The output may contain all the component frequencies >>of the input, but shape isn't necessarily maintained. > > > Did I say or imply the _shape_ was maintained? In fact I did not.
I thought he asked about shape, at least by implication.
> But beyond the question of what I said or didn't say, your comments seem > to be aimed at how one should explain something, and THAT depends on > style and technique. This is my style. Making the truth sharp, in my > experience, usually dispells bad conclusions and sheds light on wrong > thinking.
Till wants to know what unique property of sine waves makes them come through a linear circuit unaltered (presumably in shape) while square waves do not. However sharp your truths, I don't think your answer to him was transparent at his level of understanding. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
"Till Crueger" <TillFC@gmx.net> writes:

> On Fri, 26 Mar 2004 15:28:12 +0000, Randy Yates wrote: > >> Hey Till, >> >> You're right - it isn't just sine waves that a linear system will not >> produce new frequencies of - ANY waveform will be unaltered in frequency >> by a linear system. >> >> The special thing about sine waves are that they provide a fundamental >> set of functions from which ANY other waveform can be constructed, so if >> they aren't moved in frequency by a linear system, neither are any other >> waveforms. > > Hmm, I thought about that too. > However shouldn't it also be possible to construct any waveform from > square waves?
Yes, that is possible.
> If this is so, how come the shape of square waves is altered > in a linear system?
You can think of a square wave as the sum of many sine waves of specific amplitudes and phases. Even though a linear system passes the sine waves without changing their frequencies, it can change their amplitudes and/or phases, and therefore it can change their shape. I gave you the wrong reason before about why sine waves were special in this sense. A linear system can't change the shape of a sine wave (other than its amplitude or phase, that is) because a sine wave only consists of one frequency. In any other waveform, multiple frequencies are involved, and the shape of the waveform is determined by not only the frequencies but the phases and amplitudes as well. Thus a linear system may change the shape of such a waveform since it may change the phases and amplitudes of each frequency component.
> Furthermore I thought about a system which only would double any frequency > component present in a given Signal, and I fail to see how this system is > non-linear.
I've thought about this a bit and can't answer. Clay? Robert? -- % Randy Yates % "My Shangri-la has gone away, fading like %% Fuquay-Varina, NC % the Beatles on 'Hey Jude'" %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Shangri-La', *A New World Record*, ELO http://home.earthlink.net/~yatescr
Jerry Avins <jya@ieee.org> writes:
> [...] > Till wants to know what unique property of sine waves makes them come > through a linear circuit unaltered (presumably in shape) while square > waves do not. However sharp your truths, I don't think your answer to > him was transparent at his level of understanding.
You're right - I had provided the wrong reason to him, but not for the reason you stated (that sine waves don't change in frequency). --Randy <all black-and-blue from that beating...> -- % Randy Yates % "...the answer lies within your soul %% Fuquay-Varina, NC % 'cause no one knows which side %%% 919-577-9882 % the coin will fall." %%%% <yates@ieee.org> % 'Big Wheels', *Out of the Blue*, ELO http://home.earthlink.net/~yatescr
Randy Yates wrote:

(someone wrote)

>>For example, you can't expect >>the output of a linear system to be a square wave just because the input >>is excited by one.
> I do not expect that. What I do expect is that the output > will not contain any frequencies that weren't in the > input.
The original question included: "In a DSP book I read that from these properties you can conclude that a sine wave as input will always produce a sine wave on the systems output with the same frequency." Sine in sine out.
>>The output may contain all the component frequencies >>of the input, but shape isn't necessarily maintained.
> Did I say or imply the _shape_ was maintained? In fact I did not.
Sine IS special, in that the shape is maintained. If the shape isn't maintained, you are answering a different question, and that could be misleading. It doesn't seem to have come out yet that sine is special because it is the solution to the simplest linear second order differential equation. It is also important that such equations are the result of many important systems, including RLC filters and transmission lines. -- glen
Randy Yates <yates@ieee.org> writes:
> [...] > Even though a linear system passes the sine waves > without changing their frequencies, it can change their amplitudes and/or > phases, and therefore it can change their shape.
^^^^^ Aaaah! Sorry about that sentence, Till and everyone. A correct and clear wording would have been: Even though a linear system passes the sine waves without changing their frequencies, it can change their amplitudes and/or phases, and therefore it can change the shape of the square wave. -- % Randy Yates % "I met someone who looks alot like you, %% Fuquay-Varina, NC % she does the things you do, %%% 919-577-9882 % but she is an IBM." %%%% <yates@ieee.org> % 'Yours Truly, 2095', *Time*, ELO http://home.earthlink.net/~yatescr
In article <65crwh5t.fsf@ieee.org>, Randy Yates  <yates@ieee.org> wrote:
>Jerry Avins <jya@ieee.org> writes: >> Randy Yates wrote: >>> You're right - it isn't just sine waves that a linear system will not >>> produce new frequencies of - ANY waveform will be unaltered in >>> frequency by a linear system. >> This is true in a sense, >This is true absolutely. No "sense" or interpretation required.
Sense is absolutely required.
>> The output may contain all the component frequencies >> of the input, but shape isn't necessarily maintained. >Did I say or imply the _shape_ was maintained? In fact I did not.
The sense required is that a waveform altered in shape can be somehow said to be the "same" waveform unaltered in frequency, which is the result of assumptions and definitions that likely only mathematicians would find intuitive. However the English language wasn't invented by mathematicians, so I feel free to find statements such as these silly, unless the words are subscripted or footnoted to the effect that weird and not the common definitions are being used. So define your terms, and only then do you get "Sense" instead of nonsense. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
In article <c41jep$g2s$1@f1node01.rhrz.uni-bonn.de>,
Till Crueger <TillFC@gmx.net> wrote:
>Furthermore I thought about a system which only would double any frequency >component present in a given Signal, and I fail to see how this system is >non-linear.
This is not both linear and time-invarient. Shift the input by half the period and the output will shift by a full period. This is equivalent to inverting the gain depending on the starting phase. So either you get non-linear gain, or you lose time invariance. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
Randy Yates wrote:
> Jerry Avins <jya@ieee.org> writes: > >>[...] >>Till wants to know what unique property of sine waves makes them come >>through a linear circuit unaltered (presumably in shape) while square >>waves do not. However sharp your truths, I don't think your answer to >>him was transparent at his level of understanding. > > > You're right - I had provided the wrong reason to him, but not for the > reason you stated (that sine waves don't change in frequency). > > --Randy <all black-and-blue from that beating...> >
I think we're in this together, trying to help Till understand. However much we might disagree about which words are best to do that, we're not likely to confuse each other. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
In article <c41ho7$13u4$1@f1node01.rhrz.uni-bonn.de>,
Till Crueger <TillFC@gmx.net> wrote:
>Hi, >I have some simple questions about the properties of linear systems. In a >linear System we have to assume the properties of homogeneity, additivity >and shift invariance. >In a DSP book I read that from these properties you can conclude that a >sine wave as input will always produce a sine wave on the systems output >with the same frequency.
The trick is in knowing that this really isn't true. The output may look like a sine wave of the same frequency, but it's really the linear combination of a sine wave and a cosine wave of the same frequency which haven't been shifted in time. Try to find another set of basis functions where an arbitrary time shift of one looks just like a different linear combination of the same basis functions, or vice versa. IMHO. YMMV. -- Ron Nicholson rhn AT nicholson DOT com http://www.nicholson.com/rhn/ #include <canonical.disclaimer> // only my own opinions, etc.
Till Crueger wrote:

> On Fri, 26 Mar 2004 15:28:12 +0000, Randy Yates wrote: > > >>Hey Till, >> >>You're right - it isn't just sine waves that a linear system will not >>produce new frequencies of - ANY waveform will be unaltered in frequency >>by a linear system. >> >>The special thing about sine waves are that they provide a fundamental >>set of functions from which ANY other waveform can be constructed, so if >>they aren't moved in frequency by a linear system, neither are any other >>waveforms. > > > Hmm, I thought about that too. > However shouldn't it also be possible to construct any waveform from > square waves?
See the Harr wavelet. > If this is so, how come the shape of square waves is > altered in a linear system? Don't get the problem. The decomposition afterward will also be square waves but a different set.
> Furthermore I thought about a system which only would double any frequency > component present in a given Signal, and I fail to see how this system is > non-linear.
Is it realizable? Bob -- "Things should be described as simply as possible, but no simpler." A. Einstein