Folks, Can someone let me know what one means by frequency domain. Initially, I was trying hard to understand what exactly is a frequency component in a time domain signal (could not see any convergence of the two) When I read through the materials, I get to understand that it is just representation of a waveform in another form. It is more of a convenient way to represent in a different way. For example (dont laugh at this): I can represent 5 as 3+2 I hope someone will respond to this question since this is so fundamental to understanding most of the DSP topics ...
Frequency domain representation
Started by ●July 21, 2011
Reply by ●July 21, 20112011-07-21
On Jul 21, 12:12�pm, "manishp" <manishp.p18@n_o_s_p_a_m.gmail.com> wrote:> Folks, > > Can someone let me know what one means by frequency domain. > Initially, I was trying hard to understand what exactly is a frequency > component in a time domain signal (could not see any convergence of the > two) > > When I read through the materials, I get to understand that it is just > representation of a waveform in another form. It is more of a convenient > way to represent in a different way. > > For example (dont laugh at this): I can represent 5 as 3+2 > > I hope someone will respond to this question since this is so fundamental > to understanding most of the DSP topics ...I think that I already answered this question in your other thread. Let us know if I left anything unclear and I'll try to fix that. Jerry -- Engineering is the art of making what you want from things you can get.
Reply by ●July 21, 20112011-07-21
On Jul 21, 12:34�pm, Jerry Avins <j...@ieee.org> wrote:> On Jul 21, 12:12�pm, "manishp" <manishp.p18@n_o_s_p_a_m.gmail.com> > wrote: > > > Folks, > > > Can someone let me know what one means by frequency domain. > > Initially, I was trying hard to understand what exactly is a frequency > > component in a time domain signal (could not see any convergence of the > > two) > > > When I read through the materials, I get to understand that it is just > > representation of a waveform in another form. It is more of a convenient > > way to represent in a different way. > > > For example (dont laugh at this): I can represent 5 as 3+2 > > > I hope someone will respond to this question since this is so fundamental > > to understanding most of the DSP topics ... > > I think that I already answered this question in your other thread. > Let us know if I left anything unclear and I'll try to fix that.I should add that one reason we use "domain" instead of "representation" is that the applicable tools of analysis are so different from one domain to the other. A few tools, Fourier analysis, for example, allow us to move between domains, but others are useful only in one domain or the other. Jerry -- Engineering is the art of making what you want from things you can get.
Reply by ●July 21, 20112011-07-21
On Thu, 21 Jul 2011 11:12:58 -0500, manishp wrote:> Folks, > > Can someone let me know what one means by frequency domain. Initially, I > was trying hard to understand what exactly is a frequency component in a > time domain signal (could not see any convergence of the two) > > When I read through the materials, I get to understand that it is just > representation of a waveform in another form. It is more of a convenient > way to represent in a different way. > > For example (dont laugh at this): I can represent 5 as 3+2 > > I hope someone will respond to this question since this is so > fundamental to understanding most of the DSP topics ...I should write a little white paper for my web site. Most people aren't smart enough to realize that there's two distinct domains in which to think, absorb a bunch of frequency domain rules, mix them with what they already 'know', add glue, shake, and get all confused... I say: The frequency domain is the mathematical domain you work in when you're using the Fourier transform or its cousins to solve problems in dynamic systems analysis or signal processing. You say: Uh, that sounds nice, but what does it _mean_? I: Waaaay back in the way back (with a phase shift of about -10^-9 radians/ Hz), Joseph Fourier (http://en.wikipedia.org/wiki/Joseph_Fourier) figured out that you could solve linear differential equations much easier if you changed the representation of the variables you were interested in. Basically, he asserted that you could express a function of a variable as a sum of sines of the variable. Then, if you were dealing with a linear system of differential equations, you could find the system's response to each of those sine waves, and use the superposition property of linear systems to put those individual solutions together into a 'real' solution. So what all this frequency domain fuss boils down to is that 'under the hood' you're making an assumption that your signal is going to be processed by a linear system and that it'll be easier (far easier!) to do the math if things are in terms of frequency, rather than time. http://en.wikipedia.org/wiki/Fourier_Transform http://en.wikipedia.org/wiki/Laplace_transform http://en.wikipedia.org/wiki/Z_transform You: So because there are so many linear systems in the world, we use the frequency domain? I: Wrong-o! There are _no_ truly linear systems in the world. A linear system is a platonic ideal, like a perfect sphere, an infinite plane, or an honest politician*. It's something that you can define, but never see. You: So this is all just academic wanking??!!?? I: No. There are a lot of systems that come quite close to being linear. Common enough and close enough that the whole frequency domain thing can be used quite profitably. But if you ignore their limitations, there are some pretty deep holes that you can fall into. You: So what can I _do_ with this? I: Pretty much what Fourier did, only with more attention paid to the niceties**. Because it's pretty easy to build a linear resonator that discriminates frequencies, because antennas are roughly linear systems in their response to electromagnetic waves _and_ are frequency sensitive, because hearing and sound transmission and baseband electronic signal transmission is all frequency sensitive, expressing signals in terms of the frequency domain is about as close to natural as you can get. Using the Fourier transform and its cousins then makes the math a lot easier. You: So that's _it_? All that trouble running my brain through a blender to understand the Four Flavors of the Fourier Transform (and Laplace, and z), and it's just convenience to make the math easier? I: Yup. But that's a pretty big "it" -- solving linear differential equations without resort to the frequency domain is difficult, and must be re-done for each specific system encountered. Solving them with Fourier &c is not only easy, but easy to do in generalizations -- so you can talk about a bandpass filter, without having to specify the _exact_ filter. So, looking at things in the frequency domain is just a way of turning the whole world 90 degrees and taking another look (and, initially, not having the mental tools to perceive what you're looking at). It's a bunch of trouble, there's all these picky rules that you have to pay attention to lest you fall into a hole, when you're done you have to think about your results before they'll make sense, you'll be constantly getting details wrong even after you've been doing it for thirty years -- and it's still infinitely easier than trying to get by without. ========================= * I'm slandering my own father here, who was chairman of the board of directors of the Boring Volunteer Fire Department in Oregon for years, then mayor of Damascus Oregon for a while, all the time being 99.44% straight up with his public -- and if you're going to quibble about the 0.56%, consider that it mostly came about with carefully coming right against the public meetings laws without stepping over, so that necessary things could get done in a hurry. Then think about whether _you_ manage to be honest more than 99.44% of the time, and how your relations with your friends and family would be if you always answered honestly when asked "does this dress make my butt look big?". But I digress. ** Fourier got some details wrong. Interestingly enough, if I'm reading Wikipedia right and if they've got it right, it's details that Fourier newbies are still getting wrong today. Joseph Fourier: the father of the misuse of the Fourier Transform. -- www.wescottdesign.com
Reply by ●July 21, 20112011-07-21
On Jul 21, 1:05�pm, Tim Wescott <t...@seemywebsite.com> wrote:> ... �Joseph Fourier: the father of the misuse of the Fourier Transform.That was a great example of the kind of math that newcomers need. We get so accustomed to knowing what we mean that we miss the occasions when it needs to be made explicit. Jerry -- Engineering is the art of making what you want from things you can get.
Reply by ●July 21, 20112011-07-21
On Jul 21, 12:12�pm, "manishp" <manishp.p18@n_o_s_p_a_m.gmail.com> wrote:> Folks, > > Can someone let me know what one means by frequency domain. > Initially, I was trying hard to understand what exactly is a frequency > component in a time domain signal (could not see any convergence of the > two) > > When I read through the materials, I get to understand that it is just > representation of a waveform in another form. It is more of a convenient > way to represent in a different way. > > For example (dont laugh at this): I can represent 5 as 3+2 > > I hope someone will respond to this question since this is so fundamental > to understanding most of the DSP topics ...as 5 can be broken down into 3 + 2, a time domain function x(t) can also be broken down as a sum of components like x(t) = y(t) + z(t) if y(t) and z(t) happen to sinusoids that particular representation is considered to be a "frequency domain" representation (even though in reality the time domain function x(t) is still being represented by time domain functions y(t) and z(t)). y(t) and z(t) can be written as A cos(wt+ phase), so in the "frequency domain" you work with an array of three variables A's, w's and phases. And are given a set of tools to solve problems that manipulate this array. This simplifies analysis, in some cases. but in reality your still working in the time domain....
Reply by ●July 21, 20112011-07-21
On Thu, 21 Jul 2011 10:32:14 -0700, steve wrote:> On Jul 21, 12:12 pm, "manishp" <manishp.p18@n_o_s_p_a_m.gmail.com> > wrote: >> Folks, >> >> Can someone let me know what one means by frequency domain. Initially, >> I was trying hard to understand what exactly is a frequency component >> in a time domain signal (could not see any convergence of the two) >> >> When I read through the materials, I get to understand that it is just >> representation of a waveform in another form. It is more of a >> convenient way to represent in a different way. >> >> For example (dont laugh at this): I can represent 5 as 3+2 >> >> I hope someone will respond to this question since this is so >> fundamental to understanding most of the DSP topics ... > > as 5 can be broken down into 3 + 2, a time domain function x(t) can > also be broken down as a sum of components like x(t) = y(t) + z(t) > > if y(t) and z(t) happen to sinusoids that particular representation is > considered to be a "frequency domain" representation (even though in > reality the time domain function x(t) is still being represented by > time domain functions y(t) and z(t)). > > y(t) and z(t) can be written as A cos(wt+ phase), > > so in the "frequency domain" you work with an array of three variables > A's, w's and phases. And are given a set of tools to solve problems that > manipulate this array. This simplifies analysis, in some cases. > > but in reality your still working in the time domain....There are two things that seem hardest to keep in mind. First is that you really are stuck in the time domain, even if you're doing your analysis in the frequency domain. Second is that -- strictly speaking -- frequency domain analysis only applies to linear system (and Laplace only to linear time-invariant systems). Lose sight of these and you won't fail to analyze your system -- you'll just come up with answers that appear perfectly reasonable yet are totally wrong. -- www.wescottdesign.com
Reply by ●July 21, 20112011-07-21
Jerry Avins <jya@ieee.org> wrote: (snip on frequency domain)> I should add that one reason we use "domain" instead of > "representation" is that the applicable tools of analysis are so > different from one domain to the other. A few tools, Fourier analysis, > for example, allow us to move between domains, but others are useful > only in one domain or the other.It seems that physics often uses "space" instead of "domain" or "representation." That is especially true in more than one dimension. The Fourier transform of position (after multiplying by hbar) is momentum, (in three dimensions), and so such a description is called momentum space. Pretty much the same tools as used in DSP work in the position <--> momentum case. In only one dimension, time and energy (again, a factor of hbar) are Fourier transforms of each other. -- glen
Reply by ●July 21, 20112011-07-21
On 21/07/2011 17:12, manishp wrote:> Folks, > > Can someone let me know what one means by frequency domain. > Initially, I was trying hard to understand what exactly is a frequency > component in a time domain signal (could not see any convergence of the > two) > > When I read through the materials, I get to understand that it is just > representation of a waveform in another form. It is more of a convenient > way to represent in a different way. > > For example (dont laugh at this): I can represent 5 as 3+2 > > I hope someone will respond to this question since this is so fundamental > to understanding most of the DSP topics ...Sometime a picture is worth 1000 words (or formulae), and an animated applet even more. Try: http://www.falstad.com/fourier/ (being Java it may glitch on slow machines). And use the "magnitude + phase" option. You see the waveform, and the components by which is it constructed. The waveform is the time-domain picture of the signal, and the mag/phase plot is the frequency-domain picture. And of course you can listen to it as you make changes. Here is a (non-rigorous) description with ~no maths~. The frequency domain has (by definition) no time dimension. This is a bit like defining "Concert A = 440Hz" - it does not say how long it is, simply because however long it is (even infinitely long) it is always Concert A. To analyse the sound of an instrument playing Concert A (oboe or trumpet?) we don't care how long the tone is so long as we have enough, which in practice is a small fraction of a second (in theory, just one complete cycle - dsp can cope with this easily but the human ear cannot - it needs quite a long chunk of sound to recognise the pitch). To tell the difference, we can either inspect the waveform (time-domain - tricky, the eye is easily deceived), or analyse it to find out how much of each harmonic there is, as that is what makes the instruments distinct (frequency domain - a much better description, and is really how our ears work). A 2-second Concert A gives you virtually no more information than a 0.1-second Concert A. The frequency domain picture (excluding all that tiresome transient stuff), = the description of the intensity and phase of each harmonic, is the same in each case. So in a sense the frequency domain is the "natural" and most compact way to describe a sound (or of course something visual - the rainbow; it is irrelevant how long you look at it for). The time domain is needed however (including your decision about how long it will be) when you want to actually play the note and record it; or when you have reason to be interested in the waveform shape. A standard demonstration is to take a classic waveform such as the square wave and mess with the component phases (as you can do in the applet above). The shape changes dramatically (nothing like a square any more), but it sounds identical. HTH, Richard Dobson
Reply by ●July 21, 20112011-07-21
>On 21/07/2011 17:12, manishp wrote: >> Folks, >> >> Can someone let me know what one means by frequency domain. >> Initially, I was trying hard to understand what exactly is a frequency >> component in a time domain signal (could not see any convergence of the >> two) >> >> When I read through the materials, I get to understand that it is just >> representation of a waveform in another form. It is more of aconvenient>> way to represent in a different way. >> >> For example (dont laugh at this): I can represent 5 as 3+2 >> >> I hope someone will respond to this question since this is sofundamental>> to understanding most of the DSP topics ... > >Sometime a picture is worth 1000 words (or formulae), and an animated >applet even more. Try: > >http://www.falstad.com/fourier/ > >(being Java it may glitch on slow machines). > >And use the "magnitude + phase" option. You see the waveform, and the >components by which is it constructed. The waveform is the time-domain >picture of the signal, and the mag/phase plot is the frequency-domain >picture. And of course you can listen to it as you make changes. > >Here is a (non-rigorous) description with ~no maths~. > >The frequency domain has (by definition) no time dimension. This is a >bit like defining "Concert A = 440Hz" - it does not say how long it is, >simply because however long it is (even infinitely long) it is always >Concert A. To analyse the sound of an instrument playing Concert A (oboe >or trumpet?) we don't care how long the tone is so long as we have >enough, which in practice is a small fraction of a second (in theory, >just one complete cycle - dsp can cope with this easily but the human >ear cannot - it needs quite a long chunk of sound to recognise the >pitch). To tell the difference, we can either inspect the waveform >(time-domain - tricky, the eye is easily deceived), or analyse it to >find out how much of each harmonic there is, as that is what makes the >instruments distinct (frequency domain - a much better description, and >is really how our ears work). > >A 2-second Concert A gives you virtually no more information than a >0.1-second Concert A. The frequency domain picture (excluding all that >tiresome transient stuff), = the description of the intensity and phase >of each harmonic, is the same in each case. So in a sense the frequency >domain is the "natural" and most compact way to describe a sound (or of >course something visual - the rainbow; it is irrelevant how long you >look at it for). The time domain is needed however (including your >decision about how long it will be) when you want to actually play the >note and record it; or when you have reason to be interested in the >waveform shape. A standard demonstration is to take a classic waveform >such as the square wave and mess with the component phases (as you can >do in the applet above). The shape changes dramatically (nothing like a >square any more), but it sounds identical. > >HTH, > >Richard Dobson > >Thanks to everyone. I can't say I have understood everything that was posted on this thread. So, I will go through once/twice/thrice and get back if I have any more questions ...
