hi everyone, I am brand new to communication/DSP and I have a real silly question regarding Fourier Series and sinusoids. If I have a signal: v(t) = cos(2pi x f_1 x t) + cos(2pi x f_2 x t) where f_1 != f_2 Then would the exponential Fourier expansion of this just be v(t) = (0.5)exp(j(2pi x f_1x t)) + (0.5)exp(-j(2pi x f_1x t)) + (0.5)exp(j(2pi x f_2x t)) + (0.5)exp(-j(2pi x f_2x t)) ? Therefore the magnitude of each term is 0.5, phase is 0deg and values would only be present at +/-f_1 and +/-f_2 for the amplitude spectrum. I know this is a silly quesiton, but my instructor gave a problem that stated "find the Fourier series expression of" a signal similar to that. It didn't really make sense to me. Thanks in advance for the help.
Real silly Fourier Series question
Started by ●September 20, 2006
Reply by ●September 20, 20062006-09-20
tricard@gmail.com writes:> hi everyone, > > I am brand new to communication/DSP and I have a real silly question > regarding Fourier Series and sinusoids. If I have a signal: > > v(t) = cos(2pi x f_1 x t) + cos(2pi x f_2 x t) where f_1 != f_2 > > Then would the exponential Fourier expansion of this just be > > v(t) = (0.5)exp(j(2pi x f_1x t)) + (0.5)exp(-j(2pi x f_1x t)) + > (0.5)exp(j(2pi x f_2x t)) + (0.5)exp(-j(2pi x f_2x t)) ? > > Therefore the magnitude of each term is 0.5, phase is 0deg and values > would only be present at +/-f_1 and +/-f_2 for the amplitude spectrum. > > I know this is a silly quesiton, but my instructor gave a problem that > stated "find the Fourier series expression of" a signal similar to > that. It didn't really make sense to me.Hi, It appears to me you understand it perfectly. Your answer is perfectly correct. Please do not demean your questions. While there are folks who look down their noses on questions they think are "simple," many others (such as myself) understand that significant learning goes on at this level. (What other better place to begin that at the begninning?) It is a fine question, and there are many knowledgable and kind people here who would be willing to help. -- % Randy Yates % "Midnight, on the water... %% Fuquay-Varina, NC % I saw... the ocean's daughter." %%% 919-577-9882 % 'Can't Get It Out Of My Head' %%%% <yates@ieee.org> % *El Dorado*, Electric Light Orchestra http://home.earthlink.net/~yatescr
Reply by ●September 20, 20062006-09-20
Randy Yates <yates@ieee.org> writes:> tricard@gmail.com writes: > >> hi everyone, >> >> I am brand new to communication/DSP and I have a real silly question >> regarding Fourier Series and sinusoids. If I have a signal: >> >> v(t) = cos(2pi x f_1 x t) + cos(2pi x f_2 x t) where f_1 != f_2 >> >> Then would the exponential Fourier expansion of this just be >> >> v(t) = (0.5)exp(j(2pi x f_1x t)) + (0.5)exp(-j(2pi x f_1x t)) + >> (0.5)exp(j(2pi x f_2x t)) + (0.5)exp(-j(2pi x f_2x t)) ? >> >> Therefore the magnitude of each term is 0.5, phase is 0deg and values >> would only be present at +/-f_1 and +/-f_2 for the amplitude spectrum. >> >> I know this is a silly quesiton, but my instructor gave a problem that >> stated "find the Fourier series expression of" a signal similar to >> that. It didn't really make sense to me. > > Hi, > > It appears to me you understand it perfectly. Your answer is perfectly > correct. > > Please do not demean your questions. While there are folks who look > down their noses on questions they think are "simple," many others > (such as myself) understand that significant learning goes on at this > level. (What other better place to begin that at the begninning?) > > It is a fine question, and there are many knowledgable and kind people > here who would be willing to help.Arrrg! Wait! I think I need to back-peddle here a bit. If the instructor asked for the Fourier *SERIES* representation of this signal, then I think this might be a "trick" question and you're answer would be wrong. Recall that a Fourier series representation is only valid for a PERIODIC signal. Unless f_1 and f_2 are harmonically related, i.e., f_1 = n*f_2, the combination of the two cosines will not be periodic. Remember that a Fourier series is a set of integer indexed coefficients a_k, -\infty < k < +\infty, so that x(t) = \sum_{k=-\infty}^{+\infty} a_k e^{j\omega_0 t}. So there is an implicit fundamental frequency \omega_0 associated with the series. If the f_1 and f_2 aren't harmonically related, then there is no such fundamental frequency. -- % Randy Yates % "The dreamer, the unwoken fool - %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." %%% 919-577-9882 % %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO http://home.earthlink.net/~yatescr
Reply by ●September 20, 20062006-09-20
Randy Yates <yates@ieee.org> writes:> [...] > x(t) = \sum_{k=-\infty}^{+\infty} a_k e^{j\omega_0 t}.Correction: x(t) = \sum_{k=-\infty}^{+\infty} a_k e^{j*k*\omega_0 t}. -- % Randy Yates % "She's sweet on Wagner-I think she'd die for Beethoven. %% Fuquay-Varina, NC % She love the way Puccini lays down a tune, and %%% 919-577-9882 % Verdi's always creepin' from her room." %%%% <yates@ieee.org> % "Rockaria", *A New World Record*, ELO http://home.earthlink.net/~yatescr
Reply by ●September 20, 20062006-09-20
Thank you for the help; I am very glad that there are people who want to help :). I have been on many usenet sites in my day and have found many groups to be very unhelpful (and sometimes rude) when questions are asked.I am glad this group is not like that :). Tim Randy Yates wrote:> tricard@gmail.com writes: > > > hi everyone, > > > > I am brand new to communication/DSP and I have a real silly question > > regarding Fourier Series and sinusoids. If I have a signal: > > > > v(t) = cos(2pi x f_1 x t) + cos(2pi x f_2 x t) where f_1 != f_2 > > > > Then would the exponential Fourier expansion of this just be > > > > v(t) = (0.5)exp(j(2pi x f_1x t)) + (0.5)exp(-j(2pi x f_1x t)) + > > (0.5)exp(j(2pi x f_2x t)) + (0.5)exp(-j(2pi x f_2x t)) ? > > > > Therefore the magnitude of each term is 0.5, phase is 0deg and values > > would only be present at +/-f_1 and +/-f_2 for the amplitude spectrum. > > > > I know this is a silly quesiton, but my instructor gave a problem that > > stated "find the Fourier series expression of" a signal similar to > > that. It didn't really make sense to me. > > Hi, > > It appears to me you understand it perfectly. Your answer is perfectly > correct. > > Please do not demean your questions. While there are folks who look > down their noses on questions they think are "simple," many others > (such as myself) understand that significant learning goes on at this > level. (What other better place to begin that at the begninning?) > > It is a fine question, and there are many knowledgable and kind people > here who would be willing to help. > -- > % Randy Yates % "Midnight, on the water... > %% Fuquay-Varina, NC % I saw... the ocean's daughter." > %%% 919-577-9882 % 'Can't Get It Out Of My Head' > %%%% <yates@ieee.org> % *El Dorado*, Electric Light Orchestra > http://home.earthlink.net/~yatescr
Reply by ●September 20, 20062006-09-20
I think the non-periodicity of the two waves is one of the reasons why I got stuck :). The signal is x(t) = cos(t) + sin(2.5t) which means f_1 and f_2 are NOT harmonically related, so this would not have a Fourier Series representation. Very interesting. I suppose power couldn't be calculated from non-periodic signals either, since there would be no average area under a non-periodic wave? Tim> If the instructor asked for the Fourier *SERIES* representation > of this signal, then I think this might be a "trick" question > and you're answer would be wrong. > > Recall that a Fourier series representation is only valid for a > PERIODIC signal. Unless f_1 and f_2 are harmonically related, i.e., > f_1 = n*f_2, the combination of the two cosines will not be > periodic. > > Remember that a Fourier series is a set of integer indexed > coefficients a_k, -\infty < k < +\infty, so that > > x(t) = \sum_{k=-\infty}^{+\infty} a_k e^{j\omega_0 t}. > > So there is an implicit fundamental frequency \omega_0 associated > with the series. If the f_1 and f_2 aren't harmonically related, then > there is no such fundamental frequency. > -- > % Randy Yates % "The dreamer, the unwoken fool - > %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." > %%% 919-577-9882 % > %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO > http://home.earthlink.net/~yatescr
Reply by ●September 20, 20062006-09-20
Hi Tim, tricard@gmail.com writes:> I think the non-periodicity of the two waves is one of the reasons why > I got stuck :). The signal is x(t) = cos(t) + sin(2.5t) which means f_1 > and f_2 are NOT harmonically related, so this would not have a Fourier > Series representation. Very interesting.Right.> I suppose power couldn't be > calculated from non-periodic signals either, since there would be no > average area under a non-periodic wave?Yes, there is an average if the signal is a finite-power signal, but the average can change over time as the waveform changes, which won't happen with a periodic signal as long as the averaging time is >= the period. There are three classes of signals: finite energy (the area of the integral of the magnitude of the signal is finite), finite power (the total energy is infinite, but there is finite energy per unit time), and infinite power (infinite energy AND infinite power). --Randy> > Tim > >> If the instructor asked for the Fourier *SERIES* representation >> of this signal, then I think this might be a "trick" question >> and you're answer would be wrong. >> >> Recall that a Fourier series representation is only valid for a >> PERIODIC signal. Unless f_1 and f_2 are harmonically related, i.e., >> f_1 = n*f_2, the combination of the two cosines will not be >> periodic. >> >> Remember that a Fourier series is a set of integer indexed >> coefficients a_k, -\infty < k < +\infty, so that >> >> x(t) = \sum_{k=-\infty}^{+\infty} a_k e^{j\omega_0 t}. >> >> So there is an implicit fundamental frequency \omega_0 associated >> with the series. If the f_1 and f_2 aren't harmonically related, then >> there is no such fundamental frequency. >> -- >> % Randy Yates % "The dreamer, the unwoken fool - >> %% Fuquay-Varina, NC % in dreams, no pain will kiss the brow..." >> %%% 919-577-9882 % >> %%%% <yates@ieee.org> % 'Eldorado Overture', *Eldorado*, ELO >> http://home.earthlink.net/~yatescr >-- % Randy Yates % "How's life on earth? %% Fuquay-Varina, NC % ... What is it worth?" %%% 919-577-9882 % 'Mission (A World Record)', %%%% <yates@ieee.org> % *A New World Record*, ELO http://home.earthlink.net/~yatescr
Reply by ●September 21, 20062006-09-21
tricard@gmail.com wrote:> I think the non-periodicity of the two waves is one of the reasons why > I got stuck :). The signal is x(t) = cos(t) + sin(2.5t) which means f_1 > and f_2 are NOT harmonically related, so this would not have a Fourier > Series representation. Very interesting. I suppose power couldn't be > calculated from non-periodic signals either, since there would be no > average area under a non-periodic wave?... F_1 and f_2 are related. F-1 = 2f_0 and f_2 = 5f_0. For there to be no periodicity, f_1 and f_2 would need to have no common multiple. That's another way to say that f_1/f_2 is an irrational number. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●September 21, 20062006-09-21
tricard@gmail.com wrote:> Thank you for the help; I am very glad that there are people who want > to help :). I have been on many usenet sites in my day and have found > many groups to be very unhelpful (and sometimes rude) when questions > are asked.I am glad this group is not like that :).The groups you've become accustomed to must have many stupid people. In contrast, comp.dsp is peopled by a few near-geniuses and most of the rest are either humble, kind, or both. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●September 21, 20062006-09-21
Jerry Avins wrote:> tricard@gmail.com wrote: >> Thank you for the help; I am very glad that there are people who want >> to help :). I have been on many usenet sites in my day and have found >> many groups to be very unhelpful (and sometimes rude) when questions >> are asked.I am glad this group is not like that :). > > The groups you've become accustomed to must have many stupid people. In > contrast, comp.dsp is peopled by a few near-geniuses and most of the > rest are either humble, kind, or both. > > JerryOr have a rather weird sense of humor. :-) -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
