Hi folks, for those who are bored on this sunny winter afternoon, a riddle: Assume you are given a continuous but not necessarily band-limited periodic function f, and some arbitrary time constant T (not related to the period of f). Is it possible to regularly sample f with sampling period T' >= T, such that f can be exactly reconstructed from the samples? Yes / No answers are not accepted. :-) Regards, Andor
DSP riddle
Started by ●February 14, 2006
Reply by ●February 14, 20062006-02-14
Andor wrote:> Hi folks, > > for those who are bored on this sunny winter afternoon, a riddle: > > Assume you are given a continuous but not necessarily band-limited > periodic function f, and some arbitrary time constant T (not related to > the period of f). Is it possible to regularly sample f with sampling > period T' >= T, such that f can be exactly reconstructed from the > samples? > > Yes / No answers are not accepted. :-)But "Is it possible ..." is a Yes / No question! The answer is no for the general case of unspecified waveform, possibly not bandlimited. Nyquist tells us so. A more interesting question would be, "What are the conditions that make it possible?" I don't have a good answer (but I have a bad one). Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 14, 20062006-02-14
How long do I get to regluarly sample for? If I know T and choose T' appropriately (for example, choose T' to be an irrational multiple of T), won't I eventually get samples that are dense over a period, and eventually be able to reconstruct f arbitrarily well? rif
Reply by ●February 14, 20062006-02-14
Andor wrote:> Hi folks, > > for those who are bored on this sunny winter afternoon, a riddle: > > Assume you are given a continuous but not necessarily band-limited > periodic function f, and some arbitrary time constant T (not related to > the period of f). Is it possible to regularly sample f with sampling > period T' >= T, such that f can be exactly reconstructed from the > samples? >In the general case the answer is NO. If it is known that your function can be exactly described with some finite parametric model, then you can determine the parameters of the model from the samples. In this case the answer is YES. Vladimir Vassilevsky DSP and Mixed-Up Signal Design Consultant http://www.abvolt.com
Reply by ●February 14, 20062006-02-14
Jerry Avins wrote:> Andor wrote: > > Hi folks, > > > > for those who are bored on this sunny winter afternoon, a riddle: > > > > Assume you are given a continuous but not necessarily band-limited > > periodic function f, and some arbitrary time constant T (not related to > > the period of f). Is it possible to regularly sample f with sampling > > period T' >= T, such that f can be exactly reconstructed from the > > samples? > > > > Yes / No answers are not accepted. :-) > > But "Is it possible ..." is a Yes / No question!It would be boring if you wrote "no" and I wrote "wrong!" :-)> The answer is no for the general case of unspecified waveform, possibly > not bandlimited. Nyquist tells us so.I disagree. As a second question I was thinking of asking: "If this were possible, would it contradict the Nyquist sampling theorem?". Hint to answer that second question: what _exactly_ is the Nyquist theorem, and what _exactly_ is its logical inversion?> A more interesting question would be, "What are the conditions that make > it possible?" I don't have a good answer (but I have a bad one).Obviously, this riddle is only fun because it is possible (under the conditions I stated) - I think I can even extend this to non-continuous and non-bandlimited functions, but for starters try to solve the riddle as it is stated! Regards, Andor
Reply by ●February 14, 20062006-02-14
Andor wrote:> Jerry Avins wrote: > >>Andor wrote: >> >>>Hi folks, >>> >>>for those who are bored on this sunny winter afternoon, a riddle: >>> >>>Assume you are given a continuous but not necessarily band-limited >>>periodic function f, and some arbitrary time constant T (not related to >>>the period of f). Is it possible to regularly sample f with sampling >>>period T' >= T, such that f can be exactly reconstructed from the >>>samples? >>> >>>Yes / No answers are not accepted. :-) >> >>But "Is it possible ..." is a Yes / No question! > > > It would be boring if you wrote "no" and I wrote "wrong!" :-) > > >>The answer is no for the general case of unspecified waveform, possibly >>not bandlimited. Nyquist tells us so. > > > I disagree. As a second question I was thinking of asking: "If this > were possible, would it contradict the Nyquist sampling theorem?". > Hint to answer that second question: what _exactly_ is the Nyquist > theorem, and what _exactly_ is its logical inversion? > > >>A more interesting question would be, "What are the conditions that make >>it possible?" I don't have a good answer (but I have a bad one). > > > Obviously, this riddle is only fun because it is possible (under the > conditions I stated) - I think I can even extend this to non-continuous > and non-bandlimited functions, but for starters try to solve the riddle > as it is stated!As I read the question, this is a paraphrase: Assume you are given any continuous function. Is it possible to reconstruct it from samples taken at any arbitrarily long repetition time? Proof of the affirmative would revolutionize signal processing and much of mathematics. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●February 14, 20062006-02-14
Don't forget 'periodic'. Jerry Avins wrote:> Andor wrote: > > Jerry Avins wrote: > > > >>Andor wrote: > >> > >>>Hi folks, > >>> > >>>for those who are bored on this sunny winter afternoon, a riddle: > >>> > >>>Assume you are given a continuous but not necessarily band-limited > >>>periodic function f, and some arbitrary time constant T (not related to > >>>the period of f). Is it possible to regularly sample f with sampling > >>>period T' >>>>samples? > >>> > >>>Yes / No answers are not accepted. :-) > >> > >>But "Is it possible ..." is a Yes / No question! > > > > > > It would be boring if you wrote "no" and I wrote "wrong!" :-) > > > > > >>The answer is no for the general case of unspecified waveform, possibly > >>not bandlimited. Nyquist tells us so. > > > > > > I disagree. As a second question I was thinking of asking: "If this > > were possible, would it contradict the Nyquist sampling theorem?". > > Hint to answer that second question: what _exactly_ is the Nyquist > > theorem, and what _exactly_ is its logical inversion? > > > > > >>A more interesting question would be, "What are the conditions that make > >>it possible?" I don't have a good answer (but I have a bad one). > > > > > > Obviously, this riddle is only fun because it is possible (under the > > conditions I stated) - I think I can even extend this to non-continuous > > and non-bandlimited functions, but for starters try to solve the riddle > > as it is stated! > > As I read the question, this is a paraphrase: Assume you are given any > continuous function. Is it possible to reconstruct it from samples taken > at any arbitrarily long repetition time? > > Proof of the affirmative would revolutionize signal processing and much > of mathematics. > > Jerry > -- > Engineering is the art of making what you want from things you can get. > =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF==AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF= =AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF=AF
Reply by ●February 14, 20062006-02-14
Andor wrote: /*snip*/> Assume you are given a continuous but not necessarily band-limited > periodic function f, and some arbitrary time constant T (not related to > the period of f)./*snip*/ Are you saying that f is not necessarily band-limited nor periodic? Or do you mean that f is periodic but not necessarily band-limited? -- Jani Huhtanen Tampere University of Technology, Pori
Reply by ●February 14, 20062006-02-14
Jani Huhtanen wrote:> Andor wrote: > /*snip*/ > > Assume you are given a continuous but not necessarily band-limited > > periodic function f, and some arbitrary time constant T (not related to > > the period of f). > /*snip*/ > > Are you saying that f is not necessarily band-limited nor periodic? Or do > you mean that f is periodic but not necessarily band-limited?The latter (a missing coma).
Reply by ●February 14, 20062006-02-14
"Andor" <andor.bariska@gmail.com> writes:> Hi folks, > > for those who are bored on this sunny winter afternoon, a riddle: > > Assume you are given a continuous but not necessarily band-limited > periodic function f, and some arbitrary time constant T (not related to > the period of f). Is it possible to regularly sample f with sampling > period T' >= T, such that f can be exactly reconstructed from the > samples?Andor, your problem statement seems ill-posed by the simultaneous presence of "not necessarily band-limited" and "is it possible." Which is it? For example, it is certainly possible if the input is band-limited. I.e., not necessarily bandlimited also means not necessarily not bandlimited. I think what you are trying to ask is this: Does a periodic, non-bandlimited signal f(t) and sampling period T exist such that f(t) sampled at T-second intervals can be exactly reconstructed? -- % Randy Yates % "With time with what you've learned, %% Fuquay-Varina, NC % they'll kiss the ground you walk %%% 919-577-9882 % upon." %%%% <yates@ieee.org> % '21st Century Man', *Time*, ELO http://home.earthlink.net/~yatescr
