Jerry Avins wrote:> Rune Allnor wrote: > > ... > >> Nyquist's criterion for a single sideband IQ signal >> (which formally is complex-valued) is Fs >= BW. > > Almost. Fs > BW. When the inequality isn't large enough, there are at > least two practical difficulties. > > JerryOne of the really nice things about complex sampling is FS < BW can be so very useful. With non-complex sampling, the folding effects make unwrapping the aliases messy. With complex sampling, correlating across the aliases at multiple sanpling rates provides very powerful solutions. In these cases, a true full 100% of the bandwidth might be available for real use. Many defence systems rely on this kind of multirate approach. Steve
Nyquist constrain and IQ represented signal
Started by ●November 14, 2007
Reply by ●November 15, 20072007-11-15
Reply by ●November 15, 20072007-11-15
Chris Bore wrote: ...> Now to the complex versus real. The Fourier transform is defined in > terms of sinusoids, specifically complex sinusoids that can be > expressed in the form exp(jwt+p) where j is sqrt(-1). I have for some > time taken this to imply that all signals processed by an FT are > assumed to have complex values.I take issue with that, Chris. Representing sine and cosine by counter-rotating complex exponentials can be wonderfully simplifying and even lead to grand insights, but it is a mistake to link that to the fundamental nature of the Fourier transform. While the FFT depends on that decomposition, the Fourier transform itself does not. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 15, 20072007-11-15
Steve Underwood wrote:> Jerry Avins wrote: >> Rune Allnor wrote: >> >> ... >> >>> Nyquist's criterion for a single sideband IQ signal >>> (which formally is complex-valued) is Fs >= BW. >> >> Almost. Fs > BW. When the inequality isn't large enough, there are at >> least two practical difficulties. >> >> Jerry > > One of the really nice things about complex sampling is FS < BW can be > so very useful. With non-complex sampling, the folding effects make > unwrapping the aliases messy. With complex sampling, correlating across > the aliases at multiple sanpling rates provides very powerful solutions. > In these cases, a true full 100% of the bandwidth might be available for > real use. Many defence systems rely on this kind of multirate approach.I'm puzzled. How is resampling at a different rate possible after aliasing already happened? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 15, 20072007-11-15
Jerry Avins wrote:> Rune Allnor wrote:>> Nyquist's criterion for a single sideband IQ signal >> (which formally is complex-valued) is Fs >= BW.> Almost. Fs > BW. When the inequality isn't large enough, > there are at least two practical difficulties.Is an SSB signal allowed to have f=0? I would guess not, since it couldn't be separated for the USB and LSB signals. -- glen
Reply by ●November 15, 20072007-11-15
glen herrmannsfeldt wrote:> Jerry Avins wrote: > >> Rune Allnor wrote: > >>> Nyquist's criterion for a single sideband IQ signal >>> (which formally is complex-valued) is Fs >= BW. > >> Almost. Fs > BW. When the inequality isn't large enough, there are at >> least two practical difficulties. > > Is an SSB signal allowed to have f=0? > > I would guess not, since it couldn't be separated for > the USB and LSB signals.I'm not sure what you mean. A complex signal can be shifted in frequency by any amount. f=0 can be outside the band, at either edge, or anywhere within it. If you don't like it there, it can always be shifted again. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 15, 20072007-11-15
Jerry Avins wrote:> Steve Underwood wrote: >> Jerry Avins wrote: >>> Rune Allnor wrote: >>> >>> ... >>> >>>> Nyquist's criterion for a single sideband IQ signal >>>> (which formally is complex-valued) is Fs >= BW. >>> >>> Almost. Fs > BW. When the inequality isn't large enough, there are at >>> least two practical difficulties. >>> >>> Jerry >> >> One of the really nice things about complex sampling is FS < BW can be >> so very useful. With non-complex sampling, the folding effects make >> unwrapping the aliases messy. With complex sampling, correlating >> across the aliases at multiple sanpling rates provides very powerful >> solutions. In these cases, a true full 100% of the bandwidth might be >> available for real use. Many defence systems rely on this kind of >> multirate approach. > > I'm puzzled. How is resampling at a different rate possible after > aliasing already happened? > > JerryYou are thinking serially. Think parallel. You sample the original domain at multiple rates, typically rates which are mutually prime. The spacing of the aliases changes with the sampling rate. If the rates are carefully chosen, you can scan for the aliases which line up across the various sampling rates, and that is the real deal. If you have things in the environment which obscure parts of your signal space (e.g. regions of clutter in radar), a few redundant sampling rates in the mix let you be more crud tolerant during the cross rate correlation procedure. Think of a pulse doppler radar. Its transmitted pulse rate is actually a sampling frequency, sampling the world around it. Try looking up how a radar like the one in AWACS works. I guess there should be enough detail on the web to get a fairly clear picture. I think it cycles around 5 different pulse rates in its main surveillance mode. Steve
Reply by ●November 16, 20072007-11-16
Jerry Avins wrote:> glen herrmannsfeldt wrote:>> Jerry Avins wrote:>>> Rune Allnor wrote:>>>> Nyquist's criterion for a single sideband IQ signal >>>> (which formally is complex-valued) is Fs >= BW.>>> Almost. Fs > BW. When the inequality isn't large enough, there are at >>> least two practical difficulties.>> Is an SSB signal allowed to have f=0?>> I would guess not, since it couldn't be separated for >> the USB and LSB signals.> I'm not sure what you mean. A complex signal can be shifted in frequency > by any amount. f=0 can be outside the band, at either edge, or anywhere > within it. If you don't like it there, it can always be shifted again.Mathematically, the bandwidth is the same for a closed interval, an open interval, or a half closed half open interval. You want to say Fs > BW to exclude the upper limit. (Fs=BW) It is equally valid to exclude zero, but this is normally not done. In the case of an SSB signal, it seems reasonable to exclude f=0 from the demodulated signal, and so allow Fs=BW. (Not that I am at all sure about SSB modulation of complex signals.) -- glen
Reply by ●November 16, 20072007-11-16
Steve Underwood wrote:> Jerry Avins wrote: >> Steve Underwood wrote: >>> Jerry Avins wrote: >>>> Rune Allnor wrote: >>>> >>>> ... >>>> >>>>> Nyquist's criterion for a single sideband IQ signal >>>>> (which formally is complex-valued) is Fs >= BW. >>>> >>>> Almost. Fs > BW. When the inequality isn't large enough, there are >>>> at least two practical difficulties. >>>> >>>> Jerry >>> >>> One of the really nice things about complex sampling is FS < BW can >>> be so very useful. With non-complex sampling, the folding effects >>> make unwrapping the aliases messy. With complex sampling, correlating >>> across the aliases at multiple sanpling rates provides very powerful >>> solutions. In these cases, a true full 100% of the bandwidth might be >>> available for real use. Many defence systems rely on this kind of >>> multirate approach. >> >> I'm puzzled. How is resampling at a different rate possible after >> aliasing already happened? >> >> Jerry > > You are thinking serially. Think parallel. > > You sample the original domain at multiple rates, typically rates which > are mutually prime. The spacing of the aliases changes with the sampling > rate. If the rates are carefully chosen, you can scan for the aliases > which line up across the various sampling rates, and that is the real > deal. If you have things in the environment which obscure parts of your > signal space (e.g. regions of clutter in radar), a few redundant > sampling rates in the mix let you be more crud tolerant during the cross > rate correlation procedure.If you count the total number of samples taken that way, aliasing shouldn't be a problem. ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●November 16, 20072007-11-16
On Nov 16, 9:38 am, glen herrmannsfeldt <g...@ugcs.caltech.edu> wrote:> Jerry Avins wrote: > > glen herrmannsfeldt wrote: > >> Jerry Avins wrote: > >>> Rune Allnor wrote: > >>>> Nyquist's criterion for a single sideband IQ signal > >>>> (which formally is complex-valued) is Fs >= BW. > >>> Almost. Fs > BW. When the inequality isn't large enough, there are at > >>> least two practical difficulties. > >> Is an SSB signal allowed to have f=0? > >> I would guess not, since it couldn't be separated for > >> the USB and LSB signals. > > I'm not sure what you mean. A complex signal can be shifted in frequency > > by any amount. f=0 can be outside the band, at either edge, or anywhere > > within it. If you don't like it there, it can always be shifted again. > > Mathematically, the bandwidth is the same for a closed interval, an > open interval, or a half closed half open interval. You want to > say Fs > BW to exclude the upper limit. (Fs=BW) It is equally > valid to exclude zero, but this is normally not done. In the > case of an SSB signal, it seems reasonable to exclude f=0 from > the demodulated signal, and so allow Fs=BW. (Not that I am > at all sure about SSB modulation of complex signals.) >i think, strictly speaking, that if you had an "audio" signal with some non-zero DC ideally doing SSB modulation, then the signal you would get for the USB and the other signal you would get for the LSB should add to precisely the DSB/SC (that is "double sideband/ suppressed carrier"). so half of the DC component should be living in both the USB and LSB. i think if you create, out of audio x(t), an ideal USB signal with "suppressed carrier" at 0 Hz, that would be half of the "analytic signal": USB = 1/2 * ( x(t) + j*Hilbert{ x(t) } ) for the LSB it's: LSB = 1/2 * ( x(t) - j*Hilbert{ x(t) } ) note that when you add the upper sideband and lower sideband together you get full baseband signal. this is also what would happen if you did this all modulated up to some RF f0. since no DC gets through the Hilbert Transformer, 1/2 of the DC exists in the USB and 1/2 is in the LSB. r b-j
Reply by ●November 16, 20072007-11-16
glen herrmannsfeldt wrote:> robert bristow-johnson wrote: > > (snip) > >>> Mathematically, the bandwidth is the same for a closed interval, an >>> open interval, or a half closed half open interval. You want to >>> say Fs > BW to exclude the upper limit. (Fs=BW) It is equally >>> valid to exclude zero, but this is normally not done. In the >>> case of an SSB signal, it seems reasonable to exclude f=0 from >>> the demodulated signal, and so allow Fs=BW. (Not that I am >>> at all sure about SSB modulation of complex signals.) > >> i think, strictly speaking, that if you had an "audio" signal with >> some non-zero DC ideally doing SSB modulation, then the signal you >> would get for the USB and the other signal you would get for the LSB >> should add to precisely the DSB/SC (that is "double sideband/ >> suppressed carrier"). so half of the DC component should be living in >> both the USB and LSB. > > Trying not to stretch this too far, what does it mean to be > a sideband? Doesn't "sideband" exclude the carrier? It is > supposed to be on (one or both) sides of the carrier.It's possible in theory to put any frequency within the sideband at 0 Hz. In practice, the necessary quadrature is hard to come by. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
