Re: Some questions on digital filter design
Is it really a breach of causality if the output that displays the breach doesn't actually occur until some processing delay after the input that caused the output? I remember discussing this stuff at length some 50 years ago in a class on linear systems.... all I remember is discussing it, not what we decided, nor how we got there. I do recall that simplicity in the math required that certain apparently non-causal things would appear in the results. There is a whole lot of blue ink underlining way too much, in the section on convolution of the impulse function, and causality, in my copy of Linear Systems Analysis by Liu & Liu... which, for me, usually meant I was hopelessly confused... -Chuck Harris On Sat, 6 Jul 2024 16:37:12 +0000 (UTC) "Reginald Beardsley via groups.io" <pulaskite@...> wrote: ?This?is?a?general?reply?to?Jeff,?Mike?and?Dave.?I?initially?looked
at:
Digital?Design?Handbook
Fred?J.?Taylor
Marcel?Dekker?1983
Digital?Signal?Processing
Oppenheim?and?Shafer
Prentice-Hall?1975
Theory?and?Application?of?Digital?Signal?Processing
Rabiner?and?Gold
Prentice-Hall?1975
Digital?Signal?Processing?with?Kernel?Method
Rojo-Alvarez?et?al
Wiley/IEEE?2018
Advanced?Digital?Signal?Processing
and?Noise?Reduction
Vaseghi
Wiley?4th?ed?2008
?Subsequent?to?the?responses?I?checked:
Spectral?Analysis?and?Time?Series
Priestly
Academic?Press?1981
Continuous?a?Signals?and?System?Analysis?and?Discrete
McGillem?and?Cooper
Holt,?Reinhart?and?Winston?1974
None?of?them?properly?discuss?the?issue?of?causality.
I?did?NOT?look?at?any?of?the?many?monographs?by?Enders?Robinson,
Robinson?&?Treitel?or?John?H.?Karl?as?I?know?they?have?the?matter
stated?very?clearly.?Robinson?and?Treitel?were?members?of?Norbert
Weiner's?Geophysical?Analysis?Group?at?MIT?and?founded?the?entire
basis?of?DSP.?I?spent?4?years?at?UT?Austin?under?another?member?of
the?GAG,?Milo?Backus.?Robinson?performed?the?first?digital
deconvolution?problem?in?1952?using?pencil,?paper?and?a?desk
calculator.??Robinson?has?pride?of?place?as?the?first?person?to?apply
DSP?to?data.
Causality?requires?that?the?real?and?imaginary?parts?be?a?Hilbert
transform?pair.?This?is?well?stated?in:
The?Fourier?Integral?and?Some?of?Its?Applications
Ronald?Bracewell
McGraw-Hill?2nd?ed?1978
Causality?simply?states?that?there?is?no?output?prior?to?the?input.
If?a?filter?is?not?causal?it?produces?output?before?the?event.?The?EE
community?appears?to?consistently?label?a?zero?phase?signal?as?linear
phase?with?a?phase?delay.?As?Bracewell?provides?a?proof?and?I?spent?2
semesters?studying?Churchill's?"Operational?Mathematics"?under?Bill
Guy?in?"Integral?Transforms"?at?Austin?in?addition?to?the?semester?I
spent?with?Bracewell?in?"Linear?Systems",?if?the?EE?community?wants
to?redefine?the?math?they?may.?I?shall?stick?to?what?the
mathematicians?wrote.
Have?Fun!
Reg
|
Re: Some questions on digital filter design
Most of my DSP work was in the early 70's using discrete chips. It was not even called DSP then. Built both FIT and IIR filters. At the time chips and DACs and ADCs were slow. That was OK as we were working on detecting and classifying submarine signals at acoustic frequencies. In fact, the first couple of filters that I built used serial arithmetic using chips from Rockwell Collins. Things got a lot nicer when AMD came out with parallel multipliers. All my filters were built with cascaded second order sections of the Elliptic design. Also, we used the Weaver method to make them tunable. Regards - Mike
Mike B. Feher, N4FS
89 Arnold Blvd.
Howell NJ 07731
908-902-3831
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-----Original Message----- From: [email protected] < [email protected]> On Behalf Of Reginald Beardsley via groups.io Sent: Saturday, July 6, 2024 12:37 PM To: [email protected]Subject: Re: [qex] Some questions on digital filter design This is a general reply to Jeff, Mike and Dave. I initially looked at: Digital Design Handbook Fred J. Taylor Marcel Dekker 1983 Digital Signal Processing Oppenheim and Shafer Prentice-Hall 1975 Theory and Application of Digital Signal Processing Rabiner and Gold Prentice-Hall 1975 Digital Signal Processing with Kernel Method Rojo-Alvarez et al Wiley/IEEE 2018 Advanced Digital Signal Processing and Noise Reduction Vaseghi Wiley 4th ed 2008 Subsequent to the responses I checked: Spectral Analysis and Time Series Priestly Academic Press 1981 Continuous a Signals and System Analysis and Discrete McGillem and Cooper Holt, Reinhart and Winston 1974 None of them properly discuss the issue of causality. I did NOT look at any of the many monographs by Enders Robinson, Robinson & Treitel or John H. Karl as I know they have the matter stated very clearly. Robinson and Treitel were members of Norbert Weiner's Geophysical Analysis Group at MIT and founded the entire basis of DSP. I spent 4 years at UT Austin under another member of the GAG, Milo Backus. Robinson performed the first digital deconvolution problem in 1952 using pencil, paper and a desk calculator. Robinson has pride of place as the first person to apply DSP to data. Causality requires that the real and imaginary parts be a Hilbert transform pair. This is well stated in: The Fourier Integral and Some of Its Applications Ronald Bracewell McGraw-Hill 2nd ed 1978 Causality simply states that there is no output prior to the input. If a filter is not causal it produces output before the event. The EE community appears to consistently label a zero phase signal as linear phase with a phase delay. As Bracewell provides a proof and I spent 2 semesters studying Churchill's "Operational Mathematics" under Bill Guy in "Integral Transforms" at Austin in addition to the semester I spent with Bracewell in "Linear Systems", if the EE community wants to redefine the math they may. I shall stick to what the mathematicians wrote. Have Fun! Reg
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Re: Some questions on digital filter design
If you want something simple to implement you can use what I did in the fldigi version of SCAMP:
look at the?
int scamp::rx_process(const double *buf, int len)
method.? It implements a simple circular buffer filter, which has a sinc-like response but its frequency may be moved around dynamically.? I use a different approach in my guitar pedal code:
for example look at the "DSP_TYPE_WAH" filter effect.? This uses a dynamically changed bandpass filter based on the position of a foot pedal, and does some simple interpolation to move the center frequency around by changing the coefficients of a digital biquad filter.?
All digital filters are implemented with a delay, and so none are "noncausal."? There is a compromise between group delay, which is the amount of lag that the filter applies to the signal and the sharpness of the filter.? A "minimum phase" filter minimizes this delay and has the property that the real and imaginary parts are related by the Hilbert transform.? A filter can be expressed as a cascade of a minimum phase filter and an all-pass filter.
Dan
On Sat, Jul 6, 2024 at 8:54?AM Reginald Beardsley via <pulaskite= [email protected]> wrote: I?am?working?on?numerical?modeling?of?CCW?and?am?at?the?point?that?I?need?to?write?the?code?to?implement?twin?pass?band?tuning.??As?this?is?NOT?seismic?processing?I?pulled?a?small?stack?of?my?EE?DSP?classics?and?started?reading?them.
I?have?always?found?the?descriptions?of?digital?filter?design?in?the?EE?literature?baffling?complex.??As?seismic?is?done?in?recorded?time?it?has?a?great?deal?of?latitude?in?implementation?that?a?real?time?DSP?implementation??lacks.??So?I?presumed?that?the?reason?lay?in?the?real?time?constraint.
Yesterday?I?realized?that?the?design??of?a?static?filter?is?completely?trivial.??Specify?the?frequency?domain?characteristics?and?back?transform?to?the?time?domain.??The?impulse?response?is?the?taps?for?any?arbitrary?filter.??I?also?noticed?that?the?filters?in?the?examples?were?zero?phase?filters.
In?seismic?we?always?use?zero?phase?(aka?"linear?phase"?in?EE?jargon)??filters?as?interpreting?a?symmetric?waveform?is?much?easier.
The?first?step?in?seismic?processing?is?called?"designature".??This?consists?of?using?a?recording?of?the??system?impulse?response?to?create?an?all?pass?filter?which?removes?the?phase?imposed?by?causality.??Typically?the?recording?system?impulse?response?is?measured?and??then?the?phase?of?the?source?is?recorded?for?marine?work?using?arrays?of?air?guns.
Analog?filters?are?minimum?phase?as?are?all?physical?processes.??But?the?digital?design?examples?I?saw?were?zero?phase?with?a?symmetric?impulse?response.??This?leads?to?a?non-physical?result.??The?filter?output?begins?*before*?the?input?arrives.??I?find?that?rather?hard?to?justify?in?a?communication?system.??A?quick?check?of?the?indices?revealed?that?except?for?a?very?brief?mention?of?minimum?phase?and?causality?the?issue?is?completely?ignored.??Oppenheim?and?Schafer?devote?a?few?page?and?Rabiner??and?Gold?a?paragraph?before?dismissing?the?matter.? Another text made no mention of it at all.
Would?a?DSO?which?responded?to?an?impulse?showing?the?trace?varying?*before*?the?peak?of?the?impulse?be?acceptable???I?can't?imagine?why?anyone?would?accept?that.
So?why?should?it?be?acceptable?in?a?communication?system???Can?anyone?elucidate?the?matter?
Have?Fun!
Reg
|
Re: Some questions on digital filter design
Good answer. But, dang, I now have to peruse those.
DaveD
KC0WJN
==============================
All spelling mistakes are the responsibilty of the reader (Rick Renz, STK, ca. 1994)
==============================
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Show quoted text
On Jul 6, 2024, at 12:37, Reginald Beardsley via groups.io <pulaskite@...> wrote:
? This is a general reply to Jeff, Mike and Dave. I initially looked at:
Digital Design Handbook
Fred J. Taylor
Marcel Dekker 1983
Digital Signal Processing
Oppenheim and Shafer
Prentice-Hall 1975
Theory and Application of Digital Signal Processing
Rabiner and Gold
Prentice-Hall 1975
Digital Signal Processing with Kernel Method
Rojo-Alvarez et al
Wiley/IEEE 2018
Advanced Digital Signal Processing
and Noise Reduction
Vaseghi
Wiley 4th ed 2008
Subsequent to the responses I checked:
Spectral Analysis and Time Series
Priestly
Academic Press 1981
Continuous a Signals and System Analysis and Discrete
McGillem and Cooper
Holt, Reinhart and Winston 1974
None of them properly discuss the issue of causality.
I did NOT look at any of the many monographs by Enders Robinson, Robinson & Treitel or John H. Karl as I know they have the matter stated very clearly. Robinson and Treitel were members of Norbert Weiner's Geophysical Analysis Group at MIT and founded the entire basis of DSP. I spent 4 years at UT Austin under another member of the GAG, Milo Backus. Robinson performed the first digital deconvolution problem in 1952 using pencil, paper and a desk calculator. Robinson has pride of place as the first person to apply DSP to data.
Causality requires that the real and imaginary parts be a Hilbert transform pair. This is well stated in:
The Fourier Integral and Some of Its Applications
Ronald Bracewell
McGraw-Hill 2nd ed 1978
Causality simply states that there is no output prior to the input. If a filter is not causal it produces output before the event. The EE community appears to consistently label a zero phase signal as linear phase with a phase delay. As Bracewell provides a proof and I spent 2 semesters studying Churchill's "Operational Mathematics" under Bill Guy in "Integral Transforms" at Austin in addition to the semester I spent with Bracewell in "Linear Systems", if the EE community wants to redefine the math they may. I shall stick to what the mathematicians wrote.
Have Fun!
Reg
|
Re: Some questions on digital filter design
This?is?a?general?reply?to?Jeff,?Mike?and?Dave.?I?initially?looked?at:
Digital?Design?Handbook
Fred?J.?Taylor
Marcel?Dekker?1983
Digital?Signal?Processing
Oppenheim?and?Shafer
Prentice-Hall?1975
Theory?and?Application?of?Digital?Signal?Processing
Rabiner?and?Gold
Prentice-Hall?1975
Digital?Signal?Processing?with?Kernel?Method
Rojo-Alvarez?et?al
Wiley/IEEE?2018
Advanced?Digital?Signal?Processing
and?Noise?Reduction
Vaseghi
Wiley?4th?ed?2008
?Subsequent?to?the?responses?I?checked:
Spectral?Analysis?and?Time?Series
Priestly
Academic?Press?1981
Continuous?a?Signals?and?System?Analysis?and?Discrete
McGillem?and?Cooper
Holt,?Reinhart?and?Winston?1974
None?of?them?properly?discuss?the?issue?of?causality.
I?did?NOT?look?at?any?of?the?many?monographs?by?Enders?Robinson,?Robinson?&?Treitel?or?John?H.?Karl?as?I?know?they?have?the?matter?stated?very?clearly.?Robinson?and?Treitel?were?members?of?Norbert?Weiner's?Geophysical?Analysis?Group?at?MIT?and?founded?the?entire?basis?of?DSP.?I?spent?4?years?at?UT?Austin?under?another?member?of?the?GAG,?Milo?Backus.?Robinson?performed?the?first?digital?deconvolution?problem?in?1952?using?pencil,?paper?and?a?desk?calculator.??Robinson?has?pride?of?place?as?the?first?person?to?apply?DSP?to?data.
Causality?requires?that?the?real?and?imaginary?parts?be?a?Hilbert?transform?pair.?This?is?well?stated?in:
The?Fourier?Integral?and?Some?of?Its?Applications
Ronald?Bracewell
McGraw-Hill?2nd?ed?1978
Causality?simply?states?that?there?is?no?output?prior?to?the?input.?If?a?filter?is?not?causal?it?produces?output?before?the?event.?The?EE?community?appears?to?consistently?label?a?zero?phase?signal?as?linear?phase?with?a?phase?delay.?As?Bracewell?provides?a?proof?and?I?spent?2?semesters?studying?Churchill's?"Operational?Mathematics"?under?Bill?Guy?in?"Integral?Transforms"?at?Austin?in?addition?to?the?semester?I?spent?with?Bracewell?in?"Linear?Systems",?if?the?EE?community?wants?to?redefine?the?math?they?may.?I?shall?stick?to?what?the?mathematicians?wrote.
Have?Fun!
Reg
|
Re: Some questions on digital filter design
Reg, my apologies, but I'm confused as to what you are trying to say.
It looks to me as though you are discussing a standard Finite-Impulse-Response (FIR) filter, which can be thought of as a physical realization of a convolution integral, in which the input signal is convolved with the impulse response of the filter being realized.
I don't know what you mean when you say "the filter output begins *before* the input arrives."
The filter output is not starting *before* the signal arrives -- it starts *the moment* the signal arrives.? If you were to connect the filter's input and output signals to two channels of a scope and then generate, as an input, a gated signal, you'd see on the output channel a filtered version of that gated signal that looks like it has been delayed, in time, by half the number of FIR taps (actually, the delay is (N-1/)2, where N is the number of taps).
My apologies if I am misinterpreting the point you are trying to make.
Best regards,
- Jeff, k6jca
On Saturday, July 6, 2024 at 06:54:36 AM PDT, Reginald Beardsley via groups.io <pulaskite@...> wrote:
I?am?working?on?numerical?modeling?of?CCW?and?am?at?the?point?that?I?need?to?write?the?code?to?implement?twin?pass?band?tuning.??As?this?is?NOT?seismic?processing?I?pulled?a?small?stack?of?my?EE?DSP?classics?and?started?reading?them.
I?have?always?found?the?descriptions?of?digital?filter?design?in?the?EE?literature?baffling?complex.??As?seismic?is?done?in?recorded?time?it?has?a?great?deal?of?latitude?in?implementation?that?a?real?time?DSP?implementation??lacks.??So?I?presumed?that?the?reason?lay?in?the?real?time?constraint.
Yesterday?I?realized?that?the?design??of?a?static?filter?is?completely?trivial.??Specify?the?frequency?domain?characteristics?and?back?transform?to?the?time?domain.??The?impulse?response?is?the?taps?for?any?arbitrary?filter.??I?also?noticed?that?the?filters?in?the?examples?were?zero?phase?filters.
In?seismic?we?always?use?zero?phase?(aka?"linear?phase"?in?EE?jargon)??filters?as?interpreting?a?symmetric?waveform?is?much?easier.
The?first?step?in?seismic?processing?is?called?"designature".??This?consists?of?using?a?recording?of?the??system?impulse?response?to?create?an?all?pass?filter?which?removes?the?phase?imposed?by?causality.??Typically?the?recording?system?impulse?response?is?measured?and??then?the?phase?of?the?source?is?recorded?for?marine?work?using?arrays?of?air?guns.
Analog?filters?are?minimum?phase?as?are?all?physical?processes.??But?the?digital?design?examples?I?saw?were?zero?phase?with?a?symmetric?impulse?response.??This?leads?to?a?non-physical?result.??The?filter?output?begins?*before*?the?input?arrives.??I?find?that?rather?hard?to?justify?in?a?communication?system.??A?quick?check?of?the?indices?revealed?that?except?for?a?very?brief?mention?of?minimum?phase?and?causality?the?issue?is?completely?ignored.??Oppenheim?and?Schafer?devote?a?few?page?and?Rabiner??and?Gold?a?paragraph?before?dismissing?the?matter.? Another text made no mention of it at all.
Would?a?DSO?which?responded?to?an?impulse?showing?the?trace?varying?*before*?the?peak?of?the?impulse?be?acceptable???I?can't?imagine?why?anyone?would?accept?that.
So?why?should?it?be?acceptable?in?a?communication?system???Can?anyone?elucidate?the?matter?
Have?Fun!
Reg
|
Re: Some questions on digital filter design
Look at the Weaver method of SSB generation. An easy way to make a tunable BP filter using LP digital filters and DDS oscillator with I/Q outputs. 73 ¨C Mike ? Mike B. Feher, N4FS 89 Arnold Blvd. Howell NJ 07731 908-902-3831 ?
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-----Original Message-----
From: [email protected] <[email protected]> On Behalf Of Reginald Beardsley via groups.io
Sent: Saturday, July 6, 2024 9:55 AM
To: [email protected]; [email protected]
Subject: [qex] Some questions on digital filter design ? I?am?working?on?numerical?modeling?of?CCW?and?am?at?the?point?that?I?need?to?write?the?code?to?implement?twin?pass?band?tuning.??As?this?is?NOT?seismic?processing?I?pulled?a?small?stack?of?my?EE?DSP?classics?and?started?reading?them. ? I?have?always?found?the?descriptions?of?digital?filter?design?in?the?EE?literature?baffling?complex.??As?seismic?is?done?in?recorded?time?it?has?a?great?deal?of?latitude?in?implementation?that?a?real?time?DSP?implementation??lacks.??So?I?presumed?that?the?reason?lay?in?the?real?time?constraint. ? Yesterday?I?realized?that?the?design??of?a?static?filter?is?completely?trivial.??Specify?the?frequency?domain?characteristics?and?back?transform?to?the?time?domain.??The?impulse?response?is?the?taps?for?any?arbitrary?filter.??I?also?noticed?that?the?filters?in?the?examples?were?zero?phase?filters. ? In?seismic?we?always?use?zero?phase?(aka?"linear?phase"?in?EE?jargon)??filters?as?interpreting?a?symmetric?waveform?is?much?easier. ? The?first?step?in?seismic?processing?is?called?"designature".??This?consists?of?using?a?recording?of?the??system?impulse?response?to?create?an?all?pass?filter?which?removes?the?phase?imposed?by?causality.??Typically?the?recording?system?impulse?response?is?measured?and??then?the?phase?of?the?source?is?recorded?for?marine?work?using?arrays?of?air?guns. ? Analog?filters?are?minimum?phase?as?are?all?physical?processes.??But?the?digital?design?examples?I?saw?were?zero?phase?with?a?symmetric?impulse?response.??This?leads?to?a?non-physical?result.??The?filter?output?begins?*before*?the?input?arrives.??I?find?that?rather?hard?to?justify?in?a?communication?system.??A?quick?check?of?the?indices?revealed?that?except?for?a?very?brief?mention?of?minimum?phase?and?causality?the?issue?is?completely?ignored.??Oppenheim?and?Schafer?devote?a?few?page?and?Rabiner??and?Gold?a?paragraph?before?dismissing?the?matter.? Another text made no mention of it at all. ? Would?a?DSO?which?responded?to?an?impulse?showing?the?trace?varying?*before*?the?peak?of?the?impulse?be?acceptable???I?can't?imagine?why?anyone?would?accept?that. ? So?why?should?it?be?acceptable?in?a?communication?system???Can?anyone?elucidate?the?matter? ? Have?Fun! Reg ? ? ?
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Re: Some questions on digital filter design
What texts are you using?
DaveD
KC0WJN
==============================
All spelling mistakes are the responsibilty of the reader (Rick Renz, STK, ca. 1994)
==============================
toggle quoted message
Show quoted text
On Jul 6, 2024, at 09:54, Reginald Beardsley via groups.io <pulaskite@...> wrote:
?I am working on numerical modeling of CCW and am at the point that I need to write the code to implement twin pass band tuning. As this is NOT seismic processing I pulled a small stack of my EE DSP classics and started reading them.
I have always found the descriptions of digital filter design in the EE literature baffling complex. As seismic is done in recorded time it has a great deal of latitude in implementation that a real time DSP implementation lacks. So I presumed that the reason lay in the real time constraint.
Yesterday I realized that the design of a static filter is completely trivial. Specify the frequency domain characteristics and back transform to the time domain. The impulse response is the taps for any arbitrary filter. I also noticed that the filters in the examples were zero phase filters.
In seismic we always use zero phase (aka "linear phase" in EE jargon) filters as interpreting a symmetric waveform is much easier.
The first step in seismic processing is called "designature". This consists of using a recording of the system impulse response to create an all pass filter which removes the phase imposed by causality. Typically the recording system impulse response is measured and then the phase of the source is recorded for marine work using arrays of air guns.
Analog filters are minimum phase as are all physical processes. But the digital design examples I saw were zero phase with a symmetric impulse response. This leads to a non-physical result. The filter output begins *before* the input arrives. I find that rather hard to justify in a communication system. A quick check of the indices revealed that except for a very brief mention of minimum phase and causality the issue is completely ignored. Oppenheim and Schafer devote a few page and Rabiner and Gold a paragraph before dismissing the matter. Another text made no mention of it at all.
Would a DSO which responded to an impulse showing the trace varying *before* the peak of the impulse be acceptable? I can't imagine why anyone would accept that.
So why should it be acceptable in a communication system? Can anyone elucidate the matter?
Have Fun!
Reg
|
Some questions on digital filter design
I?am?working?on?numerical?modeling?of?CCW?and?am?at?the?point?that?I?need?to?write?the?code?to?implement?twin?pass?band?tuning.??As?this?is?NOT?seismic?processing?I?pulled?a?small?stack?of?my?EE?DSP?classics?and?started?reading?them.
I?have?always?found?the?descriptions?of?digital?filter?design?in?the?EE?literature?baffling?complex.??As?seismic?is?done?in?recorded?time?it?has?a?great?deal?of?latitude?in?implementation?that?a?real?time?DSP?implementation??lacks.??So?I?presumed?that?the?reason?lay?in?the?real?time?constraint.
Yesterday?I?realized?that?the?design??of?a?static?filter?is?completely?trivial.??Specify?the?frequency?domain?characteristics?and?back?transform?to?the?time?domain.??The?impulse?response?is?the?taps?for?any?arbitrary?filter.??I?also?noticed?that?the?filters?in?the?examples?were?zero?phase?filters.
In?seismic?we?always?use?zero?phase?(aka?"linear?phase"?in?EE?jargon)??filters?as?interpreting?a?symmetric?waveform?is?much?easier.
The?first?step?in?seismic?processing?is?called?"designature".??This?consists?of?using?a?recording?of?the??system?impulse?response?to?create?an?all?pass?filter?which?removes?the?phase?imposed?by?causality.??Typically?the?recording?system?impulse?response?is?measured?and??then?the?phase?of?the?source?is?recorded?for?marine?work?using?arrays?of?air?guns.
Analog?filters?are?minimum?phase?as?are?all?physical?processes.??But?the?digital?design?examples?I?saw?were?zero?phase?with?a?symmetric?impulse?response.??This?leads?to?a?non-physical?result.??The?filter?output?begins?*before*?the?input?arrives.??I?find?that?rather?hard?to?justify?in?a?communication?system.??A?quick?check?of?the?indices?revealed?that?except?for?a?very?brief?mention?of?minimum?phase?and?causality?the?issue?is?completely?ignored.??Oppenheim?and?Schafer?devote?a?few?page?and?Rabiner??and?Gold?a?paragraph?before?dismissing?the?matter. Another text made no mention of it at all.
Would?a?DSO?which?responded?to?an?impulse?showing?the?trace?varying?*before*?the?peak?of?the?impulse?be?acceptable???I?can't?imagine?why?anyone?would?accept?that.
So?why?should?it?be?acceptable?in?a?communication?system???Can?anyone?elucidate?the?matter?
Have?Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
On Fri, Jun 21, 2024 at 03:53 PM, Daniel Marks wrote: Any linear encoding/decoding method boils down to which subspace does the
signal live in, and which does the noise live in. The extent of the
overlap of the two determines the error probability. If the signal and
noise are described by Gaussian statistics, then the noise is completely
described by its second-order statistics, and the ratio of the variance of
the signal estimate to that of the noise in that estimate is the
signal-to-noise ratio. If the statistics are not Gaussian, but still have
second order statistics, one can still get a SNR, though it does not
directly map to an error probability.
Daniel, I am troubled by your assertion that one cannot determine an error probability unless both processes are Gaussian because the independence of random variables requires that p(x,y) = p(x)p(y). The Gaussian case is so trivial and invoked so often I refer to it as "sprinkling Gauss water on the problem". I have observed it being invoked in many cases where there was no justification and some where it was provably wrong. But the independence of random processes imposes no constraint on the distributions that I am aware of. Would you be so kind as to explain your reasoning? Have Fun! Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
FWIW
My goal in this thread is to clearly state the mathematics. I taught myself compressive sensing in complete isolation. It took me 3 years of continuous effort in large part because I had no one with whom to discuss the subject.
The mathematics of DSSS is moderately advanced and moving from author to author it is painfully easy to misinterpret what is being said because of minor nuances of notation and naming. As this is not as daunting as compressive sensing, I'm hoping that someone else has read and understands the math well enough to catch misunderstandings on my part.
Aside from Shannon, section 5.7 of:
Spread Spectrum Communications Handbook
Simon et al
Revised Edition
McGraw-Hill 1994
explicitly treats Direct Sequence Multiple Access systems such as I am proposing. Gold originally discussed this concept in a 1967 paper.
The processing gain for 1000 Hz BPSK modulation and a 100 ms dit is 10 dB c.f. middle of p. 6 in Simon et al. That and a 5x increase in the number of CW stations which can operate in 1 kHz band segment seem very attractive to me.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
Daniel,
Is this correct? I saw no mention of it anything on the RFBitbanger that I read.
Have Fun!
Reg
On Monday, June 24, 2024 at 07:14:23 PM CDT, Erik Nelson <erik.nels0n99@...> wrote:
Hi Reg,
I believe there's still an OOK based submode of SCAMP included, not just the 2FSK. These links should go into more detail:
On Sat, Jun 22, 2024, 3:33 PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Erik,
Thanks for the video link.
SCAMP is not decodable by ear or sent with a key. It's an RTTY protocol. Very cool, and really well suited to overnight message transfer at very low power levels. However, it's also narrow BW.
My objective is to create a CW carrier which has a fixed BW and is selectable as a discreet CW channel even though it's one of many operating in the same spectrum segment.
Have Fun!
Reg
SCAMP is a mode that reminds of what it sounds like you're trying to accomplish here. Besides that there's Ebnaut, and WOLF - though those have different requirements for a transmitter.
Any linear encoding/decoding method boils down to which subspace does the signal live in, and which does the noise live in.? The extent of the overlap of the two determines the error probability.? If the signal and noise are described by Gaussian statistics, then the noise is completely described by its second-order statistics, and the ratio of the variance of the signal estimate to that of the noise in that estimate is the signal-to-noise ratio.? If the statistics are not Gaussian, but still have second order statistics, one can still get a SNR, though it does not directly map to an error probability.
When real channels are encountered, and one gets burst errors, impulse noise, and the like, error correcting codes can help reduce the probability of decoding error at the expense of greatly increased processing and often increased latency.?
If you're trying to produce a Morse-like channel, what are the qualities of a Morse channel, and what are the qualities to be preserved in the Morse-like channel??? Long latency removes the "immediacy" of communication, in that one must wait for a long decoding interval before one can respond.? Also, what part of Morse is preserved?? Should the information communicated be directly the dits, dahs, intraelement, and interelement pauses?? Should letters or words be communicated?? Should the timing of the sender be exactly preserved, or simplified to 1, 3, or 7 dit lengths??
For example, lets say you have these symbols: a "no symbol" symbol, a dit, a dah, a 1 dit length pause, a 3 dit length pause, and a 7 dit length pause.? Lets round it up to 8 symbols so that it is a power of two.? If you want the symbols to be orthogonal and time-invariant (no synchronization required) you can use 8FSK.? If time-invariant orthogonality is not necessary, you could use, for example, a Walsh-Hadamard code of 8 BPSK/BFSK symbols (see matrix below).? You can reserve one of the symbols as a sync code, for example the uniform symbol, and send that one three times, with the symbol in the middle with inverted phase (++++++++ -------- ++++++++).? The bit edges could be used for synchronization.? Alternatively, use BFSK if the channel has a very unstable phase.
![image.png]() Decoding would be as simple as correlating the received code against all 8 code words and keeping the code word with the highest correlation.? This would be an inefficient but fairly robust and easy to implement linear code.? A similar code was used on the Mariner mission.? It would also keep latency relatively low.
Dan
On Thu, Jun 20, 2024 at 2:49?PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Well, that should have been CDMA, not TDMA :-(
I've attached a short proof of the mathematics.? It assumes synchronization, but I left out that detail to keep it to
a single page.
The objective is to increase the time * power * bandwidth product using DSSS to increase the BW term.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
Hi Reg,
I believe there's still an OOK based submode of SCAMP included, not just the 2FSK. These links should go into more detail:
On Sat, Jun 22, 2024, 3:33 PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Erik,
Thanks for the video link.
SCAMP is not decodable by ear or sent with a key. It's an RTTY protocol. Very cool, and really well suited to overnight message transfer at very low power levels. However, it's also narrow BW.
My objective is to create a CW carrier which has a fixed BW and is selectable as a discreet CW channel even though it's one of many operating in the same spectrum segment.
Have Fun!
Reg
SCAMP is a mode that reminds of what it sounds like you're trying to accomplish here. Besides that there's Ebnaut, and WOLF - though those have different requirements for a transmitter.
Any linear encoding/decoding method boils down to which subspace does the signal live in, and which does the noise live in.? The extent of the overlap of the two determines the error probability.? If the signal and noise are described by Gaussian statistics, then the noise is completely described by its second-order statistics, and the ratio of the variance of the signal estimate to that of the noise in that estimate is the signal-to-noise ratio.? If the statistics are not Gaussian, but still have second order statistics, one can still get a SNR, though it does not directly map to an error probability.
When real channels are encountered, and one gets burst errors, impulse noise, and the like, error correcting codes can help reduce the probability of decoding error at the expense of greatly increased processing and often increased latency.?
If you're trying to produce a Morse-like channel, what are the qualities of a Morse channel, and what are the qualities to be preserved in the Morse-like channel??? Long latency removes the "immediacy" of communication, in that one must wait for a long decoding interval before one can respond.? Also, what part of Morse is preserved?? Should the information communicated be directly the dits, dahs, intraelement, and interelement pauses?? Should letters or words be communicated?? Should the timing of the sender be exactly preserved, or simplified to 1, 3, or 7 dit lengths??
For example, lets say you have these symbols: a "no symbol" symbol, a dit, a dah, a 1 dit length pause, a 3 dit length pause, and a 7 dit length pause.? Lets round it up to 8 symbols so that it is a power of two.? If you want the symbols to be orthogonal and time-invariant (no synchronization required) you can use 8FSK.? If time-invariant orthogonality is not necessary, you could use, for example, a Walsh-Hadamard code of 8 BPSK/BFSK symbols (see matrix below).? You can reserve one of the symbols as a sync code, for example the uniform symbol, and send that one three times, with the symbol in the middle with inverted phase (++++++++ -------- ++++++++).? The bit edges could be used for synchronization.? Alternatively, use BFSK if the channel has a very unstable phase.
![image.png]() Decoding would be as simple as correlating the received code against all 8 code words and keeping the code word with the highest correlation.? This would be an inefficient but fairly robust and easy to implement linear code.? A similar code was used on the Mariner mission.? It would also keep latency relatively low.
Dan
On Thu, Jun 20, 2024 at 2:49?PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Well, that should have been CDMA, not TDMA :-(
I've attached a short proof of the mathematics.? It assumes synchronization, but I left out that detail to keep it to
a single page.
The objective is to increase the time * power * bandwidth product using DSSS to increase the BW term.
Have Fun!
Reg
|
The Restricted Isometery Property (RIP)
As it is likely that most, if not all, list members have not read "A Mathematical Introduction to Compressive Sensing" by Foucart and Rauhut, I though it useful to provide a bit more detail.
In the solution of Ax=y, if x is sparse and A possesses the RIP, then an L1 solution for x is identical to the L0 solution. If y consists of the sum of a small number of columns selected from an A matrix with the RIP, David Donoho proved in 2004-09 that if and only if a sparse solution to Ax=y existed it was the L0 solution. A solution is not guaranteed to exist, but the probability of such a case is very low and well defined.
RIP means that *any* combination of columns in A cross correlated with *any* other combination of columns from A has zero correlation (near zero in the case of finite lengths of the series). This is inherently an NP-hard problem and thus one cannot directly prove that an A matrix possesses the RIP. But if you can find an L1 solution to Ax=y for the sum of an arbitrary choice of a small number of columns then the matrix has the RIP. Otherwise a solution would not be obtained.
By definition, an A matrix which is constructed from a purely random series possesses the RIP.
By using such sequences for the DSSS carrier modulation as described in the recent thread I started one can readily meet the detection criteria I listed in the attachment to the first post and all the DSSS sequences are orthogonal.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
I finally found my copy of Shannon. I've attached scans of 4 pages I think especially relevant to what I am proposing. The entire paper can be found online if you search on "bell system technical journal". UPenn.edu has an Internet Archive link to volume 27 in which the paper appeared.
Note the first sentence of the preface about trading bandwidth for signal to noise ratio.
Shannon helpfully gives the effective capacity of Morse telegraphy as 0.539*2*dits per second. For 12 wpm, 5 dits per second, that gives a channel capacity of ~5 bits per second.
A 1 kHz BW signal has a capacity of 1000 bits per second. That places an upper limit on the number of CW channels which a 1 kHz DSSS channel can support at 200 12 wpm channels .
Present CW practices achieve about 20 channels in the same BW as can be readily observed on a waterfall display during a contest.
Thus use of a DSSS carrier for CW should provide 5-10 times the spectral efficiency of the current 50 Hz separation between stations.
I hope to have a numerical demonstration within a week or two (dependent on other activities) with which to evaluate the degradation behavior as one approaches the BW imposed capacity.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
Erik,
Thanks for the video link.
SCAMP is not decodable by ear or sent with a key. It's an RTTY protocol. Very cool, and really well suited to overnight message transfer at very low power levels. However, it's also narrow BW.
My objective is to create a CW carrier which has a fixed BW and is selectable as a discreet CW channel even though it's one of many operating in the same spectrum segment.
Have Fun!
Reg
On Friday, June 21, 2024 at 09:01:26 PM CDT, Erik Nelson <erik.nels0n99@...> wrote:
SCAMP is a mode that reminds of what it sounds like you're trying to accomplish here. Besides that there's Ebnaut, and WOLF - though those have different requirements for a transmitter.
Any linear encoding/decoding method boils down to which subspace does the signal live in, and which does the noise live in.? The extent of the overlap of the two determines the error probability.? If the signal and noise are described by Gaussian statistics, then the noise is completely described by its second-order statistics, and the ratio of the variance of the signal estimate to that of the noise in that estimate is the signal-to-noise ratio.? If the statistics are not Gaussian, but still have second order statistics, one can still get a SNR, though it does not directly map to an error probability.
When real channels are encountered, and one gets burst errors, impulse noise, and the like, error correcting codes can help reduce the probability of decoding error at the expense of greatly increased processing and often increased latency.?
If you're trying to produce a Morse-like channel, what are the qualities of a Morse channel, and what are the qualities to be preserved in the Morse-like channel??? Long latency removes the "immediacy" of communication, in that one must wait for a long decoding interval before one can respond.? Also, what part of Morse is preserved?? Should the information communicated be directly the dits, dahs, intraelement, and interelement pauses?? Should letters or words be communicated?? Should the timing of the sender be exactly preserved, or simplified to 1, 3, or 7 dit lengths??
For example, lets say you have these symbols: a "no symbol" symbol, a dit, a dah, a 1 dit length pause, a 3 dit length pause, and a 7 dit length pause.? Lets round it up to 8 symbols so that it is a power of two.? If you want the symbols to be orthogonal and time-invariant (no synchronization required) you can use 8FSK.? If time-invariant orthogonality is not necessary, you could use, for example, a Walsh-Hadamard code of 8 BPSK/BFSK symbols (see matrix below).? You can reserve one of the symbols as a sync code, for example the uniform symbol, and send that one three times, with the symbol in the middle with inverted phase (++++++++ -------- ++++++++).? The bit edges could be used for synchronization.? Alternatively, use BFSK if the channel has a very unstable phase.
![image.png]() Decoding would be as simple as correlating the received code against all 8 code words and keeping the code word with the highest correlation.? This would be an inefficient but fairly robust and easy to implement linear code.? A similar code was used on the Mariner mission.? It would also keep latency relatively low.
Dan
On Thu, Jun 20, 2024 at 2:49?PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Well, that should have been CDMA, not TDMA :-(
I've attached a short proof of the mathematics.? It assumes synchronization, but I left out that detail to keep it to
a single page.
The objective is to increase the time * power * bandwidth product using DSSS to increase the BW term.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
Natural processes produce Gaussian noise. For the dit modulation one can choose any distribution one wishes.
Consider that you have a 100 sample PRN that represents a dit at 12 wpm.There are 2^100 possible binary sequences multiplied by however many bits of DAC drives the transmitter. That's a moderately large space.
The requirement is creating a 100 x M matrix which has the restricted isometry property. Such a matrix can be readily generated using a good PRN such as the the Merrsenne Twister. By constructing a 100 row table with many thousands of columns, selecting a random set of columns and summing the rows to generate 100 random values one can generate such a series. One can then test for the RIP by seeking a sparse L1 solution. If the solution exists, it follows that the RIP is met by the matrix and it can be used to generate an arbitrary number of sequences by summing a small number of columns.
Generate 1000 such 100 sample series. Designate one as CQ and the other 999 as open channels.
Present technology limits CW to about 20 channels per kHz of spectrum. I don't know whether using a DSSS modulated carrier can give 1000 channels in 1 kHz. My copy of Shannon is MIA.
The non-linear nature of the audio output transfer function suggests that experimental work to estimate the channel capacity is needed.
Have Fun!
Reg
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
There is " dit" and "not dit". International Morse is defined in terms of those two primitives: fixed lengths of time on and off.
If you modulate a carrier with a 1 kHz PRN sequence that will spread the signal out in a manner which can be recovered by cross correlation using a single vector sum. If one has a set of orthogonal PRNs they can each serve as discrete, non-interfering channels while occupying the same frequency space.
Basic idea is to key a DSSS dit or series of dits as required with the appropriate "not dit" spaces inserted. The RX station has to do a synchronized sum over the frame to determine if it's a dit or not dit. That's computationally very cheap. For synchronization one can use a cross correlation with the decoded Morse character to adjust for frame variation every Nth character. IIRC the longest character requires about 2 kSa/word with zero padding. That's a pretty cheap FFT.
I've also considered decode by correlation at character level but question if it's worth the cost in moving electrons. Certainly worth testing. A path with fluttering delay might break dit frame, but not character frame correlation. It introduces character duration latency, but that is not a problem in conditions which would require it. The alternative would be no channel.
If B(t) has a 1 kHz BW, then a dit can be spread across 1 kHz by DSSS. I have always thought of this in terms of hard clipping. That's simply an accident of how I initially conceived of this 40+ years ago. Sign bit (aka hard clipping) was being used in the late 60's to record 1024 channels of Vibroseis data when most systems could only handle 48 channels. The sweeps were long enough they acquired 16 bit data.or better.
In fact, considering the matter last evening I realized that hard clipping is a nuisance and that driving the PA from a DAC would greatly simplify maintaining precise and accurate BW control.
With traditional narrow BW Morse one has only time and power as the variables in the Shannon-Nyquist Time-Power-Bandwidth product. The best technology available today allows about 20 CW channels in 1 kHz. In principle, it should be possible to increase that number dramatically by using a DSSS carrier with a single CQ code and a set of orthogonal codes such that on making contact the stations switch to a different code. This is effectively applying CDMA technology to HF communications using Morse code to maximize spectrum utilization.
Morse for the 21st century.
Have Fun!
Reg
On Friday, June 21, 2024 at 03:53:51 PM CDT, Daniel Marks <profdc9@...> wrote:
Any linear encoding/decoding method boils down to which subspace does the signal live in, and which does the noise live in.? The extent of the overlap of the two determines the error probability.? If the signal and noise are described by Gaussian statistics, then the noise is completely described by its second-order statistics, and the ratio of the variance of the signal estimate to that of the noise in that estimate is the signal-to-noise ratio.? If the statistics are not Gaussian, but still have second order statistics, one can still get a SNR, though it does not directly map to an error probability.
When real channels are encountered, and one gets burst errors, impulse noise, and the like, error correcting codes can help reduce the probability of decoding error at the expense of greatly increased processing and often increased latency.?
If you're trying to produce a Morse-like channel, what are the qualities of a Morse channel, and what are the qualities to be preserved in the Morse-like channel??? Long latency removes the "immediacy" of communication, in that one must wait for a long decoding interval before one can respond.? Also, what part of Morse is preserved?? Should the information communicated be directly the dits, dahs, intraelement, and interelement pauses?? Should letters or words be communicated?? Should the timing of the sender be exactly preserved, or simplified to 1, 3, or 7 dit lengths??
For example, lets say you have these symbols: a "no symbol" symbol, a dit, a dah, a 1 dit length pause, a 3 dit length pause, and a 7 dit length pause.? Lets round it up to 8 symbols so that it is a power of two.? If you want the symbols to be orthogonal and time-invariant (no synchronization required) you can use 8FSK.? If time-invariant orthogonality is not necessary, you could use, for example, a Walsh-Hadamard code of 8 BPSK/BFSK symbols (see matrix below).? You can reserve one of the symbols as a sync code, for example the uniform symbol, and send that one three times, with the symbol in the middle with inverted phase (++++++++ -------- ++++++++).? The bit edges could be used for synchronization.? Alternatively, use BFSK if the channel has a very unstable phase.
![image.png]() Decoding would be as simple as correlating the received code against all 8 code words and keeping the code word with the highest correlation.? This would be an inefficient but fairly robust and easy to implement linear code.? A similar code was used on the Mariner mission.? It would also keep latency relatively low.
Dan
On Thu, Jun 20, 2024 at 2:49?PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Well, that should have been CDMA, not TDMA :-(
I've attached a short proof of the mathematics.? It assumes synchronization, but I left out that detail to keep it to
a single page.
The objective is to increase the time * power * bandwidth product using DSSS to increase the BW term.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
SCAMP is a mode that reminds of what it sounds like you're trying to accomplish here. Besides that there's Ebnaut, and WOLF - though those have different requirements for a transmitter.
Any linear encoding/decoding method boils down to which subspace does the signal live in, and which does the noise live in.? The extent of the overlap of the two determines the error probability.? If the signal and noise are described by Gaussian statistics, then the noise is completely described by its second-order statistics, and the ratio of the variance of the signal estimate to that of the noise in that estimate is the signal-to-noise ratio.? If the statistics are not Gaussian, but still have second order statistics, one can still get a SNR, though it does not directly map to an error probability.
When real channels are encountered, and one gets burst errors, impulse noise, and the like, error correcting codes can help reduce the probability of decoding error at the expense of greatly increased processing and often increased latency.?
If you're trying to produce a Morse-like channel, what are the qualities of a Morse channel, and what are the qualities to be preserved in the Morse-like channel??? Long latency removes the "immediacy" of communication, in that one must wait for a long decoding interval before one can respond.? Also, what part of Morse is preserved?? Should the information communicated be directly the dits, dahs, intraelement, and interelement pauses?? Should letters or words be communicated?? Should the timing of the sender be exactly preserved, or simplified to 1, 3, or 7 dit lengths??
For example, lets say you have these symbols: a "no symbol" symbol, a dit, a dah, a 1 dit length pause, a 3 dit length pause, and a 7 dit length pause.? Lets round it up to 8 symbols so that it is a power of two.? If you want the symbols to be orthogonal and time-invariant (no synchronization required) you can use 8FSK.? If time-invariant orthogonality is not necessary, you could use, for example, a Walsh-Hadamard code of 8 BPSK/BFSK symbols (see matrix below).? You can reserve one of the symbols as a sync code, for example the uniform symbol, and send that one three times, with the symbol in the middle with inverted phase (++++++++ -------- ++++++++).? The bit edges could be used for synchronization.? Alternatively, use BFSK if the channel has a very unstable phase.
 Decoding would be as simple as correlating the received code against all 8 code words and keeping the code word with the highest correlation.? This would be an inefficient but fairly robust and easy to implement linear code.? A similar code was used on the Mariner mission.? It would also keep latency relatively low.
Dan
On Thu, Jun 20, 2024 at 2:49?PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Well, that should have been CDMA, not TDMA :-(
I've attached a short proof of the mathematics.? It assumes synchronization, but I left out that detail to keep it to
a single page.
The objective is to increase the time * power * bandwidth product using DSSS to increase the BW term.
Have Fun!
Reg
|
Re: Direct Sequence Spread Spectrum CW carrier
Any linear encoding/decoding method boils down to which subspace does the signal live in, and which does the noise live in.? The extent of the overlap of the two determines the error probability.? If the signal and noise are described by Gaussian statistics, then the noise is completely described by its second-order statistics, and the ratio of the variance of the signal estimate to that of the noise in that estimate is the signal-to-noise ratio.? If the statistics are not Gaussian, but still have second order statistics, one can still get a SNR, though it does not directly map to an error probability.
When real channels are encountered, and one gets burst errors, impulse noise, and the like, error correcting codes can help reduce the probability of decoding error at the expense of greatly increased processing and often increased latency.?
If you're trying to produce a Morse-like channel, what are the qualities of a Morse channel, and what are the qualities to be preserved in the Morse-like channel??? Long latency removes the "immediacy" of communication, in that one must wait for a long decoding interval before one can respond.? Also, what part of Morse is preserved?? Should the information communicated be directly the dits, dahs, intraelement, and interelement pauses?? Should letters or words be communicated?? Should the timing of the sender be exactly preserved, or simplified to 1, 3, or 7 dit lengths??
For example, lets say you have these symbols: a "no symbol" symbol, a dit, a dah, a 1 dit length pause, a 3 dit length pause, and a 7 dit length pause.? Lets round it up to 8 symbols so that it is a power of two.? If you want the symbols to be orthogonal and time-invariant (no synchronization required) you can use 8FSK.? If time-invariant orthogonality is not necessary, you could use, for example, a Walsh-Hadamard code of 8 BPSK/BFSK symbols (see matrix below).? You can reserve one of the symbols as a sync code, for example the uniform symbol, and send that one three times, with the symbol in the middle with inverted phase (++++++++ -------- ++++++++).? The bit edges could be used for synchronization.? Alternatively, use BFSK if the channel has a very unstable phase.
 Decoding would be as simple as correlating the received code against all 8 code words and keeping the code word with the highest correlation.? This would be an inefficient but fairly robust and easy to implement linear code.? A similar code was used on the Mariner mission.? It would also keep latency relatively low.
Dan
On Thu, Jun 20, 2024 at 2:49?PM Reginald Beardsley via <pulaskite= [email protected]> wrote: Well, that should have been CDMA, not TDMA :-(
I've attached a short proof of the mathematics.? It assumes synchronization, but I left out that detail to keep it to
a single page.
The objective is to increase the time * power * bandwidth product using DSSS to increase the BW term.
Have Fun!
Reg
|
Erik Nelson