An unusual question needing a bit of maths......

Discussion in 'General Technical Questions and Answers' started by G1OJS, Jul 18, 2019.

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  1. G1OJS

    G1OJS Premium Subscriber QRZ Page

    On and off over the last 30 years since university I've wanted to see written down the transfer function for an SSB transmitter feeding (through suitable attenuation) an *AM* receiver, with the aim being of understanding the simplest possible representation and/or equivalent circuit that turns normal speech into "Donald Duck with a sore throat".

    In my final year at uni I designed, built and tested an I&Q SSB receiver and worked through some of the relevant maths then, and I've had a couple of goes since but never managed to come up with a nice, compact formula that is easy to "read" intuitively to build a mental concept of what's going on.

    Just wondering if anyone else has wondered, tried, succeeded?
  2. KA9JLM

    KA9JLM Ham Member QRZ Page

    Smoking cigarettes and working around asbestos worked for me. :(

    Why would you want that ?
  3. KM3F

    KM3F Ham Member QRZ Page

    Audio modulates a carrier to double side bands same as AM.
    Carrier is 'suppressed', one side band is passed to the RF power amplifier minus the carrier.
    The receiver provides the missing carrier 'reference'.
    If there is a phase difference it does not sound like the original signal and moves the audio higher or lower depending on the difference and which side band is being used, upper or lower.
    and it's position within the filter edge or skirt.
  4. G1OJS

    G1OJS Premium Subscriber QRZ Page

    To clarify the question I asked, I know how SSB works in that I know how it's commonly modulated and demodulated, and why. The receiver I designed at uni was based on a Weaver demodulator - and I spent a fair few hours looking at the maths of that (that was 30 years ago, however).

    I'm looking for a *mathematical* description of SSB modulation followed by AM demodulation, and a simplification (in the mathematical sense) of the result.

    Why? Interest and fun, that's all.

    So, for example, an SSB signal can be written as

    s(t).cos(wt) - s'(t).sin(wt)​

    where w is the carrier angular frequency, s(t) is the modulating signal and s'(t) is the Hilbert transform of s(t).

    An AM signal can be written as

    where this time r(t) is the signal

    If we rearrange the SSB expression into

    [s(t) - s'(t) sin(wt)/cos(wt)] . cos(wt)
    and assert that putting the AM signal above into a perfect AM demodulator would recover r(t), then by inspection putting the SSB signal into that demodulator would recover the thing in square brackets.

    We all know how this sounds - what I would like to be able to do is to look at:

    [s(t) - s'(t) sin(wt)/cos(wt)]​

    and say "Yep, I can 'hear' that in my mind and visualise what the maths is telling me.

    Now, I can visualise Fourier transforms pretty easily because I spent 3 years playing with 1D and 2D Fourier transforms of complex functions representing EM waves during my PhD. But I don't have the same level of intuition for Hilbert transforms. I'm OK with trig functions and complex exponentials for signals, and I was wondering if anyone had done this a different way that I could relate to.

    Having done that, it would be fun to see what happens when you use a real (i.e. imperfect) AM detector.
    Last edited: Jul 18, 2019
    AH7I likes this.
  5. KT1F

    KT1F Ham Member QRZ Page

    I couldn't begin to do the math but I have thought about this intuitively in idle moments.

    Long ago I heard about someone getting up to some shenanigans by listening to a SSB signal on AM, recording the duck talk and then retransmitting the audio over SSB. It would drive people nuts trying to tune it in :)

    I wondered if there is any way you could recover the original audio from the duck talk. I think the answer is no. There's not enough information there. I think the original audio frequencies are lost. i.e., 1 kHz and 1.5 kHz transmitted over SSB but received on AM sound the same which is silence.

    Of course that's not totally true because we all know that it is possible to somewhat understand it especially when they're saying something recognizable like CQ. That raises the question of what are you really hearing in the duck talk. I think you're basically hearing amplitude variations only all on the same audio frequency but I'm not sure what determines the frequency of the duck talk.
  6. SM0AOM

    SM0AOM Ham Member QRZ Page

    A way of addressing this is to break up the signal chain in sections;
    1. There is a message function or waveform expressed as a power spectrum at each instant in time G(t);

    2. Taking the inverse Fourier transform of this creates a waveform or message function s(t)

    3. Creating an SSB signal, which is equal to frequency translating the message function s(t), comprises use of the Hilbert transform:

    4. This can be multiplied with the transfer function for an AM detector
    Vo(t) = 0 for sssb(t) < o and
    Vo(t) = K*sssb(t) for sssb(t) > 0

    5. Band-limit or low-pass filter Vo(t)

    6. Make the Fourier transform of Vo(t) to find the output signal in the frequency domain

    To do the actual algebraic manipulations and calculate the output
    "is left as an exercise for the reader" :)

    Last edited: Jul 18, 2019
    KA9JLM likes this.
  7. G1OJS

    G1OJS Premium Subscriber QRZ Page

    That's the kind of evil genius that makes me smile. Now of course I'm wondering how the maths for *that* would look!

    It's exactly this, almost philosophical, kind of question that gets me interested. When I solve problems using maths, I like to know (and hence be able to sanity-check) what the maths is telling me at each stage - so for example, from one line of algebra to the next ask myself "where did that bit of information go?"

    Like you, I've listened to the duck talk (before I had a receiver with a BFO!) and wondered what I was listening to (yes, SSB, but *why specifically* does it sound like that and how come it's 30% intelligible?):

    • If someone whistles on SSB, all you hear on AM is quieting of background noise, because you're effectively listening to a constant frequency carrier
    • If someone says "Aaaaaaaaah", then you're pretty much listening to a constant frequency carrier (OK spread a bit) varying in amplitude - and you can hear *something* of this demodulated, but not everything.
    So I want to see the maths tell me - this is the bit you can hear, and this is the bit that you can't hear.

    As to "can you recover the original audio" - my intuition tells me that you could piece it together if you replaced the missing information with some assumptions. It reminds me very much (and this might be almost a direct analogy) of something one of my fellow PhD students was doing in the early 90s - recovering the amplitude distribution on the aperture of a microwave antenna using *only* the *amplitude* of the far field. The amplitude (and phase) can be determined 100% from the *complex* far field via a fairly simple Fourier transform, but not having the phase information means that any one of an (infinite?) number of complex aperture distributions could have made that far field. The algorithm they developed made some assumptions to narrow down and choose from likely candidates, as I remember. So I reckon you could do the same with the audio that's missing some information. Why you'd need to though :)
    Last edited: Jul 18, 2019
  8. G1OJS

    G1OJS Premium Subscriber QRZ Page

    Indeed - it's that simple, so I feel within a very short distance of getting the intuitive perspective that I'm looking for

    Many thanks :)

  9. SM0AOM

    SM0AOM Ham Member QRZ Page

    It is the message function at the output of the SSB generator.
    The forum does not handle subscripts properly.

  10. G1OJS

    G1OJS Premium Subscriber QRZ Page

    I'm wondering if what we're listening to is as simple as

    s(t) - s'(t)
    because, if I'm right that the SSB signal passed through an ideal AM demodulator is:

    [s(t) - s'(t) sin(wt)/cos(wt)]
    then isn't the band-limited version of the second term pretty much s'(t)?

    I guess to check I need to look at convolving an audio signal with the Fourier transform of tan(wt) which would give the frequency spectrum of the second term, I think? Then see how the part in the audio spectrum is changed.

    If I'm not right precisely, I wonder how close this is approximately?

    How does s(t) - s'(t) sound?

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