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MMWAVE-DEVPACK: Doubt in your Whitepaper: The fundamentals of millimeter wave sensors

Prasanna Kannan
Prodigy 130 points
Part Number: MMWAVE-DEVPACK
Other Parts Discussed in Thread: AWR1642

Dear Texas Instruments Team,

Firstly, many thanks for publishing detailed whitepaper, the literatures published by TI are excellent and hold a very high value for my day to day work.
I have a certain doubt in one of your publication, "The fundamentals of millimeter wave sensors" (www.ti.com/.../spyy005.pdf).
There is a part which explains the working of a mixer with the help of two sinasoids. As mentioned in the equations (1), (2) and (3) it is not clear to me how two signals
x1 = sin(w1.t + phi1) &
x2 = sin(w2.t + phi2) combine here to give a signal

xout = sin[(w1-w2).t + (phi1-phi2)]

Somehow, this is not clear to me. Is it pure trigonometric relation or is there a deeper meaning to this.
It would be great if you could elaborate this, may be a link to the literature where this is derived could be great.

Thank you in Advance.

Best Regards from Karlsruhe,
Prasanna Kannan

  • 0
    TI__Mastermind 24955 points

    Prasanna,

    Below you will find some helpful links that provides more information about mixers.

    https://en.wikipedia.org/wiki/Frequency_mixer

    Essentially, when you mix the transmitted signal and the received signal, the end result is a signal that contains a difference between the two signal and the sum of the two signals. Given the frequencies of the two signals, it does not make sense to use the sum of the two signals as they cannot be easily sampled with an ADC. They are filtered out using a low-pass filter, leaving only the difference between the two input signals. This signal falls within a range where an ADC is capable of sampling.

    Regards,

    Kyle

  • 0 in reply to Kyle Cousino
    Prodigy 130 points

    Dear Kyle,

    Thanks for your explanation, I understand that the response is a difference in frequency and difference in phase. What remains unclear is the mathematical expression, how did two sinusoids result in a third sinusoid.

    Let me put the other way around.

    a. What happens when two cosine signals are mixed

    x1 = cos(w1.t + phi1) &
    x2 = cos(w2.t + phi2)

    or even better would be what happens when

    b. cosine and a sine signal are mixed

    x1 = cos(w1.t + phi1) &
    x2 = sin(w2.t + phi2)

    Somehow I missing the mathematical relation here.

    Many Thanks

    Prasanna

  • 0 in reply to Prasanna Kannan
    Expert 1310 points

    Hi Prasanna,

    mixing means multiplication, hence the "x" symbol for a mixer. Refer https://en.wikipedia.org/wiki/Prosthaphaeresis for appropriate identities.

    Best regards

  • 0 in reply to user6096953
    Prodigy 130 points

    Hi,

    thanks for the link. Sorry I am missing something here.

    x1 = sin(w1.t + phi1)
    x2 = sin(w2.t + phi2)

    Going by the trigonometrical relation

    xout = x1 * x2  = (cos((w1-w2).t + phi1-phi2) - cos(w1+w2).t + phi1+phi2))/2

    how does this inturn become

    xout = sin[(w1-w2).t + (phi1-phi2)] in your paper.

    Best Regards

    Prasanna

  • 0 in reply to Prasanna Kannan
    TI__Mastermind 24955 points

    Prasanna,

    If you take a look at the AWR1642 Block Diagram, you will see that the output of the mixer is then sent to through an IF filtering stage before the ADC sampling block. It is in this IF filtering stage that that the summed frequency signal is filtered out through a low pass filter. With regards to the divide by 2 factor, this can in a practical sense be ignored since the the received signal goes through a LNA gain stage first before going through the mixer. This is effectively what I said in my previous post.

    The (cos(w1+w2)t + (phi1+phi2)) is filtered out by the low pass filter because the resulting frequency would be in the ~150 GHz range

    http://www.ti.com/data-sheets/diagram.tsp?genericPartNumber=AWR1642&diagramId=SWRS203A

    With all of these points, we can now arrive at the final equation (3) that you referred to in the white paper. 

    Regards,
    Kyle

  • 0 in reply to Kyle Cousino
    Prodigy 130 points

    Hi Kyle,

    Thanks for the detailed explanation. Can we summarize this as follows:

    a. The mixer stage depicted in the while paper includes the Low Pass filter, which essentially filters the w1+w2 term

    b. The equation for xout should be

        xout =  1/2 * (cos((w1-w2).t + phi1-phi2) or without the 1/2

    but it is for the sake of simplicity presented as

        xout = sin((w1-w2).t + (phi1-phi2))

    Best Regards
    Prasanna

  • 0 in reply to Prasanna Kannan
    TI__Mastermind 24955 points

    Prasanna,

    Technically, the low pass filter happens in the IF stage, not at the mixer stage. Otherwise, your understanding is correct. I will consider this thread resolved and close it.

    Regards,

    Kyle

  • 0 in reply to Kyle Cousino
    Prodigy 130 points

    Thanks Kyle, and the Texas team for the detailed literatures