Hello, The mathematics of frequency mixing seems so simple, but I always end up getting confused when I am trying to relate the math to what I observe in the lab - I don't get the correct products - have to change a sign or something to get things to work out correctly. Right now, I am trying to come up with the power transfer function of sinewave that is delayed and frequency, mixed with itself. One way I think of an "ideal mixer" in the time domain is as a multipler - in the frequency domain I think of it as a convolver. But if I think of the mixing process as a difference in the frequency domain (which seems intuitively obvious), I get the power transfer function that corresponds with my observations: H(w)=1-e^jwT, or |H(w)|^2=2*(1-cos(w*T)), where T is the delay. I tried to prove to myself that a mixer performs a difference in the frequency domain by taking the simplest case, but couldn't work it out. If I difference the fourier transform of two complex functions, e^jw1t and e^jw2t (to inspect the resulting difference frequency w1-w2) I have to difference two delta functions, and, embarassingly I don't know how to do that. I then got even more confused when I tried to just sit down and prove to myself that the mixer does indeed perform a convolution, by convolving the aforementioned delta functions, but ended up with a delta function at w1+w2 - not w1-w2. Can someone help me out with this? Is there a paper or book that lines out frequency mixing so I can understand the basic math of mixing? I would assume this is basic linear systems stuff. And is it valid to think of mixing as a difference, not a convolution, in the frequency domain? Thanks, John M.
mixing a sinewave with time delayed replica of itself
Started by ●July 27, 2006
Reply by ●July 27, 20062006-07-27
JohnReno wrote:> Hello, > The mathematics of frequency mixing seems so simple, but I always end up > getting confused when I am trying to relate the math to what I observe in > the lab - I don't get the correct products - have to change a sign or > something to get things to work out correctly. Right now, I am trying to > come up with the power transfer function of sinewave that is delayed and > frequency, mixed with itself. One way I think of an "ideal mixer" in the > time domain is as a multipler - in the frequency domain I think of it as a > convolver. But if I think of the mixing process as a difference in the > frequency domain (which seems intuitively obvious), I get the power > transfer function that corresponds with my observations: H(w)=1-e^jwT, or > |H(w)|^2=2*(1-cos(w*T)), where T is the delay. I tried to prove to myself > that a mixer performs a difference in the frequency domain by taking the > simplest case, but couldn't work it out. If I difference the fourier > transform of two complex functions, e^jw1t and e^jw2t (to inspect the > resulting difference frequency w1-w2) I have to difference two delta > functions, and, embarassingly I don't know how to do that. I then got > even more confused when I tried to just sit down and prove to myself that > the mixer does indeed perform a convolution, by convolving the > aforementioned delta functions, but ended up with a delta function at > w1+w2 - not w1-w2. Can someone help me out with this? Is there a paper > or book that lines out frequency mixing so I can understand the basic math > of mixing? I would assume this is basic linear systems stuff. And is it > valid to think of mixing as a difference, not a convolution, in the > frequency domain?Mathematically, mixing is usually treated as a multiplication in the time domain. In practice, it is often accomplished with analog signals by running the signals simultaneously through a nonlinear element. Sometimes, as with balanced (and doubly balanced) mixers and AM modulators, each signal is applied to different "ports" of the circuit. Addition or subtraction play no part in mixing, other than as a means to get the signals simultaneously into the nonlinear element. The actual mixing -- "intermodulation" is another name -- is a result of the nonlinearity. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●July 27, 20062006-07-27
"Jerry Avins" <jya@ieee.org> wrote in message news:FsCdnbkmZ7MIXlXZnZ2dnUVZ_sydnZ2d@rcn.net...> > Mathematically, mixing is usually treated as a multiplication in the > time domain. In practice, it is often accomplished with analog signals > by running the signals simultaneously through a nonlinear element.That's a long time ago.Nowadays we have linear multipliers.In fact they only need to be switching multipliers 1 or -1 since the higher harmonics get filtered out. M.P -- Posted via a free Usenet account from http://www.teranews.com
Reply by ●July 28, 20062006-07-28
Mad Prof wrote:> "Jerry Avins" <jya@ieee.org> wrote in message > news:FsCdnbkmZ7MIXlXZnZ2dnUVZ_sydnZ2d@rcn.net... >> Mathematically, mixing is usually treated as a multiplication in the >> time domain. In practice, it is often accomplished with analog signals >> by running the signals simultaneously through a nonlinear element. > > That's a long time ago.Nowadays we have linear multipliers.In fact they only > need to be switching multipliers 1 or -1 since the higher harmonics get > filtered out.Do you mean that multipliers with gains that switch from +1 to -1 aren't nonlinear devices? If not, how does anything you wrote contradict what I did? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
