Finding the frequency of sinuosid in a constant + sinusoid?

[khurram usman]

Hi everyone,

I am working on some problem relating to radars. The problem boils down to finding the frequency of the complex exponential in a constant + complex exponential + noise model. I found some papers on sinusoid recognition but they use the sinusoid + noise model only. I tried to come up with an approach myself. It worked fine without noise but with noise it crashes. My approach actually increases the variance of the noise and being in a SNR limited region I can't work with that. Just wanted to inquire whether this problem has been explored by anybody else in any field.

Thanks
Khurram

 

[olivermsun]

Hi everyone,

I am working on some problem relating to radars. The problem boils down to finding the frequency of the complex exponential in a constant + complex exponential + noise model...Just wanted to inquire whether this problem has been explored by anybody else in any field.

Just about anybody else working on radars, for example?

Do you have any example data and/or description of the method(s) you have tried?

 

[FactChecker]

You say that the SNR is limited but that an increasing noise variance is a problem. Is the noise proportional to the magnitude of the complex exponential? If so, might it be a multiplier of the exponential and taking a logarithm might turn it into a linear model with a limited noise variance?

 

[khurram usman]

Just about anybody else working on radars, for example?

Do you have any example data and/or description of the method(s) you have tried?

No i don't have any data as such. I have my own simulation setup. But the end result of the simulation from which i am trying to compute the frequency is the model i described above, that is a constant + complex exponential + white noise (most probably white noise). I can provide a brief description of the method that i was trying to use. I took sample by sample difference of the data array. That removes the constant but doubles the noise variance. From here onward, i ideally wanted to extract the unwrapped phase of the leftover exponential and noise and use least squares fit on the phase to find the frequency. This was a very basic algorithm but because of already low SNRs and noise enhancement of my method its not working very well

 

[khurram usman]

You say that the SNR is limited but that an increasing noise variance is a problem. Is the noise proportional to the magnitude of the complex exponential? If so, might it be a multiplier of the exponential and taking a logarithm might turn it into a linear model with a limited noise variance?

Please read by reply just above this post. No noise doesn't increase proportionally wioth the exponential amgnitude. Its just that i am not operating at high SNRs and when i extract the phase after differencing as described above its not very reliable. It jumps around a lot due to which unwrapping doesn't work well and finally LS fails

 

[olivermsun]

The differencing step amplifies noise with increasing frequency, as you found already. What if you just remove the mean of the signal?

Have you explored using either FFTs or the autocorrelation of the data?

 

[khurram usman]

The differencing step amplifies noise with increasing frequency, as you found already. What if you just remove the mean of the signal?

Have you explored using either FFTs or the autocorrelation of the data?

Well removing the mean will most probably not work because there is no guarantee that the exponential is going to an integer number of cycles. That will bias the mean. FFTs might work. I thought a little about using it. Don't have any idea about using autocorrelation. Expand on these two points if you have any idea.

 

[FactChecker]

I have only a vague understanding of what you are doing, but I have used something that might help. Please forgive me if this idea is not relevant.
It's about unwrapping an angle time series in the presence of noise. I had to take the deltas and increment or decrement a "winding number" when the angle jumped more than +-180 degrees. Then I added 2π*(winding_number) to the final output. That removed the large jumps 2π jumps when it crossed +- 180 degrees and allowed me to turn the angle into a continuous (unbounded) angle. It worked as long as the noise of the angle was not so large that it caused 180 degree noise jumps.

 

[khurram usman]

I have only a vague understanding of what you are doing, but I have used something that might help. Please forgive me if this idea is not relevant.
It's about unwrapping an angle time series in the presence of noise. I had to take the deltas and increment or decrement a "winding number" when the angle jumped more than +-180 degrees. Then I added 2π*(winding_number) to the final output. That removed the large jumps 2π jumps when it crossed +- 180 degrees and allowed me to turn the angle into a continuous (unbounded) angle. It worked as long as the noise of the angle was not so large that it caused 180 degree noise jumps.

Your reply is helpful. But unwrapping comes after I have removed the DC or the constant term. Because otherwise the phase of the whole thing itself stays small if the constant term is large in magnitude. It does not make sense to take the phase of the whole thing. I somehow need to get rid of the constant