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Beat signal of a single target will be a sinusoid in the idealized world, so theoretically the signal processing gain of an FMCW pulse correlated with Tx waveform in this way should be analogous to FFT processing gain of a single tone, right?

Just trying to wrap my head around what to expect in the field when playing with waveform parameters and what kind of SNR before/after I can expect to see. Is it really just a product of the duration of wave form and bandwidth? Looking into buying a textbook on the subject of FMCW so if you have any recommendations feel free to drop em here

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  • $\begingroup$ Beat signal of a single target will be a sinusoid only for a sawtooth frequency trajectory of your FMCW radar; $\endgroup$ Apr 8 '21 at 15:28
  • $\begingroup$ Just to be clear, OP you want to compare the difference between match-filtering an LFM pulse (or any FM modulation for that matter) vs generating a beat frequency and taking the DFT? $\endgroup$
    – Envidia
    Apr 8 '21 at 17:40
  • $\begingroup$ Correct, I just don't have a good feel for how to go about approximating processing gain for generating a beat frequency. Is it just a function of length of sawtooth and bandwidth in the LFM case? And how does that translate to a gain in dB? $\endgroup$ Apr 8 '21 at 20:16
  • $\begingroup$ @kit_traverse Also, I'm guessing you're trying to plug in this factor in the radar range equation? $\endgroup$
    – Envidia
    Apr 11 '21 at 4:17
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Let's start with your main question

Is there processing gain for FMCW using heterodyne-style receiver as opposed to matched filter?

The answer is a "yes" with a big asterisk next to it. It's important to note that there are two main ways to determine range in a radar system:

  1. Matched-filter: This is an ideally auto-correlation based method where range is found based on the signal return's delay.
  2. Fourier transform: Uses the Fourier transform to extract beat frequencies generated from the mixing process. These frequencies can then be mapped to a range as shown here.

The Fourier transform of a signal can be thought of as a matched-filter for a particular frequency. There is an important result that states that only the energy of the pulse determines the SNR after matched-filter processing. Given a signal $x(t)$ with Fourier transform $X(f)$, it's energy $E$ is given by

$$E = \int_{-\infty}^{\infty} |X(f)|^2df$$

And in an environment with white-noise variance (power) $\sigma_w$ the achieved SNR is

$$SNR = \frac{E}{\sigma_w^2}$$

You might have seen that the "processing gain" of an LFM pulse is given by the time-bandwidth product $\beta\tau$. If you were to plug this into the radar range equation for SNR, and assuming that your receiver also has a bandwidth of $\beta$, then that term cancels and you're left with just the pulse length $\tau$, just as predicted by the equation above.

When it comes to DFT processing, there is a similar result where given a signal $x(t)$ and white-noise $n(t)$ you can apply statistical expectations to both signals and take the ratio. Without going into the details, the result is given by

$$SNR = \frac{\sigma_x^2N^2}{\sigma_n^2N} = \frac{\sigma_x^2N}{\sigma_n^2}$$

There are to major caveats here that are very important and are difficult to achieve practically:

  1. The length of the signal is also $N$. This is usually not the case since padding is usually done to improve the frequency bin size and performance, by usually padding to a power of 2. If the total length of the padding is larger than $N$, you are still capped at an SNR increase by the factor $N$.
  2. The frequency bin size, given by $f_s/N$, is exactly an interger multiple of the frequency to be estimated. If this is not realized, you do not achieve the maximum SNR. This is the hardest of the two to achieve in real life.

Assuming that the FMCW signal's final bandwidth is exactly at Nyquist, just as before with the matched-filter, you will see that the factor of $N$ after some manipulations gets cancelled out in the radar range equation.

In either case of the matched-filter or the DFT, the maximum SNR comes down to the energy provided in the pulse, just as matched-filter theory predicts.

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  • $\begingroup$ This is great and just what I was looking for. Thank you for your answer. I do not expect caveat 2 to be attainable here as I don't have a way to predict/control the frequency to be estimated, however caveat 1 - you're basically saying padding zeros will not count towards my signal length, N, right? This makes sense as I can't get something for nothing. Thank you again for your answer $\endgroup$ Apr 12 '21 at 14:22
  • $\begingroup$ @kit_traverse That's right, once you go beyond N you're just coherently integrating zeros. There's a few more caveats here but the takeaway is that these best-case gain values are for very exact signal conditions but are used as a good starting point. You can then start determining your losses and get realistic SNR values. $\endgroup$
    – Envidia
    Apr 12 '21 at 15:27
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Beat signal of a single target will be a sinusoid in the idealized world,

Only for a TX waveform that has a constant slope in frequency domain (that means only with sawtooth-shaped frequency trajectory).

so theoretically the signal processing gain of an FMCW pulse correlated with Tx waveform in this way should be analogous to FFT processing gain of a single tone, right?

No, because the FFT gain is below the theoretically maximum processing gain, in general (it is the optimal estimator only for beat frequencies that are multiples of (sample rate)/(FFT length))

The FFT is usually not a good frequency estimator if you're just looking for a single tone.

Just trying to wrap my head around what to expect in the field when playing with waveform parameters and what kind of SNR before/after I can expect to see. Is it really just a product of the duration of wave form and bandwidth?

Yeah, the integration gain basically collects all received energy.

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  • $\begingroup$ Should have specified, yes this is a LFM, sawtooth application I am considering here. I am not expecting a single tone, but thought for ease of conceptualizing I'd start there. $\endgroup$ Apr 8 '21 at 15:35
  • $\begingroup$ It's not the number of tones – it's the fact that the FFT is simply a bad frequency estimator if the frequency isn't actually in the frequency grid spanned by the FFT's length. There's simply better ways to estimate the frequency of a signal composed of one or a number of tones. $\endgroup$ Apr 9 '21 at 12:04
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Is there processing gain for anything at all using heterodyne-style receiver as opposed to matched filter?

All else being equal -- no. By "all else being equal", I mean that you're not throwing away information.

If you don't decimate, then heterodyning + post processing is going to, in the end, come up with the same result including signal/noise ratio as a matched filter.

If you do decimate, and you're careful with saving the extra bits resulting from process gain, you still come up with substantially the same result, including signal/noise ratio. And I only say "substantially the same" because you may have some tiny amount of extra quantization noise in the mix; as long as you're careful with word lengths, than will be buried in the noise and make no discernible difference.

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