How to interpret an Ambiguity function I need to evaluate several different waveforms regarding their doppler tolerance, for use in a radar application. So I came across something called the Ambiguity function.

The following picture shows the Ambiguity function contour-plot of a Barker-Code with 7 Chips. 



My problem is that I don't fully understand how to read it. 

If I take the 0Hz Doppler-cut, I get the expected, ideal barker autocorrelation. Now consider a target which moves with a fixed velocity. In order to calculate the doppler frequency $f_d$ I can apply the following formula:

$$ f_d = \frac{\Delta v}{c} f_0$$

with $\Delta v$ being the (relative) velocity of the target, $c$ being the velocity of the medium (or the waves in the medium) and $f_0$ being the emitted frequency.

Now my problem is, that a barker code is not just built up by a single frequency which I could use as value for $f_0$. The barker code has a whole spectrum of frequencies (fft). This also means that I have to look at a bunch of doppler frequencies istead of looking just at one, right?.... 

So I cannot just take, for instance the 20kHz doppler cut, to see what correlation result I can expect for a target with a certain velocity. 

What frequency of my signal-spectrum do I need to use to calculate the doppler frequency?

In my particular case I do a BPSK-Modulation of a 13x13 (nested) barker code, before sending it out and calculate the matched-filter response on the received signal. In order to get an idea of how the matched-filter output looks, I would like to take a "doppler cut" at $f_d = ..$. But should I take the carrier-frequency as $f_0$ ? Or should I take the maximal frequency that I have in my spectrum as $f_0$? ($f_0$ is needed to calculate $f_d$)
The ambiguity function is used in radar systems to get the distance and the relative speed of a moving object with respect to the transmitter. Is called ambiguity because it also tells about the ability to distinguish objects that are close between them and with a similar vector velocity. 

The ideal ambiguity function is a Delta function in both domains. However there is no waveform with this ambiguity function. With your Barker-Code, you could have troubles to identify different objects that are at a distance +/- &lt;10us, and whose Doppler is +/- &lt;10 kHz. But you could distinguish two objects that are at 5us, one with 5 KHz and the other with 27 kHz.

In order to estimate the Doppler and the delay, you first compute the ambiguity function between the Barker-Code and the reflected signal. You will get a noisy version of the above figure but shifted in both domains, depending on the delay and Doppler of the target. If there are more targets, you will obtain overlapped ambiguity functions, each one shifted in a different place on the correlation plane. 

You then just find the position of the maximum. From the obtained Doppler you can get the speed with the equation you wrote.