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I am confused about the delay concept in the ambiguity function. The FFT of slow time is the doppler frequency. However, people usually relate the range-doppler domain with the ambiguity function domain. However, the range corresponds to the fast time.

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Your question is a little bit unclear, but I'm going to try to answer according to my understanding of your question:

So it sounds like you're asking about pulse-Doppler radar processing, where we have two time dimensions: fast-time and slow-time. Let us review those terms really quick.

In pulse-Doppler processing, multiple pulses are sent out by the radar at a steady rate/frequency (PRI/PRF, pulse repetition interval/frequency). Each time a new pulse is sent out, the radar receives signals for a period of time, and then it switches back to a transmit operation and starts the cycle over again. The data that is received is stored in a 2D matrix. Each time the radar receives a new pulse, a new row is added to the matrix, and the data that is recorded at each receive interval is placed in this row.

This means that, like you said before, we have two time dimensions: fast-time and slow-time. Fast-time corresponds to a small-scale time measurement of radar data at each pulse; slow-time is a (typically) much larger time scale, and corresponds to the dimension where each pulse was sent out.

Note that in this explanation so far, we have not taken the DFT of anything, so our 2D matrix represents time and time. Like you alluded to before, if you take the DFT of the slow time dimension's IQ data, what you will get is a transformation to the Doppler domain. This happens because over the slow-time dimension, your target signal's phase will be changing if it is moving. Assuming the target is moving at a constant rate, this means that the phase of the target signal will linearly change, and a liner change in phase is equivalent to oscillation at some frequency; this is why we use the DFT for this: it allows for an easy way/convenient way to measure phase.

So now that we've reviewed that, lets talk about ambiguity functions:

Think of how range-Doppler data recording is structured. There are multiple transmit/receive options, and in order to keep the target's phase stable, every transmit pulse is (typically) exactly the same. Therefore, how can I tell whether a target signal appeared in range during my reception of the first pulse, or if the target appeared at any other range? This is the concept of range ambiguity. When you look at an ambiguity function, typically the x-axis is labeled as Tau, representing a time delay: this is equivalent to the range dimension.

Similarly, let's think about how Doppler is being measured: with pulse-Doppler processing, we're counting on being able to adequately measure the rate of change of the phase in question using a set sampling interval (our PRI/PRF from earlier). It turns out that our PRF/2 is our effective Nyquist frequency for our slow-time sampling operation Well, just like in sampling say, digital audio signals, we have to worry about aliasing any signals we record. For audio signals, if my Nyquist frequency is, for example, 1kHz, any signals with frequency content above 1kHz will be aliased to look as if they are at another frequency. This is exactly what can happen in the Doppler domain when using pulse-Doppler processing: if I have target velocity that is greater than my PRF, my DFT cannot accurately represent it, and that target signal "appears" somewhere else in the Doppler domain ambiguously. This is called Doppler ambiguity.

So how does the range-Doppler map equate to the ambiguity function? Well, they've both effectively measuring the same thing! The ambiguity function is a measurement of Doppler speed and range/time. The range-Doppler map (RDM) from pulse-Doppler processing is a measurement of... Doppler speed and range/time!

There are two types of ambiguity functions: single pulse, and multi-pulse. A multi-pulse ambiguity function simple shows you the "full" picture of both range/Doppler that you cannot see on the RDM due to measurement ambiguities.

Now a single pulse ambiguity function is also a 2D image, but we have no slow-time dimension. Instead, our Doppler dimension is derived by using multiple hypotheses of the target speed and baking our guesses in to the pulse-compression operation that gets us our range dimension.

So hopefully all of that helps answer your question a bit. If you'd like to learn more and have access to MATLAB, try walking through the examples on this MATLAB page

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    $\begingroup$ Thank you very much for your explanation. Could I understand as follows: We do not need to specify the time delay in the ambiguity function domain as the fast time or the slow time. If we use a single pulse, then it is a fast time. However, usually, the Doppler frequency resolution for a single pulse is low. The utilization of multiple pulses can help us to improve the Doppler frequency resolution. In the case of multiple pulses, we can refer to the time delay as a slow time. $\endgroup$
    – shui
    Apr 25, 2020 at 1:38
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    $\begingroup$ Thanks again for your time. It is really helpful. I think I should not be struggling with the definition of time delay as a fast time or a slow time. Time itself is more suitable. $\endgroup$
    – shui
    Apr 25, 2020 at 1:44
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    $\begingroup$ Fast time means I’m measuring a range, I.e. what the radar receiver sees. Slow time is when I look across each sample of each fast time measurement from a number of samples. As for the comment about Doppler resolution for single pulse, that’s not true with phase/frequency coded waveforms! These present processing challenges, but their Doppler resolution in the ambiguity function is defined only by their pulse width (their bandwidth defines their range resolution), and we can basically make a RDM from them. You are correct with multi pulse waveforms, more pulses means better Doppler resolution. $\endgroup$
    – vintagevogue
    Apr 25, 2020 at 1:47
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    $\begingroup$ Thank you very much for your correction. I checked your suggested MathWorks page regarding the LFM, stepped FM, and Backer Code. I should have been more accurate. Could you help me to check is the following saying correct? In the case of multiple pulses, even though it uses multiple pulses, but the information of the range is contained in the time duration of the first pulse. So fast time will be more suitable to describe the time delay in the ambiguity function domain. $\endgroup$
    – shui
    Apr 25, 2020 at 2:13
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    $\begingroup$ I got it. Thank you very much for your explanation. In the past, I did not get out of the loop: when we use multiple pulses, it should refer to the slow time, but why it refers to the range. I think when we use multiple pulses, both the information of fast time and slow time are included. I have one additional question: when we use the coherent pulse train why the Doppler resolution is decided by the reciprocal of the entire pulse train (N*PRT) instead of the pulse width? Thank you very much for your time. $\endgroup$
    – shui
    Apr 26, 2020 at 2:42

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