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my book about RADAR says that:

If a Continuous wave radar sends a sine wave at frequency $f_T$ to a moving object (at speed V), a frequency $f_R$ is received. Their difference is called Doppler frequency and it is equal to $f_D = f_R - f_T = - 2 \frac{V}{\lambda}$.

It is quite clear. Then, it says that the situation is slightly different if a radar which sends rectangular pulses modulated with a sine wave at frequency $f_T$ is used. In such a case, the continuous wave is sampled by a train of strict rectangular pulses spaced by period PRT (Pulse Repetition Time): enter image description here

Then, it says that:

If $PRF = \frac{1}{PRT}$ (Pulse Repetition Frequency) is less than $2f_D$, spectral aliasing will occur (Nyquist Theorem).

First Question: why is the Nyquist theorem applied to the doppler frequency $f_D$? It should be applied to the sampled sine wave, which is at frequency $f_T$.

Then, it shows the aliased spectrum:

enter image description here

and says that:

The spectral aliasing causes spectrum folding at $f_D > \frac{PRF}{2}$ in the main interval ranging from $-\frac{PRF}{2}$ to $+\frac{PRF}{2}$

Second Question: why does aliasing cause folding? As far as I'm concerned, spectral aliasing means that the original (correct) spectrum is replicated and those spectra overlap (and hence it is not possible to separate them correctly).

enter image description here

I don't understand the link between my last two pictures.

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  • $\begingroup$ What book is this by the way? I want to take a look at it so my answer can be within the book's context to make things hopefully easier to understand. $\endgroup$ – Envidia May 17 at 18:13
  • $\begingroup$ @Envidia I've not mentioned the book since it's in italian and there isn't any pdf on the web. If you want to take a look to some slides extracted from it, you may look at pages 90-93 here: files.materiale-didattico.com/200000137-aecc2afc64/… $\endgroup$ – Kinka-Byo May 17 at 19:00
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So I took a look at the slides and it seems to be giving a cursory overview of how to measure Doppler with multiple pulses, so it's very generic. Hopefully this is a pretty straight forward thing to discuss given that your questions are suspicious of their statements, that's good!

First Question: why is the Nyquist theorem applied to the doppler frequency $f_D$? It should be applied to the sampled sine wave, which is at frequency $f_T$.

The received frequency $f_R$ is the transmitted frequency $f_T$ with the added Doppler shift $f_D$ as $f_R = f_T + f_D$. So in order to measure Doppler unambiguously, you need to be able to sample $f_R$ above Nyquist. But the question still stands as to why the text states that we must appropriately sample $f_D$ instead of $f_R$?

As is done in modern radar systems, and as implied in the text, the received pulses are mixed-down to baseband. That is, the carrier is centered around zero frequency. This means that $f_T = 0$ and so

$$f_R=f_D$$

Where the Nyquist criterion for the PRF on sampling Doppler is now more apparent.

It is possible to undersample around carrier frequencies that are not zero and still extract the desired signal, but I don't think that's necessarily important for this discussion. It's a slightly more advanced technique that simply achieves the same thing. You can look up "bandpass sampling" for more information.

Second Question: why does aliasing cause folding? As far as I'm concerned, spectral aliasing means that the original (correct) spectrum is replicated and those spectra overlap (and hence it is not possible to separate them correctly).

As you already stated, sampling produces spectral copies that are separated apart by your sampling frequency, in this case which is the $PRF$. Aliasing causes "folding" because as the frequency being sampled goes past Nyquist, in this case $\frac{PRF}{2}$, there is a copy incoming from the left into your un-aliased window that spans $[-\frac{PRF}{2}, \frac{PRF}{2})$.

Let's do a simple example where we have the following properties:

  1. $PRF = 50 \space kHz$

  2. Unambiguous spectrum span: $[-\frac{PRF}{2}, \frac{PRF}{2}) = [-25 \space kHz, 25 \space kHz)$

  3. Doppler frequency $f_D = 10 \space kHz$

Everything is fine here, we're well below the Nyquist rate and we can recover the Doppler frequency nicely:

enter image description here

Due to sampling, the adjacent copies to the left and right this sinusoid exist at $f_D \pm PRF$. Meaning we have two tones that exist at the frequencies

$$f_{left} = 10 \space kHz - 50 \space kHz = -40 \space kHz$$ $$f_{right} = 10 \space kHz + 50 \space kHz = 60 \space kHz$$

As expected, the copies are outside our Nyquist zone achieved by properly sampling.

Now, what if the Doppler frequency was beyond Nyquist? This is where aliasing causes "folding". Let's make the Doppler frequency be $f_D = 35 \space kHz$. The two adjacent frequencies are now

$$f_{left} = 35 \space kHz - 50 \space kHz = -15 \space kHz$$ $$f_{right} = 35 \space kHz + 50 \space kHz = 85 \space kHz$$

We can see here that the left copy has now moved, or folded, into our Nyquist zone! Then due to this aliasing, we measure the wrong Doppler:

enter image description here

A note on your last image:

This is a generic representation of signals that have some finite instantaneous bandwidth. In our case, we're dealing with sinusoids which have some peculiarities when it comes to aliasing since they are ideally delta functions in the frequency domain that don't have a "width" per se.

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  • $\begingroup$ Thank you for your answer. My last doubt is about fT, how can it be zero (DC transmitting EM wave)? $\endgroup$ – Kinka-Byo May 20 at 8:22
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    $\begingroup$ @Kinka-Byo That is done via a process called mixing. A mixer shifts the center frequency to another. In modern systems, "complex" or I/Q mixing is performed to mix down the signal to be centered around zero frequency. You can search on this site to learn more about that. $\endgroup$ – Envidia May 20 at 14:51
  • $\begingroup$ Clear. Last question: the concept of aliasing described is based on the fact that a frequency shift inside the "unambiguous interval" is problematic while if it lies outside that interval it is still good. Why? Why should we look at the interval -PRF/2; + PRF/2? $\endgroup$ – Kinka-Byo May 20 at 15:00
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    $\begingroup$ @Kinka-Byo Because that frequency span is your unaliased portion of the spectrum due to sampling. As long as the signals you sample have bandwidths that fit in this range, you completely describe the signal by its samples. This is the fundamental aspect behind the Nyquist sampling theorem. There's also filtering involved which is another story. Again, I would recommend to search the site to learn a bit more! $\endgroup$ – Envidia May 20 at 15:06

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