I've come across an interesting question in the course of doing some exam review in a quantum mechanics book and thought I'd share it here.

"What must be the frequency bandwidth of the detecting and amplifying stages of a radar system operating at pulse widths of 0.1usec? If the radar is used for ranging, what is the uncertainty in the range?"

I don't know the solution; although I'd guess that it could involve a Fourier transform (a generic first crack at anything with waves) and the uncertainty relation (as the problem does call out uncertainty in range; which could be interpreted as uncertainty in x).

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Since you're talking about a quantum mechanics book rather than a radar design engineering book, I assume that they're talking about a basic textbook radar system, rather than one with all the advanced features.

The receiver needs to have a bandwidth great enough to receive most of the energy in the reflected pulses. Since they're sine waves modulated by a pulse shape, they will have spectral splatter and the receiver needs to be able to process most of this energy for each pulse, which is now spread over a range of frequencies. You're right in that, to compute this spectral spread, you need to do a Fourier transform - the spread you'll get depends on the pulse shape.

The longer the pulse, the narrower the bandwidth of the spectrum - sometimes people use a "rule of thumb" whereby the bandwidth is estimated as the reciprocal of the pulse duration.

With respect to range, I suppose if you imagine two targets close to each other(let's assume same azimuth), one behind the other, then you can resolve them if the reflected pulse from the nearest has finished being received before the reflected pulse from the furthest starts to be received. It's fairly easy to see that this implies that the distance between the targets must be greater than 0.5*PulseTime*c

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Can't see why it would be a QM question though - pretty standard classical EM – Martin Beckett Nov 28 '11 at 1:47
@Martin: No, neither can I really. Unless it's just illustrating the properties of the Fourier transform: shorter pulse time implies wider bandwidth, and drawing some analog with the HUP ? – twistor59 Nov 28 '11 at 7:53

The first part of the question deals with the so called "Transform Limited" pulses. A transform limited pulse is one where the time-bandwidth product is a minimum (unity). This can be loosely thought of as a manifestation of the energy time uncertainty relationship.

So:

$\Delta t\times B=1$. From this you can find the bandwidth.

The next part of the question seems to be related to the coherence length of the radiation, which is defined as:

$L_{coherence}=\frac{c}{\pi \Delta f}$

Note that $\Delta f$ is the bandwidth B from the first part of your question. So, anything beyond this range will have an associated error in position measurement.

Edit: Also take a look at Wiki, where there is an expression for coherence length in terms of wavelength. What you need is the $\Delta\lambda$ term.

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