# efficient spectrum/fft analysis of sparse signals (found in radar applications)

consider the classic scenario of identifying a radar frequency. You typically have a very short pulse of very high RF frequency. For example, a 50ns pulse repeating itself every 1MHz or so (PRF 1MHz and Pulse Repitition Interval 1us) and let's say there are 50 such pulses in each (radar) interrogation so that each set of coherent pulses takes 50us.

The RF frequency is generally in Gigahertz but let's assume it has been downconverted to lie somewhere between 200MHz to 400MHz. Now you want to analyze this signal after subsampling it at a frequency of 100MHz. The straightforward approach would be to sample the whole 50 us worth of data but in reality the only time is signal is on during the 50us period is 50 times 50 ns which is just 2.5 us. The rest of the signal is just zeros. My question therefore is:

1. is there a 'smart' way of getting rid of this apparently useless information and hence be able to do a much shorter fft?

2. is it possible to throw away long strings of zeros in between active pulses, stick the pulses together and perform fft? would that cause any sort of distortion in the spectrum?

One solution for this kind of problem is called an Instantaneous Frequency Measurement (IFM) Receiver. The pulses are located with an envelope detector and a threshold. The average phase change per sample is computed across each pulse; this is proportional to frequency of the pulse.

• yes, but generally IFM techniques relying on phase-only do not work well under multiple simultaneous signal conditions. This is where fft analysis comes in – user4673 Jan 5 '14 at 1:48
• Are you talking about multiple simultaneous signals with overlapping pulses? If they don't overlap, the IFM will find all the pulses/frequencies. – John Jan 5 '14 at 1:52
• yes, I am referring to the overlapping pulses case – user4673 Jan 5 '14 at 2:04

I'm not sure I'm understanding your application. Are you trying to figure out the radar's frequency from the observations.

If you know the radar's PRF a priori - you can use that information to coherently add the pulses together. If you don't know the PRF you won't be able to eliminate the capture of the zero data.

Is the radar platform moving? If so there is a spatial Doppler effect which may need to be compensated for. The situation is similar to Synthetic Aperture Radar (SAR) for coherently adding pulses together. 50 pulses is not a lot - whether you'll need to compensate for the spatial Doppler will depend on the geometry of the problem.

• yes, I am trying to estimate the frequency. Let's assume the PRF is known or can be calculated, then I am not sure what you meant by 'coherently adding' the pulses. If you simply 'stitch' or concatenate them, a phase discontinuity will be introduced which will manifest itself as a strong component in the FFT with frequency equal to PRF and bandwidth as 1/PulseWidth?? – user4673 Jan 17 '14 at 0:39
• @user4673 Adding coherently means adding the pulses so that the phase differences are taken into account. So if a series of pulses are transmitted with relative phases of 0, $\pi/4$, $\pi/2$ ... , then to add these pulses coherently means to account for their relative phase shift. Adding them incoherently just takes their absolute value and adds them together. – David Jan 21 '14 at 13:29
• thanks for the clarification. Let me now rephrase what you said in paragraph 2 of your original response: – user4673 Jan 22 '14 at 1:39
• (continued) basically you are suggesting that individual pulses are stored as they arrive. And because the phase will most likely be discontinuous at pulse boundaries, shift all but the first pulse in phase by (1) finding the phase of the last sample, (2) adding a phase component to the next pulse samples taking into account phase change per sample plus the phase computed in (1)? NOTE: I do understand coherent integration used in radar receivers but that is not directly applicable here because phase/freq is unknown. Also, PRF is clearly known in this scenario. – user4673 Jan 22 '14 at 1:49
• @user4673 Yes. In the situation you're describing you may not be able to coherently integrate the pulses. – David Jan 22 '14 at 13:57