Please help me out with this one. I think I've been given an impossible task.

I'm working on a system that transmits an active signal and looks for strong reflections. The transmitter is actually composite: it is a three-dimensional array of approximately point-like transmitters. Each point=like transmitter transmits a slightly different signal. The beampattern of the composite transmitter has been empirically characterized in azimuth and declination. This beampattern samples the transmitted time series along a sample grid of (azimuth, declination) points. Knowing this beampattern is essential to the signal processing.

The goal is to look for coherent signal returns using an associated receiver and pulse compression (among other processing techniques). The intention is to use the characterized beampattern as the template for pulse compression. The trouble is, the beampattern characterization was done using a shortened form of the transmitted waveform.

Specifically, the transmitted waveform is a simple frequency modulated one. Say it's a linear frequency modulated (LFM) waveform. The shortened waveform has a start frequency of $f_0$ and an end frequency of $f_1$ and a duration of $T$.

The real waveform is a lengthened version of this but with the same start and stop frequencies: the long waveform has the same start frequency of $f_0$, and the same end frequency of $f_1$, and a longer duration of say $kT$.

My trouble is that I need to use the empirically-measured shortened beampattern to pulse compress (in particular via matched filtering) the long waveform which is actually transmitted. I don't think there is a transformation of the short waveform that will generate the long waveform. I think this is an impossible task.

This can be stated as a resampling question: I need to resample the short waveform into the long waveform but keep its frequency content intact.

The rescaling property of the Fourier transform (see the Similarity Theorem) means that if I lengthen the time domain signal then I compress the frequency domain signal and vice versa.

Can anyone help me out? I would love to be proven wrong!

I thought I would come back and update with an interesting approach that I'm trying. It's hard to imagine what future reader would possibly be interested in this question, but for them and for posterity here is the method I'm trying.

The idea is to treat the transmitter array as a linear system. I know the analytic waveform that is being sent to individual transmitter elements. I have the characterization of the overall output beampattern. I simply assume that for every point in (azimuth, declination) space the transmitter acts as a linear system on the input waveform. Moving ahead anyway, I can solve for the transfer function of the transmitter array by deconvolving the measured output by the analytic input. Now that I have the transfer function, I can convolve it with the analytic long form of the signal. This should give me the expected transmitter array output for the long signal. Indeed, in principle this should work for any arbitrary input signal. Assuming that the transmitter acts as a linear system allows me to compute the beampattern for the long waveform.

Of course, it is another matter entirely to verify the validity of this assumption ...

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    $\begingroup$ if I understand right, you still have alternatives,this can be done using time scale to expand your signal without change your frequencies, some ways here this question is applied to audio, but I believe it can be adapted to your problem $\endgroup$ – ederwander Jul 23 '16 at 2:07
  • $\begingroup$ Hm, yes you do understand my question. Yeah, I'll have to look into some of those techniques. $\endgroup$ – Austin A. Jul 23 '16 at 2:25
  • $\begingroup$ Can you add a bit of info on your transmitter? How does the beam pattern manifest? I.e. do you achieve it through digital beamforming with multiple transmit antennas, or is it just one antenna with directivity? $\endgroup$ – Marcus Müller Jul 23 '16 at 7:53
  • $\begingroup$ @Marcus I added some details on the transmitter. It is a complex transmitter and simulating the beampattern probably won't be accurate enough. $\endgroup$ – Austin A. Jul 23 '16 at 12:56
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    $\begingroup$ @Marcus each transmits something slightly different. I edited the post to include this detail too. $\endgroup$ – Austin A. Jul 23 '16 at 13:07

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