Signal Processing for SAIL

The Aerospace SAIL experiments used a series of pulses in which the optical frequency was swept quasi-linearly in time over a bandwidth greater than 1000 gigahertz. The linearity and stability of such broadly tunable sources is quite poor, leading to significant phase errors. To handle these errors, Aerospace researchers developed new digital signal processing techniques for mitigating the waveform instability problem and applied nonparametric phase gradient techniques for the pulse-to-pulse phase errors.

First, for each pulse, a Fourier transform is applied to the real values obtained from both the target and reference channels. The result is a conjugate symmetric spectrum whose sample index corresponds to the range or travel time relative to the propagation time through the "local oscillator" fiber path. Only those frequencies corresponding to echoes from the target and the reference channels are saved. This is known as a windowing operation. An inverse Fourier transform is applied to convert the windowed data back to the time or range wave number domain. A sequence of candidate phase-error corrections are applied to the target channel derived from the reference channel. For each candidate, a range-domain "sharpness" metric is computed to measure the range focus. The peak of the sharpness metric curve corresponds to the reference-channel phase-error scale factor where best focus is observed.

Next, the sharpness metric curves are averaged over the pulse index to obtain a composite sharpness metric curve. The peak in the composite curve corresponds to the reference-channel phase-error scale factor providing the best range focus for all of the pulses in aggregate. For each pulse, the scaled phase-error correction from the reference channel is applied to the target channel, using the scale factor determined from the composite sharpness metric. At this point, most of the range-phase error has been removed for each pulse.

A range Fourier transform is then applied to each phase-error-corrected pulse from the target channel. At this point, the data is range compressed. Phase-gradient autofocus techniques are then applied to the range-compressed target data to obtain a nonparametric estimate of the pulse-to-pulse phase errors. The correction is then applied to the range-compressed target data. The azimuth Fourier transform is then applied to the range-compressed data to obtain a SAIL image. The azimuth focus can now be further refined via reestimation of the azimuth quadratic phase error. Similarly, the range focus can be refined via reestimation of the range quadratic phase error, resulting in the final focused SAIL image.

	A Fourier transform applied to a single pulse would ideally produce a single narrow peak, the theoretical system impulse response function, as shown by the red curve.
	Instead, a broad (highly defocused) curve is obtained (orange curve). This discrepancy is caused by the many tens-of-thousands of degrees of phase error.
	Applying a single-pulse phase-error correction algorithm results in a well-focused one-dimensional "range image." The upper curve is the range image before phase-error correction, and the lower curve is the well-focused result after. At this point, although the range is well focused, the azimuth direction is completely out of focus because of pulse-to-pulse phase errors.
	These errors, as shown here, can be measured and thus corrected for, leading to the high-quality SAIL images.

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