2.2 The design
2.2.3 Related literature
2.2.3.2 Related systems
Though calibration is not discussed in detail, the paper suggests that non-ideal integrators should be acceptable as long as they can be characterized well, which agrees with the findings presented in this chapter. Some Matlab simulations are presented using IRLS.
(a.k.a. “blind”) bands.
If the band locations are unknown, then after collecting a reasonable number of samples (for example, 1 to 10 µs worth), an `1 block sparsity problem is solved to find locations of non-zero blocks of spectrum; this is referred to as the “continuous-to-finite” (CTF) block. The computational problem may be reasonably sized, though perhaps not possible for real-time implementation. This processing is cheapest when the bands are large; if there are many bands that are very small and evenly spread out, the computational problem may be expensive. In the follow-up paper [MEE09], the authors describe a digital processing step that recovers the base-band versions of the occupied frequency bands.
The methodology behind Xampling is motivated by working directly with analog signals, and doing all processing at base-band, rather than Nyquist rate. The work seems promising, though it may be best adapted for spectrum sensing, such as in cognitive radio, rather than full recovery; or suited for recovering signals with known types of modulation (such as QPSK). Another key feature of the model is that the system is relatively simple to implement and does not rely on delicate hardware; to this extent, they seem to have achieved the goal admirably.
The hardware of both the MWC and the RMPI are not too dissimilar, the main difference being the chipping sequence rate, which is much higher for the RMPI case. The essential difference of the system is the processing. The RMPI makes full digital processing, while the MWC uses digital processing only to identify the support, and then works with base-band digital processing. Our view is that this methodology is exciting and worth further investigation. However, below we detail a few minor criticisms of the MWC and address some comments from [MEE09] about the RMPI.
We first note the difference in signal models, which is simply a matter of application. Both the MWC and RMPI do poorly when analyzed using the wrong signal model. The MWC assumes a static spectral occupancy. Whenever the spectrum changes significantly, the CTF block must recompute the spectral support. Furthermore, this spectral estimation requires many time samples, and would give inaccurate estimates for time-dependent inputs like a radar pulse. Similarly, the RMPI adds unnecessary complexity and processing if applied to the problem of finding signals with infinite support. It doesn’t exploit continuity in the spectrum from one reconstruction window to the next, and it does not automatically offer full analog resolution since it works with finite discrete samples.
It is mentioned that RMPI systems have a high computational load, and this is true. Real- time processing of the RMPI is not yet achievable, though as this thesis shows, the processing time is reasonable. The standard RMPI reconstruction works at the Nyquist rate, although the compressive matched filter doesn’t rely on this, so is closer to the baseband processing suggested in Xampling. Other valid criticisms are the lack of flexibility for changing the bandwidthfson-the-fly, and sensitivity to windowing issues. In particular, the on-grid tone model is viewed as too academic.
For the radar signals studied in this thesis, windowing and on-grid frequencies are not a major issue since the radar envelope is itself a window with finite bandwidth. More about windowing is discussed in §2.7.2.6. It is true that if our RMPI were restricted to measuring on-grid pure tones, it would perform much better, but we find adequate performance without this assumption.
It is suggested that in the RMPI, any deviation of the integrator filter from being a true integrator will introduce signal-dependent errors, and the system really must be calibratedcontinuously! This is quite misleading. The RMPI system certainly needs to be calibrated, but this can be done once.
In particular, the post-processing does not rely in any way on the ideal integrator model. The calibration will not be exact, and the error due to this will be signal dependent, but this is not necessarily dangerous, and is inherent in many hardware devices, including the MWC. For example, the system may be mis-calibrated because of an absolute timing shift, but this is harmless because the reconstruction will just be a time-shifted version of the input.
Cognitive radio. Cognitive radio systems are designed to sense a broad range of RF spectrum to detect unoccupied channels, and as such, this technique is not tied to a particular hardware implementation. For example, both the MWC and the RMPI can be used for this. Many papers on this area have appeared recently; see [ME10] for a discussion.
Nyquist Folding Receiver (NYFR). Proposed in [FBC+08] and also part of the same DARPA project as the RMPI, this design samples at a time-varying rate. For example, the sampling rate averagesfavg= 2 GHz, but varies sinuisoidally in time between 1950 and 2050 MHz. For input tones of high frequency, sayfin= 10 GHz, this induces aliasing. The aliased signal will change frequency over time since the sampling rate is changing over time. The amount this aliased signal changes over time is determined by what “Nyquist zone” the carrier signal was in. The Nyquist zone L is the smallest integer such that favgL > fin. Thus even from aliased signals, it is possible to determine what the original absolute frequency was, since the time dependence introduces a splitting.
This design is examined in [FBC+08] for a few narrowband tones, and it is shown experimentally that their original frequencies can be successfully recovered; this is compared to a standard time- independent sampler, which cannot recover the original frequency. However, the implications are not clear, since many applications are not actually concerned with recovery of absolute frequency information and only need the narrowband content. It has not been shown that this design is robust to recovering arbitrary sparse tones, e.g., tones that are close in frequency, or tones with significant bandwidth that overlap two Nyquist zones. The method is claimed not to rely on digital Nyquist-rate
`1 recovery, though the actually recovery algorithm is not discussed.