The Big Picture
Signal Acquisition and Processing in Union of Subspaces
Moshe Mishali and Yonina C. Eldar
What is Xampling ?

Xampling is a system architecture designed for sampling and processing of analog inputs at rates far below the Nyquist rate, whose underlying structure can be modeled as a union of subspaces. This website provides a brief  introduction to union modeling and the Xampling framework, through several examples of engineering applications.
More detailed information on Xampling can be found on our Xampling workshop page and in the tutorial slides below.
You can learn more about Xampling by viewing several tutorials presented at the Xampling Workshop at the Technion, You can also see some Xampling Demos on our hardware page. A short video which demonstrates sub-Nyquist sampling and processing for ultrasound imaging can be found here, and a demonstration of sub-Nyquist radar can be found here. Finally, the links under the Xampling tab to the left contain more information on specific applications.


Consider a communication receiver which intercepts multiple radio-frequency (RF) transmissions, but is not provided with their carrier frequencies fi. In this setting, the input x(t) has multiband spectra with energy that concentrates on N frequency intervals of individual widths B located anywhere below some maximal frequency fmax. Such a receiver faces a challenging sampling problem, since classic acquisition methods, such as RF demodulation or bandpass undersampling, require knowledge of the values fi. At first sight, it may seem that sampling at the Nyquist rate, namely twice fmax is necessary, since every frequency interval below fmax can potentially contain a transmission of interest. On the other hand, since each specific x(t) fills only a portion of the Nyquist range (only NB Hz), one would intuitively expect to be able to reduce the sampling rate below 2fmax.

Another interesting application is estimation of time delays of an input signal that consists of several echoes of a given pulse shape. This scenario is encountered, for example, in a communication channel that introduces multipath fading. Another example is radar, where the delays correspond to target locations, while the attenuations of each pulse encode Doppler shifts indicating target speeds. Medical imaging techniques, e.g. ultrasound, record signals of this form to probe density changes in human tissues as a vital tool in medical diagnosis. Underwater acoustics also conform with this signal model. Since in all these applications, the pulse shape is short in time, sampling according to its Nyquist bandwidth results in unnecessary large sampling rates. In contrast, it follows intuitively that the actual number of unknowns is 2L where L is the number of echoes, which in all the above applications can be substantially lower than the Nyquist rate.


Union of Subspaces

The model of union of subspaces is illustrated in the following figure. The input signal x(t) lies in either of the subspaces A_lambda. However, apriori, the exact signal subspace is unknown. This is different from traditional models studied in classic sampling theory, which assume that the input lies in a predefined subspace. Bandlimited signals are perhaps the most prevalent example of classic modeling.


Mathematically, the union is formulated with the following equations:

A union model can readily capture the motiving applications of multiband communication or time-delay echoes, as we shall see in the relevant pages of this website.

Unified Architecture
The Xampling framework we propose has the high-level architecture presented below, which unifies the treatment in all the example applications considered hereafter. The first two blocks, termed X-ADC, perform the conversion of x(t) to digital. An operator P compresses the high-bandwidth input x(t) into a signal with lower bandwidth, effectively capturing the entire union U by a subspace S with substantially lower sampling requirements. A commercial ADC device then takes pointwise samples of the compressed signal, resulting in the sequence of samples y[n].
The role of P in Xampling is to narrow down the analog bandwidth that proceeds acquisition, so that lowrate ADC devices can be used. As in digital compression, the goal is to capture all vital information of the input in the compressed version, though here this functionality is achieved by hardware rather than software. The design of P therefore needs to wisely exploit the union structure, in order not to lose any essential information while reducing the bandwidth.
Two example prototype boards that are currently used in cognitive radio and radar are shown below:


In the digital domain, Xampling consists of three computational blocks. A nonlinear step detects the signal subspace from the lowrate samples. Once the index lambda^* is determined, we gain backward compatibility, meaning that standard DSP methods apply and commercial DAC devices can be used for signal reconstruction. The combination of nonlinear detection and standard DSP is referred to as X-DSP. As we shall demonstrate, besides backward compatibility, the nonlinear detection decreases computational loads, since the subsequent DSP and DAC stages need to treat only that subspace. The important point is that the detection stage can be performed efficiently from the lowrate samples.


What can be found on this website?


Xampling enables sampling and processing at rates lower than Nyquist. We describe several sampling systems for interesting union models in the following pages:


A tutorial on Xampling can be found:


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