In digital signal analysis, a signal is typically reconstructed from discrete measurements. Typical approaches to sampling signals or images follow Shannon's theorem wherein a sufficient sampling rate is at least twice the bandwidth (i.e., a lowpass or bandpass bandwidth depending on the signal type) present in the signal. This minimal sampling rate is known as the Nyquist rate or frequency.
This principle underlies nearly all signal acquisition protocols used in consumer audio and video electronics, medical imaging devices, radio receivers, etc. In reference to data conversions, for example, a standard analog-to-digital converter (ADC) implements the quantized Shannon representation in that the signal is uniformly sampled at or above the Nyquist rate.
Even though traditional approaches to data acquisition depend on a Nyquist rate, in many applications the Nyquist rate far exceeds the necessary sample rate needed (or the number of samples over the time record) to accurately reconstruct the signal. For efficient storage or communications of the signal information content, data compression is an important processing step prior to storage or transmission.
However, there are penalties associated with implementing data compression. One such penalty is that the original information bearing signal will likely be sampled above the Nyquist rate, and thus produce many samples with redundant information content about the signal. This process can at times require an ADC with a very high sample rate. High sample rate converters tend to require higher power and are often limited in resolution with respect to their lower sample rate counterparts. This induces a system penalty of low resolution and high power. In some cases it is difficult to even purchase or construct an ADC with the required sample rate given the Nyquist viewpoint of the signal.
Another penalty is induced by the data compression step, as performed with a Karhunen-Loève transform, for example. The data compression step attempts to remove the information redundancy in the samples, induced by the Nyquist view of the conversion, so that a minimal set of samples result to sufficiently represent the information content of the original signal. Implementing compression prior to storage or transmission requires some type of computing resource, which in turn increases the power drawn by the sensor system. The larger the oversampling factor in a Nyquist paradigm, the more severe these penalties become.
Recent developments have shown that compressive sampling or compressive sensing can provide sub-Nyquist rate sampling for communications systems. One such approach is disclosed in U.S. Patent No. 2011/0090394 to Tian et al. A disclosed method of signal processing includes receiving at a processor a data packet comprising compressively sensed or measured data of a signal, with the compressively measured data comprising wavelet transform coefficients. The received signal is a discrete signal, which in turn, requires transform coding before being processed by the processor. The processor reconstructs the signal using a clustering property of the wavelet transform coefficients. A disadvantage of this approach is that transform coding of the discrete signal before being compressively sampled requires additional processing, which in turn consumes power. Additional power consumption may be undesirable, particularly for battery-powered systems.