Multi-coset Sampling and Recovery of Sparse Multiband Signals

Multi-coset sampling is a periodic nonuniform sub-Nyquist sampling technique for acquiring continuous-time spectrally-sparse signals. This report presents a concise mathematical analysis of the multi-coset sampling system as originally proposed by Feng and Bresler and serves a reference for the accompanying MATLAB software [1] that simulates multi-coset sampling and signal reconstruction. Emphasis is placed on the derivation of the linear relationship between the spectrums of the input and output signals, the conditions of perfect reconstruction, and the reconstruction algorithm. 1 Signal Model and System Description Sparse multiband signals. A multiband signal x(t) is a bandlimited, continuous-time, squared integrable signal that has all of its energy concentrated in one or more disjoint frequency bands (of positive Lebesque measure). Denoting the Fourier transform of x(t) by X(jω), X(jω) = ∫ ∞ −∞ x(t) e dt, a bandlimited signal is one whose spectrum is bounded, i.e., X(jω) = 0 for −πW ≤ ω < πW radians per second, for some positive real number W . Here, W/2 is the bandwidth of x(t) and W is therefore the Nyquist frequency. The spectral support of a multiband signal is the union of the frequency intervals that contain the signal’s energy. A sparse multiband signal is thus a multiband signal whose spectral support has Lebesgue measure that is small relative to the overall signal bandwidth [2]. If, for instance, all the active bands have equal bandwidth B Hz and the signal is composed of K disjoint frequency bands, then a sparse multiband signal is one satisfying KB << W . Multi-coset sampler. Multi-coset sampling (MC) is a periodic nonuniform sub-Nyquist sampling technique for acquiring sparse multiband signals [2–6]. For a fixed time interval T that is less than or equal to the Nyquist period and for a suitable positive integer L, MC samplers sample x(t) at the time instants t = (kL+ ci)T for 1 ≤ i ≤ q, k = 0, 1, . . . . The time offsets ci are distinct, positive real numbers less than L and are known collectively as the multi-coset sampling pattern. The system thus collects q ≤ L samples in LT seconds, or equivalently, exhibits an average sampling rate of q/LT Hz. Here we set T equal to the Nyquist period T = 1/W , thereby referencing the system’s sampling rate to the Nyquist rate. Multi-coset samplers are parameterized by q, L, and {ci}, and the system design depends on conditioning them properly to ensure successful recovery of x(t) from the output samples. MC samplers are most easily implemented as multichannel systems where channel i shifts x(t) by ci/W seconds and then samples uniformly at W/L Hz (see Figure 1). 2 System Analysis The following analysis yields a basic time and frequency domain description of the MC sampler. We employ standard Fourier transform properties without explicit explanation for the sake of conciseness.