It has been widely recognized that in compressed sensing, many restricted isometry property (RIP) conditions can be easily obtained by using the null space property (NSP) with its null space constant (NSC) 0<θ≤1 to construct a contradicted method for sparse signal recovery. However, the traditional NSP with θ=1 will lead to conservative RIP conditions. In this paper, we extend the NSP with 0<θ<1 to a scale NSP, which uses a factor τ to scale down all vectors belonged to the Null space of a sensing matrix. Following the popular proof procedure and using the scale NSP, we establish more relaxed RIP conditions with the scale factor τ, which guarantee the bounded approximation recovery of all sparse signals in the bounded noisy through the constrained l1 minimization. An application verifies the advantages of the scale factor in the number of measurements.

In this paper, we propose applying our previously reported adaptive peak-to-average power ratio (PAPR) reduction method using null space in a multiple-input multiple-output (MIMO) channel for orthogonal frequency division multiplexing (OFDM) signals to the downlink MIMO amplify-and-forward (AF) relaying transmission. Assuming MIMO-OFDM transmission, mitigating its high PAPR not only at the base station (BS) but also at the relay station (RS) transmitters is essential to achieve sufficient coverage enhancement from the RSs by minimizing the transmission power backoff levels at the nonlinear power amplifier. In this study, we assume an AF-type RS with multiple antennas. In the proposed method, the BS suppresses the PAPR of the transmitted signal through adaptive PAPR reduction utilizing the null space of the integrated overall MIMO channel that combines the channel between the BS and RS and the channel between the RS and a set of user equipment (UE). However, the PAPR of the received signal at each RS antenna is increased again due to the MIMO channel between the BS and RS. The proposed method reduces this increased PAPR at the AF-type RS transmitter by PAPR reduction processing that utilizes the null space in the MIMO channel between the RS and UE. Since the in-band PAPR reduction signal added at the RS transmitter is transmitted only in the null space of the MIMO channel between the RS and UE, interference at the UE receiver is mitigated. Computer simulation results show that the proposed method significantly improves the PAPR-vs.-throughput performance compared to that for the conventional one thanks to the reduced interference levels from the PAPR reduction signal observed at the UE receiver.

Compressive sensing is a promising technique in data acquisition field. A central problem in compressive sensing is that for a given sparse signal, we wish to recover it accurately, efficiently and stably from very few measurements. Inspired by mathematical analysis, we introduce a combining scheme between stability and robustness in reconstruction problems using compressive sensing. By choosing appropriate parameters, we are able to construct a condition for reconstruction map to perform properly.

In this letter, we propose an efficient user selection algorithm aiming to select users with less spatially correlation and meet the user number limit of zero-forcing beamforming in downlink multiuser MIMO systems. This algorithm yields a considerable complexity reduction with only a small loss in performance and it only needs partial users' CSI feedback. Coupled with the algorithm, a null space updating method in O(K2) time and a modified proportional fair scheduling algorithm are also proposed.