Measuring Method Based On Random Projection And Its Applications For Ultra Wideband Signal Sampling

In modern signal processing, since the requirement of information amountincreases continually, the band of signal for carrying information becomes wider andwider, which brings a huge challenge for signal discretization. Therefore the research ondata acquisition for ultra wideband (UWB) signal became very important in the area ofsignal processing. If the sampling of UWB signal is implemented based on the Nyquisttheorem, the properties of high speed and high resolution for the samples cannot bepossessed at the same time, considering the technology of current ADC chips. Thispaper studies the data acquisition of UWB signal for two different situations. First,considering a cooperative environment in which some prior knowledge can be obtained,we study the measuring method for the echo signal in UWB active radar based on thetheory of compressed sensing, in which the prior knowledge of active radar is usedappropriately. In order to effectively measure the echo signal of UWB radar at low rate,we focus on the study of the method for representing the echo signal and the measuringmethod for analog signal. Second, for a non-cooperative environment in which no priorknowledge can be obtained besides the signal’s bandwidth, we study the approach tosampling and reconstructing UWB signal with properties of high speed and highresolution.1. We start with the analog-to-information converter (AIC) by analyzing itscomponents and equivalent matrices, and then we apply it to measure the echo signal ofUWB radar at low rate. If the echo signal can be sparsely represented in a dictionary, theproposed system can achieve the random measuring of signal and the recovering ofsignal by optimization. The scene of this problem is set to be utilizing an active radar todetect static scattered targets. So the mathematical model of the echo signal can beproperly designed based on the feature of the interested targets. The conventional modelfor active detection is employed in this paper, in which the echo signal containingmultiple target echoes can be represented as the linear summation of the shifted versionof the transmitted signal. A dictionary can be constructed with the shifted versions of aprototype atom which is generated by discretizing and normalizing the transmittedsignal. Therefore, the constructed dictionary can represent the echo signal sparsely. As aresult, the precondition of applying compressed sensing theory on the echo signal inUWB active radar is satisfied, and the successful reconstruction of signal with highprobability can be guaranteed. 2. Considering the disadvantages of the AIC structure, i.e., the lack of universality forthe transform space of sparse signal representation, in order to get a measurementmatrix with universality for signal representation, that is to say, the equivalentmeasurement matrix for the sparse coefficients vector will have good properties, nomatter how long the duration of the basis functions is, we propose a parallel samplingstructure based on random projection (PSRP). Here the equivalent measurement matrixfor the coefficients vector is the product of the measurement matrix and the transformmatrix for the signal. Each channel of the proposed PSRP structure is composed with arandom modulator, an integrator and an ADC. The output of each channel is ameasurement of input signal. Since the equivalent measurement matrix of the PSRPstructure is with good properties, when the PSPR structure is used as a measuringstructure in a data acquisition system for the echo signal in radar, compared with theacquisition system using AIC structure, the acquisition system using the PSRP structurerequires less number of measurements to achieve the same performance of signalreconstruction. In addition, the latter is more robust to noise.3. In order to reduce the complexity of the measuring structure and make itcompatible with the conventional sampling circuit, the direct sub-Nyquist randomsampling (DSRS) method is presented. In the proposed DSRS method, the randommeasurement of signal can be realized by adjusting the trigger time of the conventionalNyquist-based sampling circuit. For a conventional sampling circuit, besides theredesign of sampling clock, there is no need to change the other components to measurea signal based on compressed sensing theory. When the proposed DSRS structure isutilized in the measuring process in the framework of compressed sensing, the size,weight, energy consumption and cost of signal measuring will decrease greatly. Whenthe DSRS structure is employed in the data acquisition system for the echo signal basedon compressed sensing, under certain conditions, it can accomplish the low-ratemeasurement of signal, which is demonstrated by the simulation results.4. In non-cooperative (or passive) environment, we research the method for acquiringgeneral UWB signal with high speed and high resolution. For general UWB signal, wefollow the conventional signal model in which the signal to be sampled is assumed to beband-limited. In order to get the high sampling rate and high resolution simultaneously,the sampling of the UWB signal in this paper is achieved by a sampling system with theparallel sampling structure based on random projection (PSRP). When M-channel PSRPstructure is used to sample signal, since the speed of ADC in channel is1/M of thespeed of the whole structure, the low-rate high-resolution ADC chip can be adopted, but the output of the PSRP structure is with properties of high rate and high resolution.Different from the data acquisition system based on compressed sensing, for theproposed system, no prior knowledge on signal besides the signal’s bandwidth is usedand the reduction of sample amount is not the purpose. So the number of themeasurement equals to the number of unknown Nyquist samples. Therefore the signalreconstruction can be accomplished by solving a linear equation, which can beimplemented very fast.In order to make the channel number in the PSRP structure as small as possible andmake the complexity of the sampling system as low as possible, the signal is segmentedinto pieces with equal length, which will cause the segmentation error. In this paper, weprovide two strategies to suppress the segmentation error. One is software-basedanti-symmetric signal extension technique which will not bring any burden on hardwareimplement. However, the segmentation error cannot be eliminated completely. Theother is to preprocess the input signal with a high-rate sampling-and-hold circuit, whichcan eliminate the segmentation error completely. But the system cost will increase atsome extent due to the exact device. According to the practical situation and needs, thechoice of parallel sampling system using the ASE technique or the high rate S/H canmeet the different precision of high-rate data acquisition.