60 Pass Pass Pass Pass Pass Fail Fail. 65 Pass Pass Pass Pass Pass Pass Pass alpha = 0.01. 70 Pass Pass Pass Pass Pass Pass Pass. 75 Pass Pass Pass ...
SPECTRUM SENSING BASED ON COMPRESSED SAMPLING
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A Thesis Presented to the Faculty of San Diego State University
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In Partial Fulfillment of the Requirements for the Degree Master of Science in Electrical Engineering
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by Ismail Rashid Alkhouri Fall 2013
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Copyright © 2013 by Ismail Rashid Alkhouri All Rights Reserved
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ABSTRACT OF THE THESIS Spectrum Sensing Based on Compressed Sampling by Ismail Rashid Alkhouri Master of Science in Electrical Engineering San Diego State University, 2013 Compressed Sensing technology is used in many applications and is a widely growing field that attracts researches. In this thesis, spectrum sensing is proposed and compared to existing HW and SW techniques that performs this function. The theory of Compressed Sensing will be covered and the limitations and conditions that play a big role are going to be presented.
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TABLE OF CONTENTS PAGE ABSTRACT ............................................................................................................................. iv LIST OF TABLES .................................................................................................................. vii LIST OF FIGURES ............................................................................................................... viii ACKNOWLEDGEMENTS ..................................................................................................... ix CHAPTER 1
INTRODUCTION TO SPECTRUM SENSING AND COMPRESSED SAMPLING ...................................................................................................................1 1.1 Spectrum Sensing...............................................................................................1 1.2 Compressed Sampling .......................................................................................2
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COMPRESSED SAMPLING: THEORY, PROBLEM DESCRIPTION & SOLUTIONS .................................................................................................................6 2.1 Sparse and Compressible Vectors ......................................................................6 2.1.1 Sparse Condition .......................................................................................7 2.1.2 Experiment 2.1 ..........................................................................................7 2.1.3 Natural Sparse Signals and Sparseness Bases ..........................................9 2.1.4 Experiment 2.2 ........................................................................................10 2.2 CS Problem Description and Solutions............................................................11 2.2.1 Measurement Reduction Matrix .............................................................13 2.2.2 Experiment 2.3 ........................................................................................13 2.2.3 Experiment 2.4 ........................................................................................14 2.2.4 Reconstruction Algorithm.......................................................................18
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DISCRETE TIME & DISCRETE FREQUENCY CS THEORETICAL EXAMPLES ................................................................................................................22 3.1 Discrete Time Example....................................................................................22 3.2 Discrete Frequency Example ...........................................................................29
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THE RANDOM DEMODULATOR ...........................................................................33 4.1 System Description ..........................................................................................33 4.2 Simulation Setup ..............................................................................................34
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LIST OF TABLES PAGE Table 3.1. Results of Experiment 3.1 .......................................................................................26 Table 3.2. Results of Experiment 3.2 .......................................................................................31 Table 4.1. Results of Experiment 4.1 .......................................................................................40
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LIST OF FIGURES PAGE Figure 1.1. CS framework. .........................................................................................................3 Figure 2.1. Ratio of maximum of X to mean of X in different number of non-zero elements. ..................................................................................................................8 Figure 2.2. Ratio of maximum of X to mean of X in different amplitudes. ..............................9 Figure 2.3. Sparse signal in DCT and DFT domains. ..............................................................11 Figure 2.4. M-sparse signal of length 500. ..............................................................................14 Figure 2.5. Different matrices (with its orthonormal) versions to test the RIP property. ........15 Figure 2.6. RIP property with compressible vectors................................................................16 Figure 2.7. Incoherence condition of the CS matrix. ...............................................................17 Figure 2.8. CS detailed framework. .........................................................................................21 Figure 3.1. Original and reconstructed signals and the difference vector mean value. ...........25 Figure 3.2. Original and reconstructed signals in frequency and time domain. ......................30 Figure 4.1. Random demodulator block diagram. ...................................................................33 Figure 4.2. Baseband sparse signal. .........................................................................................35 Figure 4.3. Sparse signal in after the up conversion. ...............................................................36 Figure 4.4. PN sequence in frequency and time domains. .......................................................37 Figure 4.5. Original and reconstructed signal for fixing the parameters {fc, T, BW, and alpha} to {250, 9, 15, 0.01} respectively. .......................................................39
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ACKNOWLEDGEMENTS First, I would like to thank my professors, fred harris and Ssantosh Nagaraj for their support and help throughout the stages of this thesis. Second, I would like to thank my colleague Zhaoquan Li for his technical help in the part of reconstructing the sparse signals. Many thanks are dedicated to my parents Rashid and Manal Alkhouri, whom their love and care motivated me to perform and give my best. Also, I would like to thank my sister Fayrouz, best friends Fahmi Alsabie and Noori Alnoori for their support. Finally, I want to thank my classmates at SDSU and workmates at Broadcom Corporation.
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CHAPTER 1 INTRODUCTION TO SPECTRUM SENSING AND COMPRESSED SAMPLING This chapter will contain a brief history to Spectrum Sensing showing how it began and why it is interesting. Communication systems that use this type of wireless frequency sensing will be introduced. Then the mostly HW complexity method and the SW complexity method will be discussed with its advantages and disadvantages to introduce the main idea of spectrum sensing using Compressed Sampling. The basic idea of the very fast growing technique of Compressed Sampling will be presented with its wide applications. Finally, this chapter will briefly show how using this technique for the purpose of spectrum sensing will be very sufficient relatively and under certain conditions and limitations.
1.1 SPECTRUM SENSING We can define Spectrum Sensing as measurement and classification of the radio spectrum into used and unused bands; we can also define it as the task of identifying the frequency support for a given input signal. The electromagnetic spectrum is a natural scarce resource. The radio frequency spectrum involves electromagnetic radiation with frequencies between 3000 Hz and 300 GHz. Government licenses the use of electromagnetic spectrum for wireless and communication technologies. Spectrum scarcity is the main problem as the demand for additional bandwidth keeps increasing. The above is a great cause why in each wireless communication network; there should be an extra stage of processing in TX and RX that enhances accessibility and usability of the spectrum allocated to the application of interest. The main advantage of Spectrum Sensing is having a better knowledge for the spectral support that can be utilized for a faster link, i.e. according to Shannon theorem, bigger the BW, channel capacity will be higher, which leads to better data rate which lead to
2 a faster link. The biggest disadvantage is complexity of performing this task as discussed in the paragraph below Standard lab equipment can provide this functionality, which is the spectrum analyzer, which basically sweeps the center frequency of an analog band pass filter for the band of interest to detect any energy that is above the noise power. This method is purely analog and it requires huge hardware resources to function. A mobile device (as an example), however, cannot embed solutions based on standard lab equipment, due to size, weight, power, and cost limitations. On the other hand, any band-limited signal when an ADC samples it at Nyquist rate, it shows its spectral support. But, if our signal of interest is centered at a frequency of 50GHz with a bandwidth of 10MHz, the signal needs to be sampled at a sampling frequency more than 100GHz, and that requires a very high clock speed that is practically very far from being efficient. The point is that we need a solution that must be performed using minimal hardware and software resources. This thesis suggests one of these solutions and studies the theory, conditions, and functionality.
1.2 COMPRESSED SAMPLING We can define Compressed Sampling, Compressive Sampling, or Compressed Sensing (CS) technically as a framework for simultaneous sensing and compression of finite dimensional vectors, that relies on linear dimensionality reduction. Theoretically, we can define it as a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems taking advantage of signal sparseness or compressibility. Or, Compressed Sensing is a new data acquisition theory asserting that one can exploit sparseness or compressibility when acquiring signals of general interest, and that one can design non-adaptive sampling techniques that condense the information in a compressible signal into a small amount of data. To be more specific, let X be a sparse (that has a very few non-zero coefficients) vector of length N, we are going to reconstruct X using an M
