Testing The Number Of Signals In Spike Model

In many signal processing applications,estimating the number of signals present becomes a key issue and is often the starting point for the signal parameter estimation problem.When the number of estimated signals is known,we also need to consider the relationship between the number of estimated signals and the number of real signals.Therefore,we propose a hypothesis test H0:k = k0 vs k>k0,we use k to denote the true number of signals and use k0 to denote the number of estimated signals.When the number of signals are underestimated,some signals are treated as noise,thus losing a lot of useful information,which is not conducive to making effective decisions.In real life:there are many examples similar to the Spike model.Hence,it is very meaningful to test the number of signals in the Spike model.The concrete content is in the following:In Cha.pter 2,we mainly introduce testing of the number of signals.Firstly,the Spike model is introduced.In the Spike model,we test the number of signals.When the null hypothesis H0 holds,that is,the number of real signals equals the number of estimated signals.When the alternative hypothesis H1 holds,that is,the number of real signals is larger than the number of estimated signals.Next,by similar projec-tion method,the null hypothesis is transformed into ?=diag{?,?,…,?},and the alternative hypothesis is transformed into ?=diag{?k0+1,…,?k,?,?,…,?}.Therefore,we transform testing of the number of signals into the spherical test prob-lem by similar projection method.Finally,from the Cauchy-Schwarz inequality,we propose test statistics to test the number of signals,and the asymptotic normal dis-tribution of the test sta.tistics is given.In chapter 3,first,we introduce the preparatory knowledge,such as Delta method,residue theorem and linear spectrum theory of large-dimensional random matrices,in order to prepare for the proof of statistics.We demonstrate by both the central limit theorem of linear spectral statistics of large-dimensional sample covari-ance matrices and Delta method that the test statistics have asymptotic normality.In chapter 4,we mainly give numerical simulation.We compare the power of test statistics T1 and T2.With the enhancement of the signal,T2 always reaches 1 before T1,indicating that T2 is better than T1.Secondly,we simulate the advantages and disadvantages of the method under different Spike models.A large number of numerical simulations show that the simulation results of T2 are better than T1.Under the null hypothesis,simulation results are close to 0.05.Under the alternative hypothesis,simulation results are close to 1.In Cha.pter 5,we summarize the main results in this paper and give some prospects for further research in the future.