Chapter 3 Cramer Rao Lower Bound What is the Cramer Rao Lower Bound Abbreviated CRLB or sometimes just CRB CRLB is a lower bound on the variance of any unbiased estimator If is an unbiased estimator of then 2 CRLB CRLB The CRLB tells us the best we can ever expect to be able to do w an unbiased estimator Some Uses of the CRLB 1 Feasibility studies e g Sensor usefulness etc Can we meet our specifications 2 Judgment of proposed estimators Estimators that don t achieve CRLB are looked down upon in the technical literature 3 Can sometimes provide form for MVU est 4 Demonstrates importance of physical and or signal parameters to the estimation problem e g We ll see that a signal s BW determines delay est accuracy Radars should use wide BW signals 3 3 Est Accuracy Consideration Q What determines how well you can estimate Recall Data vector is x samples from a random process that depends on an the PDF describes that dependence p x Clearly if p x depends strongly weakly on we should be able to estimate well poorly See surface plots vs x for 2 cases 1 Strong dependence on 2 Weak dependence on Should look at p x as a function of for fixed value of observed data x Surface Plot Examples of p x Ex 3 1 PDF Dependence for DC Level in Noise x 0 A w 0 w 0 N 0 2 Then the parameter dependent PDF of the data point x 0 is p x 0 A 1 2 2 x 0 A 2 exp 2 2 Say we observe x 0 3 So Slice at x 0 3 p x 0 3 3 A x 0 A Define Likelihood Function LF The LF the PDF p x but as a function of parameter w the data vector x fixed We will also often need the Log Likelihood Function LLF LLF ln LF ln p x LF Characteristics that Affect Accuracy Intuitively sharpness of the LF sets accuracy But How Sharpness is measured using curvature 2 ln p x 2 x given data true value Curvature PDF concentration Accuracy But this is for a particular set of data we want in general So Average over random vector to give the average curvature 2 ln p x E 2 Expected sharpness of LF true value E is w r t p x 3 4 Cramer Rao Lower Bound Theorem 3 1 CRLB for Scalar Parameter ln p x 0 Assume regularity condition is met E Then 2 1 2 ln p x E 2 true value E is w r t p x 2 ln p x 2 ln p x E p x dx 2 2 Right Hand Side is CRLB Steps to Find the CRLB 1 Write log 1ikelihood function as a function of ln p x 2 Fix x and take 2nd partial of LLF 2ln p x 2 3 If result still depends on x Fix and take expected value w r t x Otherwise skip this step 4 Result may still depend on Evaluate at each specific value of desired 5 Negate and form reciprocal Example 3 3 CRLB for DC in AWGN x n A w n n 0 1 N 1 w n N 0 2 white Need likelihood function p x A N 1 1 n 0 2 1 2 N 2 2 2 x n A 2 exp 2 2 N 1 2 x n A exp n 0 2 2 Due to whiteness Property of exp Now take ln to get LLF N N 1 1 2 ln p x A ln 2 2 2 x n A 2 2 n 0 A 0 A sample mean Now take first partial w r t A 1 ln p x A A 2 N 1 N x n A 2 x A n 0 Now take partial again 2 A 2 ln p x A N 2 Doesn t depend on x so we don t need to do E Since the result doesn t depend on x or A all we do is negate and form reciprocal to get CRLB CRLB 1 2 ln p x E 2 CRLB For fixed N 2 var A N true value Doesn t depend on A Increases linearly with 2 Decreases inversely with N 2 A CRLB CRLB Doubling Data Halves CRLB For fixed 2 For fixed N 2 N 2 N Continuation of Theorem 3 1 on CRLB There exists an unbiased estimator that attains the CRLB iff ln p x I g x for some functions I and g x Furthermore the estimator that achieves the CRLB is then given by g x 1 var with CRLB I Since no unbiased estimator can do better this is the MVU estimate This gives a possible way to find the MVU Compute ln p x need to anyway Check to see if it can be put in form like If so then g x is the MVU esimator Revisit Example 3 3 to Find MVU Estimate For DC Level in AWGN we found in that N ln p x A 2 x A A I A N 2 var A 2 N CRLB Has form of I A g x A 1 g x x N N 1 x n n 0 So for the DC Level in AWGN the sample mean is the MVUE Definition Efficient Estimator An estimator that is unbiased and attains the CRLB is said to be an Efficient Estimator Notes Not all estimators are efficient see next example Phase Est Not even all MVU estimators are efficient So there are times when our 1st partial test won t work Example 3 4 CRLB for Phase Estimation This is related to the DSB carrier estimation problem we used for motivation in the notes for Ch 1 Except here we have a pure sinusoid and we only wish to estimate only its phase Signal Model x n A cos 2 f o n o w n s n o Signal to Noise Ratio Signal Power A2 2 Noise Power 2 SNR A2 2 2 Assumptions 1 0 fo fo is in cycles sample 2 A and fo are known we ll remove this assumption later AWGN w zero mean 2 Problem Find the CRLB for estimating the phase We need the PDF p x 1 2 N 2 2 N 1 2 x n A cos 2 f o n exp n 0 2 2 Exploit Whiteness and Exp Form Now taking the log gets rid of the exponential then taking partial derivative gives see book for details A ln p x A N 1 2 x n sin 2 f o n sin 4 f o n 2 2 n 0 2 Taking partial derivative again 2 ln p x 2 A 2 N 1 x n cos 2 f o n A cos 4 f o n 2 n 0 Still depends on random vector x so need E Taking the expected value 2 ln p x A E E 2 2 x n …