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Consider the DSB carrier frequency estimation problem given in the introductory chapter to estimation theory. A message is sent across a channel modulated by a sinusoidal carrier with carrier frequency = fc and amplitude= ‘A’. The transmitted signal gets affected by zero-mean AWGN noise when it travels across the medium. The receiver receives the signal and digitizes it for further processing.

To recover the message at the receiver, one has to know every details of the sinusoid: 1) Amplitude-’A’ 2) Carrier Frequency – f_c and 3) Any uncertainty in its phase – \phi_c .

Given a set of digitized samples x[n] and assuming that both amplitude and carrier frequency are known, we are tasked with the objective of estimating the phase of the embedded sinusoid (cosine wave). For analyzing this scenario we should have a model to begin with.

The digitized samples at the receiver are modeled as

x[n] = A cos(2 \pi f_c n+ \phi_c ) + w[n] , \;\;\; n=0,1,2,\cdots,N-1

Here A and f_c are assumed to be known and w[n] is an AWGN noise with mean=0 and variance=\sigma^2 .
We will use CRLB and try to find an efficient estimator to estimate the phase component.

CRLB for Phase Estimation:

As a pre-requisite to this article, readers are advised to go through the previous chapter on “Steps to find CRLB”

In order to derive CRLB, we need to have a PDF (Probability Density Function) to begin with. Since the underlying noise is modeled as an AWGN noise with mean=0 and variance=\sigma^2 , the PDF of the observed sample that gets affected by this noise is given by a multivariate Gaussian distribution function.

Cramer Rao Lower Bound for Phase Estimation
Cramer Rao Lower Bound for Phase Estimation
Cramer Rao Lower Bound for Phase Estimation

The sample mean is given by

Cramer Rao Lower Bound for Phase Estimation

The PDF is re-written as
Cramer Rao Lower Bound for Phase Estimation

Since the observed samples x[n] are fixed in the above equation, we will use the likelihood notation instead of PDF notation. That is, p(\mathbf{x;\phi }) is simply rewritten as L(\mathbf{x;\phi }) . The log likelihood function is given by

Cramer Rao Lower Bound for Phase Estimation

For simplicity,we will denote \phi_c as \phi . Next, take the first partial derivative of log likelihood function with respect to \phi

Cramer Rao Lower Bound for Phase Estimation

Taking the second partial derivative of the log likelihood function,

Cramer Rao Lower Bound for Phase Estimation

Since the above term is still dependent on the observed samples x[n], take expectation of the entire equation to average out the variations.

Cramer Rao Lower Bound for Phase Estimation

Cramer Rao Lower Bound for Phase Estimation

Let’s derive the terms like fisher information, CRLB and find out whether we can find an efficient estimator from the equations.

Fisher Information:

The Fisher Information can be derived using

Cramer Rao Lower Bound for Phase Estimation

Cramer Rao Lower Bound:

The CRLB is the reciprocal of Fisher Information.

Cramer Rao Lower Bound for Phase Estimation

The variance of any estimator estimating the phase of the carrier for given problem will always be higher than this CRLB. That is,

var(\hat{\phi}) \geq \frac{2\sigma^2}{NA^2}

As we can see from the above result, that the variance of the estimates var(\hat{\phi}) \to CRLB as N \to \infty . Such estimators are called “Asymptotically Efficient Estimators”.

Asymptotically Efficient Estimators

An efficient estimator exists if and only if the first partial derivative of log likelihood function can be written in the form

Cramer Rao Lower Bound for Phase Estimation

Re-writing our earlier result,

Cramer Rao Lower Bound for Phase Estimation

We can clearly see that the above two equations are not having the same form. Thus, an efficient estimator does not exist for this problem.

See also:

[1]An Introduction to Estimation Theory
[2]Bias of an Estimator
[3]Minimum Variance Unbiased Estimators (MVUE)
[4]Maximum Likelihood Estimation
[5]Maximum Likelihood Decoding
[6]Probability and Random Process
[7]Likelihood Function and Maximum Likelihood Estimation (MLE)
[8]Score, Fisher Information and Estimator Sensitivity
[9]Introduction to Cramer Rao Lower Bound (CRLB)
[10]Cramer Rao Lower Bound for Scalar Parameter Estimation
[11]Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)
[12]Efficient Estimators and CRLB
[13]Cramer Rao Lower Bound for Phase Estimation
[14]Normalized CRLB - an alternate form of CRLB and its relation to estimator sensitivity
[15]The Mean Square Error – Why do we use it for estimation problems
[16]How to estimate unknown parameters using Ordinary Least Squares (OLS)
[17]Essential Preliminary Matrix Algebra for Signal Processing
[18]Why Cholesky Decomposition ? A sample case:
[19]Tests for Positive Definiteness of a Matrix
[20]Solving a Triangular Matrix using Forward & Backward Substitution
[21]Cholesky Factorization and Matlab code
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Mathuranathan

– Founder and Author @ gaussianwaves.com which has garnered worldwide readership. He is a masters in communication engineering and has 7 years of technical expertise in channel modeling and has worked in various technologies ranging from read channel design for hard drives, GSM/EDGE/GPRS, OFDM, MIMO, 3GPP PHY layer and DSL.
He also specializes in tutoring on various subjects like signal processing, random process, digital communication etc..,He can be contacted at support@gaussianwaves.com

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