MMSE (minimum mean square error)

Minimum mean square error (MMSE) is a statistical estimation technique used to find the most probable estimate of an unknown parameter or signal. It is a widely used method in signal processing, communication systems, control systems, and machine learning. The goal of MMSE is to minimize the mean square error between the estimated signal and the true signal.

MMSE estimation can be used to estimate signals in noisy environments, where the observed signals are corrupted by additive noise. The MMSE estimator takes into account both the prior information about the signal and the observation noise. This technique is based on the Bayesian framework, where the prior probability distribution of the signal is combined with the likelihood of the observed data to obtain the posterior probability distribution.

In this article, we will discuss the basic concepts of MMSE estimation, its mathematical derivation, and its applications.

Derivation of MMSE estimator:

Consider a signal X, which is to be estimated from the observed noisy signal Y. Let the noise be additive white Gaussian noise (AWGN) with zero mean and variance N0. The noisy signal can be expressed as:

Y = X + N

where N is the noise.

The MMSE estimator finds the estimate of X, which minimizes the mean square error (MSE) between the estimate and the true signal. The MSE is defined as:

MSE = E[(X - X')^2]

where X' is the estimate of X.

The MMSE estimator can be derived by finding the conditional expectation of X given Y, which minimizes the MSE. The conditional expectation of X given Y is denoted as E[X|Y]. The MMSE estimator is given by:

X' = E[X|Y]

The conditional expectation can be obtained by using the Bayes' rule:

E[X|Y] = ∫ X p(X|Y) dX

where p(X|Y) is the posterior probability density function (pdf) of X given Y. The posterior pdf is obtained by combining the prior pdf of X, p(X), and the likelihood of Y given X, p(Y|X), using the Bayes' rule:

p(X|Y) = p(Y|X) p(X) / p(Y)

The prior pdf of X represents the prior knowledge or information about X, and the likelihood of Y given X represents the probability of observing Y for a given value of X.

In the case of AWGN, the likelihood of Y given X is given by:

p(Y|X) = (1 / sqrt(2piN0)) exp(-(Y-X)^2 / (2*N0))

Substituting this expression in the Bayes' rule, we get:

p(X|Y) = (1 / sqrt(2piN0)) exp(-(Y-X)^2 / (2*N0)) p(X) / p(Y)

The prior pdf of X is assumed to be Gaussian with mean μX and variance σX^2:

p(X) = (1 / sqrt(2piσX^2)) exp(-(X-μX)^2 / (2*σX^2))

Substituting this expression in the posterior pdf, we get:

p(X|Y) = (1 / sqrt(2piσX'^2)) exp(-(X'-μX')^2 / (2*σX'^2))

where σX'^2 is the variance of X' and μX' is the mean of X'.

The mean and variance of X' can be obtained by taking the first and second moments of the posterior pdf:

μX' = ∫ X p(X|Y) dX

σX'^2 = E[(X - μX')^2]

After some algebraic manipulation, we get:

μX' = μX + σX^2 / (σX^2 + N0) (Y - μX)

σX'^2 = σX^2 - σX^4 / (σX^2 + N0)

The above equations give the MMSE estimator for the signal X given the noisy signal Y. The first equation shows that the estimate X' is a linear combination of the noisy signal Y and the prior mean μX, with the weights determined by the variances of X and the noise N0. The second equation shows that the variance of the estimate σX'^2 is less than the variance of the prior σX^2, and it decreases as the noise N0 increases.

Interpretation of MMSE estimator:

The MMSE estimator can be interpreted as a compromise between the prior knowledge and the observed data. The prior knowledge is represented by the prior pdf of X, and the observed data is represented by the likelihood of Y given X. The MMSE estimator combines these two sources of information to obtain the most probable estimate of X given Y.

The MMSE estimator is a type of linear estimator, which means that it produces an estimate that is a linear combination of the observed data and the prior knowledge. The linearity property of the MMSE estimator makes it computationally efficient and easy to implement.

The MMSE estimator is also a type of Bayesian estimator, which means that it uses the Bayesian framework to estimate the unknown parameter or signal. The Bayesian framework provides a systematic way of combining prior knowledge with observed data to obtain the posterior distribution of the unknown parameter.

Applications of MMSE estimator:

The MMSE estimator has many applications in signal processing, communication systems, control systems, and machine learning. Some of the applications are discussed below:

1. Signal denoising: The MMSE estimator can be used to denoise signals that are corrupted by additive noise. The noisy signal is first passed through a filter that estimates the signal using the MMSE estimator. The filtered signal has less noise than the original signal, and it preserves the important features of the signal.
2. Channel equalization: The MMSE estimator can be used to equalize channels that distort the transmitted signal. The channel is modeled as a linear filter with unknown coefficients, and the MMSE estimator is used to estimate the coefficients of the filter. The estimated filter coefficients are used to equalize the channel, and the received signal is reconstructed.
3. Control systems: The MMSE estimator can be used to estimate the state of a dynamic system from noisy measurements. The state of the system is modeled as a stochastic process, and the MMSE estimator is used to estimate the state from the noisy measurements. The estimated state is used to design the control law for the system.
4. Machine learning: The MMSE estimator can be used to estimate the parameters of a statistical model from observed data. The model parameters are estimated using the Bayesian framework, and the MMSE estimator is used to obtain the most probable estimate of the parameters given the observed data. The estimated parameters are used to make predictions or decisions.

Conclusion:

The MMSE estimator is a powerful statistical estimation technique that is widely used in signal processing, communication systems, control systems, and machine learning. The MMSE estimator combines the prior knowledge of the signal with the observed data to obtain the most probable estimate of the signal. The MMSE estimator is a type of linear estimator and Bayesian estimator, which makes it computationally efficient and easy to implement. The MMSE estimator has many applications, including signal denoising, channel equalization, control systems, and machine learning.