MLD Maximal Likelihood Detector

The Maximal Likelihood Detector (MLD) is a statistical technique used in signal processing and communication systems to detect the presence of a signal in the presence of noise. It is a method that determines the most likely transmission symbol or message by comparing the received signal with a set of possible transmitted signals. The MLD is widely used in various communication systems such as wireless, satellite, and optical communication systems.

The main objective of the MLD is to minimize the probability of error in detecting the transmitted signal. The MLD is a decision rule that selects the signal that has the maximum probability of being transmitted. The maximum probability is obtained by comparing the received signal with all possible transmitted signals. In order to understand the MLD, it is important to understand the concept of likelihood.

Likelihood is a measure of how well a set of parameters fits the observed data. In the context of communication systems, likelihood refers to the probability that a particular set of transmitted signals generated the received signal. The MLD calculates the likelihood of each possible transmitted signal and selects the signal with the maximum likelihood.

To understand the MLD, let's consider a simple communication system with binary signaling. The transmitter sends a sequence of bits, which are converted into a sequence of symbols. Each symbol can be transmitted over a noisy channel, which introduces noise to the signal. The received signal can be expressed as:

y = s + n

Where y is the received signal, s is the transmitted signal, and n is the noise. The MLD calculates the likelihood of each possible transmitted signal by comparing the received signal with all possible transmitted signals. For binary signaling, there are two possible transmitted signals, s1 and s2, which can be expressed as:

s1 = A s2 = -A

Where A is the amplitude of the transmitted signal. The likelihood of each possible transmitted signal can be expressed as:

L(s1) = P(y|s1) = P(s1 + n|s1) L(s2) = P(y|s2) = P(s2 + n|s2)

Where P(y|s) is the probability that the received signal is y given that the transmitted signal is s. The probability can be expressed using the probability density function (PDF) of the noise. Assuming that the noise is Gaussian with zero mean and variance σ^2, the PDF of the noise can be expressed as:

f(n) = (1/√(2πσ^2)) * exp(-n^2/(2σ^2))

The likelihood of each possible transmitted signal can be calculated by substituting the expressions for s1 and s2 and the PDF of the noise in the above equations. After some algebraic manipulation, the likelihood of each possible transmitted signal can be expressed as:

L(s1) = (1/√(2πσ^2)) * exp(-(y-A)^2/(2σ^2)) L(s2) = (1/√(2πσ^2)) * exp(-(y+A)^2/(2σ^2))

The MLD selects the signal with the maximum likelihood. In the case of binary signaling, the decision rule can be expressed as:

ŝ = arg max(L(s1), L(s2))

Where ŝ is the estimated transmitted signal. If L(s1) > L(s2), then the transmitted signal is estimated to be s1, otherwise it is estimated to be s2.

The above example illustrates the basic concept of the MLD. In practice, the MLD can be used for more complex communication systems with multiple signaling levels and multiple transmit antennas. In such systems, the likelihood of each possible transmitted signal is calculated by comparing the received signal with all possible transmitted signals. The decision rule selects the signal that has the maximum likelihood.

The MLD is an optimal detector in the sense that it provides the minimum probability of error. However, it has a high computational complexity, especially for large signal constellations and high data rates. The complexity of the MLD increases exponentially with the number of signaling levels and transmit antennas. Therefore, it is not practical for many communication systems.

To reduce the computational complexity of the MLD, various suboptimal detectors have been developed. These detectors provide a good trade-off between complexity and performance. One such detector is the Sphere Decoder, which is a suboptimal detector that is widely used in wireless communication systems.

The Sphere Decoder is a suboptimal detector that uses a sphere to limit the search space of the MLD. The sphere is centered on the received signal and has a radius that is equal to the maximum distance between the received signal and any possible transmitted signal. The Sphere Decoder searches for the transmitted signal within the sphere, which reduces the computational complexity of the MLD.

In summary, the MLD is a statistical technique used in signal processing and communication systems to detect the presence of a signal in the presence of noise. The MLD calculates the likelihood of each possible transmitted signal by comparing the received signal with all possible transmitted signals. The decision rule selects the signal that has the maximum likelihood. The MLD is an optimal detector that provides the minimum probability of error, but it has a high computational complexity. To reduce the computational complexity, various suboptimal detectors such as the Sphere Decoder have been developed.