16-QAM (16 Quadrature Amplitude Modulation) is a popular digital modulation scheme used in many communication systems, including Wi-Fi, cellular, and satellite communications. In this modulation technique, 16 different symbols are used to represent a 4-bit sequence.
1. What is Maximum Likelihood Detection?
Maximum Likelihood (ML) Detection is a decision rule used in communication systems to detect the transmitted signal based on the received signal. The goal is to choose the symbol that maximizes the likelihood of observing the received signal, considering the noise in the system.
In the context of 16-QAM, ML detection involves choosing the closest point in the 16-QAM constellation to the received signal, where the closeness is measured by the Euclidean distance.
2. 16-QAM Modulation
16-QAM uses both amplitude and phase to represent 16 different combinations of 4 bits. The constellation points are arranged in a grid, where each point corresponds to a unique combination of bits (e.g., 0000, 0001, 0010, ..., 1111).
The 16-QAM constellation diagram can be thought of as a 4x4 grid of points, with each point having both a real (in-phase) and an imaginary (quadrature) component. The constellation typically looks like this:
+3 +3i +3 +1i +3 -1i +3 -3i
+1 +3i +1 +1i +1 -1i +1 -3i
-1 +3i -1 +1i -1 -1i -1 -3i
-3 +3i -3 +1i -3 -1i -3 -3i
Each point in the constellation represents a unique 4-bit sequence, allowing the transmission of 4 bits per symbol.
3. The Likelihood Function
In the presence of noise, the received signal \( y \) is a noisy version of the transmitted symbol \( x \), where \( x \) is one of the 16 points in the 16-QAM constellation. The received signal can be modeled as:
y = x + n
Where:
- \( y \) is the received signal (which may contain noise),
- \( x \) is the transmitted symbol (one of the 16 constellation points),
- \( n \) is the noise, typically modeled as Gaussian with zero mean and variance \( \sigma^2 \).
The likelihood function \( L(y | x) \) describes the probability of receiving \( y \), given that the transmitted symbol was \( x \). Since the noise is Gaussian, the likelihood function is given by:
\( L(y | x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left( - \frac{|y - x|^2}{2 \sigma^2} \right) \)
Where:
- \( L(y | x) \) is the likelihood of receiving \( y \) given that \( x \) was transmitted,
- \( |y - x|^2 \) is the squared Euclidean distance between the received signal \( y \) and the constellation point \( x \),
- \( \sigma^2 \) is the noise variance.
The goal of Maximum Likelihood Detection is to choose the transmitted symbol \( x \) that maximizes \( L(y | x) \). However, since the constant factor in the Gaussian function is the same for all symbols, the problem simplifies to minimizing the squared distance \( |y - x|^2 \).
4. Maximum Likelihood Decision Rule
To detect the transmitted symbol, we compute the squared Euclidean distance between the received signal \( y \) and each of the 16 possible symbols \( x \) in the 16-QAM constellation. The transmitted symbol is the one that minimizes the squared distance to the received signal:
\( x_{\text{ML}} = \arg \min_x |y - x|^2 \)
In other words, we choose the symbol \( x \) in the constellation that is closest to \( y \), as this is the most likely symbol to have been transmitted.
5. Simplified ML Decoding for 16-QAM
To detect the transmitted symbol, we simply compute the Euclidean distance between the received signal \( y \) and all possible constellation points. The closest constellation point corresponds to the most likely transmitted symbol.
For example, if the received signal \( y \) is \( 1.5 + 2i \), we calculate the squared Euclidean distance to each point in the 16-QAM constellation:
\( d^2(y, x) = \left( \text{Re}(y) - \text{Re}(x) \right)^2 + \left( \text{Im}(y) - \text{Im}(x) \right)^2 \)
After calculating the distances for all 16 points, we choose the symbol \( x \) that minimizes this distance as the transmitted symbol.
6. Summary of Maximum Likelihood Detection in 16-QAM
In 16-QAM, Maximum Likelihood Detection works as follows:
- The received signal \( y \) is compared with all 16 possible transmitted symbols \( x \) in the constellation.
- The symbol that minimizes the squared Euclidean distance between \( y \) and \( x \) is chosen as the most likely transmitted symbol.
- This decision rule is equivalent to choosing the symbol that is closest to the received signal in the 16-QAM constellation.
7. Practical Considerations
In real communication systems, Maximum Likelihood Detection is widely used because it provides the most accurate decision about the transmitted symbol, given the noisy received signal. However, it can be computationally expensive, as it requires comparing the received signal with all possible constellation points.
In some applications, approximate or simplified detection methods (such as nearest neighbor detection) may be used to reduce the computational complexity.