Binary Phase Shift Keying (BPSK) is a form of digital modulation in which each bit of data is represented by one of two phases of a carrier wave. These phases typically represent values like `0` and `1`. In this note, we will focus on Maximum Likelihood (ML) Detection in BPSK and how it helps in recovering the transmitted bits from a noisy signal.
1. What is Maximum Likelihood Detection?
Maximum Likelihood (ML) detection is a statistical approach used to estimate the transmitted signal based on the received signal. The principle is to choose the signal that maximizes the likelihood of observing the received data, given a set of possible transmitted symbols.
In the context of BPSK, ML detection is about determining which of the two possible transmitted symbols (\(+A\) or \(-A\)) is most likely to have been transmitted based on the received signal.
2. The BPSK Modulation Scheme
In BPSK, there are two possible symbols representing binary data:
- Bit `0` is represented by symbol \(+A\)
- Bit `1` is represented by symbol \(-A\)
The received signal \(y\) is a noisy version of the transmitted symbol, i.e., \( y = x + n \), where:
- \( x \) is the transmitted symbol (\(+A\) or \(-A\)),
- \( n \) is the Gaussian noise with mean 0 and variance \( \sigma^2 \),
- \( y \) is the received signal.
3. The Likelihood Function
The likelihood function \( L(y | x) \) represents the probability of receiving \( y \) given that the transmitted symbol was \( x \). Since the noise is assumed to be Gaussian, the likelihood function for each symbol is given by:
\( L(y | x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp\left( - \frac{|y - x|^2}{2 \sigma^2} \right) \)
Where:
- \( y \) is the received signal,
- \( x \) is the transmitted symbol (either \( +A \) or \( -A \)),
- \( \sigma^2 \) is the variance of the noise.
The goal of ML detection is to maximize the likelihood function to determine the most probable transmitted symbol. However, instead of directly computing the likelihood, we can simplify the decision rule by minimizing the squared Euclidean distance between the received signal and each possible transmitted symbol.
4. Maximum Likelihood Decision Rule for BPSK
In BPSK, the received signal \( y \) will either be close to \( +A \) or \( -A \). The Maximum Likelihood decision rule says that we should choose the symbol \( x \) that minimizes the Euclidean distance between the received signal \( y \) and \( x \).
Mathematically, this means we should choose \( x \) that minimizes the distance \( |y - x|^2 \). This can be simplified to:
\(\text{Choose } x = +A \text{ if } |y - A|^2 < |y + A|^2\)
\(\text{Choose } x = -A \text{ if } |y + A|^2 < |y - A|^2\)
5. Simplified ML Decoding for BPSK
Since we are comparing two possible symbols \( +A \) and \( -A \), the decision rule can be simplified to the following:
\(\text{If } y > 0, \text{ choose } x = +A \quad (\text{Bit } 0)\)
\(\text{If } y < 0, \text{ choose } x = -A \quad (\text{Bit } 1)\)
This is because the symbol closest to \( y \) (in terms of the Euclidean distance) is the one that is either positive or negative, depending on the value of \( y \).
6. Summary of Maximum Likelihood Detection in BPSK
In **BPSK**, Maximum Likelihood Detection works as follows:
- The received signal \( y \) is compared with the possible transmitted symbols \( +A \) and \( -A \).
- The symbol that minimizes the squared Euclidean distance to the received signal is chosen as the transmitted symbol.
- This decision can be simplified to checking the sign of the received signal: if \( y > 0 \), decide \( +A \); if \( y < 0 \), decide \( -A \).
7. Practical Considerations
In real systems, this method works effectively because the noise is typically modeled as **Gaussian**, and the decision rule based on minimizing the Euclidean distance is equivalent to choosing the most likely symbol. The main benefit of Maximum Likelihood Detection is its ability to make the most accurate decisions about which symbol was transmitted, given the noise in the system.