Correlation Receiver

Correlation Receiver#

The optimal receiver for the AWGN channel follows the Maximum A Posteriori (MAP) decision rule, given by:

\[ \hat{m} = \arg \max_{1 \leq m \leq M} [\eta_m + \vec{r} \cdot \vec{s}_m] \]

where the bias term is:

\[ \eta_m = \frac{N_0}{2} \ln P_m - \frac{1}{2} \mathcal{E}_m. \]

Here:

  • \( \vec{r} \) is the \( N \)-dimensional received vector.

  • \( \vec{s}_m \) is the \( m \)-th transmitted signal vector.

  • \( P_m \) is the prior probability of message \( m \).

  • \( \mathcal{E}_m = \|\vec{s}_m\|^2 \) is the signal energy.

  • \( N_0/2 \) is the noise power spectral density per dimension.

This rule selects the message \( m \) that maximizes the sum of a bias term (reflecting prior knowledge and signal energy) and the inner product \( \vec{r} \cdot \vec{s}_m \) (which measures alignment between the received and transmitted signals).

Implementing the MAP Decision Rule#

Extracting \( \vec{r} \) from the Received Waveform

A practical challenge arises because the receiver does not directly observe the vector \( \vec{r} \). Instead, it receives the continuous-time waveform:

\[ r(t) = s_m(t) + n(t), \]

where:

  • \( s_m(t) \) is the transmitted signal waveform.

  • \( n(t) \) is additive white Gaussian noise (AWGN).

To apply the MAP rule, the receiver must first extract the vector \( \vec{r} \) from \( r(t) \). This is achieved by projecting \( r(t) \) onto an orthonormal basis \( \{ \phi_j(t), 1 \leq j \leq N \} \), as established earlier using the Gram-Schmidt procedure.

The components of \( \vec{r} \) are computed as:

\[ r_j = \int_{-\infty}^{\infty} r(t) \phi_j(t) \, dt \]

For each \( j \) (\( 1 \leq j \leq N \)), the receiver:

  1. Multiplies the received signal \( r(t) \) by the basis function \( \phi_j(t) \).

  2. Integrates over time, computing the projection of \( r(t) \) onto \( \phi_j(t) \).

Since the received signal is:

\[ r(t) = s_m(t) + n(t), \]

we obtain:

\[ r_j = \int_{-\infty}^{\infty} [s_m(t) + n(t)] \phi_j(t) \, dt. \]

Breaking it down:

\[ r_j = s_{mj} + n_j \]

where:

  • \( s_{mj} = \int_{-\infty}^{\infty} s_m(t) \phi_j(t) \, dt \) is the \( j \)-th component of \( \vec{s}_m \).

  • \( n_j = \int_{-\infty}^{\infty} n(t) \phi_j(t) \, dt \) is the noise component, which is a Gaussian random variable with:

    • Zero mean.

    • Variance \( N_0/2 \).

This step transforms the continuous-time problem into an \( N \)-dimensional vector problem, making it compatible with the MAP detection rule.

Role of the Correlation Receiver#

The receiver’s integration process effectively correlates \( r(t) \) with each basis function \( \phi_j(t) \), extracting the necessary signal components. This explains why the optimal receiver in an AWGN channel is often referred to as a correlation receiver—it processes the received signal by computing inner products with known basis functions, enabling vector-based decision-making.

This correlation-based transformation is a fundamental concept in digital communication systems, linking waveform detection to vector-based statistical decision theory.

Operation of The Correlation Receiver#

Once the received vector is obtained as

\[ \vec{r} = [r_1, r_2, \ldots, r_N], \]

the receiver executes the following sequence of operations:

Compute Inner Products

For each possible message \( m \) (from \( 1 \) to \( M \)), the receiver calculates the inner product between the received vector and each possible transmitted signal vector:

\[ \vec{r} \cdot \vec{s}_m = \sum_{j=1}^N r_j s_{mj}. \]

This dot product quantifies how well the received signal aligns with each potential transmitted signal. A higher inner product suggests a stronger match between \( \vec{r} \) and \( \vec{s}_m \).

Add Bias Terms

Each inner product is adjusted by adding a precomputed bias term:

\[ \eta_m = \frac{N_0}{2} \ln P_m - \frac{1}{2} \mathcal{E}_m. \]

This adjustment accounts for prior probabilities and signal energy differences. The final score for each possible transmitted message is:

\[ \eta_m + \vec{r} \cdot \vec{s}_m. \]

Make a Decision

The receiver compares the scores across all \( M \) possible messages and selects the message index that maximizes the decision metric:

\[ \hat{m} = \arg \max_{1 \leq m \leq M} [\eta_m + \vec{r} \cdot \vec{s}_m]. \]

Illustration of a correlation receiver with \(N\) correlators.

Physical Meaning of The Correlation Receiver#

The first step involves correlating the received waveform \( r(t) \) with each basis function \( \phi_j(t) \), extracting the relevant signal components. The term “correlation receiver” reflects the use of inner products (or correlations) in this process.

Structural Overview of the Receiver

  • The receiver consists of \( N \) parallel correlators, each corresponding to a basis function \( \phi_j(t) \).

  • Each correlator computes \( r_j = \int_{-\infty}^{\infty} r(t) \phi_j(t) \, dt \), forming the vector \( \vec{r} \).

  • A computation stage then combines these outputs with the signal vectors \( \vec{s}_m \) and bias terms \( \eta_m \).

  • Finally, a maximum-selection operation determines the most likely transmitted message.

This correlation-based approach efficiently extracts the optimal decision statistics from the noisy received signal, making it fundamental in digital communication receivers.

Alternative Implementation#

A key practical advantage of the correlation receiver is that many components can be precomputed, significantly reducing the real-time computational load.

  • The bias terms

    \[ \eta_m = \frac{N_0}{2} \ln P_m - \frac{1}{2} \mathcal{E}_m \]

    depend only on prior probabilities (\( P_m \)), noise power spectral density (\( N_0 \)), and signal energies (\( \mathcal{E}_m \)), all of which are system parameters independent of the received signal \( r(t) \).

  • The signal vectors

    \[ \vec{s}_m = [s_{m1}, s_{m2}, \ldots, s_{mN}] \]

    where

    \[ s_{mj} = \int_{-\infty}^{\infty} s_m(t) \phi_j(t) \, dt, \]

    are fixed for a given signal set and basis. These values can be precomputed and stored in memory, reducing real-time computation.

Dynamic Computation in the Receiver#

The only real-time computations required for each received \( r(t) \) are:

  1. Generating \( \vec{r} \) via correlators, computing

    \[ r_j = \int_{-\infty}^{\infty} r(t) \phi_j(t) \, dt \]

    for each basis function \( \phi_j(t) \).

  2. Computing inner products

    \[ \vec{r} \cdot \vec{s}_m = \sum_{j=1}^N r_j s_{mj} \]

    for each possible transmitted signal \( m \).

Alternative Implementation: Direct Time-Domain Correlation#

An alternative, yet equivalent, implementation bypasses the explicit computation of \( \vec{r} \cdot \vec{s}_m \) by directly correlating \( r(t) \) with each signal waveform \( s_m(t) \).

Starting from the inner product definition:

\[ \vec{r} \cdot \vec{s}_m = \sum_{j=1}^N r_j s_{mj} \]

and substituting \( r_j \):

\[ \vec{r} \cdot \vec{s}_m = \sum_{j=1}^N \left( \int_{-\infty}^{\infty} r(t) \phi_j(t) \, dt \right) s_{mj}. \]

Using the signal expansion:

\[ s_m(t) = \sum_{j=1}^N s_{mj} \phi_j(t), \]

we obtain:

\[ \vec{r} \cdot \vec{s}_m = \int_{-\infty}^{\infty} r(t) \left( \sum_{j=1}^N s_{mj} \phi_j(t) \right) dt = \int_{-\infty}^{\infty} r(t) s_m(t) \, dt. \]

This follows from the linearity of integration and the orthonormality of the basis functions.

Illustration of a correlation receiver with \(M\) correlators in time domain.

MAP Decision Rule in Time Domain#

Instead of computing \( \vec{r} \) explicitly, the MAP rule can be rewritten as:

\[ \hat{m} = \arg \max_{1 \leq m \leq M} \left[ \eta_m + \int_{-\infty}^{\infty} r(t) s_m(t) \, dt \right]. \]

where

\[ \eta_m = \frac{N_0}{2} \ln P_m - \frac{1}{2} \mathcal{E}_m. \]

Comparison of Implementations

We can see that

  • Both approaches are mathematically equivalent but differ in implementation efficiency.

  • The vector-based approach is more structured and leverages precomputed signal vectors.

  • The direct time-domain correlation approach bypasses the explicit construction of \( \vec{r} \), potentially reducing computational complexity in some scenarios.

Second Version of the Correlation Receiver#

This alternative formulation leads to a second version of the correlation receiver, which restructures the detection process. Instead of using \( N \) correlators (one per basis function) followed by vector operations, this version employs \( M \) correlators—one for each possible transmitted signal waveform \( s_m(t) \).

Each correlator computes:

\[ \int_{-\infty}^{\infty} r(t) s_m(t) \, dt. \]

The resulting outputs are then adjusted by adding the corresponding bias terms \( \eta_m \), and the maximum value determines the detected message:

\[ \hat{m} = \arg \max_{1 \leq m \leq M} \left[ \eta_m + \int_{-\infty}^{\infty} r(t) s_m(t) \, dt \right]. \]

Intuition Behind the Second Version

This design is intuitive:

  • It directly matches the received signal against each possible transmitted waveform, without explicitly forming the received vector \( \vec{r} \).

  • It leverages the fact that each waveform \( s_m(t) \) is a linear combination of the basis functions weighted by the signal vector \( \vec{s}_m \).

Comparison with the First Version

Both implementations are mathematically equivalent, as they compute the same decision metric. However, the second version may be more efficient under certain conditions:

  1. When \( M < N \) (fewer signals than dimensions):

    • The second version requires \( M \) correlators (one per signal) rather than \( N \) correlators (one per basis function).

    • This can reduce hardware complexity when the number of signals is smaller than the signal space dimension.

  2. When the signal waveforms \( s_m(t) \) are readily available:

    • Some systems store entire waveforms rather than decomposing them into basis components.

    • The second version avoids precomputing and storing basis functions, simplifying implementation.