sdr.biawgn_capacity

sdr.biawgn_capacity(snr: ArrayLike) → NDArray[float64]

Calculates the capacity of a binary-input additive white Gaussian noise (BI-AWGN) channel.

Parameters:¶

snr: ArrayLike¶

The signal-to-noise ratio $S / N = A^{2} / σ^{2}$ at the output of the channel in dB.

Note

This SNR is for a real signal. In the case of soft-decision BPSK, the SNR after coherent detection is $S / N = 2 E_{s} / N_{0}$ , where $E_{s}$ is the energy per symbol and $N_{0}$ is the noise power spectral density.

Returns:¶

The capacity $C$ of the channel in bits/1D.

Notes¶

The BI-AWGN channel is defined as

Y = A \cdot X + W,

where $X$ is the input with $x_{i} \in {- 1, 1}$ , $A$ is the signal amplitude, $W \sim N (0, σ^{2})$ is the AWGN noise, the SNR is $A^{2} / σ^{2}$ , and $Y$ is the output with $y_{i} \in R$ .

The capacity of the BI-AWGN channel is

C = max_{f_{X}} I (X; Y)

I (X; Y) = H (Y) - H (Y | X),

where $I (X; Y)$ is the mutual information between the input $X$ and output $Y$ , $H (Y)$ is the differential entropy of the output $Y$ , and $H (Y | X)$ is the conditional entropy of the output $Y$ given the input $X$ . The maximum mutual information is achieved when $X$ is equally likely to be $- 1$ or $1$ .

The conditional probability density function (PDF) of $Y$ given $X = x$ is

f_{Y | X} (y | x) = \frac{1}{\sqrt{2 π σ^{2}}} \exp (- \frac{(y - A x)^{2}}{2 σ^{2}}) .

The marginal PDF of $Y$ is

f_{Y} (y) = \frac{1}{2} f_{Y | X} (y | 1) + \frac{1}{2} f_{Y | X} (y | - 1) .

The differential entropy of the output $Y$ is

H (Y) = - \int_{- \infty}^{\infty} f_{Y} (y) \log f_{Y} (y) d y .

The conditional entropy of the output $Y$ given the input $X$ is

H (Y | X) = - \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f_{Y | X} (y | x) \log f_{Y | X} (y | x) d y d x .

In this case, since $Y = X + W$ , the conditional entropy simplifies to

H (Y | X) = H (W) = \frac{1}{2} \log (2 π e σ^{2}) .

The capacity of the BI-AWGN channel is

\begin{array}{r} \begin{aligned} C & = H (Y) - H (Y | X) \\ = - \int_{- \infty}^{\infty} f_{Y} (y) \log f_{Y} (y) d y - \frac{1}{2} \log (2 π e σ^{2}) . \end{aligned} \end{array}

However, the integral must be solved using numerical methods. A convenient closed-form upper bound, provided by Yang, is

C \leq C_{u b} = \log (2) - \log (1 + e^{- SNR}),

where $SNR = A^{2} / σ^{2}$ in linear units.

References¶

Examples¶

In [1]: snr = np.linspace(-30, 20, 51)

In [2]: C_ub = np.log2(2) - np.log2(1 + np.exp(-sdr.linear(snr)))

In [3]: C = sdr.biawgn_capacity(snr)

In [4]: plt.figure(); \
   ...: plt.plot(snr, C_ub, linestyle="--", label="$C_{ub}$"); \
   ...: plt.plot(snr, C, label="$C$"); \
   ...: plt.legend(); \
   ...: plt.xlabel("Signal-to-noise ratio (dB), $A^2/\sigma^2$"); \
   ...: plt.ylabel("Capacity (bits/1D), $C$"); \
   ...: plt.title("Capacity of the BI-AWGN Channel");
   ...: 

In [5]: snr = np.linspace(-30, 20, 51)

In [6]: p = sdr.Q(np.sqrt(sdr.linear(snr)))

In [7]: C_hard = sdr.bsc_capacity(p)

In [8]: C_soft = sdr.biawgn_capacity(snr)

In [9]: plt.figure(); \
   ...: plt.plot(snr, C_hard, label="BSC (hard)"); \
   ...: plt.plot(snr, C_soft, label="BI-AWGN (soft)"); \
   ...: plt.legend(); \
   ...: plt.xlabel("Signal-to-noise ratio (dB), $A^2/\sigma^2$"); \
   ...: plt.ylabel("Capacity (bits/1D), $C$"); \
   ...: plt.title("Capacity of the BSC (hard-decision) and BI-AWGN (soft-decision) Channels");
   ...: 