Calculates the capacity of an additive white Gaussian noise (AWGN) channel.

Parameters:¶

snr: ArrayLike¶

The signal-to-noise ratio $S/N$ in dB of the channel.

bandwidth: float | None = None¶

The bandwidth $B$ of the channel in Hz. If specified, the capacity is calculated in bits/s. If None, the capacity is calculated in bits/2D.

Returns:¶

The capacity $C$ of the channel in bits/2D, or bits/s if bandwidth was specified.

Notes¶

The inputs to the AWGN channel are $x_i\in \mathbb{C}$ and the outputs are $y_i\in \mathbb{C}$. The capacity of the AWGN channel is

where $S=\frac{1}{N}\sum _{i=0}^{N-1}{\left|x_i\right|}^2$ is the average signal power and $N={\sigma }^2$ is the complex noise power. The units are bits/2D, which is equivalent to bits per complex channel use.

If the channel bandwidth $B$ is specified, the channel capacity is

Examples¶

The capacity monotonically decreases as the SNR decreases. In the limit as the SNR approaches 0 ($-\mathrm{\infty }$ dB), the capacity approaches 0.

In [1]: snr = np.linspace(-20, 20, 100); \
   ...: C = sdr.awgn_capacity(snr)
   ...: 

In [2]: plt.figure(figsize=(8, 4)); \
   ...: plt.plot(snr, C); \
   ...: plt.xlabel("Signal-to-noise ratio (dB), $S/N$"); \
   ...: plt.ylabel("Capacity (bits/2D), $C$"); \
   ...: plt.title("Capacity of the AWGN Channel"); \
   ...: plt.grid(True); \
   ...: plt.tight_layout()
   ...: 

At capacity, which occurs when $R=C$, $E_b/N_0$ is related to $E_s/N_0$ by

When viewing the capacity as a function of $E_b/N_0$, the capacity approaches 0 as $E_b/N_0$ approaches -1.59 dB. This is the absolute Shannon limit.

In [3]: ebn0 = sdr.snr_to_ebn0(snr, C)

In [4]: plt.figure(figsize=(8, 4)); \
   ...: plt.semilogy(ebn0, C); \
   ...: plt.xlabel("Bit energy to noise PSD ratio (dB), $E_b/N_0$"); \
   ...: plt.ylabel("Capacity (bits/2D), $C$"); \
   ...: plt.title("Capacity of the AWGN Channel"); \
   ...: plt.grid(True, which="both"); \
   ...: plt.tight_layout()
   ...: