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 (\(-\infty\) 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()
   ...: