Implements a binary erasure channel (BEC).

Notes

The inputs to the BEC are \(x_i \in \mathcal{X} = \{0, 1\}\) and the outputs are \(y_i \in \mathcal{Y} = \{0, 1, e\}\). The capacity of the BEC is

References

• John Proakis, Digital Communications, Chapter 6.5-1: Channel Models.

Examples

When 20 bits are passed through a BEC with erasure probability \(p=0.25\), roughly 5 bits are erased at the output.

In [1]: bec = sdr.BinaryErasureChannel(0.25, seed=1)

In [2]: x = np.random.randint(0, 2, 20); x
Out[2]: array([0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1])

In [3]: y = bec(x); y
Out[3]: 
array([ 0,  1, -1,  0,  1,  1,  1,  1,  1, -1,  1,  0,  0,  1,  0,  0, -1,
        0, -1,  1])

In [4]: np.count_nonzero(x != y)
Out[4]: 4

The capacity of this BEC is 0.75 bits/channel use.

In [5]: bec.capacity
Out[5]: 0.75

When the probability \(p\) of bit erasure is 0, the capacity of the channel is 1 bit/channel use. However, as the probability of erasure approaches 1, the capacity of the channel linearly approaches 0.

In [6]: p = np.linspace(0, 1, 101); \
   ...: C = sdr.BinaryErasureChannel.capacities(p)
   ...: 

In [7]: plt.figure(); \
   ...: plt.plot(p, C); \
   ...: plt.xlabel("Erasure probability, $p$"); \
   ...: plt.ylabel("Capacity (bits/channel use), $C$"); \
   ...: plt.title("Capacity of the Binary Erasure Channel");
   ...: 

Constructors¶

BinaryErasureChannel(p: float, seed: int | None = None)

Creates a new binary erasure channel (BEC).

Special methods¶

__call__(x: ArrayLike) → NDArray[int_]

Passes the binary input sequence \(x\) through the channel.

Methods¶

static capacities(p: ArrayLike) → NDArray[float64]

Calculates the capacity of BEC channels.

reset(seed: int | None = None)

Resets the channel with a new seed.

Properties¶

property X : NDArray[np.int_]

The input alphabet \(\mathcal{X} = \{0, 1\}\) of the BEC channel.

property Y : NDArray[np.int_]

The output alphabet \(\mathcal{Y} = \{0, 1, e\}\) of the BEC channel. Erasures \(e\) are represented by -1.

property p : float

The erasure probability \(p\) of the BEC channel.

property capacity : float

The capacity \(C\) of the instantiated channel in bits/channel use.