Skip to content

komm.BinaryErasureChannel

Binary erasure channel (BEC). It is a discrete memoryless channel with input alphabet $\mathcal{X} = \{ 0, 1 \}$ and output alphabet $\mathcal{Y} = \{ 0, 1, 2 \}$. The channel is characterized by a parameter $\epsilon$, called the erasure probability. With probability $1 - \epsilon$, the output symbol is identical to the input symbol, and with probability $\epsilon$, the output symbol is replaced by an erasure symbol (denoted by $2$). For more details, see CT06, Sec. 7.1.5.

Attributes:

  • erasure_probability (float) –

    The channel erasure probability $\epsilon$. Must satisfy $0 \leq \epsilon \leq 1$. Default value is 0.0, which corresponds to a noiseless channel.

input_cardinality: int property

The channel input cardinality $|\mathcal{X}|$.

For the BEC, it is given by $|\mathcal{X}| = 2$.

output_cardinality: int property

The channel output cardinality $|\mathcal{Y}|$.

For the BEC, it is given by $|\mathcal{Y}| = 3$.

transition_matrix: npt.NDArray[np.floating] property

The channel transition probability matrix $p_{Y \mid X}$.

For the BEC, it is given by $$ p_{Y \mid X} = \begin{bmatrix} 1 - \epsilon & 0 & \epsilon \\ 0 & 1 - \epsilon & \epsilon \end{bmatrix}. $$

Examples:

>>> bec = komm.BinaryErasureChannel(0.2)
>>> bec.transition_matrix
array([[0.8, 0. , 0.2],
       [0. , 0.8, 0.2]])

mutual_information

Returns the mutual information $\mathrm{I}(X ; Y)$ between the input $X$ and the output $Y$ of the channel.

Parameters:

  • input_pmf (ArrayLike) –

    The probability mass function $p_X$ of the channel input $X$. It must be a valid pmf, that is, all of its values must be non-negative and sum up to $1$.

  • base (LogBase) –

    The base of the logarithm to be used. It must be a positive float or the string 'e'. The default value is 2.0.

Returns:

  • float –

    The mutual information $\mathrm{I}(X ; Y)$ between the input $X$ and the output $Y$.

For the BEC, it is given by $$ \mathrm{I}(X ; Y) = (1 - \epsilon) \, \Hb(\pi), $$ in bits, where $\pi = \Pr[X = 1]$, and $\Hb$ is the binary entropy function.

Examples:

>>> bec = komm.BinaryErasureChannel(0.2)
>>> bec.mutual_information([0.45, 0.55])
np.float64(0.7942195631902467)

capacity

Returns the channel capacity $C$.

Parameters:

  • base (LogBase) –

    The base of the logarithm to be used. It must be a positive float or the string 'e'. The default value is 2.0.

Returns:

  • float –

    The channel capacity $C$.

For the BEC, it is given by $$ C = 1 - \epsilon, $$ in bits.

Examples:

>>> bec = komm.BinaryErasureChannel(0.2)
>>> bec.capacity()
np.float64(0.8)

__call__

Transmits the input sequence through the channel and returns the output sequence.

Parameters:

  • input (ArrayLike) –

    The input sequence.

Returns:

  • output (NDArray[integer]) –

    The output sequence.

Examples:

>>> rng = np.random.default_rng(seed=42)
>>> bec = komm.BinaryErasureChannel(0.2, rng=rng)
>>> bec([1, 1, 1, 0, 0, 0, 1, 0, 1, 0])
array([1, 1, 1, 0, 2, 0, 1, 0, 2, 0])