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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 (Optional[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:

  • input_sequence (Array1D[int]) –

    The input sequence.

Output:

  • output_sequence (Array1D[int]) –

    The output sequence.

Examples:

>>> np.random.seed(1)
>>> bec = komm.BinaryErasureChannel(0.1)
>>> bec([1, 1, 1, 0, 0, 0, 1, 0, 1, 0])
array([1, 1, 2, 0, 0, 2, 1, 0, 1, 0])

transition_matrix: npt.NDArray[np.float64] property

The transition probability matrix of the channel, given by $$ p_{Y \mid X} = \begin{bmatrix} 1 - \epsilon & 0 & \epsilon \\ 0 & 1 - \epsilon & \epsilon \end{bmatrix}. $$

Examples:

>>> bec = komm.BinaryErasureChannel(0.1)
>>> bec.transition_matrix
array([[0.9, 0. , 0.1],
       [0. , 0.9, 0.1]])

mutual_information

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

Parameters:

Same as the corresponding method of the general class.

Examples:

>>> bec = komm.BinaryErasureChannel(0.1)
>>> bec.mutual_information([0.45, 0.55])
np.float64(0.8934970085890275)

capacity

Returns the channel capacity $C$. It is given by $$ C = 1 - \epsilon, $$ in bits.

Examples:

>>> bec = komm.BinaryErasureChannel(0.1)
>>> bec.capacity()
np.float64(0.9)