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komm.BinarySymmetricChannel

Binary symmetric channel (BSC). It is a discrete memoryless channel with input and output alphabets $\mathcal{X} = \mathcal{Y} = \{ 0, 1 \}$. The channel is characterized by a parameter $p$, called the crossover probability. With probability $1 - p$, the output symbol is identical to the input symbol, and with probability $p$, the output symbol is flipped. Equivalently, the channel can be modeled as $$ Y_n = X_n + Z_n, $$ where $Z_n$ are iid Bernoulli random variables with $\Pr[Z_n = 1] = p$. For more details, see CT06, Sec. 7.1.4.

Attributes:

  • crossover_probability (float) –

    The channel crossover probability $p$. Must satisfy $0 \leq p \leq 1$. The default value is 0.0, which corresponds to a noiseless channel.

input_cardinality: int property

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

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

output_cardinality: int property

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

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

transition_matrix: npt.NDArray[np.floating] property

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

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

Examples:

>>> bsc = komm.BinarySymmetricChannel(0.2)
>>> bsc.transition_matrix
array([[0.8, 0.2],
       [0.2, 0.8]])

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 BSC, it is given by $$ \mathrm{I}(X ; Y) = \Hb(p + \pi - 2 p \pi) - \Hb(p), $$ in bits, where $\pi = \Pr[X = 1]$, and $\Hb$ is the binary entropy function.

Examples:

>>> bsc = komm.BinarySymmetricChannel(0.2)
>>> bsc.mutual_information([0.45, 0.55])
np.float64(0.2754734936803773)

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 BSC, it is given by $$ C = 1 - \Hb(p), $$ in bits, where $\Hb$ is the binary entropy function.

Examples:

>>> bsc = komm.BinarySymmetricChannel(0.2)
>>> bsc.capacity()
np.float64(0.2780719051126377)

__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)
>>> bsc = komm.BinarySymmetricChannel(0.2, rng=rng)
>>> bsc([1, 1, 1, 0, 0, 0, 1, 0, 1, 0])
array([1, 1, 1, 0, 1, 0, 1, 0, 0, 0])