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komm.ZChannel

Z-channel. 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 decay probability. Bit $0$ is always received correctly, but bit $1$ turns into $0$ with probability $p$. Equivalently, the channel can be modeled as $$ Y_n = A_n X_n, $$ where $A_n$ are iid Bernoulli random variables with $\Pr[A_n = 0] = p$.

Attributes:

  • decay_probability (float) –

    The channel decay 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 Z-channel, it is given by $|\mathcal{X}| = 2$.

output_cardinality: int property

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

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

Examples:

>>> zc = komm.ZChannel(0.2)
>>> zc.transition_matrix
array([[1. , 0. ],
       [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 Z-channel, it is given by $$ \mathrm{I}(X ; Y) = \Hb ( \pi (1-p) ) - \pi \Hb(p), $$ in bits, where $\pi = \Pr[X = 1]$, and $\Hb$ is the binary entropy function.

Examples:

>>> zc = komm.ZChannel(0.2)
>>> zc.mutual_information([0.5, 0.5])
np.float64(0.6099865470109874)

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 Z-channel, it is given by $$ C = \log_2 ( 1 + (1-p) p^{p / (1-p)} ), $$ in bits.

Examples:

>>> zc = komm.ZChannel(0.2)
>>> zc.capacity()
np.float64(0.6182313659549211)

__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)
>>> zc = komm.ZChannel(0.2, rng=rng)
>>> zc([1, 1, 1, 0, 0, 0, 1, 0, 1, 0])
array([1, 1, 1, 0, 0, 0, 1, 0, 0, 0])