Discrete Channel Models

This module provides layers and functions that implement channelmodels with discrete input/output alphabets.

All channel models support binary inputs\(x \in \{0, 1\}\) andbipolarinputs\(x \in \{-1, 1\}\), respectively. In the later case, it is assumedthat each0 is mapped to-1.

The channels can either return discrete values or log-likelihood ratios (LLRs).These LLRs describe the channel transition probabilities\(L(y|X=1)=L(X=1|y)+L_a(X=1)\) where\(L_a(X=1)=\operatorname{log} \frac{P(X=1)}{P(X=0)}\) depends only on thea priori probability of\(X=1\). These LLRs equal thea posteriori probability if\(P(X=1)=P(X=0)=0.5\).

Further, the channel reliability parameter\(p_b\) can be either a scalarvalue or a tensor of any shape that can be broadcasted to the input. Thisallows for the efficient implementation ofchannels with non-uniform error probabilities.

The channel models are based on theGumble-softmax trick[GumbleSoftmax] toensure differentiability of the channel w.r.t. to the channel reliabilityparameter. Please see[LearningShaping] for further details.

Setting-up:

>>>bsc=BinarySymmetricChannel(return_llrs=False,bipolar_input=False)

Running:

>>>x=tf.zeros((128,))# x is the channel input>>>pb=0.1# pb is the bit flipping probability>>>y=bsc((x,pb))

classsionna.phy.channel.BinaryErasureChannel(return_llrs=False,bipolar_input=False,llr_max=100.0,precision=None,**kwargs)[source]

Binary erasure channel (BEC) where a bit is either correctly receivedor erased.

In the binary erasure channel, bits are always correctly received or erasedwith erasure probability\(p_\text{b}\).

This block supports binary inputs (\(x \in \{0, 1\}\)) andbipolarinputs (\(x \in \{-1, 1\}\)).

If activated, the channel directly returns log-likelihood ratios (LLRs)defined as

\[\begin{split}\ell =\begin{cases} -\infty, \qquad \text{if} \, y=0 \\ 0, \qquad \quad \,\, \text{if} \, y=? \\ \infty, \qquad \quad \text{if} \, y=1 \\\end{cases}\end{split}\]

The erasure probability\(p_\text{b}\) can be either a scalar or atensor (broadcastable to the shape of the input). This allowsdifferent erasure probabilities per bit position.

Please note that the output of the BEC is ternary. Hereby,-1 indicates anerasure for the binary configuration and0 for the bipolar mode,respectively.

Parameters:

• return_llrs (bool, (defaultFalse)) – IfTrue, the layer returns log-likelihood ratiosinstead of binary values based onpb.

• bipolar_input (bool, (defaultFalse)) – IfTrue, the expected input is given as\(\{-1,1\}\) instead of\(\{0,1\}\).

• llr_max (tf.float, (default 100)) – Clipping value of the LLRs

• precision (None (default) | “single” | “double”) – Precision used for internal calculations and outputs.If set toNone,precision is used.

Input:

• x ([…,n],tf.float) – Input sequence to the channel

• pb (tf.float) – Erasure probability. Can be a scalar or of any shape that can bebroadcasted to the shape ofx.

Output:

[…,n],tf.float – Output sequence of same length as the inputx. Ifreturn_llrs isFalse, the output is ternary where each-1and each0 indicate an erasure for the binary and bipolar input,respectively.

classsionna.phy.channel.BinaryMemorylessChannel(return_llrs=False,bipolar_input=False,llr_max=100.0,precision=None,**kwargs)[source]

Discrete binary memory less channel with (possibly) asymmetric bit flippingprobabilities

Inputs bits are flipped with probability\(p_\text{b,0}\) and\(p_\text{b,1}\), respectively.

This block supports binary inputs (\(x \in \{0, 1\}\)) andbipolarinputs (\(x \in \{-1, 1\}\)).

If activated, the channel directly returns log-likelihood ratios (LLRs)defined as

\[\begin{split}\ell =\begin{cases} \operatorname{log} \frac{p_{b,1}}{1-p_{b,0}}, \qquad \text{if} \, y=0 \\ \operatorname{log} \frac{1-p_{b,1}}{p_{b,0}}, \qquad \text{if} \, y=1 \\\end{cases}\end{split}\]

The error probability\(p_\text{b}\) can be either scalar or atensor (broadcastable to the shape of the input). This allowsdifferent erasure probabilities per bit position. In any case, its lastdimension must be of length 2 and is interpreted as\(p_\text{b,0}\) and\(p_\text{b,1}\).

Parameters:

• return_llrs (bool, (defaultFalse)) – IfTrue, the layer returns log-likelihood ratiosinstead of binary values based onpb.

• bipolar_input (bool, (defaultFalse)) – IfTrue, the expected input is given as\(\{-1,1\}\) instead of\(\{0,1\}\).

• llr_max (tf.float, (default 100)) – Clipping value of the LLRs

• precision (None (default) | “single” | “double”) – Precision used for internal calculations and outputs.If set toNone,precision is used.

Input:

• x ([…,n],tf.float32) – Input sequence to the channel consisting of binary values\(\{0,1\}` or :math:\){-1,1}`, respectively

• pb ([…,2],tf.float32) – Error probability. Can be a tuple of two scalars or of anyshape that can be broadcasted to the shape ofx. It has anadditional last dimension which is interpreted as\(p_\text{b,0}\)and\(p_\text{b,1}\).

Output:

[…,n],tf.float32 – Output sequence of same length as the inputx. Ifreturn_llrs isFalse, the output is ternary where a-1 and0 indicate an erasure for the binary and bipolar input,respectively.

propertyllr_max

Get/set maximum value used forLLR calculations

Type:

tf.float

propertytemperature

Get/set temperature for Gumble-softmax trick

Type:

tf.float32

classsionna.phy.channel.BinarySymmetricChannel(return_llrs=False,bipolar_input=False,llr_max=100.0,precision=None,**kwargs)[source]

Discrete binary symmetric channel which randomly flips bits with probability\(p_\text{b}\)

This layer supports binary inputs (\(x \in \{0, 1\}\)) andbipolarinputs (\(x \in \{-1, 1\}\)).

If activated, the channel directly returns log-likelihood ratios (LLRs)defined as

\[\begin{split}\ell =\begin{cases} \operatorname{log} \frac{p_{b}}{1-p_{b}}, \qquad \text{if}\, y=0 \\ \operatorname{log} \frac{1-p_{b}}{p_{b}}, \qquad \text{if}\, y=1 \\\end{cases}\end{split}\]

where\(y\) denotes the binary output of the channel.

The bit flipping probability\(p_\text{b}\) can be either a scalar or atensor (broadcastable to the shape of the input). This allowsdifferent bit flipping probabilities per bit position.

Parameters:

• return_llrs (bool, (defaultFalse)) – IfTrue, the layer returns log-likelihood ratiosinstead of binary values based onpb.

• bipolar_input (bool, (defaultFalse)) – IfTrue, the expected input is given as\(\{-1,1\}\) instead of\(\{0,1\}\).

• llr_max (tf.float, (default 100)) – Clipping value of the LLRs

• precision (None (default) | “single” | “double”) – Precision used for internal calculations and outputs.If set toNone,precision is used.

Input:

• x ([…,n],tf.float32) – Input sequence to the channel

• pb ([…,2],tf.float32) – Bit flipping probability. Can be a scalar or of any shape thatcan be broadcasted to the shape ofx.

Output:

[…,n],tf.float32 – Output sequence of same length as the inputx. Ifreturn_llrs isFalse, the output is ternary where a-1 and0 indicate an erasure for the binary and bipolar input,respectively.

classsionna.phy.channel.BinaryZChannel(return_llrs=False,bipolar_input=False,llr_max=100.0,precision=None,**kwargs)[source]

Block that implements the binary Z-channel

In the Z-channel, transmission errors only occur for the transmission ofsecond input element (i.e., if a1 is transmitted) with error probabilityprobability\(p_\text{b}\) but the first element is always correctlyreceived.

This block supports binary inputs (\(x \in \{0, 1\}\)) andbipolarinputs (\(x \in \{-1, 1\}\)).

If activated, the channel directly returns log-likelihood ratios (LLRs)defined as

\[\begin{split}\ell =\begin{cases} \operatorname{log} \left( p_b \right), \qquad \text{if} \, y=0 \\ \infty, \qquad \qquad \text{if} \, y=1 \\\end{cases}\end{split}\]

assuming equal probable inputs\(P(X=0) = P(X=1) = 0.5\).

The error probability\(p_\text{b}\) can be either a scalar or atensor (broadcastable to the shape of the input). This allowsdifferent error probabilities per bit position.

Parameters:

• return_llrs (bool, (defaultFalse)) – IfTrue, the layer returns log-likelihood ratiosinstead of binary values based onpb.

• bipolar_input (bool, (defaultFalse)) – IfTrue, the expected input is given as\(\{-1,1\}\) instead of\(\{0,1\}\).

• llr_max (tf.float, (default 100)) – Clipping value of the LLRs

• precision (None (default) | “single” | “double”) – Precision used for internal calculations and outputs.If set toNone,precision is used.

Input:

• x ([…,n],tf.float32) – Input sequence to the channel

• pb (tf.float32) – Error probability. Can be a scalar or of any shape that can bebroadcasted to the shape ofx.

Output:

[…,n],tf.float32 – Output sequence of same length as the inputx. Ifreturn_llrs isFalse, the output is binary and otherwisesoft-values are returned.

References:

[GumbleSoftmax]

E. Jang, G. Shixiang, and B. Poole.“Categorical reparameterization with gumbel-softmax,” arXiv preprint arXiv:1611.01144 (2016).

[LearningShaping]

M. Stark, F. Ait Aoudia, and J. Hoydis.“Joint learning of geometric and probabilistic constellation shaping,” 2019 IEEE Globecom Workshops (GC Wkshps). IEEE, 2019.