A polar code of length $N=2^{n}$ is encoded using the polar code generator matrix $\mathrm{\mathbf{G}}$ of size $N\times N$. Thus a block of length $N$, consisting of $N-K$ frozen and $K$ nonfrozen bits, is multiplied by $\mathrm{\mathbf{G}}$ to produce the polar codeword. The matrix $\mathrm{\mathbf{G}}$ is based on the kernel used to construct the code, with $\boldsymbol{\mathrm{G}}=\boldsymbol{\mathrm{F}}^{\otimes n}$

where $\boldsymbol{\mathrm{F}}^{\otimes n}$ denotes the $n^{th}$ Kronecker power of $\mathrm{\boldsymbol{F}}$, and where $\boldsymbol{\mathrm{F}}=\begin{bmatrix}1 & 0\\1 & 1 \end{bmatrix}$ is based on the used kernel.

A polar encoding circuit, equivalent to $\mathrm{\mathbf{G}}$, can also be used as a polar encoder, as shown in the next slide.

$\bf{\text{Systematic polar encoding}}$

Polar codes were introduced as a type of non-systematic codes.

Any linear code can be changed from a non-systematic to a systematic code.

A systematic code is very beneficial, if the decoder is to iteratively exchange information with other decoders.

The systematic polar encoding can be performed using the standard non-systematic polar encoding circuit. It is simply a three phase operation:

1. The vector $\mathbf u=(u_\mathbb{A},u_{\mathbb{A}^c})$ is encoded in the standard non-systematic fashion producing the vector $\bar{\mathbf u}$.

2. The frozen bit positions $\mathbb{A}^c$ in the vector $\bar{\mathbf u}$ are set to zero ($\bar{\mathbf u}_{\mathbb{A}^c}=0$) (the frozen bits are always set to zeros here).

3. The modified vector $\bar{\mathbf u}$ is encoded in the standard non-systematic fashion producing the codeword $\mathbf x$, which is a systematic polar codeword, in the sense that the information bits $\mathbf u_{\mathbb{A}}$ appear in the final codeword $\mathbf x$ in the information bits position ($\mathbf x_\mathbb{A}$).

The main advantage of the systematic polar codes is that its BER performance is better than that in the standard non-systematic polar codes. However, both systematic and non-systematic polar codes have the same BLER performance.

It is observed that the systematic polar coding is more robust to error propagation in its SC decoder when compared to that in the non-systematic polar coding.