Multi-Rate Polar Code Evaluation

The way how polar codes work provides an interesting property, which allows to speed up experimental testbed measurements significantly. Experimental measurements are compared to numerical simulations rather slow due to the reprogramming of the testbed hardware requires significantly more time, then the baseband signal processing operations. Another problem is that the experiment can not be easily parallelized. Hence, it is always beneficial when acquired measurement data can be used in multiple different contexts.

Polar codes allow transmission to be performed without the need to set the rate beforehand. For a code-word size N every random generated sequence of N bits is a valid code-word for an arbitrary rate given that the values of the frozen set can be arbitrarily chosen. This allows the evaluation of arbitrary many data rates using a single acquired data sample set. A precondition for this scheme is that transmitter and receiver have a sidechannel to communicate the values of the frozen bits. While this is not a viable strategy for a real communication scenario it does speed up experimental measurements significantly and is in terms of coding performance equivalent to a frozen bit set which is fixed to zero.

The sucessive interference cancellation based decoder described in [156] can be readily ex- tended to accept a frozen bit set with arbitrary fixed value. Hence, for the multi-rate measure- ment procedure a frame of multiple code-words is transmitted. Each code-word is a sequence of random bits of length N that is known at the receiver. Initially the receiver demodulates the frame to a series of LLRs completely agnostic to the intended rate. The same sequence of LLRs can now be evaluated multiple times with different rates in the polar decoder. For each rate the framework provides the polar decoder with the indices and values of the frozen set at the target rate. The non-frozen bits are then used to evaluate the resulting bit-error-rate after decoding.

As outlined before the set of frozen bits should not be a function of the current SNR, but only of the code-rate. Hence, choosing the best frozen bits set for each rate is an optimization problem, which could be solved for each target rate on it own. For each rate the waterfall region is iteratively determined. The polar code is trained based on an inital assumption of the position of the waterfall region, afterwards the code is used for BER simulation. If the waterfall of the simulated BER is different from the initially assumed waterfall region, the code is regenerated according to the new assumption. This procedure is repeated until the waterfall region stops moving to a lower SNR. Unfortunately such a procedure significantly impairs the flexibility of the polar codes in our research environment, as the process needs to be performed

(a) BER (b) PER

Figure 5.16 – Numerical multi-rate simulation of an APSK256 constellation in a mmWave LOS channel with polar coding based on 2000 simulated frames. The code-rates have been chosen to always result in integer bits for the spectral efficiency.

for each used rate. Hence, a more flexible simplified scheme is used for the simulations. The frozen bit sets are pre-generated for a regular grid of SNR points. From rate and the modulation order a target spectral efficiency is calculated. Based on the spectral efficiency the minimal required SNR can be calculated from the Shannon Capacity (see Appendix A.1). This allows to set choose a frozen bit set purely from the rate and the modulation order. The results of a numerical multi-rate simulation of an APSK trained polar code are shown in Fig. 5.16. The packet error rate results confirm the very steep error rate decay of polar codes in this environment. Despite the fact that 2000 frames have been simulated per SNR point nearly no data points below 0.5· 10−1are captured. The problem of this simulation is that the SNR grid that was necessary to perform the simulation over a wide SNR range was too coarse to properly capture the steep slope of the packet error rate. This finding underlines the importance to precisely adjust the rate to the operating conditions in polar coded systems.

In the testbed the multi-rate approach significantly reduces the time required for a BER and PER measurements. The time required for a single complete measurement and for a block-RX measurement with 10 frames are listed in Tbl. 5.2. When comparing the results with the single rate measurement times in Tbl. 4.2 we can see that the average time for the measurement of one frame with one rate can be reduced from 14 s to 5 s seconds for a short frame and from 24 s to 8 s seconds for a long frame. Hence combining block-RX mode and multi-rate code evaluation reduces the average time required to evaluate one frame at a single rate from 72 s to only 8 s.

Multi-rate simulations allow BER and PER results for multiple configurations (with different spectral efficiency) to be generated with a single set of measurements. Hence, this technique is well suited for comparisons between different modulation orders. In Fig. 5.17 the BER and PER of different APSK256 and APSK1024 configurations with the same spectral efficiency are

Short 46 s 143 s 14 s 46 s 178 s 5 s 64%

Long 72 s 240 s 24 s 81 s 334 s 8 s 66%

Table 5.2 – Multi-rate testbed measurements for single and block-RX mode with 4 rates. (The frame parameters are identical to the one used for Tbl. 4.2)

(a) BER (b) PER

Figure 5.17 – Comparison of APSK256 and APSK1024 configurations with the same spectral efficiency.

compared. Using multi-rate simulation such a comparison requires only a single simulation run per modulation. For a spectral efficiency of 5-6 bits/symbol the APSK256 has small but visible advantage, while for 7 bits/symbol APSK1024 seems to perform slightly better. This is consistent with the modulation thresholds derived from the MI in Fig. 5.12.