Telecommunication technology today
is playing an increasingly important role in the whole industry,
especially the coming generation of cellular mobile communication: 5g
It will boost up many other domains.
such as truly immersive AR and VR applications
Automated driving vehicles
ultimately the tactile internet for 5g communication.
so, in order to achieve
one aspect that of 5g communication
: the enhanced mobile broadband communication service
the polar code has been chosen as channel coding algorithm for the control channel
by the 3GPP conference.
The main purpose of our project is to explore and
inspect the properties and performance of the polar code.
Furthermore grab a perspective of the channel coding algorithm
In our system a code construction an encoder
and adecoder algorithms were desinged on Matlab.
for the code construction algorithm
It is just combining the binary tree and the Bhattacharyya bounds of
AWGN channel. As for the encoder it is mainly based on the Kronecker product
As for decoder the SCD algorithm was used
since it is the most fundamental
and important decoding method of polar codes which achieves very decent
time complexity of nearly N*log(N) it can be separated into two parts one is log
likelihood calculation the other one is bit broadcasting.
they are interacting with each other, the decoding root bits are actually what we want.
you will be demonstrated as following
After completing the polar code program
we found that the channel performance will be affected by
varying the code rate, code length and design SNR.
The longer code length or smaller code rate will result in the better
performance and the best design SNR was found to be zero in our project
we would like to give an overview of the significant contribution.
Firstly, we simulated and evaluated the polar code for the AWGN channel.
Then we implemented and evaluated the code construction algorithm
using the Bhattacharyya parameters
After that we implemented the channel encoder using the Kronecker product.
Moreover we derived a logarithmic likelihood ratio by channel observation
Then, we implemented the successive cancellation algorithm into decoding procedure.
Next we evaluated the Polar code performance in terms of different design SNR
Then we evaluated the polar code performance in terms of different code rate code length.
Lastly we reconstructed a noisy image by implementing the polar code.
Lets start with the code construction
Firstly,here are two bits channel specifically
the binary erasable channel with an erasure probability e.
Then we combine them using mode two operation
This can be seen in the lower figure.
Therefore at the decoder side
The received bits are supposed to be exactly X1, X2.
However due to the property of BEC
there is probability e that the transmitted information will be erased
therefore there are four possible situations as a decoder side for y 1 y 2
These are the four situations.
The question mark means the information is erased already.
In order to derive the polarization outcome
one of the important properties of successive cancellation decoding algorithm is used here
: the outputs of these two channel are (y1, y2) and( y1, y2, u 1) which means that
the extra information u1 is needed to decode y2.
It can be noticed that only in s4 where the knowledge about
(u1 +u2) mod 2 and u 2 is known simultaneously
in other words, neither X1 nor X2 are not erased the Y1 can be decoded correctly.
Consequently, the probability of the error occurring in the upper channel is E1 equals to 2*e-e^2.
However due to a piece of extra information is added to decode Y 2
there are 2 more situations where Y2 can be decoded successfully.
These are the three situations
Consequently, the error probability of the lower channel is E2 which is e^2.
It can be proven that the E1 is always larger than E2
given that the error probability e is a number within range from 0 to 1
with performing this operation recursively the better channel becomes
better marked by more '+' sign, the worse channel even becomes worse marked by more '-' sign.
Consequently a portion of the channel
will become nearly completely noisy while some other channels become nearly
noseless which exactly clarifies some Matthew effect in communication aspect.
The channel evolution can be summarized as a binary tree the initial value is a
Bhattacharyya bounds of a AWGN channel. We stop the iterative
process until the last stage hold N values which is the code length. Then we select
out the leaves holding K least values and it is the number of information bits.
After that we output the indices of these leaves, the rest indices are the
frozen bits indices. These 3 figure illustrates the channel capacity
when thestage equals 3, 6, 11, corresponding to the theory before more channels are polarized
with increasing the code length , Consequently the Shannon capacity will be achieved
when the code length equals infinity
Let's move on to the encoder now. The polar coder is parameterized the by
the code length: N, the information bits length: K, and the indices of information bits: I
Once we have known them, a transmitted vector can be formed
in this vector the information bits are stored in the leaves which holds the K least values,
the complementary indices are frozen bits indices, this is corresponding to the code construction
Since the polar code is one type of linear block code
a generator matrix is needed.
In order to describe the operation in the
butterfly diagram.
The Kronecker product is used to generate a generator matrix
Once we have it,
the code word is generated by multiplying the transmitted
vector and the generator matrix,
this slide gives an example for the code length equals to 8.
The butterfly diagram is a basic unit of decoding algorithm
the u 1, u 2 y 1, y 2 are the bits and L1 L2 function f and g are likelihood ratios
the bits are determined by likelihood ratios,
here is a formula for function G and F.
The proof process of this formula are demonstrated on Arikan's paper and
our final report.
So it is very reliable
the time complexity for decoding is N
times( Log(N)+1), since there is N binary tree and each binary has
(logN)+1 layers.
The channel observations are given here
likelihood ratio can be derived by channel observation,
here is a result
bits and likelihood ratios
they are set as negative 1 as initial value.
The right most likelihood ratios are based on channel observation
which is given
and the frozen bit are picked because of the lowest channel capacity that has stated previously
they are forced to be 0 and initial and both decoder and encoder would notice
then the index 0 can be calculated recursively by using function f
after root likelihood ratio is calculated here the bit can be decided
if likelihood ratio is larger than 1 then the bit is 0, otherwise the bit is 1
then the likelihood ratio of index 4 can be decided by function G
when u1 here equals to 0, G equals to L 1 times L2,
when u 1 equals to 1, G equals to L 2 divided by L1
Then u 1 and u 2 can be broadcasted right towards together.
Once bits reached
a layer here the process stopped because the layer branch bits have not decided
so the G function would be a applied here
since the bit is known upper branch
then using these two likelihood ratios,
likelihood ratio of index 2 can be calculated
using function f,
then it bit can be decided
same procedure can be implemented as the following
function G can be applied here determine the bits
broadcasting tell the upper branch
One trick made in our program is we assign the search order as bit reverse sequence
It needs to be noticed that the B reverse permutation in our case that not
rearranged our original bid order
it just changed the searching sequence of decoder
in order to improve the searching efficiency
so the encoder does not need to do anything with it
finally all bits are decided
the bits at the leftmost here are the decoded bits.
The yellow frame indicated that
there are some conflict between likelihood ratio and real bits
because of the existence of frozen bits
And when longer code length is implemented
log likelihood ratio always performs better than likelihood ratio.
It should be noticed that
During the whole decoding process the bit decision can only happen the root index here,
because the root index get all information from the nodes
which would result in better bit prediction than other nodes
this is also one reason why longer code length has better performance which is a beauty of polar codes
it is time to evaluate the performance after converting all the
theory and knowledge about polar code into MATLAB coding,
as we can see here
we dynamically changing design SNR, in the end we obtain the decent bit error rate
which is up to 10 to the power minus 4 after that due to the
non-universality of polar code
the different polar code are generated by the
different design SNR.
In this section we are committed to select out of the
best design SNR for the code length equals 1024 and 2048.
The results shows that the best design SNR is around 0 dB for both cases
these two graphs illustrate the influence of code on channel performance.
So smaller code rate has best performance in testing samples.
It is reasonable since the redundancy in the polar code is assigned to be frozen bits .
Fixed values won't be wrongly decoded.
Besides they would affect and
correct the decoding procedure comparing to the last redundancy situation.
As we stated previously longer code length has better performance because of the channel polarization.
Furthermore in the bad channel condition the performance of all
code length tend to have the same value, at 0.5 dB when R equals to 0.1.
1.5 dB when R equals to 0.5.
After these points
the trend of the performance of the code length versus SNR is becoming more obvious
In the end we would like to thank our supervisor Dr. Yi Hong for giving us an opportunity to complete
such an interesting project, also thanks Harish for giving us a lot of help
in terms of future work we have more students and put efforts on implementing
different code construction such as Monte Carlo and more complex decoding algorithms
such as BP and folded SCD to explore more properties of polar codes.
Thank you
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