Viterbi Decoder

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Viterbi decoder

by Krishna Sankar on January 4, 2009 by Krishna Sankar on January 4, 2009

Coding is a technique where redundancy is added to original bit sequence to increase the reliability of the Coding is a technique where redundancy is added to original bit sequence to increase the reliability of the communication. Lets discuss a simple

communication. Lets discuss a simple binary convolutional codingbinary convolutional coding schemscheme at e at the tthe transmransmitter and the itter and the associassociatedated

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Viterbi

Viterbi (maxim(maximum likelihood) decodingum likelihood) decodingschemscheme at e at the recethe receiver.iver. Update:

Update: For some reason, the blog For some reason, the blog is unable to display the is unable to display the article article which discuss bwhich discuss both Convolutional coding and oth Convolutional coding and Viter

Viterbi decoding. As a work around, the articbi decoding. As a work around, the article was broken upto into two posts.le was broken upto into two posts.

This post descrbes the Viterbi decoding algorithm for a simple Binary Convolutional Code with rate 1/2, This post descrbes the Viterbi decoding algorithm for a simple Binary Convolutional Code with rate 1/2, constraint

constraint length length and and having having generator generator polynomipolynomial al . . For For more more details details on on the the Binary Binary convolutionalconvolutional code, please refer to the post –

code, please refer to the post – Convolutional codeConvolutional code

Viterbi algorithm

As explained in Chapter 5.1.4 of

As explained in Chapter 5.1.4 of DigDigital Communications by John Pital Communications by John Proakisroakis for optimal decoding for a modulationfor optimal decoding for a modulation scheme

scheme with with memory memory (as (as is is the the case case here), here), if if there there are are coded coded bits, bits, we we need need to to search search from from possiblepossible combinations.

combinations. This This becomes becomes prohibitively prohibitively complex complex as as becomes becomes large. large. However,However, Mr. Andrew J ViterbiMr. Andrew J Viterbi in hisin his landmark paper

landmark paper Error bounds for convolutional codes and an asymptotically optimum decoding algorithmError bounds for convolutional codes and an asymptotically optimum decoding algorithm ,, IEEE IEEE Transactions on Information Theory

Transactions on Information Theory 13(2):260–269, April 1967 described a scheme for reducing the complexity13(2):260–269, April 1967 described a scheme for reducing the complexity to more managable levels. Some of the key assumptions are as follows:

to more managable levels. Some of the key assumptions are as follows: (a) As shown in

(a) As shown in Table 1Table 1 andand Figure 2Figure 2(in the article Convoutional Code(in the articleConvoutional Code) any state ) any state can be reached can be reached from onlfrom only 2y 2 possible previous state.

possible previous state.

(b) Though we reach each state from 2 possible states, only one of the transition in valid. We can find the (b) Though we reach each state from 2 possible states, only one of the transition in valid. We can find the transition which is more likely (based on the received coded bits) and ignore the other transition.

transition which is more likely (based on the received coded bits) and ignore the other transition.

(c) The errors in the received coded sequence are randomly distributed and the probability of error is small. (c) The errors in the received coded sequence are randomly distributed and the probability of error is small. Based

Based on on the the above above assumptions, assumptions, the the decoding decoding scheme scheme proceeproceed d as as follows: follows: Assume Assume that there that there are are coded coded bits.bits. Take two coded bits at a time for processing and compute

Take two coded bits at a time for processing and compute Hamming distanceHamming distance,, Branch MetricBranch Metric,, Path MetricPath Metric and

and SurvivSurvivor Por Path indexath indexfor for times. times. Let Let be be the the index index varying varying from from 1 1 till till ..

Hamming Distance Computation Hamming Distance Computation For

For decoding, decoding, consider consider two two received received coded coded bits bits at at a a time time and and compute compute thethe Hamming distanceHamming distance between allbetween all possible

possible combinations combinations of two of two bits. bits. The numThe number of ber of differing differing bits bits can be can be computed by computed by XOR-ing XOR-ing with 00, 01, with 00, 01, 10,10, 11 and then counting the number of ones.

11 and then counting the number of ones.

is the number of 1′s in is the number of 1′s in is the number of 1′s in is the number of 1′s in is the number of 1′s in is the number of 1′s in is the number of 1′s in is the number of 1′s in Branch Metric

Branch Metric and Path Metric Computaand Path Metric Computationtion

As stated prior, each state can be reached from two possible states (shown by red and blue lines respectively). As stated prior, each state can be reached from two possible states (shown by red and blue lines respectively). Branch metric is the sum of the path metric of the previous state and the hamming distance required for the Branch metric is the sum of the path metric of the previous state and the hamming distance required for the transition. From the two avaiable branch metrices, the one with minimum branch metric value

transition. From the two avaiable branch metrices, the one with minimum branch metric value is chosenis chosen.. ThisThis operation is als

operation is also referred o referred to asto as Add Compare and Select (ACS)Add Compare and Select (ACS) unitunit.. Note:

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1. Typically, Convolutional coder always starts from State 00. The Viterbi decoder also assumes the same. 2. For index = 1, branch metric for State 00 (from State 00) branch and State 10 (from State 00) can only be computed. In this case, path metric for each state is equal to branch metric as the other branch is not valid. 3. For index = 2, branch metric for State 00 (from State 00) branch, State 01 (from State 10), State 10 (from State 00) and State 11 (from State 10) can only be computed. In this case too, path metric for each state is equal to branch metric as the other branch is not valid.

4. Starting from index =3, each state has two branches and the need to do Add Compare and Select arises. 5. Its possible that two branch metrices might have the same value. In that scenario, we can chose one among the the branches and proceed.

Branch Metric and P ath Metric computation for Viterbi decoder

Figure: Branch Metric and Path Metric computation for Viterbi decoder State 00 can be reached from two branches

(a) State 00 with output 00. The branch metric for this transition is, .

(b) State 01 with output 11. The branch metric for this transition is, .

The path metric for state 00 is chosen based which is minimum out of the two.

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State 01 can be reached from two branches:

(c) State 10 with output 10. The branch metric for this transition is,

(d) State 11 with output 01. The branch metric for this transition is,

The path metric for state 01 is chosen based which is minimum out of the two. .

The survivor path for state 01 is stored in survivor path metric.

.

(e) State 00 with output 11. The branch metric for this transition is,

(f) State 01 with output 00. The branch metric for this transition is,

The path metric for state 10 is chosen based which is minimum out of the two. .

.

(g) State 10 with output 01. The branch metric for this transition is,

(h) State 11 with output 10. The branch metric for this transition is, .

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.

Traceback Unit

Once the survivor path is computed times, the decoding algorithm can start trying to estimate the input sequence. Thanks to presence of tail bits (additional zeros) , it is known that the final state following Convolutional code is State 00.

So, start from the last computed survivor path at index for State 00. From the survivor path, find the previous state corresponding to the current state. From the knowledge of current state and previous state, the input sequence can be determined (Ref: Table 2 Input given current state and previous state). Continue tracking back through the survivor path and estimate the input sequence till index = 1.

ip, if previous state current state 00 01 10 11 00 0 0 x x 01 x x 0 0 10 1 1 x x 11 x x 1 1

Table 2: Input given current state and previous state

Simulation Model

Octave/Matlab source code for computing the bit error rate for BPSK modulation in AWGN using the

convolutional coding and Viterbi decoding is provided. Refer to the post on Bit Error Rate (BER) for BPSK modulation for signal and channel model.

Note: Since 2 coded bits are required for transmission of each data bit, the relation between coded bits to noise ratio with bits to noise ratio is

.

The simulation model performs the following:

(a) Generation of random BPSK modulated symbols +1’s and -1’s

(b) Convolutionally encode them using rate -1/2, generator polynomial [7,5] octal code (c) Passing them through Additive White Gaussian Noise channel

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(d) Hard decision demodulation on the received coded symbols (e) Received coded bits are passed to Viterbi decoder

(f) Counting the number of errors from the output of Viterbi decoder (g) Repeating the same for multiple Eb/No value.

Click here to download Matlab/Octave script for computing BER with Binary Convolutional code and Viterbi decodig for BPSK modulation in AWGN channel

Figure 3: BER plot for BPSK modulation in AWGN channel with Binary Convolutional code and hard decision Viterbi decoding

Observations

1. For lower regions, the error rate with Viterbi decoding is higher uncoded bit error rate. This is because, Viterbi decoder prefers the error to be randomly distributed for it work effectively. At lower values, there are more chances of multiple received coded bits in errors, and the Viterbi algorithm is unable to recover.

2. There can be further optimizations like – using soft decision decoding, limiting the traceback length etc. Those can be discussed in future posts.

References

Tutorial on Convolutional Coding with Viterbi Decoding – Mr. Chip Fleming Digital Communications by John Proakis

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Please click here to SUBSCRIBE to newsletter and download the FREE e-Book on probability of error in AWGN. Thanks for visiting! Happy learning.

Tagged as: AWGN, BPSK, Viterbi

1. If you recall, the decoding complexity is reduced in Viterbi algorithm by assuming that each state can be reached from only two possible states and only one of them is valid. However, if there are burst errors (or if the probability of error is high) then the above assumption becomes invalid. So, when we traceback we might not be able to decode correctly.

2. For the current simulation, I assumed that we wait till the end of all the coded bits to start

traceback. As you said, in typical systems, we do not do that. We typically start the traceback much before. I will write a post on traceback length.

3. The Viterbi algorithm described in the post takes two coded bits and gives one data bit per

iteration. However, we can write the transition table to take four coded bits and give out 2 data bits per iteration. This architecture allows one with a clock frequency limited system to have a higher throughput at the expense of extra hardware.

Hope this helps. Reply

8 Lealem January 21, 2009 at 5:20 pm Hello! kirishna

what can i do to write a simulation of convolutionaly coded QAM, QPSK,16-QAM and 64-QAM signals and also Trellis coded MOdulation?if u have a matlab code please try to provide me.

Thanks Reply

@Lealem: Trellis coded modulation – not yet. Hope to write soon. Reply

10 Ricky January 27, 2009 at 7:09 pm

can i get the vhdl code for k=7 and the inputs being 171 and 133 octal, with the rate=1/2 Reply

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Reply

12 Dhruv April 10, 2009 at 10:41 am

hey man i am looking for the same did you get it??? Reply

13 Lealem February 10, 2009 at 7:15 pm

hello!Kirishina u did it for BPSk modulation but how can i write a matlab code when the modulation is changed to QAM, QPSK ,16-QAM and 64-QAM?

Thank you! Reply

14 scorswiss February 17, 2009 at 12:33 pm Hello Krishna,

I want to develop a verification environment in Verilog for Viterbi decoder. I want a tested C model or Verilog model for the same.From where can I get it?

Reply

15 Krishna Pillai February 21, 2009 at 7:29 am

@scorswiss: In the posts I have provided a Matlab code for soft decision and hard decision Viterbi decoder for the rate 1/2 K=3 convolutional encoder. Maybe you can convert that to C and use. Reply

16 setareh August 16, 2009 at 6:04 pm

hello , I am also looking for matlab code for k=3 and r=1/2 , I am writing VHDL code for it I cant find the code you post it for matlab could you tell me where exactly I can find it

thank you in advance Reply

17 Krishna Sankar August 18, 2009 at 3:48 am

@setareh: The URL for the code is provided just above the BER plot. Reply

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18 scorswiss February 25, 2009 at 3:41 pm Hi Krishna,

Thanks for the reply. I am still facing an issue in the encoder part- When the bits are convolutionally coded, the coded bits together with the uncoded bits are mapped to a space to generate symbols. Till this point we deal wih the digital bitstream,but when this bitstream passes through the channel, AWGN noise(whose value is between 0 and 1 eg: 0.8)is added which is analog in nature. The problem is how to add analog noise to digital stream.Is it through D/A and A/D converters or quantization concept or the AWGN output is converted to digital stream?

Reply

19 Krishna Pillai February 27, 2009 at 6:01 am

@scorswiss: The AWGN noise (which can theoreticall be from -inf to +inf) is added between the DA/ and A/D. If I may

the chain will be as follows:

Bits -> convolutional coded bits -> constellation mapping -> filtering -> D/A –> Add Noise –> A/D –> constellation to bits -> Viterbi decoding

Alternatively we can use Soft decision decoding: http://www.dsplog.com/2009/01/14/soft-viterbi/

Bits -> convolutional coded bits -> constellation mapping -> filtering -> D/A –> Add Noise –> A/D –> soft decision Viterbi decoding

Hope this helps. Reply

20 Allyson March 16, 2009 at 1:16 pm Hi,

I use the Matlab function (bercoding) to simulate the coded-AWGN-BER. and for the decoding and encoding method I use matlab function as well (vitdec and convenc)… My simulated result is near to uncoded-AWGN-BER but not near to coded-AWGN.

Why is it so? Reply

21 Krishna Pillai March 21, 2009 at 8:24 am @Allyson: Quite likely a power normalization issue. Reply

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22 Allyson March 16, 2009 at 1:27 pm Hi,

It’s me again…. I hope to see the explanation on the soft decoding which can improve the situation. Thanks.

Reply

23 Krishna Pillai March 21, 2009 at 8:26 am @Allyson: You may refer to the post @

http://www.dsplog.com/2009/01/14/soft-viterbi/ Reply

24 ahamed March 21, 2009 at 8:43 pm Hi krishna

I am doing viterbi decoder in verilog can you tell me from where i can get a code for reference Reply

25 Krishna Pillai March 25, 2009 at 5:42 am @ahmed: You may check

http://home.netcom.com/%7Echip.f/viterbi/fecbiblio.html Reply

26 chintu March 30, 2009 at 2:38 am Hi

I want to know for R=1/2 K=7 convolutional code which generator matrix is best. Please tell me how can I get the answer.

Also I want to simulate convolutional coding with FSK. Please tell me how to do it.

Reply

27 Krishna Pillai April 4, 2009 at 8:54 am @chintu: My replies:

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generator polynomial is {133,171}_octal. I think, reasearchers long back have tried simulating with different generator polynomails and the table listed in the book lists the best for each constraint length and rate combination.

(2) I have written a post on BER with FSK

http://www.dsplog.com/2007/08/30/bit-error-rate-for-frequency-shift-keying-with-coherent-demodulation/

Hope it should be reasonably easy for you to add convolutional coding + viterbi decoding to that script.

Reply

28 chintu April 7, 2009 at 3:16 pm

The generator matrix for the optimal code for above query if given a different value in Lin and costello ” error control coding”

which is (117, 155) octal.Please clarify.

Also how do I implement this R=1/2 K=7 convolution code in your convolution coding(r=1/2,k=3) script as it is designed for K=3 and can’t be generalized for K=7. Please post a generalized script.

Reply

29 Krishna Pillai April 11, 2009 at 6:20 am

@chintu: My objective was to provide an example code for Viterbi decoder to help understand the concept. Am reasonably sure that once you understand the concept, the reader can easily extended to other cases. Good luck in your project work.

Reply

30 Zeeshan Sattar June 3, 2009 at 4:16 pm Superb Explanation. Thumbs Up bro

Reply

31 Krishna Pillai June 7, 2009 at 2:05 pm @Zeeshan Sattar: Thanks pal.

Reply

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What changes should I make for k=5, r=1/3 decoder? Reply

33 Krishna Pillai July 6, 2009 at 5:51 pm

@einstein: For k=1/5, r=1/3 the state transitions change. The branch metric and path metric computations needs to be changed accordingly.

Reply

34 arun August 9, 2009 at 8:18 pm Hi krishna,

Can you give the list of generator polynomials for various constraints lengths. Ex: k=7 the Generator poly is 133 ,171

Reply

@arun: You may check the Chapter 8.2.5 in Digital Communications by John Proakis . That chapter has a list of generator polynomials for various code rates from 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8 … for constraint length from K=3 to K=14.

Reply

36 arun August 9, 2009 at 8:20 pm Hi,

Can you give me the best generator polynomials for various constraint lengths. Reply

@arun: You may check the Chapter 8.2.5 in Digital Communications by John Proakis. That chapter has a list of generator polynomials for various code rates from 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8 … for constraint length from K=3 to K=14.

Reply

38 fateme September 2, 2009 at 6:25 pm Hi kirishna

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What can i do to write a simulation of Space Time Trellis code ? if u have a matlab code please try to provide me.

Best Regards Fateme Reply

39 Krishna Sankar September 8, 2009 at 5:21 am

@fateme: Sorry, I have not tried modeling space time trellis code. Reply

40 amudhan October 30, 2009 at 11:34 am

can u send the vhdl and verilog coding for low-latency low-complexity architectres for viterbi decoder Reply

41 Krishna Sankar November 8, 2009 at 8:06 am @amudhan: sorry, i do not have vhdl/verilog code. Reply

42 Neo Cambel November 25, 2009 at 11:37 pm Excellent tutorial. Thanks a lot.

BR, Neo Reply

43 Hyunjin.Jang November 26, 2009 at 10:36 pm It’s difficult to viterbi…

Help me, mt graduation of thesis.

Image(photo) processing/ Simulation to C++ Reply

44 Krishna Sankar December 7, 2009 at 4:42 am @Hyunjin.Jang: Hope this post helps. Good luck.

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Reply

45 Harish December 4, 2009 at 3:05 am Hi,

In your code I see that the survivor path metric index you have calculated as minimum of the 2 paths. Some thing like this

case I : says idx

case II; you say it is idx+2.

I don’t understand this. What would it be if I am using 8PSK with 8 states ? I am not able to arrive at the idx thing for storing the path index. Could you please help me with that ?

Thanks, Harish Reply

46 Krishna Sankar December 7, 2009 at 5:11 am

@Harish: Well, I have assumed BPSK in this simulation. Even if you are using BPSK, I believe you might be passing on the received hard decision bits to the Viterbi decoder. So the underlying path metrics computation should still hold good. Agree?

Reply

47 sujata.kamlapur April 10, 2010 at 9:26 am hi sir

i am a mtech student.. and i’m working on d project hand writing recognition using hidden markov model. here in this project 1st we extract the hand writing features and then should build the HMM model, after this we use viterbi algorithm for recognition and baum-welch alogrithm for training. so please can u help me in viterbi algorithm part…….?

Reply

@sujata.kamlapur: Does the Viterbi decoder code provided in this post help? Reply

49 Keshav Kishore Jha April 23, 2010 at 11:34 pm Hi Krishna,

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I need your help in implementing this code when channel is AWGN+Rayleigh fading. Kindly help me to tell what modifications are required in this program for soft decoding. ….

Keshav Reply

@Keshav Kishore Jha: Need to have a multiplicative noise to model the Rayleigh channel. You may refer to examples @ http://www.dsplog.com/tag/rayleigh/

Reply

51 guzeltayyar October 8, 2010 at 10:53 pm Hi Krishna,

Appreciate your efforts in the first place.

I have a question for you.. I would like to replace the Euclidean distance with another metric for non-gaussian noise cases.. for instance, I would like to use alpha-penalty function or known state maximum likelihood distance metric to enhance my decoder performance against impulsice noise..

how would you go about it?? does it change the way how you compute the soft inputs?? or changing only the distance metric is enough??

have you ever tried another metric for non-gaussian noise for your decoder?? thanx in advance,

TayyaR. Reply

52 Krishna Sankar November 19, 2010 at 5:57 am

@guzeltayyar: I guess, one way is to give less weights to softbits affected by impulse noise. This can be done by scaling down those softbits. I have not tried any metric for non-gaussian noise.

Reply

53 Sulav Paudel November 16, 2010 at 7:52 pm Hi Krishna…………..

i am trying to plot the graph of BER vs SNR for bayesian demodulation and QPSK demodulation. I have to analysis the performance for these two techniques. Could you please help me in this matter.

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And, can you help in the matlab code for theoritical and pratical BER for QPSK. With Regards,

sulav paudel Reply

54 Krishna Sankar November 17, 2010 at 4:54 am @Sulav: Hopefully the following post might be of help a) BER of BPSK in AWGN

http://www.dsplog.com/2007/08/05/bit-error-probability-for-bpsk-modulation/ b) Symbol Error rate of QPSK in AWGN

http://www.dsplog.com/2007/11/06/symbol-error-rate-for-4-qam/ Reply

55 Rodrigo Mendoza May 25, 2011 at 9:00 am amm……..

I don`t know why but the code I download from this page doesnt work. nErrViterbi(yy) = size(find([ip- ipHat_v(1:N)]),2);

the dimensions of ipHat_v are smaller than N

Can you tell me how can I resolve this problem I just cant Reply

56 Krishna Sankar May 26, 2011 at 5:33 am

@Rodrigo: Are you sure? I re-ran the code. It’s working. Reply 57 Ajeesh December 31, 2009 at 2:27 pm Sir, Reply 58 Ajeesh December 31, 2009 at 2:29 pm Sir,

We want to compute with binary convolutional code of rate=1/2, generator polynomial- [171, 133] Octal…..

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