Convolutional code

Coding is a technique where redundancy is added to original bit sequence to increase the reliability of the communication. In this article, lets discuss a simple binary convolutional coding scheme at the transmitter and the associated Viterbi (maximum likelihood) decoding scheme at the receiver.

Update: For some reason, the blog is unable to display the article which discuss both Convolutional coding and Viterbi decoding. As a work around, the article was broken upto into two posts. This post descrbes a simple Binary Convolutional Coding scheme. For details on the Viterbi decoding algorithm, please refer to the post – Viterbi decoder.

Chapter 8, Table 8.2-1 of Digital Communications by John Proakis lists the various rate 1/2 convolutional coding schemes. The simplest among them has constraint length with generator polynomial . There are three parameters which define the convolotional code:

(a) Rate : Ratio of the number of input bits to the number of output bits. In this example, rate is 1/2 which means there are two output bits for each input bit.

(b) Constraint length : The number of delay elements in the convolutional coding. In this example, with there are two delay elements.

(c) Generator polynomial : Wiring of the input sequence with the delay elements to form the output. In this example, generator polynomial is . The output from the arm uses the XOR of the current input, previous input and the previous to previous input. The output from the uses the XOR of the current input and the previous to previous input.

Rate 1/2 Convolutional code

Figure 1: Convolutional code with Rate 1/2, K=3, Generator Polynomial [7,5] octal

From the Figure 1, it can be seen that the operation on each arm is like a FIR filtering (aka convolution) with modulo-2 sum at the end (instead of a normal sum). Hence the name Convolutional code. 🙂

State transition

For understanding the Viterbi way of decoding the convolutional coded sequence, lets understand the relation between the input and output bits and the state transition.

if ip = 0 if ip = 1
current state next state (op) next state (op)
00 00 (00) 10 (11)
01 00 (11) 10 (00)
10 01 (10) 11 (01)
11 01 (01) 11 (10)

Table 1: State transition table and the output values

State transition for K=3, rate = 1/2 convolutional code
State transition for K=3, rate = 1/2 convolutional code
Figure 2: State transition for K=3, rate = 1/2 convolutional code

For details on the Viterbi decoding algorithm, please refer to the post – Viterbi decoder.


Tutorial on Convolutional Coding with Viterbi Decoding – Mr. Chip Fleming

Digital Communications by John Proakis

40 thoughts on “Convolutional code

  1. salut, please Krishna I have to do a script to simulate QAM16 with convolutional code and viterbi decoder !

    Please help me ! it is very important !

  2. sir pls give me suggestion regarding this how to implement in matlab:

    MATLAB: Simulate the performance of a Viterbi decoder for the en-
    1 + D2 1 + D + D2
    over an additive Gaussian noise channel
    for BPSK modulated symbols. In your simulations assume the initial
    state is the all-zero state, and run blocks of 1000 information bits per
    block, terminating each block with two 0’s to drive the encoder into
    the all-zero state. Plot the BER vs SNR.

  3. Hi

    An ISI channel can be viewed as a convolutional encoder. when MMSE LE is used at the receiver the cascade of Channel and Equalizer is still not ISI free .
    How can we apply viterbi decoder after the equalizer in this case.


  4. sir im doing project on error detection and correction using convolution code in fpga,,, can u pls help me … thanking u 🙂

  5. Hi Krishna,

    I’m totally messed up with Convolutional code.

    I want to implement convo. encoder using K=3 with (7 , 5) generator. What is the rate of my problem. I have no idea what is my n an k.

    I have refered many books and ended up being confused among symbols.

    Kindly make me clear with K,k,n,M symbols.

  6. hi,
    I m doing project on implementation of turbo encoder using VHDL . Can i get information about turbo encoder?

  7. I found in various places a table containing generator polynomials for convolutional encoding, specifically Table 1-“Generator Polynomials found by Busgang for good rate ½ codes” posted at
    That table contains on the line Constraint length = 7 two identical polynomials, G1 = 110101 and G2 = 110101.
    I guess this is not possible. The two polynomials must be different.
    Does anybody knows the correct polynomials?

  8. Hi Krishna
    I’ve seen the program about the convolutional code. What if the rate is replaced 2 / 3 with constraint length 3, kira2x parameters which must be changed? listings that you make there is the theory of maximum likelihood or not? If there is where do? Why should use the AWGN in the form of the complex?
    Thank you for your attention.
    I look forward to your assistance.

  9. Hi Dr.Krishna
    I am trying to implement you paper “A novel Arq technique using the turbo coding principle”

    I got the turbo decoder working , but in your paper you mention: when a retransmission takes place the LLR’s are used as a priori information.
    before the LLR’s of the previous iteration are used in the current iteration what changes are to be done to the previous LLR;s.

    I am doing this:
    deinterleave to get original sequence and then interleave according to the interleaver pattern of second turbo enocder .

    Is there anything wrong in what i am doing to the LLR’s.

    thank you,

    1. @arun showry: Firstly, am not a Dr. 🙂
      Secondly, the paper which you mention is written by Krishna R Narayan and not me
      Thirdly, I have not studied Turbo code in much detail. Hence unable to answer

  10. Hello All,
    I am working on recursive convolution codes and viterbi decoding.

    I am able to implement viterbi decoding with convolution encoder.
    what changes do i need to make in the decoding section when i use recursive encoder.

    I am doing this for viterbi decoding with a convolution encoder.

    systematic part: Xs
    Parity: Xp
    tx_sig=[Xs Xp]

    viterbi decoing

  11. how to solve this
    A convolutional code is described by the generator polynomials g1, g2, g3 givenas
    g1 = 1
    g2 = 1+z inverse-2
    g3 = 1+z inverse-1+z inverse-2

    a)What is the code rate of this code?
    b)Draw the encoder corresponding to this code
    c)Draw the state diagram corresponding to this code
    d)Find the minimum distance dmin of this code

  12. I am doing a project on fpga implementataion of convolutional encoder and viterbi decoder.i am new to vhdl.can you give the the code for encoder.

  13. Hey Krishna, its me again. You have given simple and concise description of the convolutional code, that easy to understand.

    However, can you talk about how to “BUILD” our code so that it can correct a specified number of errors.

      1. sir i am doing project on ofdmpapr reduction
        can u give me please mail me some codes for papr reduction

        1. @manisha: Some relevant links which I picked up from a quick Googling.
          (a) Tone reservation : where some of the unused subcarriers in an OFDM system is modulated to reduce the PAPR.
          (b) MERL has a paper on PAPR reduction by oversampling, soft clipping and filtering.
          PAPR Reduction for WiMAX OFDM Systems

          (c) Nice comp.dsp thread on this topic
          Has The peak-to-average ratio problem of OFDM been solved?

  14. Pingback: Viterbi

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