Trellis-based algorithms for GNU Radio

....

The basic goal of the implementation is to have a generic way of describing an FSM that is decoupled from whether it describes a convolutional code (CC), a trellis code (TC), an inter-symbol interference (ISI) channel, or any other communication system that can be modeled with an FSM. To achieve this goal, we need to separate the pure FSM descrition from the rest of the model details. For instance, in the case of a rate 2/3 TC, the FSM should not involve details about the modulation used (it can be an 8-ary PAM, or 8-PSK, etc). Similarly, when attempting maximum likelihood sequence detection (MLSD)--using for instance the Viterbi algorithm (VA)-- the VA implementation should not be concerned with the channel details (such as modulations, channel type, hard or soft inputs, etc). Clearly, having generality as the primary goal implies some penalty on the code efficiency, as compared to fully custom implementations.

We will now describe the implementation of the basic ingedient, the FSM.

An FSM describes the evolution of a system with inputs x_k, states s_k and outputs y_k. At time k the FSM state is s_k. Upon reception of a new input symbol x_k, it outputs an output symbol y_k which is a function of both x_k and s_k. It will then move to a next state s_k+1. An FSM has a finite number of states, input and output symbols. All these are formally described as follows:

Thus, a complete description of the FSM is given by the the five-tuple (I,S,O,NS,OS). Observe that implementation details are hidden in how the outside world interprets these input and output integer symbols. Here is an example of an FSM describing the (2,1) CC with constraint length 3 and generator polynomial matrix (1+D+D² , 1+D²) from Proakis-Salehi pg. 779.

sk	xk	sk+1
0	0	0
0	1	2
1	0	0
1	1	2
2	0	1
2	1	3
3	0	1
3	1	3

sk	xk	yk
0	0	0
0	1	3
1	0	3
1	1	0
2	0	1
2	1	2
3	0	2
3	1	1

The C++ implementation of the FSM class keeps private information about I,S,O,NS,OS and public methods to read and write them. The NS and OS matrices are implemented as STL 1-dimensional vectors.

class fsm {
private:
  int d_I;
  int d_S;
  int d_O;
  std::vector<int> d_NS;
  std::vector<int> d_OS;
  std::vector<int> d_PS;
  std::vector<int> d_PI;
public:
  fsm();
  fsm(const fsm &FSM);
  fsm(const int I, const int S, const int O, const std::vector<int> &NS, const std::vector<int> &OS);
  fsm(const char *name);
  fsm(const int mod_size, const int ch_length);
  int I () const { return d_I; }
  int S () const { return d_S; }
  int O () const { return d_O; }
  const std::vector<int> & NS () const { return d_NS; }
  const std::vector<int> & OS () const { return d_OS; }
  const std::vector<int> & PS () const { return d_PS; }
  const std::vector<int> & PI () const { return d_PI; }
};

As can be seen, other than the trivial and the copy constructor, there are three additional ways to construct an FSM.

Finally, as can be seen from the above description, there are two more variables included in the FSM class implementation, the PS and the PI matrices. These are computed internally when an FSM is instantiated and their meaning is as follows. Sometimes (eg, in the traceback operation of the VA) we need to trace the history of the state or the input sequence. To do this we would like to know for a given state s_k, what are the possible previous states s_k-1 and what input symbols x_k-1 will get us from s_k-1 to s_k. This information can be derived from NS; however we want to have it ready in a convenient format. In the following we assume that for any state, the number of incoming transitions is the same as the number of outgoing transitions, ie, equal to I. All applications of interest have FSMs satisfying this requirement. If we arbitrarily index the incoming transitions to the current state by "i", then as i goes from 0 to I-1, PS(s_k,i) gives all previous states s_k-1, and PI(s_k,i) gives all previous inputs x_k-1. In other words, for any given s_k and any index i=0,1,...I-1, starting from s_k-1=PS(s_k,i) with input x_k-1=PI(s_k,i) will get us to the state s_k. More formally, for any i=0,1,...I-1 we have s_k = NS(PS(s_k,i),PI(s_k,i)).

We now discuss through a concrete example how the above FSM model can be used in GNU Radio. The communication system that we want to simulate consists of a source generating the input information in packets, a CC encoding each packet separately, a memoryless modulator, an additive white Gaussian noise (AWGN) channel, and the VA performing MLSD. The program source is as follows.

  1  #!/usr/bin/env python
  2  
  3  from gnuradio import gr
  4  from gnuradio import audio
  5  from gnuradio import trellis
  6  from gnuradio import eng_notation
  7  import math
  8  import sys
  9  import random
 10  import fsm_utils
 11  
 12  def run_test (f,Kb,bitspersymbol,K,dimensionality,constellation,N0,seed):
 13      fg = gr.flow_graph ()
 14  
 15      # TX
 16      src = gr.lfsr_32k_source_s()
 17      src_head = gr.head (gr.sizeof_short,Kb/16) # packet size in shorts
 18      s2fsmi = gr.packed_to_unpacked_ss(bitspersymbol,gr.GR_MSB_FIRST) # unpack shorts to symbols compatible with the FSM input cardinality
 19      enc = trellis.encoder_ss(f,0) # initial state = 0
 20      mod = gr.chunks_to_symbols_sf(constellation,dimensionality)
 21  
 22      # CHANNEL
 23      add = gr.add_ff()
 24      noise = gr.noise_source_f(gr.GR_GAUSSIAN,math.sqrt(N0/2),seed)
 25  
 26      # RX
 27      metrics = trellis.metrics_f(f.O(),dimensionality,constellation,trellis.TRELLIS_EUCLIDEAN) # data preprocessing to generate metrics for Viterbi
 28      va = trellis.viterbi_s(f,K,0,-1) # Put -1 if the Initial/Final states are not set.
 29      fsmi2s = gr.unpacked_to_packed_ss(bitspersymbol,gr.GR_MSB_FIRST) # pack FSM input symbols to shorts
 30      dst = gr.check_lfsr_32k_s(); 
 31  
 32      fg.connect (src,src_head,s2fsmi,enc,mod)
 33      fg.connect (mod,(add,0))
 34      fg.connect (noise,(add,1))
 35      fg.connect (add,metrics)
 36      fg.connect (metrics,va,fsmi2s,dst)
 37      
 38      fg.run()
 39      
 40      # A bit of cheating: run the program once and print the 
 41      # final encoder state.
 42      # Then put it as the last argument in the viterbi block
 43      #print "final state = " , enc.ST()
 44  
 45      ntotal = dst.ntotal ()
 46      nright = dst.nright ()
 47      runlength = dst.runlength ()
 48      return (ntotal,ntotal-nright)
 49  
 50  
 51  def main(args):
 52      nargs = len (args)
 53      if nargs == 3:
 54          fname=args[0]
 55          esn0_db=float(args[1]) # Es/No in dB
 56          rep=int(args[2]) # number of times the experiment is run to collect enough errors
 57      else:
 58          sys.stderr.write ('usage: test_tcm.py fsm_fname Es/No_db  repetitions\n')
 59          sys.exit (1)
 60  
 61      # system parameters
 62      f=trellis.fsm(fname) # get the FSM specification from a file (will hopefully be automated in the future...)
 63      Kb=1024*16  # packet size in bits (make it multiple of 16 so it can be packed in a short)
 64      bitspersymbol = int(round(math.log(f.I())/math.log(2))) # bits per FSM input symbol
 65      K=Kb/bitspersymbol # packet size in trellis steps
 66      modulation = fsm_utils.psk4 # see fsm_utlis.py for available predefined modulations
 67      dimensionality = modulation[0]
 68      constellation = modulation[1] 
 69      if len(constellation)/dimensionality != f.O():
 70          sys.stderr.write ('Incompatible FSM output cardinality and modulation size.\n')
 71          sys.exit (1)
 72      # calculate average symbol energy
 73      Es = 0
 74      for i in range(len(constellation)):
 75          Es = Es + constellation[i]**2
 76      Es = Es / (len(constellation)/dimensionality)
 77      N0=Es/pow(10.0,esn0_db/10.0); # noise variance
 78      
 79      tot_s=0
 80      terr_s=0
 81      for i in range(rep):
 82          (s,e)=run_test(f,Kb,bitspersymbol,K,dimensionality,constellation,N0,-long(666+i)) # run experiment with different seed to get different noise realizations
 83          tot_s=tot_s+s
 84          terr_s=terr_s+e
 85          if (i%100==0):
 86              print i,s,e,tot_s,terr_s, '%e' % ((1.0*terr_s)/tot_s)
 87      # estimate of the (short) error rate
 88      print tot_s,terr_s, '%e' % ((1.0*terr_s)/tot_s)
 89  
 90  
 91  if __name__ == '__main__':
 92      main (sys.argv[1:])

The program is called by

where "fsm_fname" is the file containing the FSM specification of the tested TCM code, "Es/No_db" is the SNR in dB, and "repetitions" are the number of packets to be transmitted and received in order to collect sufficient number of errors for an accurate estimate of the error rate.

The FSM is first intantiated in "main" by

 62      f=trellis.fsm(fname) # get the FSM specification from a file (will hopefully be automated in the future...)

Each packet has size Kb bits (we choose Kb to be a multiple of 16 so that all bits fit nicely into shorts and can be generated by the lfsr GNU Radio). Assuming that the FSM input has cardinality I, each input symbol consists of bitspersymbol=log₂( I ). The Kb/16 shorts are now unpacked to K=Kb/bitspersymbol input symbols that will drive the FSM encoder.

 63      Kb=1024*16  # packet size in bits (make it multiple of 16 so it can be packed in a short)
 64      bitspersymbol = int(round(math.log(f.I())/math.log(2))) # bits per FSM input symbol
 65      K=Kb/bitspersymbol # packet size in trellis steps

The FSM will produce K output symbols (remeber the FSM produces always one output symbol for each input symbol). Each of these symbols needs to be modulated. Since we are simulating the communication system, we need not simulate the actual waveforms. An M-ary, N-dimensional modulation is completely specified by a set of M, N-dimensional real vectors. In "fsm_utils.py" file we give a number of useful modulations with the following format: modulation = (N,constellation), where constellation=[c11,c12,...,c1N,c21,c22,...,c2N,...,cM1,cM2,...cMN]. The meaning of the above is that every constellation point c_i is an N-dimnsional vector c_i=(ci1,ci2,...,ciN) For instance, 4-ary PAM is represented as (1,[-3, -1, 1, 3]), while QPSK is represented as (2,[1, 0, 0, 1, 0, -1, -1, 0]). In our example we choose QPSK modulation. Clearly, M should be equal to the cardinality of the FSM output, O. Finally the average symbol energy and noise variance are calculated.

 66      modulation = fsm_utils.psk4 # see fsm_utlis.py for available predefined modulations
 67      dimensionality = modulation[0]
 68      constellation = modulation[1]
 69      if len(constellation)/dimensionality != f.O():
 70          sys.stderr.write ('Incompatible FSM output cardinality and modulation size.\n')
 71          sys.exit (1)
 72      # calculate average symbol energy
 73      Es = 0
 74      for i in range(len(constellation)):
 75          Es = Es + constellation[i]**2
 76      Es = Es / (len(constellation)/dimensionality)
 77      N0=Es/pow(10.0,esn0_db/10.0); # noise variance

Then, "run_test" is called with a different "seed" so that we get different noise realizations.

 82          (s,e)=run_test(f,Kb,bitspersymbol,K,dimensionality,constellation,N0,-long(666+i)) # run experiment with different seed to get different noise realizations

Let us examine now the "run_test" function. First we set up the transmitter blocks. The Kb/16 shorts are first unpacked to symbols consistent with the FSM input alphabet. The FSm encoder requires the FSM specification, and an initial state (which is set to 0 in this example).

 15      # TX
 16      src = gr.lfsr_32k_source_s()
 17      src_head = gr.head (gr.sizeof_short,Kb/16) # packet size in shorts
 18      s2fsmi = gr.packed_to_unpacked_ss(bitspersymbol,gr.GR_MSB_FIRST) # unpack shorts to symbols compatible with the FSM input cardinality
 19      enc = trellis.encoder_ss(f,0) # initial state = 0

We now need to modulate the FSM output symbols. The "chunks_to_symbols_sf" is essentially a memoryless mapper which for each input symbol y_k outputs a sequence of N numbers ci1,ci2,...,ciN representing the coordianates of the constellation symbol c_i with i=y_k.

 20      mod = gr.chunks_to_symbols_sf(constellation,dimensionality)

The channel is AWGN with appropriate noise variance. For each transmitted symbol c_k=(ck1,ck2,...,ckN) we receive a noisy version r_k=(rk1,rk2,...,rkN).

 22      # CHANNEL
 23      add = gr.add_ff()
 24      noise = gr.noise_source_f(gr.GR_GAUSSIAN,math.sqrt(N0/2),seed)

Part of the design methodology was to decouple the FSM and VA from the details of the modulation, channel, receiver front-end etc. In order for the VA to run, we only need to provide it with a number representing a cost associated with each transition in the trellis. Then the VA will find the sequence with the smallest total cost through the trellis. The cost associated with a transition (s_k,x_k) is only a function of the output y_k = OS(s_k,x_k), and the observation vector r_k. Thus, for each time period, k, we need to label each of the SxI transitions with such a cost. This means that for each time period we need to evaluate O such numbers (one for each possible output symbol y_k). This is done in "metrics_f". In particular, metrics_f is a memoryless device taking N inputs at a time and producing O outputs. The N inputs are rk1,rk2,...,rkN. The O outputs are the costs associated with observations rk1,rk2,...,rkN and hypothesized output symbols c_1,c_2,...,c_M. For instance, if we choose to perform soft-input VA, we need to evaluate the Euclidean distance between r_k and each of c_1,c_2,...,c_M, for each of the K transmitted symbols. Other options are available as well; for instance, we can do hard decision demodulation and feed the VA with symbol Hamming distances, or even bit Hamming distances, etc. These are all implemented in "metrics_f".

 26      # RX
 27      metrics = trellis.metrics_f(f.O(),dimensionality,constellation,trellis.TRELLIS_EUCLIDEAN) # data preprocessing to generate metrics for Viterbi

Now the VA can run once it is supplied by the initial and final states. The initial state is known to be 0; the final state is usually forced to some value by padding the information sequence appropriately. In this example, we always send the the same info sequence (we only randomize noise) so we can evaluate off line the final state and then provide it to the VA (a value of -1 signifies that there is no fixed initial or final state). The VA outputs the estimates of the symbols x_k which are then packed to shorts and compared with the transmitted sequence.

 28      va = trellis.viterbi_s(f,K,0,-1) # Put -1 if the Initial/Final states are not set.
 29      fsmi2s = gr.unpacked_to_packed_ss(bitspersymbol,gr.GR_MSB_FIRST) # pack FSM input symbols to shorts
 30      dst = gr.check_lfsr_32k_s();

The function returns the number of shorts and the number of shorts in error. Observe that this way the estimated error rate refers to 16-bit-symbol error rate.

 48      return (ntotal,ntotal-nright)