Communications Toolbox

Error Rates

Comparing messages before and after transmission can help you evaluate the quality of a communication system design or the performance of a special technique or algorithm. If your communication system uses several bits to represent a single symbol, then counting bit errors is different from counting symbol errors. In either the bit- or symbol-counting case, the error rate is the number of errors divided by the total number (of bits or symbols) transmitted.

The biterr function compares two messages and computes the number of bit errors and the bit error rate. The symerr function compares two messages and computes the number of symbol errors and the symbol error rate.

Example: Computing Error Rates

The script below uses the symerr function to compute the symbol error rates for a noisy linear block code. After artificially adding noise to the encoded message, it compares the resulting noisy code to the original code. Then it decodes and compares the decoded message to the original one.

• m = 3; n = 2^m-1; k = n-m; % Prepare to use Hamming code.
  msg = randint(k*200,1,2); % 200 messages of k bits each
  code = encode(msg,n,k,'hamming');
  codenoisy = rem(code+(rand(n*200,1)>.95),2); % Add noise.
  % Decode and correct some errors.
  newmsg = decode(codenoisy,n,k,'hamming');
  % Compute and display symbol error rates.
  [codenum,coderate] = symerr(code,codenoisy);
  [msgnum,msgrate] = symerr(msg,newmsg);
  disp(['Error rate in the received code: ',num2str(coderate)])
  disp(['Error rate after decoding: ',num2str(msgrate)])
  

The output is below. The error rate decreases after decoding because the Hamming decoder corrects some of the errors. Your results might vary because the example uses random numbers.

• Error rate in the received code: 0.054286
  Error rate after decoding: 0.03
  

Comparison of Symbol Error Rate and Bit Error Rate

In the example above, the symbol errors and bit errors are the same because each symbol is a bit. The commands below illustrate the difference between symbol errors and bit errors in other situations.

• a = [1 2 3]'; b = [1 4 4]';
  format rat % Display fractions instead of decimals.
  [snum,srate] = symerr(a,b)
  
  snum =
  
        2      
  
  srate =
  
       2/3 
  
  [bnum,brate] = biterr(a,b)
  
  bnum =
  
       5
  
  brate =
  
       5/9 
  

bnum is 5 because the second entries differ in two bits and the third entries differ in three bits. brate is 5/9 because the total number of bits is nine. The total number of bits is, by definition, the number of entries in a or b times the maximum number of bits among all entries of a and b.

Random Signals

Eye Diagrams