### It is now possible to calculate tight bounds at high SNR.

A method has been developed of computing bounds on the performance of a code comprised of two linear binary codes generated by two encoders serially concatenated through an interleaver. Originally intended for use in evaluating the performances of some codes proposed for deep-space communication links, the method can also be used in evaluating the performances of short-block-length codes in other applications.

The method applies, more specifically, to a communication system in which following processes take place:

- At the transmitter, the original binary information that one seeks to transmit is first processed by an encoder into an outer code (C
_{o}) characterized by, among other things, a pair of numbers (*n*,*k*), where*n*(*n*>*k*)is the total number of code bits associated with k information bits and*n*–*k*bits are used for correcting or at least detecting errors. Next, the outer code is processed through either a block or a convolutional interleaver. In the block inter-leaver, the words of the outer code are processed in blocks of*I*words. In the convolutional interleaver, the inter-leaving operation is performed bit-wise in*N*rows with delays that are multiples of*B*bits. The output of the interleaver is processed through a second encoder to obtain an inner code (C_{i}) characterized by (*n*,_{i}*k*)._{i} - The output of the inner code is transmitted over an additive-white-Gaussian- noise channel characterized by a symbol signal-to-noise ratio (SNR) E
_{s}/N_{o}and a bit SNR E_{b}/N_{o}. - At the receiver, an inner decoder generates estimates of bits. Depending on whether a block or a convolutional interleaver is used at the transmitter, the sequence of estimated bits is processed through a block or a convolutional de-interleaver, respectively, to obtain estimates of code words. Then the estimates of the code words are processed through an outer decoder, which generates estimates of the original information along with flags indicating which estimates are presumed to be correct and which are found to be erroneous.

From the perspective of the present method, the topic of major interest is the performance of the communication system as quantified in the word-error rate and the undetected-error rate as functions of the SNRs and the total latency of the interleaver and inner code. The method is embodied in equations that describe bounds on these functions. Throughout the derivation of the equations that embody the method, it is assumed that the decoder for the outer code corrects any error pattern of *t* or fewer errors, detects any error pattern of *s* or fewer errors, may detect some error patterns of more than *s* errors, and does not correct any patterns of more than *t* errors. Because a mathematically complete description of the equations that embody the method and of the derivation of the equations would greatly exceed the space available for this article, it must suffice to summarize by reporting that the derivation includes consideration of several complex issues, including relationships between latency and memory requirements for block and convolutional codes, burst error statistics, enumeration of error-event intersections, and effects of different interleaving depths.

In a demonstration, the method was used to calculate bounds on the performances of several communication systems, each based on serial concatenation of a (63,56) expurgated Hamming code with a convolutional inner code through a convolutional interleaver. The bounds calculated by use of the method were compared with results of numerical simulations of performances of the systems to show the regions where the bounds are tight (see figure).

*This work was done by Bruce Moision and Samuel Dolinar of Caltech for NASA’s Jet Propulsion Laboratory. For more information, download the Technical Support Package (free white paper) at www.techbriefs.com/tsp under the Information Sciences category. NPO-44652*

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** Performance Bounds on Two Concatenated, Interleaved Codes** (reference NPO-44652) is currently available for download from the TSP library.

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