The cost of a number of sequential coding search algorithms is analyzed in a systematic manner. These algorithms search code trees, and find use in data compression, error correction, and maximum likelihood sequence estimation. The cost function is made up of the size of and number of accesses to storage. It is found that algorithms that utilize sorting are much more expensive to use than those that do not; metric-first searching regimes are less efficient than breadth-first or depth-first regimes. Cost functions are evaluated using experimental data obtained from data compression and error correction studies.

Sequential Coding Algorithms: A Survey and Cost Analysis

Coherent optical transmission systems have a four-dimensional (4-D) signal space (two quadratures in two polarizations). These four dimensions can be used to create modulation formats that have a better power efficiency (higher sensitivity) than the conventional binary phase shift keying/quadrature phase shift keying (BPSK/QPSK) signals. Several examples are given, with some emphasis on a 24-level format and an 8-level format, including descriptions of how they can be realized and expressions for their symbol and bit error probabilities. These formats are, respectively, an extension and a subset of the commonly used 16-level dual-polarization QPSK format. Sphere packing simulations in 2, 3, and 4 dimensions, up to 32 levels, are used to verify their optimality. The numerical results, as the number of levels increases, are shown to agree with lattice-theoretical results. Finally, we point out that the use of these constellations will lead to improved fundamental sensitivity limits for optical communication systems, and they may also be relevant as a way of reducing power demands and/or nonlinear influence.

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Power-Efficient Modulation Formats in Coherent Transmission Systems

Speech coding

A nonuniform sampling approach to digital encoding of analog sources is proposed. The nonuniform sampler is basically a level crossing detector (LCD) which produces a sample whenever the input to the LCD crosses a threshold level. The information about the source signal is contained in the time intervals between level crossings and in the directions of level crossings. By assigning strings of the 2-tuple "00" to represent the time between level crossings and "01" and "10" to denote the directions of level crossings, the output binary sequence of the nonuniform sampling encoder (NSE) contains a high probability of the 0 symbol, which makes it suitable for further simple run-length encoding (RLE) to attain a "good" overall compression ratio. Introduction of prediction converts the NSE to a nonuniform sampling predictive coding (NSPC) scheme, which, depending on the source, can potentially improve the compression ratio. Results obtained in the encoding of a band-limied Gaussian source and a rasterscanned black and white still image reveal that an NSE/RLE or NSPC/ RLE system exhibits performance superior to that of an adaptive delta modulation system.

A Nonuniform Sampling Approach to Data Compression

We analyze the performance of a one-bit digital matched i t e r (DMF) responding to binary signaling through a noisy multipath channel. The output signal-to-noise ratio (SNR,) plays a key role in the analysis. The S N b performance is fully studied for a two-path channel, both fading and non-fading. For the general N-path channel, results are obtained only for the fading case. All results are conditioned on the knowledge of the time delays of all paths, and are valid only for smaIl input signal-to-noise ratio (SNRi). Most of the results have been verified by computer simulations.

Speech Coding

Data transmission with combined amplitude and phase modulation is considered with signal design carried out in a four-dimensional space of two amplitude and two phase variables. The energy for each signal is fixed. Quaternions are used to represent signal vectors and an algebra for quaternions is introduced which allows distance between signal vectors and equivalence between signal codes to be defined. It is also used to characterize the structure of codes. Code construction is based on regular four-dimensional polytopes and particular interest is focused on group codes with 8, 24, 48 and 120 elements. Two classes of phase modulated signals have interest as reference codes. Code evaluation is carried out by calculation of the distance distribution and the error probability assuming opitmum coherent detection. The group codes are found to be between 1.3 and 3.4 dB better than the best phase modulated codes. By using a suitable representation for a signal code it may be realized by using a few amplitude and several phase values.

Codes for Combined Phase and Amplitude Modulated Signals in a Four-Dimensional Space

A class of adaptive schemes for delta modulation is considered whereby the decision about step size is a function of the present and the last k symbols. Such schemes are studied by computer simulations using speech-like signals through evaluation of SNR. Occasional large overshoots appear in the reconstructed signal after the integrater, and to avoid such events it is proposed to store a few signal samples and decide on the proper bit stream taking into account the adaptive logic. Two quadratic criteria have been tried, and improvements of a few decibels have been established with six stored signal samples and a sampling rate of 48 and 24 kHz. Results indicate that other criteria functions may achieve still higher values of SNR.

Adaptive Delta Modulation with Delayed Decision

This concise paper is concerned with the problem of improved delta-coding by using delayed decision instead of bit-bit-by-bit decision. It is found that delayed encoding allows a fairly general predictor to be used without causing instability problems. Simulations indicate that considerable improvement can be achieved by matching the feedback filter to the input process. Delayed encoding requires a search algorithm for making the decisions. Some proposals of algorithms that are efficient from a computational point of view are presented. Particular interest is attached to a highly truncated version of the Viterbi algorithm, which seems very promising.

Algorithms for Delayed Encoding in Delta Modulation with Speech-Like Signals

The object of this work has been to study encoding of monochrome pictures with a rate of about 1 bit per picture element (pel). Differential pulse code modulation, DPCM, has been chosen to make the system reasonably simple. This motivates the choice of a fixed three-dimensional predictor made separable in time and space. It essentially amounts to encoding the frame difference using a two dimensional predictor. To improve performance, the quantizer is made adaptive and the encoding is combined with delayed decision. A three-level quantizer is being used with an adaptive scheme that works with either forward or backward estimation. The output from the DPCM unit is redundant and we have adopted entropy coding to further reduce the bit rate. The entropy measure is also used to adjust the quantizer parameters to achieve the desired bit rate of 1 bit/pel. Encoding experiments were carried out on a complex videophone scene by means of computer simulations.

Interframe DPCM with Adaptive Quantization and Entropy Coding

With DPCM, it is advantageous to adapt the quantizer characteristic to the signal. We have studied several schemes where a gain constant is controlled by either forward or backward estimation. With forward estimation, the gain is calculated on a two-dimensional block of data, and we have studied how the form and size of the block affects performance. With backward estimation, the Jayant algorithm has been extended from one to two dimensions by letting the change in gain constant depend upon prediction errors from neighboring elements on the present and previous line. Simulations are performed on a sequence of pictures from a monochrome videophone scene with very active movement. The improvement in mean square error with adaptive control is between 2 and 5 dB compared to the performance with a uniform fixed quantizer. The subjective improvement is substantial.

L. Zetterberg

Papers

Codes for Combined Phase and Amplitude Modulated Signals in a Four-Dimensional Space

Adaptive Delta Modulation with Delayed Decision

Algorithms for Delayed Encoding in Delta Modulation with Speech-Like Signals

Interframe DPCM with Adaptive Quantization and Entropy Coding

DPCM Picture Coding with Two-Dimensional Control of Adaptive Quantization