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Journal Article•DOI•

Spectral Characteristics of Convolutionally Coded Digital Signals

Dariush Divsalar1, Marvin K. Simon2•
Jet Propulsion Laboratory1, California Institute of Technology2
01 Feb 1980-IEEE Transactions on Communications (IEEE)-Vol. 28, Iss: 2, pp 173-186
TL;DR: In this article, the power spectral density of the output symbol sequence of a convolutional encoder is computed for two different input symbol stream source models, namely, an NRZ signaling format and a first-order Markov source.
Abstract: The power spectral density of the output symbol sequence of a convolutional encoder is computed for two different input symbol stream source models, namely, an NRZ signaling format and a first-order Markov source. In the former, the two signaling states of the binary waveform are not necessarily assumed to occur with equal probability. The effects of alternate symbol inversion on this spectrum are also considered. The mathematical results are illustrated with many examples corresponding to optimal performance codes. It is demonstrated that only for the case of a purely random input source (e.g., NRZ data with equiprobable symbols) and a particular class of codes is the output spectrum identical to the input spectrum except for a frequency scaling (expansion) by the reciprocal of the code rate. In all other cases, the output spectrum is sufficiently changed relative to the input spectrum that the commonly quoted statement "a convolutional encoder produces a bandwidth expansion by a factor equal to the reciprocal of the code rate" must be exercised with care.

Summary (2 min read)

Jump to: [Introduction] – [II.] – [°° sin Irk] – [VI. Encoder Output Spectrum in the Presence of Alternate Symbol Inversion] – [VII. Experimental Results] and [VIII. Observations and Conclusions]

Introduction

  • While much attention has been paid toward constructing convolutional codes that achieve optimum error probability performance, very little attention, if any, has been paid toward examining the spectral properties of the <:°orresponding encoder outputs.
  • The authors first examine the conditions (class of codes) for which the above is not a true statement.
  • In these instances, the bandwidth expansion produced by the encoder can be considerably less than the reciprocal of the code rate.
  • A discussion of the implications of this statement will be given in Section VII titled "Observations and Conclusions.
  • Following the above considerations, the authors determine the spectral characteristics of the convolutional encoder output when the input is not a purely random NRZ source.

II.

  • Convolutional Encoder Model where g. j is either one or zero depending, respectively, on whether the ith modulo summer is connected to the jth shift register stage.
  • For mathematical convenience, the authors shall assume that both the input symbols la mb + j } and the output symbols {Xmn-ip} take on values plus and minus one.

°° sin Irk

  • Thus, in order for (26) to equal zero, each of these summations must itself be zero.
  • Moreover, since the elements in each of the sums are non-negative, then the sum can be zero only if each element in each sum is zero.
  • This defines a class of convolutional codes whose significance will shortly become apparent.
  • This last statement is crucial to the results which now follow.
  • The authors observe that the spectrum becomes more and more concentrated as p, decreases.

VI. Encoder Output Spectrum in the Presence of Alternate Symbol Inversion

  • Alternate symbol inversion (Ref. 5) is a technique in which alternate symbols of the encoder output are inverted to provide the sufficient richness of bol transitions necessary for adequate symbol synchronizer performance.
  • Instead, the authors can look at the alternate symbol inverted output as having been obtained by adding (modulo 2) an external alternating binary sequence of ones and zeros to the encoder (0, 1) output sequence.
  • An equivalent procedure, which is more convenient for computation of the power spectrum is described as follows.
  • Thus for the class of uncorrelated convolutional codes, alternate symbol inversion has no effect on the encoder output spectrum.

VII. Experimental Results

  • To support the analytical results derived in this report, an optimum constraint length 3, rate 1/4 convolutional encoder was implemented, and its output spectrum was observed on a spectrum analyzer in response to a PN sequence at its input.
  • Analytical results contained in this report [see ( 33) and ( 34)'] have shown that this code belongs to the class of correlated convolutional codes and thus one would expect an output spectrum which differed from a frequency scaled version of the input spectrum [see Figure 41 .
  • Indeed this result is confirmed by the experimental results illustrated in Figure 14 (logarithmic scale) and Figure 15 (linear scale).

VIII. Observations and Conclusions

  • The authors have observed that there exists a class of convolutional codes (designated correlated convolutional codes) which have the property that a purely random input results in an encoder output sequence with correlated symbols.
  • When the power spectrum of the waveform produced by this correlated output sequence is computed, it is observed that its effective bandwidth is narrower than that produced by an uncorrelated (purely random) sequence.
  • An equivalent statement is that the power spectrum of the output waveform corresponding to the entire (doubly infinite) sequence is narrower than that for the individual pulse, the latter being identical to the power spectrum of an equivalent pulse stream with uncorrelated symbols.
  • Before leaving this subject, the authors remind the reader that if alternate symbol inversion is employed to improve the encoder-output symbol transition density, then this acts in such a way as to once again expand bandwidth and the above spectral advantage associated with correlated convolutional codes tends to disappear.

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Produced by the NASA Center for Aerospace Information (CASI)
JPL PUBLICATION 79-93
Spectral Characteristics
of
C.onvol utionally
Coded
Digital Signals
Dariush Divsalar
Marvin K. Simon
(NASA
-
CR-162295) SPECTRAL CHARACTERISTICS
OF CONVOLt1TIONALLY CODED DIGITAL SIGNALS
(Jet Propulsion Lab.)
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August 1, 1979
National Aeronautics and
Space Administration
Jet Propulsion Laboratory
California Institute of Technology
Pasadena,.
California
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JPL PUBLICATION 79-•93
.:.
Spectral
Characteristics
of Convolutionally
Cooed Digital Signals
Dariush Divsalar
Marvin K. Simon
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3
August 1, 1979
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National Aeronautics and
Space Administration
Jet Propulsion Laboratory
California Institute of Technology
Pasadena, California
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TABLE OF CONTENTS
I.
Introduction
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II, '
Convolutional Encoder Model
• . • • . • • • . . . . . .. • • . • •
• • • .
3
III.
Spectrum of a Cyclostationary Pulse Stream
• • • • • • • • . . • • • • • •
5
IV.
';Encoder Output Spectrum for Independent Binary Symbol Input
• • • •
8
A
The Case of a Purely Random Data Input (a = 0, p* a. 1/2)
13
I
4Y
B.
The Case of an Unbalanced NRZ Input (a
0,
P*
1/2)
. .. . . .
18
V. !
Encoder Output Spectrum for First Order Markov Input . . . . .. .
33
VI.
.Encoder Output Spectrum in the Presence of Alternate
Sy
mbol Inversion
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VII.
Experimental Results .
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VIII.
Observations and Conclusions
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Ref
erences
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61
Appendix A: The Computation of Power Spectral Density for
Synchronous Data Pulse Streams
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62
Appendix B: Costas Loop Track'ag Performance fora
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Convolutionally Encoded Suppressed Carrier Input Modulation
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77
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LIST OF FIGURES
11
Figures
1.
A General Constraint Length K, Rate b/n Convolutional
Code
,
.........................................
4
2.
An Illustration of the Code Constraints of Equation (27) .
.
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12
3.
Spectrum for Best Rate 1/3; Constraint Length 3
Convolutional Code; Dotted Curve is Spectrum of NRZ
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16
4.
Spectrum for Best Rate 1/4; Constraint Length 3
Convolutional Code; Dotted Curve is Spectrum of NRZ
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17
5.
Spectrum for Best Rate 1/3, Constraint Length 3
Convolutional Code
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6a.
Spectrum for Best Rate 1/2, Constraint Length 3
Convolutional Code;
p* = Oa 1 .
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6b.
Spectrum for Best Rate 1/2, Constraint Length 3
Convolutional Code; p* = 0.3
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6c.
Spectrum for Best Rate 1/2, Constraint Length 3
Convolutional Code; p* = 0.5
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24
7a.
Spectrum for Best Rate 1/2, Constraint Length 7
Convolutional Code; p* = 0.1
. . .
. . . . .
,
,
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,
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,
,
,
26
7b.
Spectrum for Best Rate 1/2, Constraint Length 7
Convolutional Code; p* = 0.2
.. .
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27
7c.
Spectrum for Best Rate 1/2, Constraint Length 7
Convolutional Code; p* = 0.3
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28
8a.
Spectrum for Best Rate 1/2, Constraint Length 7
Convolutional Code; Sampler Reversed; p* = 0.1
. . , ,
,
,
,
,
,
,
30
8b.
Spectrum
for
Best Rate 1/2, Constraint Length 7
Convolutional Code; Sampler Reversed; p* = 0.2 .
. . ,
.
,
.
,
,
31
8c.
Spectrum for Best Rate 1/2, Constraint Length 7
Convolutional Code; Sampler Reversed; p* = 0.3
, , , ,
,
,
,
,
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32
9a.
Power Spectrum of First Order Markov Source; p
t
= 0.1 ,
,
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39
9b.
Power Spectrum of First Order Markov Source; pt = 0, 3 ,
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9c.
Power Spectrum of First Order Markov Source; pt = 0.5 ,
,
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9d. ,
Power Spectrum of First Order Markov Source; p
t
= 0.7 ,
42
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References
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Book•
Telecommunication systems engineering

[...]

William C. Lindsey1, Marvin K. Simon2•
University of Southern California1, California Institute of Technology2
01 Jan 1973
TL;DR: This classic graduate- and research-level texty by two leading experts in the field of telecommunications is essential reading for anyone workign today in space and satellite digital communicatiions and those seeking a wider background in statistical communication theory and its applications.
Abstract: From the Publisher: This classic graduate- and research-level texty by two leading experts in the field of telecommunications is essential reading for anyone workign today in space and satellite digital communicatiions and those seeking a wider background in statistical communication theory and its applications. Ideal for practicing engineers as well as graduate students in communication systems courses, the book clearly presents and develops theory that can be used in the design and planning of telecommunication systems operating with either small or large performance margins. The book includes in its coverage a theory for use in the design of one-way and two-way phase-coherent and communication systems; and analysis and comparison of carrier and suppressed carrier synchronization techniques; treatment of the band-pass limiter theory; unification of phase-coherent detection with perfect and noisy synchronization reference signals. Convolutional codes, symbol synchronization, and noncoherent detection of M-ary signals are among the otehr subjects addressed in this comprehensive study. Dr. Lindsey, who is with the Communication Sciences Institute at the University of Southern California, and Dr. Simon, who is with the Jet Propulsion Laboratory at the California Institute of Technology, include at the end of each chapter a comprehensive set of problems that demonstrate the application of the theory developed. Unabridged Dover (1991) republication of the edition published by Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973.265 line illustrations. 3 photographs. References at chapter ends. Problems. Index. xviii + 574pp. 5 3/8 x 8 1/2. Paperbound.

387 citations

Journal Article•DOI•
Short convolutional codes with maximal free distance for rates 1/2, 1/3, and 1/4 (Corresp.)

[...]

K. Larsen
01 May 1973-IEEE Transactions on Information Theory
TL;DR: In this paper, the authors gave a tabulation of binary convolutional codes with maximum free distance for rates of 1/2, 1/3, and 1/4 for all constraint lengths up to and including nu = 14.
Abstract: This paper gives a tabulation of binary convolutional codes with maximum free distance for rates \frac{1}{2}, \frac{1}{3} , and \frac{1}{4} for all constraint lengths (measured in information digits) u up to and including nu = 14 . These codes should be of practical interest in connection with Viterbi decoders.

188 citations

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California Institute of Technology1, University of Southern California2
01 Feb 1977-IEEE Transactions on Communications
TL;DR: The performance of suppressed carrier receivers with Costas loop tracking is optimized by proper choice of loop arm filter bandwidth, and it is shown that for a variety of passive arm filter types, there exists an optimum filter bandwidth in the sense of minimizing the loop's squaring loss.
Abstract: The performance of suppressed carrier receivers with Costas loop tracking is optimized by proper choice of loop arm filter bandwidth. In particular, it is shown that for a variety of passive arm filter types, there exists, for a given data rate and data signal-signal-to-noise ratio, an optimum filter bandwidth in the sense of minimizing the loop's squaring loss. For the linear theory case, this is equivalent to minimizing the loop's tracking jitter. When symbol synchronization is known, it is shown that by replacing the passive arm filters with active filters, i.e., integrate-and-dump circuits, one can achieve an improvement in carrier-to-noise ratio of as much as 4 to 6 dB depending on the passive arm filter type used for comparison and the value of data signal-to-noise ratio (SNR).

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