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Journal ArticleDOI

Spectral Characteristics of Convolutionally Coded Digital Signals

01 Feb 1980-IEEE Transactions on Communications (IEEE)-Vol. 28, Iss: 2, pp 173-186

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.
Topics: Convolutional code (59%), Encoder (54%), Code rate (53%), Bandwidth expansion (51%), Spectral density (50%)

Summary (3 min read)

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.

IF-1

  • From (2) and (3) the authors note that cov (Xp) XnP+q ) -cov (Xp -nR ' Xq) (11) Thus, using (11), they can rewrite (10T in the more compact form 21 11 00 SX(f) n !.^ E e2 cov (XpXn2+q cos [2ir(nf +q -p) fT] p=1 -1q-1 i=0 (12) where c, is the Neumann factor defined by 1 Q-0 (13) 2 otherwise =The cross correlation iserformed only on th p y e parts cf the nth and shifted qth rows whose elements overlap.

°° 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.

44)

  • Comparing; these two figures, it is difficult to draw any decisive conclusions regarding their relative spectral width other than to note that the two spectra are : quite different,.
  • The authors conclude this section by noting that reversing the sampler for the .rate 1 /2, constraint length 3 code characterized by G of (37) would not alter the spectrum.
  • The reason for this conclusion is that the two rows of G have even symmetry about their midpoint (i.e.. the elements in column 2).
  • C/1 0.8 1 0.4 0.00 where E denotes modulo 2 summation.
  • Note that (50) defines a "modified generator matrix" H with (0, 1) elements which is uniquely related to the connection matrix G.

{61)

  • Note the similarity in form between ( 20) and ( 61).
  • The prinicpal difference between the two is that (20) corresponds to a cyclostationary process with memory equal to K•1 whereas the memory of the process characterized by ( 61) is infinite.
  • Thus Note that a discrete spectrums can potentially exist at the encoder output despite the fact that the encoder input has only a continuous spectrum.

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.
  • 2n) is even, the authors can once again use fixed alternating symbol inverters on these taps and the general results of ( 68), (69), and ( 70) apply with n replaced by 2n, b replaced by 2b, gp, i replaced by g^p , i, and using (50) h replaced by h= p , i p,i Before presenting a specific example, they can once again make some general statements for the various input source models previously considered.
  • Thus for the class of uncorrelated convolutional codes, alternate symbol inversion has no effect on the encoder output spectrum.

52

  • For correlated convolutional codes alternate symbol inversion will, in general, have a spreading effect on the output spectrum.
  • If, however, the authors again consider the subclass whose generator matrix has at least two identical rows, and if the identical rows are spaced such that p+q is always even, then alternate syia-ibol inversion will not affect the encoder output spectrum.
  • The spectrum f,;preading effect of alternate symbol inversion, consider the rate 1/4 code described by the generator matrix in (33), also known as To demonstrat#'.
  • This figure should be compared to Figure 4 which characterizes the same encoder output without alternate symbol inversion.
  • Similarly, Figure 13b has the discrete spectrum superimposed.

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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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.)
85 p AC A^5
/MF A^1
CSCL 17B
G3/32
N79- 32412
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35794
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A, „ *
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August 1, 1979
National Aeronautics and
Space Administration
Jet Propulsion Laboratory
California Institute of Technology
Pasadena,.
California
l
I^

JPL PUBLICATION 79-•93
.:.
Spectral
Characteristics
of Convolutionally
Cooed Digital Signals
Dariush Divsalar
Marvin K. Simon
r
3
August 1, 1979
µ
National Aeronautics and
Space Administration
Jet Propulsion Laboratory
California Institute of Technology
Pasadena, California

{
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
"
Convolutionally Encoded Suppressed Carrier Input Modulation
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77
^
9
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w
l
ti • n
^,
J

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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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 .
. . ,
.
,
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,
,
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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40
9c.
Power Spectrum of First Order Markov Source; pt = 0.5 ,
,
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41
9d. ,
Power Spectrum of First Order Markov Source; p
t
= 0.7 ,
42

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