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EUSIPCO 2000, The European Signal Processing Conference,
Tampere, Finland, September 58, 2000.
(preprint)
Jeffrey O. Coleman^{1}
jeffc@alum.mit.edu
Naval Research Laboratory
Radar Division
Washington, DC
Data signal has a simply expressed power spectrum when uncorrelated data are used to choose waveforms from a finite waveform alphabet. This result is well known for the common case . Presented in the preliminaries below is the extension from a finite waveform alphabet to an arbitrary finitedimensional waveform alphabet. For either form, the signal is constructed by shifting consecutive waveform symbols to place them sequentially in time. Figure 1 shows such a symbol with two timeaxis ``attachment points.'' In effect, the construction places the first attachment point of a given symbol on the second attachment point of the previous symbol, continues the process symbol by symbol, and then takes the sum.
Now suppose an arbitrary complex waveform is cut into segments of duration as in fig. 2(left). Each segment can be shifted to a standard time, rotated in the complex plane until the endpoint angles are symmetric about the real axis, and amplitude scaled until the end points have reciprocal magnitudes. Each transformed segment trajectory takes the form shown in fig. 2(right). If this decomposition is reversed with segments restricted to a finite waveform alphabet, a generalization of the fig. 1 construction results. The attachment points remain temporally separated by but now also have reciprocally related complex amplitudes. To place one attachment point over another, the second waveform must be both timeshifted and scaled in complex amplitude.

The next section develops the power spectrum of such a signal when waveform symbols (segments in the above) are chosen in i.i.d. fashion from a finite waveform alphabet. The derivation does not actually require waveform segments to be of finite length nor to go through their attachment points. So the commonly known special case of fullresponse continuous phase modulation (CPM), whose defining characteristics are listed in Table 1, is actually rather restrictive. Indeed, of these characteristics, only finite support is required by the typical implementation structure used to realize a coherent CPM system. This suggests one might optimize signal characteristics within the larger class.
CPM's constant envelope is important in radar and communication systems requiring maximum efficiency of the transmitter power amplifier. A preliminary, CPMspecific version of the present derivation was presented in [1]. That one and this one both are more elegant and less restrictive than the mostsimilar previous result in the literature [2], which was limited to integer .
Begin with a mixed continuous and discretetime matrixvector
convolution that generalizes on PAM signaling. Notations
and are equivalent.
In 1974 Prabhu and Rowe [3] used a mixedconvolution
version to derive the spectra of communication signals, but no
connection was mentioned to the familiar fact that it generalized: An
LTI filter operating on a random process scales its power spectrum by
the squared magnitude of the transfer function. Indeed, Fourier
transformation yields
The change of variable from normalized to unnormalized frequency in
gives this mixedconvolution version a
slightly different look from the others. The scalar version applies
to the introduction's
form. The extension to
with drawn from a finitedimensional space
is covered by the vector version of the proposition with
, with row vector
containing the waveform basis and the products with coefficient
vectors
forming the symbol waveforms.
The key is the attachmentpoint placement recursion. Suppose the
scaling of segment has mapped point ``1'' in fig. 3 to
. The second segment attachment point is then at
, and the segment ``1''
must map to
,
yielding
Using
then, along with the
automatic
symmetry,
Taking the Fourier transform of
with care (to get a key
special case right), let
, so that
and
. When
, for all , so
. If
however,
for some frequency
for which
, and any additional component
must be discovered some
other way:
This paper's complex data signal comprised complex waveform symbols chosen independently with arbitrary probabilities from a finitedimensional function space and sequentially attached through time shifting and complex scaling of Markov character. One example is fullresponse CPM, elsewhere represented with Markov encoding of the signalinginterval endpoints. Encoding the geometric mean of the endpoints instead is fundamentally responsible for this simpler result, the first in a clean vector/matrix form. The final spectral expression is a simple function of two vectors, one the basis for the waveformsymbol space and one the basis for corresponding Markov changes in complex amplitude.
In this appendix, and , subscripted or not,
are continuoustime and discretetime random processes respectively,
the latter with implicitly referring to time . These mixed
continuous and discretetime results parallel and sometimes depend on
familiar results in both continuous time and discrete time. As usual,
doublesubscripted crosscorrelations become singlesubscripted
autocorrelations when the two processes are one. The simple proofs of
the first two propositions are omitted. Fourierpair notations used:
Those definitions and results were completely parallel to the familiar ones, but these next few involve a minor, natural extension to average out the cyclostationarity.
It does not matter what interval is chosen, because
decomposition provides a onetoone map
from onto with almost everywhere.
Consequently, the right side of definition