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EUSIPCO 2000, The European Signal Processing Conference,
Tampere, Finland, September 5-8, 2000.
Jeffrey O. Coleman1
Naval Research Laboratory
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 finite-dimensional 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 time-axis ``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 end-point 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 time-shifted 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 full-response 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, CPM-specific version of the present derivation was presented in . That one and this one both are more elegant and less restrictive than the most-similar previous result in the literature , which was limited to integer .
Begin with a mixed continuous- and discrete-time matrix-vector
convolution that generalizes on PAM signaling. Notations
and are equivalent.
In 1974 Prabhu and Rowe  used a mixed-convolution
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
The change of variable from normalized to unnormalized frequency in gives this mixed-convolution version a slightly different look from the others. The scalar version applies to the introduction's form. The extension to with drawn from a finite-dimensional 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 attachment-point 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
then, along with the
Taking the Fourier transform of
with care (to get a key
special case right), let
, so that
, for all , so
for some frequency
, and any additional component
must be discovered some
This paper's complex data signal comprised complex waveform symbols chosen independently with arbitrary probabilities from a finite-dimensional function space and sequentially attached through time shifting and complex scaling of Markov character. One example is full-response CPM, elsewhere represented with Markov encoding of the signaling-interval 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 waveform-symbol space and one the basis for corresponding Markov changes in complex amplitude.
In this appendix, and , subscripted or not,
are continuous-time and discrete-time random processes respectively,
the latter with implicitly referring to time . These mixed
continuous- and discrete-time results parallel and sometimes depend on
familiar results in both continuous time and discrete time. As usual,
double-subscripted crosscorrelations become single-subscripted
autocorrelations when the two processes are one. The simple proofs of
the first two propositions are omitted. Fourier-pair 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 one-to-one map from onto with almost everywhere. Consequently, the right side of definition