Mean square estimntioa of complex and normal data is not linear as in the real case but widely linear. The purpose of this correspondence is to calculate the optimum widely linear mean square estimate and to present its main properties. The advantage with respect to linear procedure is espedplly analyzed. I. INTRODUCTION Mean square estimation (MSE) is one of the most fundamental techniques of statistical signal processing. The basic problem can be stated as follows: Let y be a scalar random variable to be estimated (estimandum) in terms of an observation that is a random vector x. The estimateŷ that minimizes the MS error is then the regression or the conditional expectation value E[y|x]. This result is usually given Manuscript received B. P. and P. C. are with the Laboratoire des Signaux et Systèmes (L2S), a joint laboratory of the C.N.R.S. and theÉcole

Abstract-In many questions of signal processing, it is important to use the concepts of instantaneous amplitude or phase of signals. This is especially the case in communication systems with amplitude or frequency modulation. These concepts are often introduced empirically. However, it is well known that the correct approach for this purpose is to use the concept of analytic signal. Starting from this point, we show some examples of contradictions appearing when using other definitions of instantaneous amplitude or frequency that are commonly admitted. This introduces the problem of characterizing pure amplitudemodulated or pure phase-modulated signals. It is especially shown that whereas amplitude modulated signals can be characterized by spectral considerations, this is no longer the case for phasemodulated signals. Furthermore, signals with constant amplitude have very specific properties, which are analyzed in detail. Some consequences and extensions to random signals are finally discussed.

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Abstract-The second-order statistical properties of complex signals are usually characterized by the covariance function. However, this is not sufficient for a complete second-order description, and it is necessary to introduce another moment called the relation function. Its properties, and especially the conditions that it must satisfy, are analyzed both for stationary and nonstationary signals. This leads to a new perspective concerning the concept of complex white noise as well as the modeling of any signal as the output of a linear system driven by a white noise. Finally, this is applied to complex autoregressive signals, and it is shown that the classical prediction problem must be reformulated when the relation function is taken into consideration.