The Operational PSD

2nd International Workshop on Advances in Communications, Boppard am Rhein, Germany, May 13–15, 2009.

The power spectral density of quadrature amplitude
modulation done right


The signals produced in Pulse Amplitude Modulation (PAM) and in Quadrature Amplitude Modulation (QAM) are not typically wide-sense stationary (WSS) stochastic processes. Consequently, such signals do not have a Power Spectral Density (PSD) in the classical sense. To overcome this difficulty, one can introduce artificial random time-jitters and random phase-jitters, which stationarize the signals. In this talk I shall propose a different approach, which does not require any artificial randomization. The new approach is based on the “operational power spectral density,” which I define as follows:

Definition 1 (Operational PSD). We shall say that the continuous-tine real stochastic process (X(t), t ∈ ℜ) is of operational power spectral density SXX if (X(t), t ∈ ℜ) is a measurable SP; the mapping SXX: is integrable and symmetric; and for every stable real filter of impulse response h1 the average power at the filter's output when it is fed (X(t), t ∈ ℜ) is given by

Power in X h = (-∞,∞) SXX(f) |ĥ(f)|2 d f.

I will show that the operational PSD generalizes the classical PSD in the sense that the operational PSD of a WSS stochastic process is equal to its classical PSD. I will then show that PAM and QAM signals typically have an operational PSD, which can be computed very easily. Finally, I will show that my approach facilitates the statement and proof of a key result on passband communication, namely, that the operational PSD of a passband signal does not depend on the pseudo-covariance of its baseband representation.

The video is 28 minutes long.

Last modified: Mon Apr 11 08:04:28 CEST 2022