The Expected Peak-to-Average Power Ratio of White Gaussian Noise in Sampled I/Q Data

One of the fundamental endeavors in radio frequency (RF) metrology is to measure the power of signals, where a common aim is to estimate the peak-to-average power ratio (PAPR), which quantifies the ratio of the maximum (peak) to the mean value. For a finite number of discrete-time samples of baseband in-phase and quadrature (I/Q) white Gaussian noise (WGN) that are independent and identically distributed (i.i.d.) with zero mean, we derive a closed-form, exact formula for mean PAPR that is well-approximated by the natural logarithm of the number of samples plus Euler’s constant. In addition, we give related theoretical results for the mean crest factor (CF). After comparing our main result to previously published approximate formulas, we examine how violations of the WGN assumptions in sampled I/Q data result in deviations from the expected value of PAPR. Finally, utilizing a measured RF I/Q acquisition, we illustrate how our formula for mean PAPR can be applied to spectral analysis with spectrograms to verify when measured RF emissions are WGN in a given frequency band.

I. Introduction

In radio frequency (RF) electronic measurements, it is common to perform peak detection, where the maximum amplitude or power of data collected over a given time interval is recorded. In particular, the peak-to-average power ratio (PAPR) and crest factor (CF), equal to the square root of PAPR, arise in the design of communication signals and power amplifiers for RF transmitters [1], [2], [3]. While it is important to understand the PAPR and CF for transmitted continuous-time communication signals and their impact on power amplifiers, it is equally important to characterize PAPR and CF for sampled RF measurements of received signals, e.g., with a signal analyzer. Notable applications of received signals include spectrum monitoring [4], [5] and the detection of transient electromagnetic interference (EMI) [6]. The appropriate selection of settings for measurement instrumentation in these domains requires foreknowledge of the peak level that can be expected over a given time period.