Center frequency and bandwidth are two generic parameters used to characterize transmitted pulse profiles in B-mode ultrasonic imaging. Increasing either is generally thought to improve spatial resolution in the final image, but at a potential cost of lower signal-to-noise ratio, with no general understanding of where they are optimal. In this work we investigate their role in converting the acquired radio-frequency signal from a linear array into an envelope image. Statistics of the backscattered signal, based on Rayleigh-Sommerfeld diffraction theory, are used in an ideal observer calculation that quantifies the task-relevant information contained in the radio-frequency (RF) signal. We then compare two approaches to computing an envelope image. The first is a standard B-mode envelope from the complex analytic signal. The second approach processes RF through a Wiener filter before forming an analytic signal. Effects of envelope detection are measured by computing the ideal observer in the envelope domain using Smith-Wagner approximations. Over frequencies ranging from 3-15MHz and fractional bandwidths ranging from 20% to 80%, we find that information transfer in the envelope varies widely with task. There is a substantial loss of information in all conditions in the formation of a standard envelope. Efficiency relative to the RF ranges from 60% to less than 5%. The Weinerfiltered envelope images substantially improve efficiency in two of the three tasks investigated. In the third task, the results are mixed, but we argue that the Weiner filter may be improved substantially by retuning it to the interior of a lesion.