Transfer functions of acoustic systems often exhibit wide dynamic ranges and very long impulse responses. A ‘‘time-stretched’’ pulse as proposed by Aoshima (ATSP), though originally given in a very specific form seems to be one of the most promising signals to measure transfer functions with characteristics of acoustic system mentioned as above. In this paper, this pulse (ATSP) is first generalized and then optimized for the measurement of long impulse responses. This optimized ATSP (OATSP) has an almost ideal characteristic to measure impulse responses shorter than its specific length N. Moreover, it is newly shown in this paper that OATSP has also a good characteristic to measure impulse responses longer than N. Discussion is presented on how to design OATSP suitable for a specific situation of measurement by analyzing errors, when the pulse is used to measure impulse responses longer than N.