Published in Viswanathan, R. (2006). On the autocorrelation of complex envelope of white noise. IEEE Transactions on Information Theory, 4298-4299. doi: 10.1109/TIT.2006.880018 ©2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.


About four decades ago, in an article in this transactions, Thomas Kailath pointed out that the autocorrelation of the complex envelope of white noise is not strictly an impulse function, even though when treated as an impulse in practical problems, it does lead to correct results. However, it is commonly assumed that by simply letting the bandwidth of a flat-bandlimited noise process to go to infinity, one obtains the result that the autocorrelation of the complex envelope of white noise equals an impulse function. In this correspondence, we show that 1) the limit operation has to be done carefully and 2) when done properly, it leads to the result in the Kailath’s paper, which is different from a pure impulse function.