On the Autocorrelation of Complex Envelope of White Noise

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