A Note on Gallager's Capacity Theorem for Waveform Channels

We correct an alleged contradiction to Gallager’s capacity theorem for waveform channels as presented in a poster at the 2012 IEEE International Symposium on Information Theory. 1 Brief Description of the Poster [2] Gallager’s capacity theorem [1, Theorem 8.5.1] is literally reproduced on p. 1 of the poster along with the figure [1, Figure 8.5.1] plus some explanations taken from Gallager’s book [1]. On p. 2, the so-called Gallager channel (now called Gaussian waveform channel), i.e., a Gaussian filter with additive white Gaussian noise (AWGN), is treated as a special instance of Gallager’s theorem. On p. 3, the heat channel, a linear time-varying (LTV) filter with AWGN, is depicted followed by a characterization of its capacity by water-filling in the time-frequency plane [4, Theorem 2]. Finally, on p. 4, an example of a signal transmission is given aiming at underpinning the apparent contradiction to Gallager’s theorem in Figs. 3 and 4 of the poster. The curves for the heat channel in the latter two figures are incorrect; the correct curves are provided in the present publication. 2 Capacity of the Heat Channel Revisited Evaluation of the double integrals occurring in the water-filling theorem [4, Theorem 2] (cf. p. 3 of [2]) and subsequent elimination of the parameter ν results in the closed-form representation of the capacity (in nats per transmission) of the heat channel, namely C(S) = αβ 2 [