Per-Tone Precoding and Per-Tone Equalization: Duality and Optimal Resource Allocation

OFDM and DMT systems add a cyclic prefix (CP) or zero pad (ZP) to the transmitted signal. Interference-free transmission requires this CP/ZP to be of the same length as the impulse response (IR), thereby reducing the achievable data rate in highly dispersive channels. One strategy for dealing with long impulse responses without increasing the CP/ZP overhead consists of applying a channel shortening FIR filter to the received signal. More specifically, this paper considers per-tone equalization (PTEQ), which applies the channel shortening filter after the DFT operation of the ODFM receiver. Another strategy towards reducing the impact of a too short CP/ZP consists of spectral bit and power allocation to avoid interference. However, little effort has been made towards joint channel shortening and power allocation. This paper therefore proposes a new method that simultaneously optimizes the PTEQ filter coefficients and transmit power allocation, which provably converges to a stationary point of the considered rate maximization problem. Moreover, using this optimal power allocation method, it is demonstrated that iterative water filling schemes can achieve good performance when the ISI/ICI is sufficiently suppressed by the PTEQ filter. This paper also considers transmitter-side channel shortening filters, which enable the implementation of channel shortening without replacement of the customers' equipment. The channel shortening filters will again be applied in the frequency domain, resulting in a structure that is referred to as a per-tone precoding (PTPC) filter. At first glance, the FIR filter design for PTPC seems much more involved than the relatively straightforward FIR filter design for PTEQ. In addition, the more involved PTPC filter design complicates the power allocation problem. However, this paper reveals that any OFDM system employing (P)TPC is in some sense - after time-reversing the impulse response - equivalent to an OFDM system employing (P)TEQ. The obtained result is rooted in MAC-BC duality theory. With this duality result in hand, PTPC systems can take full advantage of the straightforward FIR filter design in PTEQ systems, as well as of the developed power allocation algorithm. Simulation results show that the results obtained for OFDM systems with PTEQ and the ones obtained for OFDM systems with PTPC are nearly indistinguishable, making PTPC an interesting alternative channel shortening technique in scenarios where implementing PTEQ is either difficult or costly.