Pilot-based channel estimation in OFDM systems with fractional delays

Question

Most studies on OFDM-based communication systems consider that multipath channels consists of a limited set of taps in the discrete-time domain. However, this only holds if the tap delay is a multiple of the sampling period Ts = 1/B. In case the tap delays are not integer multiples of Ts, they will leak to all neighboring samples in the discrete channel impulse response, e.g. as in Fig. 2.2. here.

When simulating such fractional delays (I also did measurements and I got very similar results), the samples from one OFDM symbol leak into the next one, causing intersymbol interference (ISI). Due to the sinc-shaped decay of this interference caused by the band-limited nature of the OFDM system, this ISI becomes relatively small in the actually evaluated OFDM symbol, that is, after removing cyclic prefix (CP).

My question is regarding channel estimation. If I use a block pilot symbol (i.e., all subcarriers are used), this interference seems to be sufficiently suppressed and I get decent simulation and measurement results. However, if I use comb pilots and interpolate them to get the full channel frequency response (CFR), the experienced ISI seems to significantly affect the edges of the obtained CFR. The effect I observed is basically the same when I cut the estimated CIR with a block pilot, extract only the first N_cp samples (N_cp is the length of the CP) of it and generate a CFR via ZP-FFT - which would obviously goes wrong since the leaking of the tap over all samples would be neglected, but correcting this fractional delay also does not help here. The only solutions I found so far are either using block pilots with all subcarriers for channel estimation, or using comb pilots and a guard band at the edges of the spectrum (composed of non-zero subcarriers) that is discarded before pilot interpolation. Any thoughts on how to avoid the need for the guard band in the comb pilot case?

Answer 1

However, this only holds if the tap delay is a multiple of the sampling period Ts = 1/B.

No. We can represent any (weakly) bandlimited system with finite temporal support with these discrete taps. The sampling theorem doesn't care about whether what you're observing is a signal or a impulse response.

In case the tap delays are not integer multiples of Ts, they will leak to all neighboring samples in the discrete channel impulse response, e.g. as in Fig. 2.2. here.

We still represent the channel, it's just not sparse anymore!

When simulating such fractional delays (I also did measurements and I got very similar results), the samples from one OFDM symbol leak into the next one

Ah! Good observation: That only happens when the space between symbols is shorter than impulse response – and you're right, in theory, due to how we model sampling, the sinc interpolation for impulses between sampling instants will have infinite duration.

However: the whole modelling of channels as being finite in length is always a thing of approximation. Your wifi signal is most definitely also reflected off the mountains in 10 km distance - but the amount of power in these taps would be so small, it has no practical effect (or the effect can well be included in a AWGN floor model).

So: it's not that bad. The whole OFDM channel model already accounts for this effect, and things like the cyclic prefix are dimensioned exactly such that the side lobes have declined enough when the next symbol starts that the performance effect is negligible.

If I use a block pilot symbol (i.e., all subcarriers are used), this interference seems to be sufficiently suppressed and I get decent simulation and measurement results. However, if I use comb pilots and interpolate them to get the full channel frequency response (CFR), the experienced ISI seems to significantly affect the edges of the obtained CFR.

Then we need to talk about how exactly your interpolation happens. My guess here is that you assume the values you get from "standard" channels models are for "sample-synchronous" taps, but in reality they are averaged/expected amplitudes, gather from a model where the actual reflectors are uniformly distributed in time, and hence the models already incorporate the effect you're looking at.

Intuitively, the choice of pilot positions cannot have any effect on the effective length of the channel, so there's really something wrong with your simulation!