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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?

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  • $\begingroup$ What do you mean by "using comb pilots with a guard band" and how does it solve your problem? $\endgroup$
    – AlexTP
    Mar 10 at 21:46
  • $\begingroup$ What I mean is that I use com-type pilot OFDM symbols for channel estimation, and those symbols contain guard bands at the edges of the spectrum. If I have fractional delays, a Gibbs-like effect happens at the spectrum edges of the receive OFDM symbols and this distorts both the channel estimation and the subsequent evaluation of data at the subcarriers in this region. The workaround I found so far was using these subcarriers at the edges as guard bands and discarding them in the end. $\endgroup$ Mar 11 at 9:08
  • $\begingroup$ Sorry, I still don't see how fractional delays could worsen your channel estimation, and still not sure the guard band (zero subcarriers) could help. There are two things. First, if your CP is greater than the channel delay spread, no interference between two adjacent OFDM symbols. $\endgroup$
    – AlexTP
    Mar 11 at 11:24
  • $\begingroup$ Second, using some subcarriers as pilots is sampling the CFR, and as soon as the pilot spacing is smaller than the reciprocal of the channel delay spread, Nyquist theorem holds and you can reconstruct perfectly your CIR, at least theoretically. I think your guard band behaves like Fourier transform interpolation (I still don't understand your method, sorry). $\endgroup$
    – AlexTP
    Mar 11 at 11:26
  • $\begingroup$ For the question "what is the best way to interpolate comb-type pilot?", first you need to define what you mean by "best". If it is to minimize the mean square error, the answer is to use an MMSE channel estimator that will minimize the error by construction. $\endgroup$
    – AlexTP
    Mar 11 at 11:30

1 Answer 1

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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!

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  • $\begingroup$ Thanks for the answer and for the corrections as well. The limited number of discrete taps will indeed support the impulse response - what I wanted to mean is that you will not get a single discrete tap, but rather many taps to express a single fractional delay. $\endgroup$ Mar 10 at 16:52
  • $\begingroup$ As for the channel model, I have a model for an arbitrary number of paths that may contain fractional delays. I apply it to my signal (containing multiple OFDM symbols) in the frequency domain by multiplying the DFT of the whole OFDM signal with exp(-1i*2*pi*(0:N_samples-1)*(delay_samples/N)), where N_samples is the total number of samples of the OFDM signal and delay_samples is a normalized delay by the sampling period. $\endgroup$ Mar 10 at 17:03
  • $\begingroup$ All in all, I don't seem to have any special assumptions on the channel model and the position of the pilots also does not influence the result of the interpolation (I currently use spline), as long as I have a guard band. $\endgroup$ Mar 10 at 17:03
  • $\begingroup$ ups, posted a text to the wrong question at the end of this answers – wasn't meant for you :) $\endgroup$ Mar 11 at 11:00

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