Re: Lowpass filtering for OFDM introduces interference?

lanbaba said the following on 18/04/2006 22:21:
To show this, say your received freq-domain values are:

Y[k] = DFT{y[n]}
= DFT{x[n] + v[n]}

= X[k] + V[k]

[Where v[n] is the time-domain noise sequence, and V[k] is the freq-domain noise sequence].

Evaluating this at some arbitrary k = k_0, this relationship is clearly not affected by the size of the DFT, nor by the amount of noise at any other k.

Therefore, there is no performance advantage in low-pass filtering and downsampling at baseband.


Assume v[n] to be a wideband white noise. By lowpass filtering the
variance of v[n] should be reduced leading to a reduced noise variance of
V[k] in the frequency domain.

That doesn't follow. Consider:

y[n] = exp(j*2*pi*k1*n/N) + exp(j*2*pi*k2*n/N)

The variance of the total signal y[n] is 2, and the value of the DFT at k1 and k2 is:

Y[k1] = Y[k2] = 1

If we now filter y[n] so that we completely suppress the component at k1, we get:

y'[n] = exp(j*2*pi*k2*n/N)

The variance of this is now 1. However, the value of the DFT at k2 is still:

Y'[k2] = 1

i.e. a reduction in signal time-domain variance does not imply a reduction in variance of all frequency-domain components.


Why is there no SNR gain?

Because each bin of the DFT is a matched correlation filter. By definition, it is already optimal.

Try it in MATLAB (or whatever). Do a simple OFDM simulation (IFFT->AWGN channel->FFT), and get a BER measure. Then do it again, but pad your frequency-domain vector with zeros, which is equivalent to time-domain oversampling. The results should be identical.


I think you have no doubt that two receive antennas, even if fully
correlated, give you certain antenna gain if you combine them. I mean
there is no essential difference between the equal gain combining of two
RX antennas and low pass filtering oversampled time-domain signals. Both
can denoise.

You're right, these two scenarios are mathematically identical.

But equal gain combining isn't, in general, the optimal solution. For any linear combination scenario, the optimal solution is given by the Wiener-Hopf equation. In the case of filters, Wiener-Hopf gives the matched correlation filter. In the case of (non-pulse-shaped) OFDM, the matched filter for each sub-carrier is of the form exp{-j.2.pi.k.n/N}, which is what the DFT is already doing. Any further filtering can, at best, give zero further improvement in performance.

You can consider the DFT as mixing the signal down so that the k-th sub-carrier is at DC, and then time-domain averaging across the symbol.


--
Oli
.

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