Assume a signal detector (equalizer) able to handle 2K terms of nearest-neighbor ICI. We may partition the summation over k in (1.2) into an in-band and an out-of-band term as where cm,K is the out-of-band term, i.e., residual ICI. Alternatively, using the nota-tion of (1.6),
For large enough N, the residual ICI may be modeled as Gaussian by the central limit theorem.
It turns out that the analysis can be more conveniently carried out by way of the frequency spreading functions of the propagation paths than by way of am,k. Hence consider (3.1). From it, the autocorrelation of cm,K at lag r is given by
E[cm,Kc∗m+r,K] = Es× X
where Es is the average transmitted symbol energy and we have assumed that Xk
is white. Invoking (1.3) and (1.7), we get
E[cm,Kc∗m+r,K] = Es
We show in the Sec. 3.3 that
l is the mean-square Doppler spread of path l given by σD2 l =Rfd
where the exclusion of 0 and −r from both ranges of summation is to skip over the points of singularity where the summands are null anyway. Note further that
−1/(1 − e−j2πk/N) and −1/(1 − e−j2π(k+r)/N) (as sequences in k) are the DFTs of [n − (N − 1)/2]/N and e−j2πrn/N[n − (N − 1)/2]/N (as sequences in n), respectively.
Hence, with Parseval’s theorem we get
ρ0(r, N) = 1 i.e., it is conjugate symmetric in r. Moreover, the summands in the last summation in (3.8) are symmetric over the range of summation. But the range of summation does not allow us to obtain a compact expression for ρ1(K, r, N) as that for ρ0(r, N).
As mentioned, the proposed receiver will whiten the residual I+N before equal-ization. Here we make some observations of the properties of the normalized autocor-relation of residual ICI, i.e., E[cm,Kc∗m+r,K]/E[|cm,K|2], that are relevant to whitener design and performance. For this, note from (3.6) that E[cm,Kc∗m+r,K]/E[|cm,K|2]
depends only on K and N through ρ(K, r, N); the other factors cancel out. Thus this normalized autocorrelation is independent of the average transmitted symbol energy Es and the sample period Tsa. More interestingly, it is also independent of the power-delay profile (PDP) of the channel (i.e., σl2 vs. l) and the Doppler PSD Pl(f ) of each path. While the independence of the normalized autocorrelation on the average transmitted symbol energy may be intuitively expected, its independence of the sample period, the PDP, and the Doppler PSDs of channel paths appears somewhat surprising.
Moreover, the normalized autocorrelation is also substantially independent of the DFT size N. To see this, note that for complexity reason, in a practical receiver both the whitener and the equalizer are likely short. A short equalizer implies a small K and a short whitener implies a small range of r over which the normalized autocorrelation needs to be computed. Hence, when N is large, the exponential functions in the above summations for ρ0(r, N) and ρ1(K, r, N) can all be well ap-proximated with the first two terms of their respective power series expansion (i.e., ex ≈ 1 + x when |x| ≪ 1). As a result, we have Thus the normalized autocorrelation, being essentially given by ρ(K, r, N)/ρ(K, 0, N), is substantially independent of the DFT size N.
The rules are given below.
Property 1
Assume a receiver partition the frequency channel matrix with band width K into an in-band and an out-of-band term and is out-of-band term at m-th subcarrier.
The normalized autocorrelation of cm,K at lag r is given by
that will approximate to a constant on condition that K, r are given.
Although the above observations concern ICI only, it is straightforward to extend them to the sum of ICI and AWGN channel noise.
Property 2
that will approximate to a function only depends on E[|cE[|Wm|2]
m,K|2] with K, r given.
In particular, the resulting whitening filter and its performance can also disregard a variety of system parameters and channel conditions, including the DFT size, the sample period, the system bandwidth (which is approximately proportional to the inverse of the sample period), the OFDM symbol period NTsa, the channel PDP, and the Doppler PSDs of the channel paths. They only depend on the ICI-to-noise power ratio (INR) at the receiver.
As a result, a whitener parameterized on receiver INR can be designed for all operating conditions, which is advantageous for practical system implementation.
(The estimation of ICI and noise powers is outside the scope of the present work.
Some applicable methods have been proposed in the literature, e.g., [11] for ICI power and [12] for noise power.)
The whitener performance can be understood to a substantial extent by examin-ing the above approximation to the normalized autocorrelation E[cm,Kc∗m+r,K]/E[|cm,K|2].
We leave a detailed study along this vein to potential future work. For now, we shall be content with a first-order understanding by a look at its value at lag r = 1. A
large value indicates that whitening can effectively lower the residual ICI. For this, we see from the above approximation (after some straightforward algebra) that
E[cm,Kc∗m+1,K] For example, its values for K = 0–3 are, respectively, 0.6079, 0.7753, 0.8440, and 0.8808, which are substantial indeed.
As a side remark that will be of use later, we note the following properties from (3.6) and (3.11).
Property 3
The total ICI power E[|cm,0|2] can be approximated as
σc02 , E[|cm,0|2] ≈ 4π2Tsa2Es
which is in essence the upper bound derived in [11]. Moreover, we have an approxi-mation to the partial ICI power beyond the 2K central terms.
Property 4
The total ICI power E[|cm,K|2] can be approximated as
σcK2 , E[|cm,K|2] ≈ 4π2Tsa2Es
In the following section, we provide some numerical examples to verify the above results on ICI correlation. Then, in the next section, we consider how to incorporate a whitener for residual ICI plus noise in the receiver.