After the fine frequency offset has been corrected, the channel estimation can readily be performed. Here, both the methods of LS and the ML technique will be investigated for channel estimation.

### 4.1. LS channel estimation

Given \widehat{\delta}, we can correct the offset by multiplying (5) by {e}^{-j2\pi \left(n+lN\right)\widehat{\delta}/N} to obtain {r}_{l,n}^{\prime}={\stackrel{\u0303}{r}}_{l,n}{e}^{-j2\pi \left(n+lN\right)\widehat{\delta}/N} (before {\stackrel{\u0303}{r}}_{l,n} is fed to the DFT). If the estimate is of good accuracy, the estimate error \Delta \delta =\delta -\widehat{\delta} will be very small. Then, after frequency offset correction, the *k* th subcarrier DFT output (during training mode) can be approximated as

\begin{array}{c}{R}_{l,k}^{\prime}=\frac{1}{\sqrt{N}}{e}^{j2\pi l\Delta \delta}\sum _{n=0}^{N-1}{e}^{j2\pi n\Delta \delta /N}\left[\frac{1}{\sqrt{N}}\sum _{p=0}^{N-1}{H}_{p}{P}_{p}{e}^{j2\pi np/N}\right]{e}^{-j2\pi nk/N}+{W}_{l,k}^{\prime}\\ =\frac{{e}^{j2\pi l\Delta \delta}}{N}{H}_{k}{P}_{k}\sum _{n=0}^{N-1}{e}^{j2\pi n\Delta \delta /N}+\frac{{e}^{j2\pi l\Delta \delta}}{N}\sum _{\begin{array}{c}\hfill p=0\hfill \\ \hfill p\ne k\hfill \end{array}}^{N-1}{H}_{p}{P}_{p}\sum _{n=0}^{N-1}{e}^{j2\pi \left(p-k+\Delta \delta \right)n/N}+{W}_{l,k}^{\prime}\\ \approx {H}_{k}{P}_{k}{e}^{j\pi \left(2lN+N-1\right)\Delta \delta /N}+{W}_{l,k}^{\prime}\approx {H}_{k}{P}_{k}\left(1+j{\alpha}_{l}\Delta \delta \right)+{W}_{l,k}^{\prime},\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}l=0,1,...,L,\end{array}

(12)

where *α*_{
l
} ≡ *π* (2*lN* + *N* - 1)/*N*, {W}_{l,k}^{\prime}=\sum _{n=0}^{N-1}{\stackrel{\u0303}{w}}_{l,n}{e}^{-j2\pi \left(n+lN\right)\widehat{\delta}/N}{e}^{-j2\pi nk/N}~N\left(0,{\sigma}_{W}^{2}\right), and we have used the approximations {e}^{j{\alpha}_{l}\Delta \delta}\approx 1+j{\alpha}_{l}\Delta \delta, \sum _{n=0}^{N-1}{e}^{j2\pi n\Delta \delta /N}=\frac{{e}^{j\pi \left(N-1\right)\Delta \delta /N}\text{sin}\left[\pi \Delta \delta \right]}{\text{sin}\left[\pi \Delta \delta /N\right]}\approx N{e}^{j\pi \left(N-1\right)\Delta \delta /N}, and *p* - *k* +Δ*δ* ≈ *p*-*k*.

If the frequency offset correction were perfect (offset-free), then Δ*δ* = 0 and (12) would reduce to

{\overline{R}}_{l,k}^{\prime}={H}_{k}{P}_{k}+{W}_{l,k}^{\prime}\phantom{\rule{1em}{0ex}}l=0,1,...,L,

(13)

where {\overline{R}}_{l,k}^{\prime} stands for the offset-free DFT output. There are *L* + 1 equations in (13) and they form a system of simultaneous linear equations for one variable *H*_{
k
} and we can thus easily apply the method of LS to get the channel estimators as

\widehat{{H}_{k}}={\left({\mathbf{P}}_{k}^{H}{\mathbf{P}}_{k}\right)}^{-1}{\mathbf{P}}_{k}^{H}{\overline{\mathbf{R}}}_{k}^{\prime}=\frac{\sum _{l=0}^{L}{\overline{R}}_{l,k}^{\prime}}{\left(L+1\right){P}_{k}}\phantom{\rule{1em}{0ex}}k=0,1,...,N-1,

(14)

where **P**_{
k
} = *P*_{
k
} [1,1,...,1] ^{T} is an (*L* + 1) × 1 vector and {\overline{\mathbf{R}}}_{k}^{\prime}={\left[{\overline{R}}_{0,k}^{\prime},{\overline{R}}_{1,k}^{\prime},...,{\overline{R}}_{L,k}^{\prime}\right]}^{T}.

In reality, perfect frequency offset correction is not very likely to happen. That is, it is not very likely to obtain the offset-free outputs \left\{{\overline{R}}_{l,k}^{\prime}\right\} of (13). The best one can do is to use the \left\{{R}_{l,k}^{\prime}\right\} of (12). Thus, in practice, we shall take

\widehat{{H}_{k}}=\frac{\sum _{l=0}^{L}{R}_{l,k}^{\prime}}{\left(L+1\right){P}_{k}},\phantom{\rule{1em}{0ex}}k=0,1,...,N-1,

(15)

Note that only one sample (the *k* th sample {R}_{l,k}^{\prime}) in each training block (frequency-domain OFDM block) has been selected to estimate *H*_{
k
} and there are *L* + 1 blocks resulting in only *L* + 1 samples used for the LS channel estimation. In order not to reduce data transfer rate too much, *L* is not to be too large. Thus, with the small number of *L* + 1 samples used, the LS estimator *Ĥ*_{
k
} is expected to perform not quite so satisfactorily, though perfectly workable. We will thus seek an alternative approach using the ML technique to be described next.

### 4. 2. ML channel estimation

To perform ML channel estimation, we first need to find the log-likelihood function in terms of the variables to be estimated. These variables must be independent of each other. We shall assume a frequency-selective Rayleigh fading channel of dispersion length *v* having the CIR vector **h** = [*h*_{0}, *h*_{1},..., *h*_{v-1}] ^{T} . Then {*h*_{
m
}*, m* = 0,1,...*v*} are uncorrelated and hence independent of each other. However, the CFRs {*H*_{
k
}*, k* = 0,1,..., *N*}, which are DFTs of {*h*_{
m
} }, are obviously correlated and hence dependent upon each other. Consequently, unlike the above case of LS channel estimation where CFR can be estimated directly due to the special one variable over-determined system formulation, here we must first obtain the ML CIR estimator **ĥ** from a log-likelihood function in terms of **h** from which we then get the CFR estimator **Ĥ** through DFT.

Now, assume that frequency offset has been corrected so that *δ* = 0. Using (5) and (6) with *δ* = 0, we can readily get

{\stackrel{\u0303}{\mathbf{r}}}_{l}={\mathbf{y}}_{0}+{\widehat{\mathbf{w}}}_{l}=\frac{1}{\sqrt{N}}{\mathbf{F}}_{N}^{H}\mathbf{P}{\mathbf{F}}_{v}\mathbf{h}+{\widehat{\mathbf{w}}}_{l},

(16)

where {\mathbf{y}}_{0}=\frac{1}{\sqrt{N}}{\mathbf{F}}_{N}^{H}\mathbf{P}{\mathbf{F}}_{v}\mathbf{h} is the offset-free noiseless received signal vector with **P** = *diag*{*P*_{0}, *P*_{1},..., *P*_{N-1}} being a diagonal matrix and

{\mathbf{F}}_{v}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill {e}^{-j2\pi /N}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {e}^{-j2\pi \left(v-1\right)/N}\hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill \vdots \hfill \\ \hfill 1\hfill & \hfill {e}^{-j2\pi \left(N-1\right)/N}\hfill & \hfill \cdots \phantom{\rule{0.3em}{0ex}}\hfill & \hfill {e}^{-j2\pi \left(N-1\right)\left(v-1\right)/N}\hfill \end{array}\right].

(17)

Then, stacking *L* + 1 repeated blocks, we have

\left[\begin{array}{c}\hfill {\stackrel{\u0303}{\mathbf{r}}}_{0}\hfill \\ \hfill {\stackrel{\u0303}{\mathbf{r}}}_{1}\hfill \\ \hfill \vdots \hfill \\ \hfill {\stackrel{\u0303}{\mathbf{r}}}_{L-1}\hfill \end{array}\right]={\mathbf{r}}_{B}=\left[\begin{array}{c}\hfill {\mathbf{y}}_{0}\hfill \\ \hfill {\mathbf{y}}_{0}\hfill \\ \hfill \vdots \hfill \\ \hfill {\mathbf{y}}_{0}\hfill \end{array}\right]+\left[\begin{array}{c}\hfill {\mathbf{w}}_{0}\hfill \\ \hfill {\mathbf{w}}_{1}\hfill \\ \hfill \vdots \hfill \\ \hfill {\mathbf{w}}_{L-1}\hfill \end{array}\right].

(18)

The log-likelihood function for (18) is given by

\Lambda =-N\text{ln}\pi {\sigma}_{w}^{2}-\frac{1}{{\sigma}_{w}^{2}}\sum _{l=0}^{L}{\u2225{\stackrel{\u0303}{\mathbf{r}}}_{l}-\frac{1}{\sqrt{N}}{\mathbf{F}}_{N}^{H}\mathbf{P}{\mathbf{F}}_{v}\mathbf{h}\u2225}^{2}.

(19)

From (19), it can readily be shown that the ML estimate of **h** is given by

\widehat{\mathbf{h}}=\frac{1}{\sqrt{N}}{\left({\mathbf{F}}_{v}^{H}{\mathbf{P}}^{H}\mathbf{P}{\mathbf{F}}_{v}\right)}^{-1}{\mathbf{F}}_{v}^{H}{\mathbf{P}}^{H}{\mathbf{F}}_{N}\left(\frac{1}{L+1}\sum _{l=0}^{L}{\stackrel{\u0303}{\mathbf{r}}}_{l}\right),

(20)

Training sequence with constant amplitude has been proven optimal for channel estimation [7]. Chu sequence [8], for example, falls onto this category. Using a Chu sequence {{P}_{k}={e}^{j\pi m{k}^{2}/N}, *m* being any integer relatively prime to *N*} results in **P**^{H}**P** = **I** _{
N
}. Then, (20) can be simplified to

\widehat{\mathbf{h}}=\frac{1}{N\sqrt{N}}{\mathbf{F}}_{v}^{H}{\mathbf{P}}^{H}{\mathbf{F}}_{N}\left(\frac{1}{L+1}\sum _{l=0}^{L}{\stackrel{\u0303}{\mathbf{r}}}_{l}\right),

(21)

where the fact {\mathbf{F}}_{v}^{H}{\mathbf{F}}_{v}=N{\mathbf{I}}_{v} has been used. Note that, by using a Chu sequence, (21) can avoid the computation of matrix inversion given in (20). The estimate of the CFR vector **H** = [*H*_{0}, *H*_{1},..., *H*_{N-1}] ^{T} can be readily obtained as

\widehat{\mathbf{H}}={\mathbf{F}}_{v}\widehat{\mathbf{h}}.

(22)

Since all samples in each of the entire OFDM block are utilized in this ML estimation, we have used a total of *N* (*L* + 1) samples for this ML channel estimation. As a result, we expect that the ML channel estimator will outperform the previous LS channel estimator. Later in simulations, we shall make performance comparisons between the CFR estimators given by (15) and (22).