In this section, the CFO estimation method is proposed, which rotates each phase of the transmitted PN sequences and employs the L& R method in Section 3.1. Furthermore, a CFO estimation scheme that uses the CSI is proposed in Section 3.2 for accurate estimation in the Rayleigh channel.

### 3.1 Phase rotated PN transmission method and CFO estimation method employing L&R algorithm

In this section, a transmission method that rotates each phase of the transmitted PN sequences to prevent the PN sequence cancellation problem is proposed. In addition, a CFO estimation method employing the L&R algorithm [9] is proposed.

The proposed transmission method is explained as follows: when the number of transmission is two, the phase of *PN*_{2}(*k*) is rotated to make the phase difference between *PN*_{1}(A) and *PN*_{2}(*k*) be *π*/2 for transmission, as shown in Figure 4. When the proposed method is applied, we can write *PN*_{
Tx
}(*k*) in an ideal channel state as

P{N}_{\mathit{Tx}}\left(k\right)=\left\{\begin{array}{c}\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}j,\phantom{\rule{0.75em}{0ex}}P{N}_{1}\left(k\right)=1+0j,P{N}_{2}\left(k\right)=0+1j\\ -\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}j,\phantom{\rule{0.75em}{0ex}}P{N}_{1}\left(k\right)=-1+0j,P{N}_{2}\left(k\right)=0+1j\\ \frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}j,\phantom{\rule{0.75em}{0ex}}P{N}_{1}\left(k\right)=1+0j,P{N}_{2}\left(k\right)=0-1j\\ -\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}j,\phantom{\rule{0.75em}{0ex}}P{N}_{1}\left(k\right)=-1+0j,P{N}_{2}\left(k\right)=0-1j\end{array}\right.

(6)

When the phase of the transmitted PN sequences is rotated by the proposed method, the *PN*_{
Tx
}(*k*) sequences for all cases can be used for the CFO estimation, as shown in Figure 5a. In addition, more than one auto-correlators can be used for the CFO estimation, similar to the ML methods [9, 11, 12], because all consecutive *PN*_{
Tx
}(*k*) sequences can be used, as shown in Figure 5b. Therefore, the accuracy of the estimated CFO can be increased as compared with that of the conventional method.

The frequency offset estimation method is explained as follows: the modulation is removed by multiplying the complex conjugate of the locally generated *PN*_{
Tx
}(*k*) to receive *y*_{
PN
}(*k*).

{z}_{\mathit{PN}}\left(k\right)={y}_{\mathit{PN}}\left(k\right)P{N}_{\mathit{Tx}}^{*}\left(k\right)={e}^{j\left(2\pi {f}_{c}k{T}_{s}+\theta \right)}+{n}^{\text{'}}\left(k\right),

(7)

where {n}^{\prime}\left(k\right)=n\left(k\right)P{N}_{\mathit{Tx}}^{*}\left(k\right).

After removing the modulation of the received *y*_{
PN
}(*k*), the phase difference is calculated from the argument of the multiplication between *z*_{PN}(*k*) and {Z}_{\mathit{PN}}^{*}\left(K-1\right).

\begin{array}{l}arg\left[{z}_{\mathit{PN}}\left(k\right){z}_{\mathit{PN}}^{*}\right(k-1\left)\right]\\ \phantom{\rule{1em}{0ex}}=arg\left[\right({e}^{j\left(2\pi {f}_{c}k{T}_{s}+\theta \right)}+{n}^{\text{'}}\left(k\right)\left)\right({e}^{-j\left(2\pi {f}_{c}\left(k-1\right){T}_{s}+\theta \right)}+{n}^{\text{'}}(k-1)\left)\right]\\ \phantom{\rule{1em}{0ex}}=arg[{e}^{j2\pi {f}_{c}{T}_{s}}+{n}^{\text{'}\text{'}}\left(k\right)]=2\pi {f}_{c}{T}_{s}+{n}^{\text{'}\text{'}\text{'}}\left(k\right)\end{array},

(8)

where {n}^{\u2033}={e}^{j\left[2\pi {f}_{e}\mathit{kT}+\theta \right]}{n}^{\prime}\left(k-1\right)+{e}^{-j\left[2\pi {f}_{e}\left(k-1\right){T}_{2}+\theta \right]}{n}^{\prime}\left(k\right)+{n}^{\prime}\left(k\right)n\left(k-1\right) and *n*‴ = arg[*n*″].

In Equation 8, the CFO term 2*πf*_{
c
}*T*_{
s
} is estimated in the presence of a noise term *n*^{' ' '}. To reduce the effect of such noise, multiple auto-correlators are employed for CFO estimation in the L&R algorithm. *R*(*m*), which is the auto-correlation function of *z*_{
PN
}(*k*), can be written as

R\left(m\right)=\frac{1}{{N}_{p}-m}{\displaystyle \sum _{k=m}^{{N}_{p}-1}{z}_{\mathit{PN}}\left(k\right){z}_{\mathit{PN}}^{*}\left(k-m\right)},

(9)

where *N*_{
p
} is the number of PN sequences. Finally, the CFO can be estimated as

\hat{v}=\frac{1}{\pi \left({N}_{R}+1\right){T}_{s}}arg\left[{\displaystyle \sum _{m=1}^{{N}_{R}}R\left(m\right)}\right],

(10)

where *N*_{
R
} is the number of auto-correlators. The accuracy of the estimated CFO increases while the CFO estimation range decreases as *N*_{
R
} increases [9]. In addition, as *N*_{
R
} increases, the number of *R*(*m*) in Figure 6 increases, and the hardware complexity increases.

Figure 7 shows the normalized CFO estimation range of the proposed method, and the *N*_{
R
} value of the L&R method is equal to eight. When *N*_{
R
} is set to eight, the normalized CFO estimation range is from −0.11 to +0.11, as shown in Figure 7. This estimation range is sufficient in general terrestrial digital TV broadcasting environment [13]. If a wider estimation range is required, *N*_{
R
} can be decreased or the M&M method [12], which shows an almost full estimation range, can be employed.

### 3.2 Frequency offset estimation scheme using CSI

The received *k* th PN sequence *y*_{
PN
}(*k*) at the receive antenna in the time-varying Rayleigh channel can be written as

\begin{array}{ll}\phantom{\rule{1.4em}{0ex}}{y}_{\mathit{PN}}\left(k\right)& =\left[{h}_{11}P{N}_{1}\left(k\right)+{h}_{12}P{N}_{2}\left(k\right)\right]{e}^{j\left(2\pi {f}_{c}\cdot i{T}_{s}+\theta \right)}\\ \phantom{\rule{1em}{0ex}}+n\left(k\right)\\ =P{N}_{\mathit{Tx}}\left(k\right){e}^{j\left(2\pi {f}_{c}\cdot i{T}_{s}+\theta \right)}+n\left(k\right),\end{array}

(11)

where *h*_{11} is the channel impulse response (CIR) from the first transmit antenna to the receive antenna and *h*_{12} is the CIR from the second transmit antenna to the receive antenna. In the Rayleigh channel, *h*_{11} and *h*_{12} are time varying, and the variation rate depends on the Doppler frequency.

The proposed PN sequence transmission method rotates the phase of the PN sequences to prevent the PN sequence cancellation problem. However, when the absolute phase difference between the time-varying *h*_{11} and *h*_{12} is *π*/2, the PN sequence cancellation problem occurs again, and the accuracy of the estimated CFO decreases in contrast to that in the AWGN channel. In other words, as the absolute phase difference between *h*_{11} and *h*_{12} tends closer to *π*/2, the accuracy of the estimated CFO decreases; when it becomes smaller than *π*/2, the accuracy increases. Thus, the frequency offset estimation scheme using CSI is proposed, as shown in Figure 8.

The proposed scheme consists of the coarse carrier frequency recovery (CFR), which is a feedback structure, fine CFR, which is a feedforward structure, and phase calculator that uses CSI. For the estimation scheme, the proposed PN sequence rotation and the CFO estimation method are employed for the coarse CFR, and the pilot block correlation method in [14] can be employed for the fine CFO estimation algorithm. In the proposed scheme, the fine CFR estimates the CFO and applies it to the compensator using the PN sequence at every frame. On the other hand, the coarse CFR first estimates the CFO at every frame and applies it to the compensator using the CSI. The phase estimator estimates the CSI using PN sequences over one PN block (guard interval) after the CFR and applies the estimated CFO to the phase locked loop (PLL) only when the absolute phase difference between *h*_{11} and *h*_{12} is smaller than *π*/4. If the absolute phase difference between *h*_{11} and *h*_{12} is greater than *π*/4, zero is applied to the PLL.

However, the CSI may not be acquired until CFR is roughly achieved. Thus, the proposed scheme controls the PLL using the CSI only when a rough CFR is achieved. Figure 9 shows the PLL output of a coarse CFR. CSI is not used for the CFR when the variance in the PLL output is smaller than 10^{−4} or 100,000 frames are used for the CFR (dotted line). After 100,000 frames are used, the CSI is used for the phase difference calculation. In Figure 9, the PLL converges roughly without using the CSI, and rough CFR can be achieved without using the CSI. In addition, a fine CFO can be recovered after a coarse CFR; we can therefore decide whether to apply an estimated CFO to the PLL with the aid of the CSI after the PLL has roughly converged.

The PN cancellation problem in the conventional method can also be solved by employing the proposed frequency estimation scheme. For the conventional system, if the absolute phase difference between *h*_{11} and *h*_{12} over one block is greater than *π*/4, the CFO is estimated by employing several auto-correlators and applying them to the PLL. However, the proposed PN sequence transmission method that uses the proposed scheme is suitable for both the AWGN and time-varying Rayleigh channels, whereas the conventional method that uses the proposed scheme only works well in the time-varying channel.