# Beam Centric Doppler Pre-Compensation
###### tags: `dpc`

## System Model

We Consider a multiple spot beam satellite $S$ in the LEO circular orbit and a User terminal $U$. Where the satellite knows its state vector $\mathbf{r}_{\mathrm{sat}}(t) = [x_{\mathrm{sat}}(t), y_{\mathrm{sat}}(t), z_{\mathrm{sat}}(t)]^{T}$ and $\mathbf{v}_{\mathrm{sat}}(t) = [v_{\mathrm{x,sat}}(t), v_{\mathrm{y,sat}}(t), v_{\mathrm{z,sat}}(t)]^{T}$. Similarly the user terminal knows its state vector $\mathbf{r}_{\mathrm{U}}(t) = [x_{\mathrm{U}}(t), y_{\mathrm{U}}(t), z_{\mathrm{U}}(t)]^{T}$ and $\mathbf{v}_{\mathrm{U}}(t) = [v_{\mathrm{x,U}}(t), v_{\mathrm{y,U}}(t), v_{\mathrm{z,U}}(t)]^{T}$ in Earth-Center-Earth-Fixed (ECEF) coordinate system. We assume a regenerative satellite equipped with steerable beams fixed with respect to Earth’s surface. We also assume that neither the satellite nor the user-terminal knows the location of the user-terminal and satellite in advance before the synchronization, respectively. But each beam of the satellite knows the location of the beam center (BC) $\mathbf{r}_{\mathrm{BC}} = [x_{\mathrm{BC}}, y_{\mathrm{BC}}, z_{\mathrm{BC}}]^{T}$ where it will be pointing during the course of the traverse.

## Channel Model

We consider the time-varying frequency-selective channel with large and small-scale fading, modeled as components with $Q_{p}$ multi-path for the $p$th user modeled as

\begin{align}
\mathbf{h}_{p}(t, \tau) &= \sqrt{\mathcal{P}_{p}(t)}\sum_{q = 0}^{Q_{p}(t) - 1}\mathbf{\alpha}_{p,q}(t)e^{-j2\pi f_{\mathrm{D,p,q}}(t)}\delta(\tau - \tau_{p,q}(t))\notag\\
&\approx \sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\mathbf{\alpha}_{p,q}e^{-j2\pi f_{\mathrm{D,p}}t}\delta(\tau - \tau_{p,q})
\end{align}

### Large scale fading

The large-scale fading consists of path loss due to the distance between the satellite and the user terminal. And shadowing attenuation $\mathcal{P}_{\mathrm{S}} \sim \mathcal{CN}(0, \sigma^{2}_{S})$. Where $ \sigma^{2}_{S}$ represents the shadowing variance.
\begin{align}
\mathcal{P}_{p} = 10^{-(\mathcal{P}_{\mathrm{d}} + \mathcal{P}_{\mathrm{S}})/10}
\end{align}
Where $\mathcal{P}_{p}(t) = 10^{-(\mathcal{P}_{\mathrm{d}} + \mathcal{P}_{\mathrm{S}})/10}$, $\mathcal{P}_{\mathrm{S}} \sim \mathcal{CN}(0, \sigma^{2}_{S})$ represents the large scale fading loss due to distance and shadowing. $Q_{p}, \alpha_{p,q}, \tau_{p,q}, f_{\mathrm{O,p}}$ represents the number of paths, path gain, delay, and offset in the frequency, respectively. In the above equation, we have approximated the time-varying to be negligible, which is justifiable due to the duration of the 5G NR slot.

### Doppler Shift

The Doppler shift is defined as an apparent shift in the transmitted signal frequency due to movement between the transmitter and receiver. Considering the geometry shown in the system model, the Doppler shift introduced by the satellite in the BC can be expressed as
\begin{align}
f_{\mathrm{D}}(t) &= \frac{f_{c}}{c}v(t)\cos{\theta(t)}
\end{align}

where $f_{\mathrm{D}}, f_{c}, v, c, \theta$ are Doppler frequency, the carrier frequency of the transmitted signal, the magnitude of the relative velocity between receiver and transmitter, speed of light, and the angle between the relative velocity vector and direction of arrival of the transmitted signal. In above equation $\theta(t)$ and $v(t)$ can be computed from the satellite state vector as expressed below

\begin{align}
f_{\mathrm{D,sat}}(t) &= \frac{f_{c}}{c}|\mathbf{v}_{\mathrm{BC, sat}}(t)|\frac{\mathbf{v}_{\mathrm{BC, sat}}(t)^T\mathbf{d}_{\mathrm{BC, sat}}(t)}{|\mathbf{v}_{\mathrm{BC, sat}}(t)||\mathbf{d}_{\mathrm{BC, sat}}(t)|}\\
f_{\mathrm{D,sat}} &\approx \frac{f_{c}}{c}\frac{\mathbf{v}_{\mathrm{BC, sat}}(t)^T\mathbf{d}_{\mathrm{BC, sat}}(t)}{|\mathbf{d}_{\mathrm{BC, sat}}(t)|}.
\end{align}

Where $\mathbf{v}_{\mathrm{BC, sat}}(t) = - \mathbf{v}_{\mathrm{sat}}(t)$ is the relative velocity vector between the BC and the satellite, and $\mathbf{v}_{\mathrm{sat}}(t)$ is the absolute satellite. Whereas $\mathbf{d}_{\mathrm{BC,sat}}(t) = \mathbf{r}_{\mathrm{BC}}(t) - \mathbf{r}_{\mathrm{sat}}(t)$ is the relative position vector of satellite with respect to BC. Furthermore, in the equation, we have approximated the Doppler frequency to be constant, which is justifiable due to the duration of the 5G NR slot. The figure shows the shift in Doppler frequency due to STARLINK 4085 satellite pointing towards the BC for the S-band and Ka-band.

Further more the channel impulse response in frequency domain can be expresses as

\begin{align}
\mathbf{h}_{p}(t, f) &= \int_{-\infty}^{\infty}\mathbf{h}_{p}(t, \tau)e^{-\jmath 2\pi f\tau}d\tau\\
&= \sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\mathbf{\alpha}_{p,q}e^{-j2\pi f_{\mathrm{D,p}}t}\int_{-\infty}^{\infty}\delta(\tau - \tau_{p,q})e^{-\jmath 2\pi f\tau}d\tau\\
&= \sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\mathbf{\alpha}_{p,q}e^{-j2\pi f_{\mathrm{D,p}}t}\times e^{-\jmath 2\pi f\tau_{p,q}}
\end{align}

## Satellite Transmitted Signal

We consider an OFDM signal model supporting multi-sub-carrier-spacing as that of 5G New Radio. The OFDM baseband equivalent model of the transmit signal can then be expressed as

\begin{align}
\hat{\mathbf{x}}_{p}(t) = \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi r\Delta f t}.
\end{align}
For $t = -N_{\text{CP}}T_{s},\cdots, (N - 1)T_{s}$. Where $N, N_{\text{CP}}, \mathbf{X}_{p}[k]$ denotes FFT size, length of cyclic prefix, the $k$th sub-carrier symbol.

### Doppler Pre-Compensation
Pre-compensation is a technique that allows satellites to compensate the transmitted signal such that when the user terminal receives the signal, the frequency swing reduces. Before transmission satellite pre-compensates the signal as
\begin{align}
\mathbf{x}_{p}(t) &= \hat{\mathbf{x}}_{p}(t)e^{\jmath 2\pi rf_{\mathrm{D,sat}}^{\mathrm{pc}}t}\notag\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + f_{\mathrm{D,sat}}^{\mathrm{pc}})t}\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}
\end{align}

Where $\epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} = \frac{f_{\mathrm{D,sat}}^{\mathrm{pc}}}{\Delta f} =\epsilon_{\mathrm{D,I,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{D,F,sat}}^{\mathrm{pc}}$ is the **normalized Doppler frequency** consisting of Integer and fractional part.

## User Terminal Received Sigal

At the user-terminal, we further introduce the carrier frequency offset $f_{\mathrm{o}}$ impairment due to the mismatch between the transmitter and receiver local oscillators. Hence the downlink received signal at the $p$th UT at $t$th time instant can be expressed

\begin{align}
\mathbf{y}_{p}(t) &= \big(\mathbf{h}_{p}(t, \tau)\otimes \mathbf{x}_{p}(t)\big)e^{-\jmath 2\pi f_{\mathrm{o}} t} + \mathbf{n}_{p}(t)\\
&= e^{-\jmath 2\pi f_{\mathrm{o}}t}\underbrace{\int_{-\infty}^{\infty}\mathbf{h}_{p}(t, \tau^\prime)\mathbf{x}_{p}(t - \tau^\prime)d\tau^\prime}_{\mathcal{I}} + \mathbf{n}_{p}(t).\notag\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[t, r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f]e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}\times e^{-\jmath 2\pi f_{\mathrm{o}}t} + \mathbf{n}_{p}(t)\\
\mathbf{y}_{p}[n] &= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}})n/N}\times e^{-\jmath 2\pi \epsilon_{\mathrm{o}}n/N} + \mathbf{n}_{p}[n]\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}})n/N} + \mathbf{n}_{p}[n]\\
\end{align}

Where $\mathcal{I}$ can be evaluated as

\begin{align}
\mathcal{I} &= \int_{-\infty}^{\infty}\mathbf{h}_{p}(t, \tau^\prime)\mathbf{x}_{p}(t - \tau^\prime)d\tau^\prime\\
&= \int_{-\infty}^{\infty}\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{-j2\pi f_{\mathrm{D,p}}t}\delta(\tau^\prime - \tau_{p,q})\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}})\Delta f(t - \tau^\prime)}d\tau^\prime\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}})\Delta ft}\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{-j2\pi f_{\mathrm{D,p}}t}\int_{-\infty}^{\infty}\delta(\tau^\prime - \tau_{p,q})e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)\tau^\prime}d\tau^\prime\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{-j2\pi f_{\mathrm{D,p}}t}e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)\tau_{p,q}}\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}\mathbf{H}_{p}[t, r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f]\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[t, r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f]e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}.
\end{align}

Another approach

\begin{align}
\mathcal{I} &= \int_{-\infty}^{\infty}\mathbf{h}_{p}(t, \tau^\prime)\mathbf{x}_{p}(t - \tau^\prime)d\tau^\prime\\
&= \int_{-\infty}^{\infty}\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{-j2\pi\epsilon_{\mathrm{D,sat}}\Delta ft}\delta(\tau^\prime - \tau_{p,q})\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)(t - \tau^\prime)}d\tau^\prime\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{-j2\pi \epsilon_{\mathrm{D,sat}}\Delta ft}\int_{-\infty}^{\infty}\delta(\tau^\prime - \tau_{p,q})e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)\tau^\prime}d\tau^\prime\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)t}\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{-j2\pi \epsilon_{\mathrm{D,sat}}\Delta ft}e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)\tau_{p,q}}\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\sqrt{\mathcal{P}_{p}}\sum_{q = 0}^{Q_{p} - 1}\alpha_{p,q}e^{\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f - \epsilon_{\mathrm{D,sat}}\Delta f)t}e^{-\jmath 2\pi (r\Delta f + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}\Delta f)\tau_{p,q}}\\
\end{align}

#### Frequency Domain

\begin{align}
\mathbf{Y}_{p}[r] &= \int_{-\infty}^{\infty}\mathbf{y}_{p}(t)e^{-\jmath 2\pi r\Delta f t}dt\\
&= \int_{-\infty}^{\infty}\Big(\big(\mathbf{h}_{p}(t, \tau)\otimes \mathbf{x}_{p}(t)\big)e^{-\jmath 2\pi \epsilon_{\mathrm{o}}\Delta f t} + \mathbf{n}_{p}(t)\Big)e^{-\jmath 2\pi r\Delta f t}dt\\
&= \int_{-\infty}^{\infty}\big(\mathbf{h}_{p}(t, \tau)\otimes \mathbf{x}_{p}(t)\big)e^{-\jmath 2\pi \epsilon_{\mathrm{o}}\Delta f t}e^{-\jmath 2\pi r\Delta f t}dt + \int_{-\infty}^{\infty}\mathbf{n}_{p}(t)e^{-\jmath 2\pi r\Delta f t}dt\\
&= \int_{-\infty}^{\infty}\Bigg(\int_{-\infty}^{\infty}\mathbf{h}_{p}(t, \tau^\prime)\frac{1}{N}\sum_{k = 0}^{N - 1}\mathbf{X}_{p}[k]e^{\jmath 2\pi (k + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}})\Delta f(t - \tau^\prime)}d\tau^\prime\Bigg)e^{-\jmath 2\pi (\epsilon_{\mathrm{o}} + r)\Delta f t}dt + \mathbf{N}_{p}[r]\\
\end{align}

### Estimation

Before the UT can synchronize with the satellite, the residual frequency offset need to be compensated after estimating it. So we rely on joint PSS detection and coarse frequency offset estimation.

#### Coarse Frequency Offset Estimation

\begin{align}
\mathbf{y}_{rp}[n] &= \mathbf{y}_{p}[n]e^{\jmath 2\pi f_{\mathrm{co}}nT_{s}} = \mathbf{y}_{p}[n]e^{\jmath 2\pi \epsilon_{i}n/N},~\text{where}~ \epsilon_{i}\in \mathbb{Z}\notag\\
&= \bigg(\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{o})n/N} + \mathbf{n}_{p}[n]\bigg)e^{\jmath 2\pi \epsilon_{\mathrm{i}}n/N}\notag\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}})n/N}e^{\jmath 2\pi \epsilon_{i}n/N} + \mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{i}n/N}\notag\\
&= \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}} - \epsilon_{i})n/N} + \mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{i}n/N}\notag\\
\end{align}

Then perform the cross correlation with the $\mathbf{x}_{\mathrm{m}}[n]$ is time domain 5G NR primary synchronization signal generated the UT for all possible value of physical layer cell ID, $N_{\mathrm{cell}}^{\mathrm{ID}}\in\{0,1,2\}$

\begin{align}
\mathbf{x}_{\mathrm{m}}[n] = \frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{\mathrm{m}}[r]e^{\jmath 2\pi rn/N}.
\end{align}

which can be expressed as
\begin{align}
\mathbf{\Lambda}_{m,l}[k] &= \sum_{n = 0}^{N - 1}\mathbf{y}_{rp}[n]\mathbf{x}_{\mathrm{m}}^{*}[n - k], ~ k = 2N - 1, m \in \{0,1,2\}\\
&= \sum_{n = 0}^{N - 1}\Big(\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}})n/N}e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N} + \mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\Big)\mathbf{x}_{\mathrm{m}}^{*}[n - k]\\
&= \sum_{n = 0}^{N - 1}\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}})n/N}e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\mathbf{x}_{\mathrm{m}}^{*}[n - k]\\ &+ \sum_{n = 0}^{N - 1}\mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\mathbf{x}_{\mathrm{m}}^{*}[n - k]\\
&= \sum_{n = 0}^{N - 1}\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}})n/N}e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\Bigg(\frac{1}{N}\sum_{r^{\prime} = 0}^{N - 1}\mathbf{X}_{\mathrm{m}}[r^{\prime}]e^{\jmath 2\pi r^{\prime}(n-k)/N}\Bigg)^{*}\\ &+ \sum_{n = 0}^{N - 1}\mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\mathbf{x}_{\mathrm{m}}^{*}[n - k]\\
&= \sum_{n = 0}^{N - 1}\frac{1}{N}\sum_{r = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}} -\epsilon_{\mathrm{co}})n/N}\frac{1}{N}\sum_{r^{\prime} = 0}^{N - 1}\mathbf{X}^{*}_{\mathrm{m}}[r^{\prime}]e^{-\jmath 2\pi r^{\prime}(n-k)/N}\\ &+ \sum_{n = 0}^{N - 1}\mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\mathbf{x}_{\mathrm{m}}^{*}[n - k]\\
&= \frac{1}{N^2}\sum_{n = 0}^{N - 1}\sum_{r = 0}^{N - 1}\sum_{r^{\prime} = 0}^{N - 1}\mathbf{X}_{p}[r]\mathbf{X}^{*}_{\mathrm{m}}[r^{\prime}]\mathbf{H}_{p}[n, r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}}]e^{-\jmath 2\pi (r + \epsilon_{\mathrm{D,sat}}^{\mathrm{pc}} + \epsilon_{\mathrm{o}} -\epsilon_{\mathrm{co}})n/N}e^{-\jmath 2\pi r^{\prime}(n-k)/N}\\ &+ \sum_{n = 0}^{N - 1}\mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{\mathrm{co}}n/N}\mathbf{x}_{\mathrm{m}}^{*}[n - k]\\
\end{align}

Where

\begin{align}
(\hat{m}, \hat{l}) = \arg \max_{m, l} \sum_{r = 0}^{N_{R} - 1}\sum_{t = 0}^{N_{T} - 1}\Big\vert\mathbf{\Lambda}_{m,l}[k]\Big\vert
\end{align}

##### PSS Detection Flase Alarm

False alarm means that the PSS is detected but there is not a corresponding sequence transmitted. If UT do not transmit any PSS, the received signal is Gaussian noise, i.e.

\begin{align}
\mathbf{y}_{rp}[n] &= \mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{i}n/N}
\end{align}

So the cross correlation will be
\begin{align}
\mathbf{\Lambda}_{m,l}[k] &= \frac{\sum_{n = 0}^{N - 1}\mathbf{y}_{rp}[n]\mathbf{x}_{\mathrm{m}}^{*}[n - k]}{\sum_{n = 0}^{N - 1}\vert\mathbf{x}_{\mathrm{m}}[n]\vert^{2}}\\
&= \frac{\sum_{n = 0}^{N - 1}\mathbf{n}_{p}[n]e^{\jmath 2\pi \epsilon_{i}n/N}\mathbf{x}_{\mathrm{m}}^{*}[n - k]}{E_{\mathbf{x}_{\mathrm{m}}}}\\
&= \frac{\sum_{n = 0}^{N - 1}\mathbf{n}_{p}[n]\mathbf{x}_{\mathrm{m}}^{*}[n - k]e^{\jmath 2\pi \epsilon_{i}n/N}}{E_{\mathbf{x}_{\mathrm{m}}}}\\
\end{align}

So the probability of false alarm can be evaluated as
\begin{align}
P_{\mathrm{FA}} &= \Pr\Bigg\{\sum_{r = 0}^{N_{R} - 1}\sum_{t = 0}^{N_{T} - 1}\Big\vert\mathbf{\Lambda}_{m,l}[k]\Big\vert > \beta \Bigg\}\\
&= \Pr\Bigg\{\sum_{r = 0}^{N_{R} - 1}\sum_{t = 0}^{N_{T} - 1}\mathbf{\Lambda}_{m,l}[k]\mathbf{\Lambda}_{m,l}^{*}[k] > \beta \Bigg\}\\
\end{align}

##### Compensation

Here we compensate the received signal with the estimated coarse frequency offset, and estimate the fine frequency offest.

#### Fine Frequency Offset Estimation

For fine frequency offset estimation we rely on auto correlation between and signal and its cyclic prefix part.

\begin{align}
\hat{f}_{0} &= -\frac{\angle\big\{\mathbb{E}\big[\mathbf{y}_{\mathrm{pre}}[n]\mathbf{y}_{\mathrm{pre}}^{*}[n + N]\big]\big\}}{2\pi NT_{s}}\notag\\
\end{align}