The SC-FrFDE system is shown as Figure 1. After QPSK modulation, the current block contains *N* samples *x*(*n*), *n* = 0, 1, ⋯, *N* − 1. In order to reduce the symbol interference, GI is needed to insert in. The cyclic prefix (CP) of *x*(*n*) is not just the copy of its tail data but should have the chirp periodicity. This kind of CP is called chirp-periodic circular prefix; for simplicity, it is denoted as chirp CP. The signal being added chirp CP is expressed as

{\mathit{x}}_{\mathrm{cp}}={\left[\mathit{x}\left(-{\mathit{N}}_{\mathit{c}}\right),\mathit{x}\left(-{\mathit{N}}_{\mathit{c}}+1\right),\dots ,\mathit{x}\left(0\right),\dots ,\mathit{x}\left(\mathit{N}-1\right)\right]}^{\mathit{T}}

(13)

In this paper, {⋅}^{T}, {⋅}^{H}, {⋅}* stand for transpose, conjugate transpose and conjugate. *N*_{
c
} is the length of chirp CP. According to (10), the chirp CP is written as

\begin{array}{ll}\mathit{x}\left(-{\mathit{N}}_{\mathit{c}}+\mathit{n}\right)& =\mathit{x}\left(\mathit{N}-{\mathit{N}}_{\mathit{c}}+\mathit{n}\right)\cdot {\mathit{e}}^{\mathit{j}\frac{1}{2}cot\mathit{\alpha}\left[{\left(\mathit{N}-{\mathit{N}}_{\mathit{c}}+\mathit{n}\right)}^{2}-{\left(-{\mathit{N}}_{\mathit{c}}+\mathit{n}\right)}^{2}\right]\mathit{\Delta}{\mathit{t}}^{2}},\mathit{n}\\ =0,1,\dots ,{\mathit{N}}_{\mathit{c}}-1\end{array}

(14)

To describe this process in matrix form, we finally get

{\mathit{x}}_{\mathrm{cp}}={\mathit{T}}_{\mathrm{cp}}\times \mathit{x}

(15)

in which

{\mathit{T}}_{\mathrm{cp}}={\left[\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{0}}_{{\mathit{N}}_{\mathit{c}}\times \left(\mathit{N}-{\mathit{N}}_{\mathit{c}}\right)}\hfill & \hfill {\mathit{C}}_{{\mathit{N}}_{\mathit{c}}\times {\mathit{N}}_{\mathit{c}}}\hfill \end{array}\hfill \\ \hfill {\mathit{I}}_{\mathit{N}\times \mathit{N}}\hfill \end{array}\right]}_{\left(\mathit{N}+{\mathit{N}}_{\mathit{c}}\right)\times \mathit{N}}

(16)

{0}_{{\mathit{N}}_{\mathit{c}}\times \left(\mathit{N}-{\mathit{N}}_{\mathit{c}}\right)} is a *N*_{
c
} × (*N*−*N*_{
c
}) matrix with elements of zeroes. *I*_{N × N} represents the identity matrix.

\begin{array}{ll}{\mathit{C}}_{{\mathit{N}}_{\mathit{c}}\times {\mathit{N}}_{\mathit{c}}}& =\mathrm{diag}\left[{\mathit{e}}^{\mathit{j}\frac{1}{2}cot\mathit{\alpha}\left[{\left(\mathit{N}-{\mathit{N}}_{\mathit{c}}+\mathit{N}\right)}^{2}-{\left(-{\mathit{N}}_{\mathit{c}}+\mathit{n}\right)}^{2}\right]\mathrm{\Delta}{\mathit{t}}^{2}}\right],\mathit{n}\\ =0,1,\dots ,{\mathit{N}}_{\mathit{c}}-1\end{array}

(17)

After inserting chirp CP, the signal is transmitted to the fast time-varying channels *H*. It varies during one data block, and its time domain impulse response is *h*(*n*, *l*) with length of *L*. Then the received signal is given by

{\mathit{y}}_{\mathrm{cp}}=\mathit{H}\cdot {\mathit{x}}_{\mathrm{cp}}+{\mathit{\gamma}}_{\mathrm{cp}}

(18)

*γ*_{cp} is the additive white Gaussian noise.

\mathit{H}=\left[\begin{array}{ccccc}\hfill \mathit{h}\left(-{\mathit{N}}_{\mathit{c}},0\right)\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \mathit{h}\left(-{\mathit{N}}_{\mathit{c}}+1,1\right)\hfill & \hfill \mathit{h}\left(-{\mathit{N}}_{\mathit{c}}+1,0\right)\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \vdots \hfill & \hfill \mathit{h}\left(-{\mathit{N}}_{\mathit{c}}+2,1\right)\hfill & \hfill \ddots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill \\ \hfill \mathit{h}\left(-{\mathit{N}}_{\mathit{c}}+\mathit{L}-1,\mathit{L}-1\right)\hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill \\ \hfill 0\hfill & \hfill \mathit{h}\left(-{\mathit{N}}_{\mathit{c}}+\mathit{L},\mathit{L}-1\right)\hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill \\ \hfill \vdots \hfill & \hfill 0\hfill & \hfill \vdots \hfill & \hfill 0\hfill & \hfill \vdots \hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \mathit{h}\left(\mathit{N}-2,0\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \mathit{h}\left(\mathit{N}-1,1\right)\hfill & \hfill \mathit{h}\left(\mathit{N}-1,0\right)\hfill \end{array}\right]

(19)

In SC-FrFDE system, the removal of chirp CP is the same with that in SC-FDE system and is expressed as follows:

\mathit{y}={\mathit{R}}_{\mathit{cp}}\cdot {\mathit{y}}_{\mathit{cp}}={\mathit{R}}_{\mathit{cp}}\cdot \mathit{H}\cdot {\mathit{T}}_{\mathit{cp}}\cdot \mathit{x}+\mathit{\gamma}=\overline{\mathit{\eta}}\cdot \mathit{x}+\mathit{\gamma}

(20)

{\mathit{R}}_{\mathit{cp}}={\left[\begin{array}{cc}\hfill {0}_{\mathit{N}\times {\mathit{N}}_{\mathit{c}}}\hfill & \hfill {\mathit{I}}_{\mathit{N}}\hfill \end{array}\right]}_{\mathit{N}\times \left(\mathit{N}+{\mathit{N}}_{\mathit{c}}\right)}

(21)

\mathit{y}={\left[\mathit{y}\left(0\right),\mathit{y}\left(1\right),\dots ,\mathit{y}\left(\mathit{N}-1\right)\right]}^{\mathit{T}}

(22)

By adding and removing chirp CP, the time domain channel matrix *H* is transformed to (23)

\begin{array}{l}\overline{\mathit{H}}={\left[\begin{array}{ccccccc}\hfill \mathit{h}\left(0,0\right)\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill \mathit{h}\left(0,\mathit{L}-1\right){\mathit{e}}^{\mathit{j}\frac{1}{2}cot\mathit{\alpha}\left\{{\left[\mathit{N}-\left(\mathit{L}-1\right)\right]}^{2}-{\left(\mathit{L}-1\right)}^{2}\right\}\mathrm{\Delta}{\mathit{t}}^{2}}\hfill & \hfill \cdots \hfill & \hfill \mathit{h}\left(0,1\right){\mathit{e}}^{\mathit{j}\frac{1}{2}cot\mathit{\alpha}\left\{\left[{\left(\mathit{N}-1\right)}^{2}\right]-{\left(-1\right)}^{2}\right\}\mathrm{\Delta}{\mathit{t}}^{2}}\hfill \\ \hfill \vdots \hfill & \hfill \mathit{h}\left(1,0\right)\hfill & \hfill \cdots \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill \mathit{h}\left(\mathit{L}-2,\mathit{L}-2\right)\hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \mathit{h}\left(\mathit{L}-2,\mathit{L}-1\right){\mathit{e}}^{\mathit{j}\frac{1}{2}cot\mathit{\alpha}\left\{\left[{\left(\mathit{N}-1\right)}^{2}\right]-{\left(-1\right)}^{2}\right\}\mathrm{\Delta}{\mathit{t}}^{2}}\hfill \\ \hfill \mathit{h}\left(\mathit{L}-1,\mathit{L}-1\right)\hfill & \hfill \mathit{h}\left(\mathit{L}-1,\mathit{L}-2\right)\hfill & \hfill \vdots \hfill & \hfill \mathit{h}\left(\mathit{N}-\mathit{L},0\right)\hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill \mathit{h}\left(\mathit{L},\mathit{L}-1\right)\hfill & \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \mathit{h}\left(\mathit{N}-\mathit{L}+1,0\right)\hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill \vdots \hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \mathit{h}\left(\mathit{N}-2,\mathit{L}-2\right)\hfill & \hfill \vdots \hfill & \hfill \ddots \hfill & \hfill \vdots \hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \cdots \hfill & \hfill \mathit{h}\left(\mathit{N}-1,\mathit{L}-1\right)\hfill & \hfill \mathit{h}\left(\mathit{N}-1,\mathit{L}-2\right)\hfill & \hfill \cdots \hfill & \hfill \mathit{h}\left(\mathit{N}-1,0\right)\hfill \end{array}\right]}_{\mathit{N}\times \mathit{N}}\end{array}

(23)

Through the process above, the linear convolution will be turned into fractional circular convolution of the signal and the equivalent channel response. Let us unfold (20) to explain this problem.

\begin{array}{ll}\mathit{y}\left(\mathit{n}\right)& ={\displaystyle \sum _{\mathit{l}=0}^{\mathit{L}\u20121}\mathit{h}\left(\mathit{n},\mathit{l}\right)\mathit{x}\left(\mathit{N}+\mathit{n}-\mathit{l}\right)}\\ \cdot {\mathit{e}}^{\mathit{j}\frac{1}{2}cot\mathit{\alpha}{\left[\mathit{N}+\left(\mathit{n}-\mathit{l}\right)\right]}^{2}\mathrm{\Delta}{\mathit{t}}^{2}-\mathit{j}\frac{1}{2}cot\mathit{\alpha}{\left(\mathit{n}-\mathit{l}\right)}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}\\ +\mathit{\gamma}\left(\mathit{n}\right)\phantom{\rule{0.6em}{0ex}}0\le \mathit{n}\le \mathit{L}\end{array}

(24)

\mathit{y}\left(\mathit{n}\right)={\displaystyle \sum _{\mathit{l}=0}^{\mathit{L}\u20121}}\mathit{h}\left(\mathit{n},\mathit{l}\right)\mathit{x}\left(\mathit{n}-\mathit{l}\right)+\mathit{\gamma}\left(\mathit{n}\right)\phantom{\rule{0.84em}{0ex}}\mathit{L}+1\le \mathit{n}\le \mathit{N}-1

(25)

Because of the chirp periodicity shown in (10), we can combine Equations 24 and 25 into one.

\begin{array}{ll}\mathit{y}\left(\mathit{n}\right)& ={\displaystyle \sum _{\mathit{l}=0}^{\mathit{L}\u20121}\mathit{h}\left(\mathit{n},\mathit{l}\right)\cdot \mathit{x}{\left(\left(\mathit{n}-\mathit{l}\right)\right)}_{\mathit{p},\mathit{N}}\cdot {\mathit{R}}_{\mathit{N}}\left(\mathit{n}\right)}+\mathit{\gamma}\left(\mathit{n}\right)\phantom{\rule{0.2em}{0ex}}\\ =\phantom{\rule{0.3em}{0ex}}{\displaystyle \sum _{\mathit{l}=0}^{\mathit{N}-1}\mathit{h}\left(\mathit{n},\mathit{l}\right)\cdot \mathit{x}{\left(\left(\mathit{n}-\mathit{l}\right)\right)}_{\mathit{p},\mathit{N}}\cdot {\mathit{R}}_{\mathit{N}}\left(\mathit{n}\right)}+\mathit{\gamma}\left(\mathit{n}\right)\phantom{\rule{0.5em}{0ex}}0\le \mathit{n}\le \mathit{N}-1\end{array}

(26)

Equation 26 indicates that by adding and removing chirp CP, the linear convolution is turned into circular convolution (It is not fractional circular convolution). It is the circular convolution about *h*(*n*, *l*) and one chirp periodic extension sequence *x*((*n*))_{p,N}.

Now let us turn this circular convolution into fractional circular convolution.

\begin{array}{l}\mathit{y}\left(\mathit{n}\right)={\displaystyle \sum _{\mathit{l}=0}^{\mathit{N}-1}\mathit{h}\left(\mathit{n},\mathit{l}\right)\cdot \mathit{x}{\left(\left(\mathit{n}-\mathit{l}\right)\right)}_{\mathit{p},\mathit{N}}\cdot {\mathit{R}}_{\mathit{N}}\left(\mathit{n}\right)}\phantom{\rule{0.3em}{0ex}}+\mathit{\gamma}\left(\mathit{n}\right)\\ \phantom{\rule{0.3em}{0ex}}={\mathit{e}}^{\frac{-\mathit{j}cot\mathit{a}\cdot {\mathit{n}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}{\displaystyle \sum _{\mathit{l}=0}^{\mathit{N}-1}\phantom{\rule{0.1em}{0ex}}\left[{\mathit{e}}^{\frac{\mathit{j}cot\mathit{a}\cdot {\mathit{n}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}\mathit{h}\left(\mathit{n},\mathit{l}\right)\right]\mathit{x}{\left(\left(\mathit{n}-\mathit{l}\right)\right)}_{\mathit{p},\mathit{N}}\cdot {\mathit{R}}_{\mathit{N}}\left(\mathit{n}\right)}+\mathit{\gamma}\left(\mathit{n}\right)\\ ={\mathit{e}}^{\frac{-\mathit{j}cot\mathit{a}\cdot {\mathit{n}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}{\displaystyle \sum _{\mathit{l}=0}^{\mathit{N}-1}\left[\phantom{\rule{0.1em}{0ex}}{\mathit{e}}^{\frac{\mathit{j}cot\mathit{a}\cdot {\mathit{n}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}\mathit{h}\left(\mathit{n},\mathit{l}\right)\phantom{\rule{0.1em}{0ex}}{\mathit{e}}^{\frac{-\mathit{j}cot\mathit{a}\cdot {\left(\mathit{n}-\mathit{l}\right)}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}{\mathit{e}}^{\frac{-\mathit{j}cot\mathit{a}\cdot {\mathit{l}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}\right]}\phantom{\rule{0.1em}{0ex}}\\ \phantom{\rule{1em}{0ex}}{\mathit{e}}^{\frac{\mathit{j}cot\mathit{a}\cdot {\mathit{l}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}\cdot \mathit{x}{\left(\left(\mathit{n}-\mathit{l}\right)\right)}_{\mathit{p},\mathit{N}}\cdot {\mathit{R}}_{\mathit{N}}\left(\mathit{n}\right)\phantom{\rule{0.1em}{0ex}}{\mathit{e}}^{\frac{\mathit{j}cot\mathit{a}\cdot {\left(\mathit{n}-\mathit{l}\right)}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}+\mathit{\gamma}\left(\mathit{n}\right)\\ ={\mathit{e}}^{\frac{-\mathit{j}cot\mathit{a}\cdot {\mathit{n}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}{\displaystyle \sum _{\mathit{l}=0}^{\mathit{N}-1}{\mathit{h}}_{\mathit{p}}\left(\mathit{n},\mathit{l}\right){\mathit{e}}^{\frac{\mathit{j}cot\mathit{a}\cdot {\mathit{l}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}}\phantom{\rule{0.3em}{0ex}}\cdot \mathit{x}{\left(\left(\mathit{n}-\mathit{l}\right)\right)}_{\mathit{p},\mathit{N}}\cdot {\mathit{R}}_{\mathit{N}}\left(\mathit{n}\right)\phantom{\rule{0.1em}{0ex}}{\mathit{e}}^{\frac{\mathit{j}cot\mathit{a}\cdot {\left(\mathit{n}-\mathit{l}\right)}^{2}\mathrm{\Delta}{\mathit{t}}^{2}}{2}}+\mathit{\gamma}\left(\mathit{n}\right)\end{array}

(27)

Define {\mathit{h}}_{\mathit{p}}\left(\mathit{n},\mathit{l}\right)=\mathit{h}\left(\mathit{n},\mathit{l}\right){\mathit{e}}^{\mathit{j}cot\mathit{a}\cdot \mathit{nl\Delta}{\mathit{t}}^{2}}{\mathit{e}}^{-\mathit{j}cot\mathit{a}\cdot {\mathit{l}}^{2}\mathrm{\Delta}{\mathit{t}}^{2}} as the equivalent channel response. By selecting the proper FrFT order, *h*_{
p
}(*n*, *l*) will be time-invariant or varies very slowly. In this circumstances, *h*_{
p
}(*n*, *l*) ≈ *h*_{
p
}(*l*); and Equation 27 can be seen as the fractional circular convolution. According to the fractional circular convolution theorem, we can get

{\mathit{Y}}_{\mathit{p}}\left(\mathit{k}\right)=\left[{\mathit{e}}^{-\mathit{j}cot\mathit{a}\cdot {\mathit{k}}^{2}\mathrm{\Delta}{\mathit{u}}^{2}/2}\cdot {\mathit{H}}_{\mathit{p}}\left(\mathit{k}\right)\right]\cdot {\mathit{X}}_{\mathit{p}}\left(\mathit{k}\right)+~\mathit{\Upsilon}\left(\mathit{k}\right)

(28)

In which {\mathit{H}}_{\mathit{p}}\left(\mathit{k}\right)={\displaystyle \sum _{\mathit{l}=0}^{\mathit{L}-1}}{\mathit{h}}_{\mathit{p}}\left(\mathit{l}\right)\cdot {\mathit{e}}^{\mathit{j}cot\mathit{a}\cdot {\mathit{l}}^{2}\cdot \mathrm{\Delta}{\mathit{t}}^{2}/2}\cdot {\mathit{e}}^{\mathit{j}cot\mathit{a}\cdot {\mathit{k}}^{2}\cdot \mathrm{\Delta}{\mathit{u}}^{2}/2}\cdot {\mathit{e}}^{-\mathit{j}\cdot 2\mathit{\pi lk}/\mathit{N}}. *Y*_{
p
}(*k*) and *X*_{
p
}(*k*) is the fractional Fourier domain data of *y*(*n*) and *x*(*n*). The fractional circular convolution is very useful in the FrFD equalization.

Taking DFrFT of the received signal,

{\mathit{Y}}_{\mathit{p}}=\mathit{F}\cdot \mathit{y}=\mathit{F}\cdot \overline{\mathit{H}}\cdot {\mathit{F}}_{-1}\cdot {\mathit{X}}_{\mathit{p}}+\tilde{\mathit{\gamma}}=\tilde{\mathit{H}}\cdot {\mathit{X}}_{\mathit{p}}+\tilde{\mathit{\gamma}}

(29)

*F*_{−1} is the IDFrFT matrix. *F*_{−1} = *F*^{H}. According to (28), \tilde{\mathit{H}} is a diagonal matrix. And its diagonal elements are {\mathit{e}}^{-\mathit{j}cot\mathit{a}\cdot {\mathit{k}}^{2}\mathrm{\Delta}{\mathit{u}}^{2}/2}\cdot {\mathit{H}}_{\mathit{p}}\left(\mathit{k}\right). The final expression is

\begin{array}{ll}{\mathit{Y}}_{\mathit{p}}\left(\mathit{k}\right)& =\tilde{\mathit{H}}\left(\mathit{k},\mathit{k}\right)\cdot {\mathit{X}}_{\mathit{p}}\left(\mathit{k}\right)+~\mathit{\Upsilon}\left(\mathit{k}\right)\\ \phantom{\rule{1em}{0ex}}=\left[{\mathit{e}}^{-\mathit{j}cot\mathit{a}\cdot {\mathit{k}}^{2}\mathrm{\Delta}{\mathit{u}}^{2}/2}\cdot {\mathit{H}}_{\mathit{p}}\left(\mathit{k}\right)\right]\cdot {\mathit{X}}_{\mathit{p}}\left(\mathit{k}\right)+~\mathit{\Upsilon}\left(\mathit{k}\right)\end{array}

(30)

It is assumed that we know the perfect knowledge of FrFD channel response \tilde{\mathit{H}}. Then the FrFD data {\widehat{\mathit{X}}}_{\mathit{p}} can be obtained *via*

{\widehat{\mathit{X}}}_{\mathit{p}}\left(\mathit{k}\right)\approx {\mathit{Y}}_{\mathit{p}}\left(\mathit{k}\right)/\tilde{\mathit{H}}\left(\mathit{k},\mathit{k}\right)

(31)

At last, the FrFD data {\widehat{\mathit{X}}}_{\mathit{p}} is transformed to time-domain data \widehat{\mathit{x}} by taking IDFrFT. The estimation of time-domain data is

\widehat{\mathit{x}}={\mathit{F}}_{-1}\cdot {\widehat{\mathit{X}}}_{\mathit{p}}

(32)