Chirp-based Multicarrier Communication

Related Publication: (Guo et al., 2024) (Guo et al., 2024)

Disclaimers

This is a short non-academic introduction to a general topic I worked on, notations are very loose.

Motivations

The motivation is a simple one, what are the tradeoffs associated with using chirp-based multicarrier systems as supposed to sinusoidal ones like OFDM. What do we gain at the cost of linearity?

For instance, when there is narrow band interference, OFDM on principle will suffer great BER loss due to decoding errors dominated by subcarriers of a certain frequency band. However, chirps do not have such concern as all subchirps occupy the entire bandwidth and theoretically errors caused by narrow band interference are propogated across subchirps, thereby significantly improving performance.

Model

There are a number of chirp models in literature. We restrict the scope of chirp signals to the action of a special linear group \(SL_2(\mathbb{R})\) on the time-frequency plane. A \(SL_2(\mathbb{R})\) group is a group of \(2 \times 2\) real matrices with determinant one:

Each matrix uniquely defines an integral transform in the following way: \(\begin{align}\label{2.3} S(u) & =O_F^{(a,b,c,d)}(s(t)) = \begin{cases} \int_{-\infty}^\infty K^{(a,b,c,d)}(t,u) s(t) \rm{d}t \quad & \text{\(b\neq 0\)} \\ \sqrt{d}e^{\frac{j}{2} cdu^2} s(du) \quad & \text{\(b = 0\)}, \end{cases} \end{align}\) where the transform basis is defined as: \(\begin{equation} K^{(a,b,c,d)}(t,u) = \sqrt{\frac{1}{2\pi |b|}}e^{-j(\frac{1}{2}\frac{d}{b}u^2+ \frac{ut}{b}+\frac{1}{2}\frac{a}{b}t^2)}. \end{equation}\)

After some algebra and assumptions, the resulting generalizing transform is in optics referred to as linear canonical transform, in communication it’s called discrete affine Fourier transform. Assume \(\mathbf{s} \in \mathbb{S}^{N\times1}\) is the \(N \times 1\) transform domain and \(\mathbf{x}\) is the time-domain signal, then \(\mathbf{s}\) and \(\mathbf{x}\) are related in matrix-vector form as: \(\begin{equation} \mathbf{x} = \mathbf{\Lambda}_{c_1}^\mathcal{H}\mathbf{F}^\mathcal{H}\mathbf{\Lambda}_{c_2}^\mathcal{H} \mathbf{s}, \end{equation}\) where elements of \(\mathbf{\Lambda}_c\) are defined as \([\mathbf{\Lambda}_c]_{n,n}=e^{-j2\pi c(n-1)^2}\). The IDAFT can be expressed equivalently as the following: \(\begin{equation} x_n = \frac{1}{\sqrt{N}}\sum_{m=0}^{M-1} e^{j2\pi (c_1 n^2+\frac{1}{N}mn+c_2m^2)} s_m, \end{equation}\)

Problems

From now, the research problems branch into many directions. A number of transforms commonly used in wireless and optical communication can be represented accordingly. DFT, used as the transform basis for OFDM, is characterized by the \(2\times 2\) DAFT matrix \((0,\frac{1}{2\pi};-2\pi,0)\). Discrete Fresnel transform, used in OCDM, with Talbot distance \(z\), chirp period \(T\) and wavelength \(\lambda\) has the following matrix \((1,\frac{T^2}{2\pi};0,1)\), where \(z = \frac{T^2}{\lambda}\). Also seen extensive use in wireless communication and signal processing are FrFT, Laplace transform, which are among the many special cases of AFT.

My research focus majorly on OCDM-based systems with \(2Nc_1 \in \mathbb{Z}\). The specific questions tackled, and once again specifics readers should refer to published works and my thesis (which I am currently withholding release for publication reasons).

1. Demonstration that the peak-to-average power ratio (PAPR) and capacity of DAFT-based multicarrier equivalent channels are independent of the basis used. The capacity and probability distribution of PAPR are both derived and simulated.(Thesis)
2. Detailed analysis of the bit error rate (BER) performance of chirp multicarrier systems through diversity and piece-wise error probability (PEP) analysis. Our study reveals that the chirp rate significantly influences multipath diversity, coding gains, and error performance under different equalization schemes. We characterize diversity in relation to the chirp basis and propose algorithms to enhance BER performance and reduce equalization complexity.((Guo et al., 2024))
3. Impact of carrier frequency offset (CFO) on chirp multicarrier systems. Our analysis includes the derivation of mutual information and BER performance under CFO. Additionally, we study CFO estimation, compensation, and related identifiability issues for chirp systems. We propose algorithms to mitigate these challenges effectively.((Guo et al., 2024))