Fig. 1. Schematic diagram of fractional Hilbert transformer based on an integrated Kerr micro-comb source. EDFA: erbium-doped fiber amplifier. PC: polarization controller. MRR: micro-ring resonator. WS: WaveShaper. IM: Intensity modulator. SMF: single mode fiber. OSA: optical spectrum analyzer. BPD: Balanced photodetector. VNA: vector network analyzer.
In order to scale the bandwidth of the standard and fractional Hilbert transformers, we design the spectral transfer function of the Hilbert transformer through the Remez algorithm [69], and change the operating bandwidth by multiplying the corresponding impulse response with the cosine function. Therefore, the resulting discrete impulse response becomes:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h}_{\text{TBWHT}}\left(n\right)=P_{n}\bullet\cos\left(2\pi n\bullet f_{\text{BW}}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)\)
where fBW is the scalable bandwidth. To further switch the centre frequency of the Hilbert transformer, the tap coefficients were multiplied by a sine function to shift the RF transmission spectrum. The corresponding discrete impulse response is given by
\begin{equation} \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h}_{\text{TCFHT}}\left(n\right)=P_{n}\bullet\sin{\left(2\pi n\bullet f_{\text{BW}}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\nonumber \\ \end{equation}
we use the transverse method to realize the Hilbert transformer with RF center frequency and variable bandwidth. The transfer function [1, 8, 9, 70-78] can be described as:
\(\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ F}\left(\omega\right)=\sum_{n-0}^{M-1}{h\left(n\right)e^{-j\omega nT}}\)(7)
where M is the number of taps, ω is the RF angular frequency, T is the time delay between adjacent taps, and h(n) is the tap coefficient of the nth tap.