# Photonic-Assisted Scanning Receivers for Microwave Frequency Measurement
## 2 Principle of Operation  
The operation of the scanning system can be divided into four steps, as illustrated in Figure 1. First, a fast-swept tunable laser produces an optical carrier (point A, inset (A) in Figure 1). Second, an EOM modulates the unknown input RF signal onto the carrier frequency, hence generating the upper and lower sidebands, with each sideband containing an entire copy of the microwave spectral information (point B, inset (B) in Figure 1). When the carrier frequency changes with a constant sweeping step \(\Delta w\), the modulated signal is successively shifted across the filter response in the frequency domain, which can be represented by \(x(w+n\Delta w)\), where \(n\) denotes the number of steps, \(n\)=1,2,...,N. Third, an optical filter with spectral response \(h(w)\) acts as an optical spectral weight that multiplies the frequency spectrum of the modulated signal at each frequency with the spectral weight at that frequency (point C, inset (C) in Figure 1), which results in a weighted signal \(h(w)x(w+n\Delta w)\). Fourth, an optical power meter is employed to sum the power of the weighted signal across the entire frequency range, which is the integration operation (point D, inset (D) in Figure 1). The power meter readout after \(n\) sweeping steps, \(y[n]\), can be expressed mathematically as  
\[y[n]=\mathrm{h}*x=\sum_{w=-\infty}^{\infty}h(w)x(w+n\Delta w), \tag{1}\]where \(w\) is the angular frequency, \(h(w)\) is the spectral response of the optical filter, and * depicts the cross-correlation operator. It can be noted that the operation of cross-correlation can be represented as convolution simply by pre-flipping the spectrum of the modulated signal \(x(w)\). Given the typical case of a double sideband optical modulation where the sidebands are symmetrical, the operation of the scanning system can thus be directly described by the convolutional model [25]. Figure 2 illustrates the advantage of our proposed scanning system--based on deconvolution--over the conventional threshold detection method [16] by using a two-tone RF signal as an example. As shown in Equation (1), the detected optical power in the scanning receiver forms a convolutional result, as the intensity over time represents the intensity sum of shifted pulses of the filter shape where the location of each pulse is determined by the frequency of optical components. Therefore, if two RF tones are too closely spaced, the two shifted pulses would merge into one, therefore making it impossible or extremely difficult to resolve the location of each individual pulses via the conventional detection method. In contrast, the deconvolution theorem we propose eliminates the detrimental effect of filter shape on the power meter reading by using an effective matrix pencil method to reconstruct the original spectrum of an input RF signal, hence breaking the measurement limit inherent in conventional methods. Due to the sampling nature of scanning receivers, Equation (1) can be rewritten in a discrete model where the spectrum of the modulated signal and filter response can be represented by their sampled values at a constant frequency interval \(\Delta w\) as \(x[n]\) and \(h[n]\), respectively. Then, a discrete Fourier transform can be adopted to convert the operation of convolution to multiplication. The Fourier transform representation of \(x[n]\), \(x[k]\), is given by  
Figure 1: Schematic diagram of the scanning receiver system. Insets (A), (B), (C) and (D) illustrate the operation principle of the scanning system at points (A), (B), (C) and (D), respectively. Figure 2: Comparison of the scanning systems based on deconvolution and the conventional method. \[X[k]=\frac{Y[k]}{H[k]}=\frac{\sum\limits_{n=1}^{N}y[n]e^{-j2\pi(k-1)(n-1)/N}}{ \sum\limits_{n=1}^{N}h[n]e^{-j2\pi(k-1)(n-1)/N}}, \tag{2}\]  
where \(Y[k]\) and \(H[k]\) are the discrete Fourier transforms of \(y[n]\) and \(h[n]\), respectively.