# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 2 Models and Signals
### Fourier Transform and Spectral Scaling Analysis  
Fourier transform and spectral scaling relation can be employed to make fractal-based spatial signal analyses for urban evolution. Fourier analysis can be applied to both time series and spatial series analyses [19; 27]. The Fourier transform is often employed to identify periods [28]. In fact, it can also be used to make trend analysis and estimate fractal parameters [13; 14; 29]. By using the similarity principle of the Fourier transform, we can derive the spectral scaling relation between wave number and wave spectral density [19]. Using the spectral scaling relation, we can estimate the fractal dimension of the spatial signals of urban patterns [22]. Based on Equation (1), a density-density correlation function of urban form, \(C(r)\), can be expressed as  
\[C(r)=\int_{-\infty}^{\infty}\rho(r)\rho(x+r)\mathrm{d}x \tag{2}\]  
where \(\rho(x)\) refers to the urban density at the distance \(x\) from the center of city (\(x=0\)), and \(\rho(x+r)\) to the density at the distance \(r\) from the first point at x. Equation (2) is a point-point spatial autocorrelation function [22]. Suppose that \(x\) equals 0, that is, one point is fixed to the city center. Based on the inverse power-law distribution, the point-point correlation function will be reduced to a one-point correlation such as  
\[C(r)=\rho(0)\rho_{1}r^{D_{f}-d}\propto r^{-(d-D_{f})} \tag{3}\]  
where \(D_{f}\) denotes fractal dimension, which is a type of radial dimension, and \(d=2\) is a Euclidean dimension. Applying the Fourier transform to Equation (3) yields [14]  
\[F(k)=\int_{-\infty}^{\infty}C(r)e^{-2\pi kir}\mathrm{d}r=2\rho_{0}\rho_{1}\int _{0}^{\infty}r^{D_{f}-d}e^{-2\pi kir}\mathrm{d}r=a(k)+b(k)i \tag{4}\]  
where \(F(k)\) denotes the image function of the spatial correlation function, i.e., \(C(r)\), and \(k\) is wave number. As for the parameters, \(a(k)\) refers to the real part of the complex number for given wave number, \(b(k)\) to the corresponding imaginary part, and \(i=(-1)^{1/2}\) is the sign of the complex number unit. In theory, \(b(k)=0\) due to the symmetry of correlation functions. Equation (3) can be simplified as \(F\left[C(r)\right]=F(k)\), where F denotes the Fourier operator.  
One of the mathematical properties of the Fourier transform is similarity, and the essence of the similarity is symmetry. If the symmetry is associated with power function, we will meet a type of scaling symmetry. Thus, applying scaling transform to Equation (4) yields  
\[F(\zeta k)=\frac{1}{\zeta^{D_{f}-d+1}}2\rho_{0}\rho_{1}\int_{0}^{\infty}{( \zeta r)^{D_{f}-d}e^{-2\pi k(\zeta r)}\mathrm{d}(\zeta r)}=k^{d-D_{f}-1}F( \zeta k) \tag{5}\]  
where \(\zeta\) refers to scale factor of scaling transform. Equation (5) indicates that Equation (3) bears invariance under scaling transform, and the invariance is actually dilation symmetry. The solution to the functional relation, Equation (5), is a power law as follows  
\[F(k)=\sqrt{a(k)^{2}+b(k)^{2}}=F_{0}k^{-(1+D_{f}-d)}=F_{0}k^{-\alpha} \tag{6}\]  
where spectral exponent \(\alpha=1+D_{f}d\). It is a Fourier scaling exponent. If the spatial analysis is made in 2-dimensional space, then Euclidean dimension \(d=2\), and thus we have  
\[\alpha=1+D_{f}-d=D_{f}-1 \tag{7}\]  
Correspondingly, fractal dimension is  
\[D_{f}=\alpha+1 \tag{8}\]If we can obtain the spectral exponent by using the Fourier scaling relation, Equation (6), then we will get the estimated value of self-similar fractal dimension of spatial signals. In many cases, the fractal dimension values cannot be calculated directly due to incomplete patterns of data extraction or self-affine processes of urban evolution.