# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 3 Empirical Analysis
### Data and Methods
\[\rho(r)\stackrel{{\text{FFT}}}{{\rightarrow}}F(k)=\hat{a}+\hat{b}i \tag{15}\]  
where \(F(k)\) is the result of FFT, expressed by complex numbers, \(\hat{a}\) is the real part estimated value of the complex number, \(\hat{b}\) is the imaginary part estimated value. As indicated above, in theory, \(\hat{b}=0\) because of the symmetry of correlation function. However, in practice, \(\hat{b}\
eq 0\) owing to random disturbance of spatial signals. Thus, the wave spectrum can be defined as  
\[F^{*}(k)=\left(\frac{|F(k)|^{2}}{N}\right)^{1/2}=\left(\frac{F(k)\overline{F(k) }}{N}\right)^{1/2}=\left(\frac{\hat{a}^{2}+\hat{b}^{2}}{N}\right)^{1/2} \tag{16}\]  
Figure 1: Traffic networks in 10 Chinese cities based on administrative boundaries in 2016. (**a**) Beijing; (**b**) Tianjin; (**c**) Guangzhou; (**d**) Shenzhen; (**e**) Wuhan; (**f**) Nanjing; (**g**) Shanghai; (**h**) Hangzhou; (**i**) Chongqing; (**j**) Chengdu. Note: The data extraction for spatial signals is confined by each municipal boundary, so that the spatial information is not complete. In this case, it is improper to estimate the fractal parameter using simple fractal model. It is advisable to apply wave-spectrum scaling analysis to these spatial patterns. Based on Equation (16), Equation (6) can be substituted with  
\[F^{*}(k)=F_{0}^{*}k^{-\alpha} \tag{17}\]  
where the asterisk \"*\" denotes an equivalent redefinition of variable \(F(k)\) and parameter \(F_{0}\) for simplification. The redefinition does not influence the computational results and analytical process of spatial signals. Taking natural logarithm of both sides of Equation (17) yields a linear equation as below  
\[\ln F^{*}(k)=\ln F_{0}^{*}-\alpha\ln k \tag{18}\]  
Thus, by using the OLS method, we can estimate the values of the spectral exponent. In the light of Equation (8), the spectral exponent values can be converted into fractal dimension values and the Hurst exponent. Using these fractal parameters based on spatial signals, we can make spatial analysis for geographical systems. In summary, the approach of fractal parameter estimation consists of number of steps. The computing process can be implemented step by step as follows. Step 1: Defining study area. In our case studies, a study area is a spatial system of streets and roads within urban envelope. Step 2: extracting spatial datasets. As indicated above, the operation can be carried out by using ArcGIS, a Windows-based collection of Geographic Information Systems (GIS) software. The formula is Equation (13) based on concentric circles. Step 3: generating density series. The formula is Equation (14). The density distribution indicates a spatial correlation function. Step 4: Fourier transform. Applying FFT to density series to yield the wave-spectrum series or the wave-spectrum density series. The formula is Equation (4), which can be discretized by substituting integral for summation. The calculation can be realized through MS Excel. Step 5: wave-spectrum scaling. Fitting Equation (17) or Equation (18) to the datasets of relationships between wave number and wave spectrum series to yield spectral exponents. Step 6: fractal parameter estimation. In the light of Equations (7) and (8) as well as Equations (11) and (12), the spectral exponent values can be converted into fractal dimension and Hurst exponent values (Figure 2). Using these fractal parameters based on spatial signals, we can make spatial analysis for geographical systems.