# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 2 Models and Signals  
### Spatial Signals and Urban Density Models  
The similarities and differences between time signals and spatial signals should be clarified first of all. The signals based on time series may be stationary, or may be nonstationary. In contrast, the data series for spatial signals are usually nonstationary. In many cases, spatial signals take on certain trends. If the trends can be modeled by a function with characteristic lengths, it belongs to simple signals, and can be addressed by conventional mathematical methods. For example, the spatial series of urban population density follows [PERSON]'s law, a negative exponential model [16]. This model bears a scale parameter which represents characteristic length [13; 14]; it is simple and does not need fractal geometry and scaling analysis. On the contrary, if the trends cannot be modeled by a function with characteristic scales, or the trends follows power law, it belongs to complex signals and cannot be dealt with by conventional mathematical methods. This type of signals have no scale and look like the curves generated by the Weierstrass-Mandelbrot fractal function [17]. In this case, fractal-based mathematical methods may be useful for researching the scale-free signals. For example, the traffic network density of a city based on digital maps or remote sensing images can be modeled by [PERSON]'s law [18]). This is an inverse power law, which bears no parameter representing characteristic length [19]. It is complex and needs fractal parameters to describe.  
Power law relation represents the universal trends hidden behind spatial signals. Based on this type inverse power law, we can make wave-spectrum analysis. The wave spectrum exponent is associated with fractal dimension in theory [13; 20]. The standard form of [PERSON]'s model can be expressed by an inverse power law [21; 22]  
\[\rho(r)=\rho_{1}r^{-\alpha}=\rho_{1}r^{D-d} \tag{1}\]  
where \(\rho(r)\) denotes the traffic network density at the distance \(r\) from the center of city (\(r=0\)), \(d=2\) is the Euclidean dimension of the embedding space, \(D\) is the fractal dimension, \(\rho_{1}\) refers to traffic network density near city center, and \(\alpha=d\)-\(D\) to the scaling exponent of the urban traffic network [23]. This fractal dimension represents a type of local dimension, and is termed radial dimension in literature [24; 25; 26]. [PERSON]'s model can be treated as a basis for spatial signal analysis. The central density for \(r=0\) can be defined as \(\rho(0)=\rho_{0}\). If the spatial distribution of urban density is well described with Equation (1), we can calculate the fractal dimension directly using [PERSON]'s model. Thus, the wave-spectrum scaling analysis based on the Fourier transform is unnecessary. However, if a city takes on self-affine growth rather than self-similar growth, or the spatial patterns of traffic networks are incomplete due to administration boundaries, we have to make use of wave spectral analysis to address geographical signals. Equation (1) is, in fact, a density correlation function. One of the basic properties of correlation functions is symmetry. Based on the symmetry, the general Fourier transform of a correlation function can be simplified as a Fourier cosine transform [14].