# Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks
## 4 Discussion  
The empirical analyses show that the method of wave-spectrum scaling analysis, based on the Fourier transform, can be employed to estimate fractal dimension of spatial signals. Using these fractal parameters, we can make spatial analyses by means of geographical signals. Spatial signal extraction is not limited to traffic networks, but can involve many aspects of geographic systems. Note that fractal approach can only be applied to fractal phenomena. In fact, spatial signals can be divided into two categories and four subcategories (Table 4). If we meet stationary spatial series of geographical signals, the wave-spectrum scaling analysis will be useless. If we meet the trend series of spatial signals with characteristic lengths, generally speaking, the wave-spectrum scaling is useless. Sometimes, the wave-spectrum scaling analysis can be applied to the spatial series satisfying exponential distributions, but the spectral exponent indicates a Euclidean dimension [13,20]. If a self-similar spatial signal series, i.e., the spatial series without characteristic length, is encountered, the method presented in this paper is suitable. If a self-affine spatial signal series appears, the spectral analysis can also be applied to it in a proper way. In particular, if the spatial patterns for signal extraction are not incomplete, the wave-spectrum scaling is indispensable for estimating radial dimension of growing fractal processes. Incomplete spatial patterns result in power law degeneration, and the fractal dimension cannot be directly estimated by [PERSON]'s model. Fourier transform and frequency and wave-spectrum scaling is one of important approaches for spatial analysis of geographical fractal phenomena. In urban studies, this method is useful for indirect estimation of fractal dimension, especially when fractal structure is hidden by random noises. Scaling invariance is a universal law in geographical systems. The wave-spectrum scaling analysis has universal applicability. It can be used in different geographical regions to analyze different types of spatial signals. Compared with previous research, this work bears clear novelty. First, the previous works are devoted to theoretical models and fractal parameter relations [19]. In particular, using wave-spectral scaling analysis, new fractal parameter relations are derived [22]. In contrast, this work is devoted to positive analysis of wave-spectral scaling for spatial signals. Second, previous works are mainly devoted to self-affine fractal analysis by empirical approach [20], specifically the relationships between self-affine fractal parameters and self-similar fractal parameters are revealed by means of wave-spectral scaling relations. In contrast, this paper is chiefly devoted to self-similar fractal series. Third, in previous studies, the cases are based on land use patterns of cities [22,29]. In contrast, this paper is devoted to research urban traffic networks. The shortcomings of this study lies in three respects. Firstly, this paper is actually devoted to 1-dimensional spatial signals. In other words, with the help of statistical average, the 2-dimensional spatial signals are converted into 1-dimensional spatial signals. The 2-dimensional random signals in strict sense cannot be analyzed by means of the wave-spectrum scaling for the time being; Secondly, this paper is principally devoted to self-similar fractal signals, or pseudo self-affine fractal signals, for the self-affine fractal signals, in strict sense, no deep analysis is made in this work. Due to the lack of enough