Biomedical Signal Analysis: Contemporary Methods and Applications

2. Spectral Transformations

2 Spectral Transformations Pattern recognition tasks require the conversion of biosignals in features describing the collected sensor data in a compact form. Ideally, this should pertain only to relevant information. Feature extraction is an important technique in pattern recognition by determining descriptors for reducing dimensionality of pattern representation. A lower-dimensional representation of a signal is a feature. It plays a key role in determining the discriminating properties of signal classes. The choice of features, or measurements, has an important influence on (1) accuracy of classification , (2) time needed for classification, (3) number of examples needed for learning, and (4) cost of performing classification. A carefully selected feature should remain unchanged if there are variations within a signal class, and it should reveal important differences when discriminating between patterns of different signal classes. In other words, patterns are described with as little loss as possible of pertinent information. There are four known categories in the literature for extracting features [54]: 1. Nontransformed structural characteristics: moments, power, amplitude information, energy, etc. 2. Transformed signal characteristics: frequency and amplitude spectra, subspace transformation methods, etc. 3. Structural descriptions: formal languages and their grammars, parsing techniques, and string matching techniques 4. Graph descriptors: attributed graphs, relational graphs, and semantic networks Transformed signal characteristics form the most relevant category for biosignal processing and feature extraction. The basic idea employed in transformed signal characteristics is to find such transform-based features with a high information density of the original input and a low redundancy. To understand this aspect better, let us consider a radiographic image. The pixels (input samples) at the various positions have a large degree of correlation. Gray values only introduce redundant information for the subsequent classification. For example, by using the wavelet transform we obtain a feature set based on the wavelet 30 Chapter 2 coefficients which retains only the important image information residing in some few coefficients. These coefficients preserve the high correlation between the pixels. There are several methods for obtaining transformed signal characteristics . For example, Karhunen-Loeve transform and singular value decomposition are problem-dependent and the result of an optimization process [70, 264]. They are optimal in terms of decorrelation and information concentration properties, but at the same time are too computationally expensive. On the other hand, transforms which use fixed basis vectors (images), such as the Fourier and wavelet transforms, exhibit low computational complexity while being suboptimal in terms of decorrelation and redundancy. We will review the most important methods for obtaining transformed signal characteristics, such as the continuous and discrete Fourier transform, the discrete cosine and sine transform, and the wavelet transform . 2.1 Frequency Domain Representations In this section, we will show that Fourier analysis offers the rigorous language needed to define and design modern bioengineering systems. Several continuous and discrete representations derived from the Fourier transform are presented. Thus, it becomes evident that these techniques represent an important concept in the analysis and interpretation of biological signals. Continuous Fourier Transform One of the most important tasks in processing of biomedical signals is to decompose a signal intp its frequency components and to determine the corresponding amplitudes. The standard analysis for continuous time signals is performed by the classical Fourier transform. The Fourier transform is defined by the following equation: F(ω) =  ∞ −∞ f(t)e−jωt dt (2.1) while the inverse transform is given as Spectral Transformations 31 f(t) = 1 2π  ∞ −∞ F(ω)ejωt dω (2.2) The direct transform extracts spectrum information from the signal, and the inverse transform synthesizes the time-domain signal from the spectral information. Example 2.1: We consider the following exponential signal f(t) = e−5t u(t) (2.3) where u(t) is the step function. The Fourier transform is given as F(jω) =  ∞ 0 e−5t e−jωt dt =  ∞ 0 e−5+jωt dt = 1 5 + jω (2.4) For real-world problems, we employ the existing properties of the Fourier transform that help to simplify the frequency domain transformations [190]. However, the major drawback of the classical Fourier transform is its inability to deal with nonstationary signals. Since it considers the whole time domain, it misses the local changes of high-frequency components in the signal. In summary, it is assumed that the signal properties (amplitudes, frequency, and phases) will not change with time and will stay the...