# Applied Fourier Analysis: From Signal Processin...

Being an inter-disciplinary subject, Signal Processing has application in almost all scientific fields. Applied Signal Processing tries to link between the analog and digital signal processing domains. Since the digital signal processing techniques have evolved from its analog counterpart, this book begins by explaining the fundamental concepts in analog signal processing and then progresses towards the digital signal processing. This will help the reader to gain a general overview of the whole subject and establish links between the various fundamental concepts.

## Applied Fourier Analysis: From Signal Processin...

This applied course covers the theory and application of Fourier analysis, including the Fourier transform, the Fourier series, and the discrete Fourier transform. Motivation will be provided by the theory of partial differential equations arising in physics and engineering. We will also cover Fourier analysis in the more general setting of orthogonal function theory. Applications in signal processing will be discussed, including the sampling theorem and aliasing, convolution theorems, and spectral analysis. Prerequisite(s): Familiarity with differential equations, linear algebra, and real analysis.

Short-time Fourier analysis is well suited for processing tissue echographic signals which are nonstationary. We have investigated the use of short-time Fourier analysis to provide an estimation of the echographic spectral composition as a function of time. It will be shown that the time dependence of the spectral centroid of this representation allows one to deduce easily the frequency-dependent attenuation. A simple correction of the noninvariant filtering effect due to diffraction is used to unbias the attenuation slope estimation. This new signal processing technique was first tested on simulated echographic data from a 1-D tissue model. Experimental results obtained from echo signals on a tissue-like phantom and on in vivo liver tissue show the influence of diffraction and attenuation respectively.

The transformation from the time domain to the frequency domain is reversible. Once the power spectrum is displayed by one of the two previously mentioned transforms, the original signal can be reconstructed as a function of time by computing the inverse Fourier transform (IFT). Each of these transforms will be discussed individually in the following paragraphs to fill in missing background and to provide a yardstick for comparison among the various Fourier analysis software packages on the market.

Fortunately, a solution exists to minimize this leakage effect error and ensure accuracy in the frequency domain. Aside from the DFT (to be defined), the only solution is to multiply the time series by a window weighting function before the FFT is performed. Most window weighting functions (often referred to as just "windows") attenuate the discontinuity by tapering the signal to zero at both ends of the window, as shown in Figure 5d. However, if your waveform has important information appearing at the ends of the window, it will be destroyed by the tapering. In this case, a solution other than a window must be sought. With the window approach, the periodically incorrect signal as processed by the FFT will have a smooth transition at the end points which results in a more accurate power spectrum representation. A number of windows exist. Each has different characteristics that make one window better than the others at separating spectral components near each other in frequency, or at isolating one spectral component that is much smaller than another, or whatever the task. Some popular windows (named after their inventors) are Hamming, Bartlett, Hanning, and Blackman. The Hamming window offers the familiar bell-shaped weighting function but does not bring the signal to zero at the edges of the window. The Hamming window produces a very good spectral peak, but features only fair spectral leakage reduction. The Bartlett window offers a triangular shaped weighting function that brings the signal to zero at the edges of the window. This window produces a good, sharp spectral peak and is good at reducing spectral leakage as well. The Hanning window offers a similar bell-shaped window (a good approximation to the shape of the Hanning window can be seen inFigure 5d) that also brings the signal to zero at the edges of the window. The Hanning window produces good spectral peak sharpness (as good as the Bartlett window), but the Hanning offers very good spectral leakage reduction (better than the Bartlett). The Blackman window offers a weighting function similar to the Hanning but narrower in shape. Because of the narrow shape, the Blackman window is the best at reducing spectral leakage, but the trade-off is only fair spectral peak sharpness. As Figure 4 illustrates, the choice of window function is an art. It depends upon your skill at manipulating the trade-offs between the various window constraints and also on what you want to get out of the power spectrum or its inverse. Obviously, a Fourier analysis software package that offers a choice of several windows is desirable to eliminate spectral leakage distortion inherent with the FFT.

As with other bilateral transformations, such as rectangular to polar coordinates, the Fourier transformation works in both directions. If the power spectrum (as a function of frequency) were to be "run backward", the original signal would be, in principle, reconstructed as a function of time. This is known as the inverse Fourier transform (IFT). You might be questioning the purpose of an IFT if all it does is get you back to where you started. The beauty of the IFT lies in its ability to get you back to the time domain after the power spectrum has been edited in the frequency domain. This capability is very useful in power spectrum filtering applications. For example, in many cases it is desirable to examine a waveform without any "noise" present to distort the true nature of the signal. This can be done by applying high-pass, low-pass, band-pass, and notch filter functions to the power spectrum before performing the IFT. A high-pass filter will remove all unwanted frequency components less than a designated point on the power spectrum and a low-pass filter removes all unwanted frequency components greater than the designated point. A band-pass filter is a combination of high-pass and low-pass filters applied to isolate a narrow band of interest on the power spectrum. A notch filter removes the unwanted frequency component at the designated point. Figure 5 illustrates the kind of power spectrum editing possible in the frequency domain. Filtering operations can be a powerful feature in a Fourier analysis software package.

Editing takes place in the frequency domain. The waveform shown by (a) is a 20 Hz signal containing undesirable 60 Hz noise. A 512-point FFT was used to generate its power spectrum shown by (b). While in the frequency domain, all undesirable frequency components greater than the 40 Hz corner frequency (including the 60 Hz noise) were edited out, or reduced to zero by applying a low pass filter as shown by (c). An IFT was then generated from this filtered power spectrum resulting in the pure 20 Hz waveform shown by (d). Note the bell-shaped appearance of the waveform. This is due to the application of a Hanning window, a solution to the spectral leakage dilemma inherent with the FFT. Note also how the Hanning window attenuates the signal to zero at the edges of the window. Had a DFT been applied, this attenuation would be eliminated and the 20 Hz signal would be displayed at its full amplitude from end to end.

To first understand the relevance of signal processing in finance, it may first be rewarding to explore the concept of a signal itself. A signal is any sequence of numerical data that varies with respect to an underlying independent variable, mostly time. Although the very abstract nature of this definition is exceedingly useful in treating problems from a diverse range of fields, it has historically been applied almost exclusively to problems in electrical engineering. In this field, one encounters a plethora of situations with linear streams of numbers that vary with time. Current and voltage, among others, are obvious examples of this. As an illustration, let us consider the data in Table 1 and its corresponding graph in Figure 1.

Here, the sequence of numerical data is voltage and it varies with the number of seconds elapsed, i.e. time. Since this data fulfils all the criteria set by the definition above, it is a signal. The table and the graph by themselves provide valuable information about this signal such as its exact value for every second and visualization of its fluctuation. But the diversity and quantity of information directly obtainable from them is fairly limited. This is where signal processing enters the scene to delve into the guts and bowels of a signal to reveal information seldom accessible via a straightforward glance at the data.

In financial investment strategy, there are two prominent schools of thought: fundamental analysis and technical analysis. The former aims to assess the true value of a business regardless of its transient market value. This approach has limited use for signal processing because it specifically avoids the troves of data such as daily share prices and uses more modest quantities of data for a somewhat subjective assessment. In contrast, at the heart of technical analysis lies the aim of using historical financial data to predict the future market value of a business. This is precisely the type of task for which signal processing is suited because the quantity of historical data is often immense and the sheer objectivity demanded in calculations is scarcely different from that seen in electrical engineering applications. 041b061a72