Applications of Linear Algebra in Signal Sampling and Modeling by Corey Brown Joshua Crawford Brett Rustemeyer and Kenny Stieferman Abstract: Many situations encountered in engineering require sampling a signal to determine a model of that signal. Such a signal could have its source in temperature probes antennas or virutally any real time scenario. This paper examines the fundamental theory behind signal processing works an example problem based on a suggested algorithm and briefly discusses other possible applications. INTRODUCTION: A signal is a time dependent numerical representation of events in the physical world. In typical applications the signal is in the form of a current or a voltage. For the signal to be useful it must be modeled. Signal processing takes time dependent data and manipulates it to create a mathematical model useful to practical problem solvers. Many techniques for signal processing exist including Fourier Transforms moving averages filtering and spectral analysis. Spectral analysis uses sampled data to reconstruct a given signal. Though conceptually simple sampling is typically impractical for most applications due to the large quantity of data involved in the calculations. However the fundamental concepts behind signal processing are better demonstrated with this method than with more advanced techniques. THEORY AND DEFINITIONS: A signal can be many different things depending on the application. An example of ideal signal would be s= k which alternates across the time axis and resembles a clock signal in a digital circuit. Typical signals are not so simple.
A complex continuous signal can be modeled by sampling the data at uniform discrete time intervals. The number of samples taken in each second is the sample rate or sampling frequency. Continuous signals can not be effectively utilized however the sampled data allows for the construction of a functional model which can be used in calculations. For the purposes of modeling the signal can be viewed as some combination of sine and cosine waves. Fourier Series require an infinite number of frequencies but the sampling frequency is subdivided into a finite number of frequency ranges to reduce calculations. One cosine and one sine function is needed to represent the signal for each subdivision of the sampling frequency. If the sampled data is represented by a vector it can be written as a linear combination of two vectors each composed of the appropriate sinusoidal entries. Let b m contain the cosine entries at frequency subdivision m and c m contain the sine entries at subdivision m. Then B b b m ] C c c m ] D B C ] and the columns of D form a basis for the vector space V. It can be shown that the columns of D are in fact an orthogonal basis because the dot product of any two vectors in D is zero. The signal s V and s=b u C v. This can be rewritten as s=d w where w= [ u v]. Given that the columns of D form an orthogonal basis the weights can be calculated using the following relation: u s b m m] m b m b v s c m m] m c m c. This discussion forms the foundation for the calculations in the following example.
EXAMPLE: SAMPLING AT 6 HZ Take the following signal: s={ 5 9 } where s is sampled at at a rate of 6 Hz. Subdividing into 6 equal frequency ranges yields the following sinusoidal vectors: b ] c ] ] cos 3 cos b 3 cos cos 4 3 c ] sin 3 sin 3 sin sin 4 3 cos 5 sin 5 3 3 ] cos 3 cos 4 b 3 cos cos 8 3 c ] sin 3 sin 4 3 cos sin cos sin b 3 cos 3 and 3. sin 8 3 cos 4 cos 5 ] c3 sin 4 sin 5 ] cos sin 3 3 Note that only four sets of vectors are required since the upper half of the frequency ranges are a reflection of the lower half. This phenomenon is known as aliasing. Putting these vectors together yields matrix B and matrix C. The sampled signal then is composed of the linear combination of these vectors: s=b u C v. As stated previously s=d w where D B C ] and w= [ u v].
] 3 3 Now D 3 3 3 3 3 3 um s bm m] b m b and v s c m m] m c m c. So u = 9 6 u = 5 6 u = 3 6 u 3 = 5 6 v = v = 3 6 v = 3 v 3 =. The amplitude for each frequency subdivision then is found by A= u m v m. A graph of these amplitudes with respect to frequency is referred to as the signal's spectrum. (Reference personal website for spreadsheet and graphs.) The spectrum allows reconstruction of the original signal. Since the number of possible frequencies is infinite six sampled points is inadequate to form an accurate model of the signal. However in real applications many more points are sampled allowing for a closer representation of the signal. CONCLUSIONS: Signal processing can use linear algebra in combination with sampling techniques to create a mathematical representation of a complete analog signal. As mentioned before there are many other methods for analyzing signals. One such algorithm uses a formula involving a covariance matrix whose entries are calculated by Laplace Transforms. A simpler but highly effective signal processing technique for filtering out noise takes a moving average of the points to eliminate irregular spikes and isolate the primary signal. In spectral analysis the
chosen sample rate is an important factor for the calculations (since it is the starting point for the algorithm) and depends primarily on the media involved. For example sounds of frequencies greater than.5 khz are not audible to the average human. So for accurate sound reproduction a sample rate of approximately 44. khz is appropriate since the upper half of this range is not present in the model due to aliasing. A large number of subdivisions would be required to create this signal model making calculation an enormous task. So to avoid tedious and prolonged hand calculations engineers assign the task of signal processing to devices such as multiplexers FPGAs and processors. These devices are hard coded with algorithms applied in similar ways as the example to create an accurate representation of the signal based on the spectrum. Regardless of which technique is used signal processing is a common and critical task in modern engineering. REFERENCES. C. Brown J. Crawford B. Rustemeyer and K. Stieferman. <http://web.umr.edu/~jmcnpc>. Freeman Randy A. Linear Algebra in Digital Audio. (Accessed on /4/4) <http://ea.mccormick.northwestern.edu/casestudies/dig.pdf>. 3. Johnson Don. A Signal's Spectrum. (Accessed on //4) <http://cnx.rice.edu/content/m4/latest>. 4. Lay David C. Linear Algebra and Its Applications. -3 rd Edition Pearson Education 3. 5. Madisetti Vijay K. and Williams Douglas B. The Digital Signal Processing Handbook. CRC Press 998.