PROJ 1002-AF Project 2

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PROJ 1002-AF Project 2

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PROJ 1002-AF Project 2

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Course Code: PROJ1002AF
University: Middle East College

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Country: Oman

Question:
1.Write a report on the study of over-sampling rate in sampling process.2.Write a report on any one of the applications of Fourier/Z-transforms
Answer:

Introduction: Oversampling
Oversampling is a method that is normally used to reduce the quantization error in analog-to-digital converters. It needs a mixture of both quantization and sampling.  It can also be defined as the signal sampling at a frequency that is at a higher rate than the Nyquist.
Reasons for oversampling
Firstly, there is a swift realization of analog anti-aliasing filters with the use of oversampling (Godwin, 2011). This is done as a result of rising the bandwidth sampling system thereby relaxing the design limits for the anti-aliasing filter.
Secondly, in most cases oversampling is employed as a means of reducing expenses and rising the performance of ADC (analog to digital converter) or DAC (digital to analog converter) converter (Ruben, 2008). In practical terms, dithering noise is usually out of frequency range and hence can only be filtered using the digital domain. This results in a final measurement that is of low noise and high resolution.
Thirdly, usually delta sigma converters employ a method termed as noise shaping in order to shift to a high frequency the noise of quantization. Some types of analog to digital converters give out a lot of quantization noise (Ronald, 2000). These converters give a final output that has less noise by low- pass filtering the oversampled signal by half.
Lastly, oversampling also represents a reconstruction stage of digital-to-analog conversion where an oversample is employed at the centre of digital input and analog output.  
Conclusion
The standard practice if one is to sample a signal involves using a sample rate that is two time greater than the bandwidth as a means of getting rid of aliasing from modulation and distortion.
Fourier Transform
The Fourier Transform (FT) is a tool used in breaking a signal or function (waveform) to form a representation that is alternating and features the cosines and sine. It takes a pattern that is time-based, measures each cycle that is a possibility, and gives a return on the overall offset, amplitude and circumnavigation speed for every found cycle.
The Fourier Transform normally involves Fourier Series in general and is applied to aperiodic, continuous as well as impulse functions (Chris, 2004). The FT makes a sum of sinusoidal basis functions using any kind of function.  
The FT has its properties that include: linearity, duality, shifting, scaling, derivative, convolution, and modulation properties.
Applications of Fourier Transforms
Firstly, the FT is employed in the design and usage of antennas. The Fourier transformation normally takes a key role in determining the features of antenna radiation (Matuir, 2011). This is by using various ways like intermediate and near-field (NF), compact antenna test ranges (CATR), and even antenna test ranges. The measured information is useful in calculating the desirable parameters.
Secondly, the Fourier Transform is useful in image processing and filters. This is by breaking down an image to get the cosine and sine components. The spatial and frequency domains are the final outputs after transformation. The Fourier Transform is also applicable in image reconstruction, analysis, and also compression.
Thirdly, it is also useful in data analysis and processing mainly in acoustics, partial differential equations, theory on probability, sonar optics, combinatorics, diffraction, number theory, geometry, processing of signals, cryptography, oceanography, forensics, option pricing as well as analysing structure of proteins (Loukas, 2004).
Furthermore, Fourier Transforms are key in seismic streamers and arrays. This is done by use of sum, delay, and also cross-correlational methods of filtering the seismic signals and represent such data as an array processing.
In addition, it is applied in the interferometer to turn actual spectrum from raw information. The spectra are gathered according to coherent measurement of sources of radiation with the aid of space-domain or even time-domain measurements.
Also, the Fourier Transform is vital in cross-correlational functions. This is by creating a matched set of algorithms like motion tracking and facial recognition using the fastest computation.
Lastly, it is also useful in estimating immediate noise. This is by the help of an algorithm that is useful in supressing and estimating the moving noise that is in the background of the speech (James, 2011). The method vital in this case is the average magnitude difference equation (AMDF).   
Conclusion
The Fourier Transform is a multipurpose tool in mathematics and has a wide range of application in engineering and science. In virtual terms, almost everything in the world can be broken down with the use of waveform. The FT presents a special and vital way of viewing the waveforms. It turns complex mathematical problems to be a lot simpler. It is the super hero of this situation.
References
Chris, C., 2004. The Analysis of Time Series. London: Chapman & Hall Press.
David, K., 2000. A First Course in Fourier Analysis. New Jersey: Prentice Hall.
Godwin, G., 2011. Sampling in Digital Signal Procesing. Basel: Birkhauser.
Hwei, H., 2013. Schaum’s Outline of Signals and System. 3 rd ed. New York: McGraw Hill.
James, F., 2011. A Students Guide to Fourier Transforms. London: Cambridge University Press.
Loukas, G., 2004. Classical and Modern Fourier Analysis. New Jersey: Prentice Hall .
Mark, P., 2002. Introduction to Fourier Analysis and Wavelets. New York: Brooks & Cole Press.
Matuir, R., 2011. Applications of Fourier Transforms to Generalized Functions. Southampton: WIT Press.
Ronald, B., 2000. The Fourier Transforms & Its Applications. Boston: McGraw Hill.
Ruben, G., 2008. Problems and Solutions in Signals and Systems. New York: NYU Press.
Shawn, R., 2010. Fundamentals of Signals and Systems. New York: McGraw Hill.
Simon, H., 2007. SIgnals and Systems. 2 nd ed. New York: Wiley.

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