Comprehensive Report: Oversampling and Fourier Transform Applications

This report provides a comprehensive overview of oversampling and Fourier Transforms, two crucial concepts in electrical engineering. It begins by explaining oversampling as a technique used to reduce quantization error in analog-to-digital converters, detailing its advantages like noise reduction, anti-aliasing filter simplification, and improved ADC/DAC performance. The report then transitions to the Fourier Transform (FT), a mathematical tool for breaking down signals into their constituent frequencies, explaining its properties and diverse applications in antenna design, image processing, data analysis, seismic analysis, and more. The document emphasizes the practical applications of both techniques, highlighting their significance in various engineering and scientific fields. The report concludes by emphasizing the versatility of the FT and its importance in simplifying complex mathematical problems. The document includes several references to support the information provided and to allow for further study.

1Oversampling and Fourier Transforms
REPORT ON OVERSAMPLING AND
FOURIER TRANSFORMS
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OVERSAMPLING
Introduction
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

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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
Introduction
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

4Oversampling and Fourier Transforms
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).

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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.

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Simon, H., 2007. SIgnals and Systems. 2 nd ed. New York: Wiley.