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Developing filters using convolution theorem and Fourier Transform. You have been hired as an Engineering Mathematician at...

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Advance Math

Developing filters using convolution theorem and FourierTransform.

You have been hired as an Engineering Mathematician at aconsulting firm located in Saint Louis. On your first job, you havebeen asked to mathematically design a frequency filter that removesfrom a standard beacon signal a periodic interference generated bya rotatory machine located in the basement of the company. Pleasesee below for more details: Let s(t) be the standard beacon signalthat is being communicated. Below you can find its Fourier Seriesrepresentation ?(?)=2.5+2 sin(?)+3cos(t)+0.5cos(2?)+ 0.3 sin (2?).The periodic interference is given by ?(?)=0.5 cos (120 ?) and themeasurement signal with noise is given as follows: ?(?)=?(?)+?(?)Let g( t) be the filter function and let ? Z(?) be the functionthat results from applying ? (?) to ?(?). Using the Fouriertransform of the convolution theorem, propose the design of afiltering function g(t) which removes from ?(?) the effect of theperiodic noise ?(?) assuming that we only know that its fundamentalperiod is equal to 2pi/120. Make sure to write the analyticalexpression of ?(?) and its Fourier transform. Also, please writethe mathematical expression that relates ? Z(?) as a function of?(?)and ?(?)

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