Question

Developing filters using convolution theorem and Fourier Transform.

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

Answer 1