Introducing the Fourier Series to LTI Systems Before looking at this module, one should be familiar with the concepts of eigenfunction and LTI systems. Recall, for ℋ LTI system we get the following relationship

Input and output signals to our LTI system. where s t is an eigenfunction of ℋ . Its corresponding eigenvalue H s can be calculated using the impulse response h t H s τ h τ s τ So, using the Fourier Series expansion for periodic f t where we input f t n c n ω 0 n t into the system,

LTI system our output y t will be y t n H ω 0 n c n ω 0 n t So we can see that by applying the fourier series expansion equations, we can go from f t to c n and vice versa, and we do the same for our output, y t

Effects of Fourier Series We can think of an LTI system as shaping the frequency content of the input. Keep in mind the basic LTI system we presented above in . The LTI system, ℋ , simply multiplies all of our Fourier coefficients and scales them. Given the Fourier coefficients c n of the input and the eigenvalues of the system H w 0 n , the Fourier series of the output is H w 0 n c n (simple term-by-term multiplication). The eigenvalues H w 0 n completely describe what a LTI system does to periodic signals with period T 2 w 0 What does this system do?

What about this system?

Examples RC Circuit h t 1 R C t R C u t What does this system do to the Fourier Series of an input f t ? Calculate the eigenvalues of this system H s τ h τ s τ τ 0 1 R C τ R C s τ 1 R C τ 0 τ 1 R C s 1 R C 1 1 R C s τ 0 τ 1 R C s 1 1 R C s Now, say we feed the RC circuit a periodic (period T 2 w 0 ) input f t . Look at the eigenvalues for s w 0 n H w 0 n 1 1 R C w 0 n 1 1 R 2 C 2 w 0 2 n 2 The RC circuit is a lowpass system: it passes low frequencies ( n around 0 ) and attenuates high frequencies (large n ). Square pulse wave through RC circuit Input Signal: Taking the fourier series of f t c n 1 2 2 n 2 n 1 t at n 0 System: eigenvalues H w 0 n 1 1 R C w 0 n Output Signal: Taking the fourier series of y t d n H w 0 n c n 1 1 R C w 0 n 1 2 2 n 2 n d n 1 1 R C w 0 n 1 2 2 n 2 n y t n d n w 0 n t What can we infer about y t from d n ? Is y t real? Is y t even symmetric? odd symmetric? Qualitatively, what does y t look like? Is it "smoother" than f t ? (decay rate of d n vs. c n ) d n 1 1 R C w 0 n 1 2 2 n 2 n d n 1 1 R C w 0 2 n 2 1 2 2 n 2 n