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Fourier Series and LTI Systems

Module by: Justin Romberg

Summary: (Blank Abstract)

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 ℋ LTI system we get the following relationship
simpleLTIsys.png
Figure 1: Input and output signals to our LTI system.
where ⅇst s t is an eigenfunction of ℋ ℋ. Its corresponding eigenvalue Hs H s can be calculated using the impulse response ht h t Hs=∫-∞∞hτⅇ-sτdτ H s τ h τ s τ
So, using the Fourier Series expansion for periodic ft f t where we input ft=∑n c n ⅇⅈ ω 0 nt f t n c n ω 0 n t into the system,
Transferfunc.png
Figure 2: LTI system
our output yt y t will be yt=∑nHⅈ ω 0 n c n ⅇⅈ ω 0 nt 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 ft f t to c n c n and vice versa, and we do the same for our output, yt 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 Figure 2. The LTI system, ℋ ℋ, simply multiplies all of our Fourier coefficients and scales them.
Given the Fourier coefficients c n c n of the input and the eigenvalues of the system Hⅈ w 0 n H w 0 n , the Fourier series of the output is Hⅈ w 0 n c n H w 0 n c n (simple term-by-term multiplication).
note: The eigenvalues Hⅈ w 0 n H w 0 n completely describe what a LTI system does to periodic signals with period T=2π w 0 T 2 w 0
Example 1 
What does this system do?
fslti_f1.png
Figure 3
Example 2 
What about this system?
fslti_f2.pngfslti_f3.png
Subfigure 4.1
Subfigure 4.2
Figure 4

Examples

Example 3: RC Circuit 
ht=1RCⅇ-tRCut h t 1 R C t R C u t
What does this system do to the Fourier Series of an input ft f t ?
Calculate the eigenvalues of this system
Hs=∫-∞∞hτⅇ-sτdτ=∫0∞1RCⅇ-τRCⅇ-sτdτ=1RC∫0∞ⅇ-τ1RC+sdτ=1RC11RC+sⅇ-τ1RC+s|τ=0∞=11+RCs 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 (1)
Now, say we feed the RC circuit a periodic (period T=2π w 0 T 2 w 0 ) input ft f t .
Look at the eigenvalues for s=ⅈ w 0 n s w 0 n |Hⅈ w 0 n|=1|1+RCⅈ w 0 n|=11+R2C2 w 0 2n2 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 n around 0 0) and attenuates high frequencies (large n n).
Example 4: Square pulse wave through RC circuit 
  • Input Signal: Taking the fourier series of ft f t c n =12sinπ2nπ2n c n 1 2 2 n 2 n 1t 1 t at n=0 n 0
  • System: eigenvalues Hⅈ w 0 n=11+ⅈRC w 0 n H w 0 n 1 1 R C w 0 n
  • Output Signal: Taking the fourier series of yt y t d n =Hⅈ w 0 n c n =11+ⅈRC w 0 n12sinπ2nπ2n d n H w 0 n c n 1 1 R C w 0 n 1 2 2 n 2 n
d n =11+ⅈRC w 0 n12sinπ2nπ2n d n 1 1 R C w 0 n 1 2 2 n 2 n yt=∑ d n ⅇⅈ w 0 nt y t n d n w 0 n t
What can we infer about yt y t from d n d n ?
  1. Is yt y t real?
  2. Is yt y t even symmetric? odd symmetric?
  3. Qualitatively, what does yt y t look like? Is it "smoother" than ft f t ? (decay rate of d n d n vs. c n c n )
d n =11+ⅈRC w 0 n12sinπ2nπ2n d n 1 1 R C w 0 n 1 2 2 n 2 n | d n |=11+RC w 0 2n212sinπ2nπ2n d n 1 1 R C w 0 2 n 2 1 2 2 n 2 n

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