# c22 - 18.03 Class 22, April 5 Fourier series and harmonic...

This preview shows pages 1–2. Sign up to view the full content.

18.03 Class 22, April 5 Fourier series and harmonic response [1] My muddy point from the last lecture: I claimed that the Fourier series for f(t) converges wherever \$f\$ is continuous. What does this really say? For example, (pi/4) sq(t) = sin(t) + (1/3) sin(3t) + (1/5) sin(5t) + . ... for any value of t which is not a whole number multiple of pi. This says that when 0 < t < pi , sin(t) + (1/3) sin(3t) + . ... = pi/4 (*) Since all these sines are odd functions, it is no additional information to say that when -pi < t < 0 , sin(t) + (1/3) sin(3t) + . ... = - pi/4 For example we could take t = pi/2 . Then sin(pi/2) = 1 , sin(3pi/2) = -1 , sin(5pi/2) = 1 , .... so 1 - (1/3) + (1/5) - (1/7) + . .. = pi/4 This is an alternating series, so we know it converges. Did you know that it converges to pi/4? And so on: there are infinitely many summations like this contained in (*) . [2] You can differentiate and integrate Fourier series. Example: Consider the function f(t) which is periodic of period 2pi and is given by f(t) = (pi/2) - t between 0 and pi. We could calculate the coefficients, using the fact that f(t) is even and integration by parts. For a start, a0/2 is the average value, which is pi/2. Or we could realize that

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## c22 - 18.03 Class 22, April 5 Fourier series and harmonic...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online