Example: the Fourier Transform of a rectangle function: rect(t)

Remarks on general Fourier series Everything we’ve done with 2ˇ-periodic Fourier series continues to hold in this case, with p replacing ˇ: We can compute general Fourier coe cients by integrating over any \convenient" interval of length 2p If p is left unspeci ed, then the formulae for a n and b n may involve p If f(x) is even, then b n

18103 Fourier Series: an outline - MIT Mathematics

18 103 Fourier Series: an outline Consider a periodic function F on the real line of period 2π, integrable on (−π,π) (or, equivalently, F ∈ L1(T) with T = R/2πZ) ) The Fourier coeﬃcients of F are deﬁn

Fourier Series & The Fourier Transform

Finding the coefficients, F’ m, in a Fourier Sine Series Fourier Sine Series: To find F m, multiply each side by sin(m’t), where m’ is another integer, and integrate: But: So: Åonly the m’ = m term contributes Dropping the ‘ from the m: Åyields the coefficients for any f(t) f (t) = 1 π F m′ sin(mt) m=0 ∑∞ 0 1

CHAPTER 4 FOURIER SERIES AND INTEGRALS

This idea started an enormous development of Fourier series Our ﬁrst step is to compute from S(x)thenumberb k that multiplies sinkx Suppose S(x)= b n sinnx Multiply both sides by sinkx Integrate from 0 to π: π 0 S(x)sinkxdx= π 0 b 1 sinxsinkx dx+···+ π 0 b k sinkx sinkxdx+···(2)

Fourier Series and Fourier Transform

6 082 Spring 2007 Fourier Series and Fourier Transform, Slide 15 Magnitude and Phase • We often want to ignore the issue of time (phase) shifts when using Fourier analysis – Unfortunately, we have seen that the A nand B n coefficients are very sensitive to time (phase) shifts • The Fourier coefficients can also be represented in

FOURIER SERIES ON ANY INTERVAL

this interval We can use the coefficients computed immediately above and write the Fourier series for this interval as: f HxL= (4) 4 p2 3 +S n=1 ¶ 4 pcos HnxL n2-4 p2 sin HnxL n This exercise shows that we can compute Fourier series for other intervals, but that we have to be careful to recompute the coefficients 2 fourierintervals nb

8 Fourier Series - Pennsylvania State University

can be integrated term by term and produce the Fourier series F(x) = Z x 0 f(y) dy ∼ C0 + X n≥1 − bn n cosnx+ an n sinnx where the constant C0 = 1 2π Rπ −π F(y) dy Example 8 8 Consider a 2π-periodic function f given by f(x) = x on (−π,π] (See Example 8 1 ) Since f is odd, its mean value over [−π,π] is equal to 0 Its

Fourier Series for Periodic Functions

BME 333 Biomedical Signals and Systems - J Schesser 15 Homework • Problem (1) – Compute the Fourier Series for the periodic functions a) f(t) = 1 for 0