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Symmetry Properties of the Fourier Series

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Connexions module: m12128 1 Symmetry Properties of the Fourier Series ∗ Richard Baraniuk This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License 1.0 † Abstract An introduction to the symmetry properties of the Fourier Series. 1 Real Signals If f (t) = f (t), then cn = c−n. Therefore, • if < (cn) = < (c−n), then f (t) is even (Figure 1(a)); • if = (cn) = −= (c−n), then f (t) is odd (Figure 1(b)); • if |cn| = |c−n|, then f (t) is even (Figure 2(a)); • if ] (cn) = −] (c−n), then f (t) is odd (Figure 2(b)). (a) (b) Figure 1: (a) even (b) odd ∗ Version 1.3: Jul 23, 2004 2:07 pm -0500 † http://creativecommons.org/licenses/by/1.0 http://cnx.org/content/m12128/1.3/ Connexions module: m12128 2 (a) (b) Figure 2: (a) even (b) odd Proof c−n = 1T ∫ T 0 f (t) eiω0ntdt = 1T ∫ T 0 f (t)e−(iω0nt)dt = 1T ∫ T 0 f (t) e−(iω0nt)dt = cn (1) where we have used the fact that f (t) = f (t). 2 Real Even Signals If f (t) is even (Figure 3: Even Signal) (f (t) = f (−t)) and real (f (t) = f (t)), then cn is even (cn = c−n) and real (cn = c−n). Even Signal Figure 3 http://cnx.org/content/m12128/1.3/ Connexions module: m12128 3 Proof cn = 1T ∫ T 2 −T2 f (t) e−(iω0nt)dt = 1T ∫ 0 −T2 f (t) e −(iω0nt)dt+ 1T ∫ T 2 0 f (t) e−(iω0nt)dt = 1T ∫ T 2 0 f (−t) eiω0ntdt+ 1T ∫ T 2 0 f (t) e−(iω0nt)dt = 1T ∫ T 2 0 f (t) eiω0ntdt+ 1T ∫ T 2 0 f (t) e−(iω0nt)dt = 1T ∫ T 2 0 f (t) ( eiω0nt + e−(iω0nt) ) dt = 1T ∫ T 2 0 f (t) 2cos (ω0nt) dt (2) where we have used the fact that f (−t) = f (t). (2) (Proof) implies that cn is real since both f (t) and cos (ω0nt) are real. Also, because of the properties of cosine, c−n = cn. This implies that cn is even. It is easy to show that: f (t) = 2 ∑ n cncos (ω0nt) (3) and f (t), cn, and cos (ω0nt) are all real and even! 3 Real Odd Signals If f (t) is odd (Figure 4: Odd!) (f (t) = −f (−t)) and real (f (t) = f (t)), then cn is odd

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