Symmetry Properties of the Fourier Series 2.3 2002/09/05 2003/07/30 11:15:32.626 GMT-5 Justin Romberg jrom@rice.edu Prashant Singh prash@ece.rice.edu Mariyah Poonawala mariyah@rice.edu symmetry fourier fourier series properties symmetry property symmetry properties even signal odd signal This module looks at the different symmetry properties of the fourier series and its fourier coefficients.

Symmetry Properties

Real Signals Real signals have a conjugate symmetric Fourier series. If f t is real it implies that f t f t ( f t is the complex conjugate of f t ), then c n c - n which implies that c n c - n , i.e. the real part of c n is even, and c n c - n , i.e. the imaginary part of c n is odd. See . It also implies that c n c - n , i.e. that magnitude is even, and that ∠ c n ∠ c - n , i.e. the phase is odd. c - n 1 T t 0 T f t ω 0 n t t f t f t 1 T t 0 T f t ω 0 n t 1 T t 0 T f t ω 0 n t c n

c n c - n , and c n c - n .

c n c - n , and ∠ c n ∠ c - n .

Real and Even Signals Real and even signals have real and even Fourier series. If f t f t and f t f t , i.e. the signal is real and even, then c n c - n and c n c n . c n 1 T t T 2 T 2 f t ω 0 n t 1 T t T 2 0 f t ω 0 n t 1 T t 0 T 2 f t ω 0 n t 1 T t 0 T 2 f t ω 0 n t 1 T t 0 T 2 f t ω 0 n t 2 T t 0 T 2 f t ω 0 n t f t and ω 0 n t are both real which implies that c n is real. Also ω 0 n t ω 0 n t so c n c - n . It is also easy to show that f t 2 n 0 c n ω 0 n t since f t , c n , and ω 0 n t are all real and even.

Real and Odd Signals Real and odd signals have Fourier Series that are odd and purely imaginary. If f t f t and f t f t , i.e. the signal is real and odd, then c n c - n and c n c n , i.e. c n is odd and purely imaginary. Do it at home. If f t is odd, then we can expand it in terms of ω 0 n t : f t n 1 2 c n ω 0 n t

Summary In summary, we can find f e t , an even function, and f o t , an odd function, such that f t f e t f o t which implies that, for any f t , we can find a n and b n such that f t n 0 a n ω 0 n t n 1 b n ω 0 n t

Triangle Wave

T 1 and ω 0 2 .

f t is real and odd. c n 4 A 2 n 2 n … -11 -7 -3 1 5 9 … 4 A 2 n 2 n … -9 -5 -1 3 7 11 … 0 n … -4 -2 0 2 4 … Does c n c - n ?

The Fourier series of a triangle wave.

We can often gather information about the smoothness of a signal by examining its Fourier coefficients. Take a look at the above examples. The pulse and sawtooth waves are not continuous and there Fourier series' fall off like 1 n . The triangle wave is continuous, but not differentiable and its Fourier series falls off like 1 n 2 . The next 3 properties will give a better feel for this.