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Symmetry Properties of the Fourier Series

Module by: Justin Romberg. E-mail the author

Summary: 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.

Theorem 1

If ft f t is real it implies that ft=ft¯ f t f t , while ( ft¯ f t is the complex conjugate of ft f t ), then c n = c - n ¯ c n c - n which implies that c n = c - n c n c - n , i.e. the real part of c n c n is even, and c n = c - n c n c - n , i.e. the imaginary part of c n c n is odd. See Figure 1. It also implies that | c n |=| c - n | c n c - n , i.e. that magnitude is even, and that c n =( c - n ) c n c - n , i.e. the phase is odd.

Proof

c - n =1T0Tftei ω 0 ntd t = t ,ft=ft¯:1T0Tft¯e(i ω 0 nt)d t ¯=1T0Tfte(i ω 0 nt)d t ¯= c n ¯ 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
(1)

Figure 1: c n = c - n c n c - n , and c n = c - n c n c - n .
(a)
Figure 1(a) (m10838ae.png)
(b)
Figure 1(b) (m10838ce.png)
Figure 2: | c n |=| c - n | c n c - n , and c n =( c - n ) c n c - n .
(a)
Figure 2(a) (m10838be.png)
(b)
Figure 2(b) (m10838de.png)

Real and Even Signals

Real and even signals have real and even Fourier series.

Theorem 2

If ft=ft¯ f t f t and ft=ft f t f t , i.e. the signal is real and even, then c n = c - n c n c - n and c n = c n ¯ c n c n .

Proof

c n =1TT2T2fte(i ω 0 nt)d t =1TT20fte(i ω 0 nt)d t +1T0T2fte(i ω 0 nt)d t =1T0T2ftei ω 0 ntd t +1T0T2fte(i ω 0 nt)d t =2T0T2ftcos ω 0 ntd t 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
(2)
ft f t and cos ω 0 nt ω 0 n t are both real which implies that c n c n is real. Also cos ω 0 nt=cos( ω 0 nt) ω 0 n t ω 0 n t so c n = c - n c n c - n . It is also easy to show that ft=2 n =0 c n cos ω 0 nt f t 2 n 0 c n ω 0 n t since ft f t , c n c n , and cos ω 0 nt 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.

Theorem 3

If ft=ft f t f t and ft=ft¯ f t f t , i.e. the signal is real and odd, then c n = c - n c n c - n and c n = c n ¯ c n c n , i.e. c n c n is odd and purely imaginary.

Proof

Do it at home.

If ft f t is odd, then we can expand it in terms of sin ω 0 nt ω 0 n t : ft= n =12 c n sin ω 0 nt f t n 1 2 c n ω 0 n t

Conclusion

  • Real Signals:
    f ( t ) = f * ( t ) then C n = C - n * R e { C n } = R e { C - n } even I m { C n } = - I m { C - n } odd | C n | = | C - n | even C n = - C - n odd f ( t ) = f * ( t ) then C n = C - n * R e { C n } = R e { C - n } even I m { C n } = - I m { C - n } odd | C n | = | C - n | even C n = - C - n odd
    (3)
  • Real Even Signals:
    f ( t ) = f ( - t ) = f * ( t ) even and real then C n = C - n = C n * even and real f ( t ) = f ( - t ) = f * ( t ) even and real then C n = C - n = C n * even and real
    (4)
  • Real Odd Signals:
    f ( t ) = - f ( - t ) = f * ( t ) odd and real then C n = - C - n = - C n * odd and purely imaginary f ( t ) = - f ( - t ) = f * ( t ) odd and real then C n = - C - n = - C n * odd and purely imaginary
    (5)

Summary

In summary, we can find f e t f e t , an even function, and f o t f o t , an odd function, such that

ft= f e t+ f o t f t f e t f o t
(6)
which implies that, for any ft f t , we can find a n a n and b n b n such that
ft= n =0 a n cos ω 0 nt+ n =1 b n sin ω 0 nt f t n 0 a n ω 0 n t n 1 b n ω 0 n t
(7)

Examples

Example 1: Triangle Wave (Real and Odd)

Figure 3: T=1 T 1 and ω 0 =2π ω 0 2 .
Figure 3 (triwave.png)

ft f t is real and odd. c n ={4Aiπ2n2  if  n=-11-7-31594Aiπ2n2  if  n=-9-5-137110  if  n=-4-2024 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 c n c - n ?

Figure 4: The Fourier series of a triangle wave.
Figure 4 (m10838ee.png)

Example 2: Sawtooth Wave (Real and Odd)

Figure 5
Figure 5 (saw.png)
C n = - A j π n ( - 1 ) n , n > 0 odd A j π n ( - 1 ) n , n < 0 odd 0 , n = 0 even note: C n = - C - n odd for real signal C n = - C n * purely imaginary for odd signal C n = - A j π n ( - 1 ) n , n > 0 odd A j π n ( - 1 ) n , n < 0 odd 0 , n = 0 even note: C n = - C - n odd for real signal C n = - C n * purely imaginary for odd signal
(8)
Figure 6
Figure 6 (sawF.png)

Example 3: Square Pulse (Real and Even)

Figure 7
Figure 7 (pulse.png)
C n = 1 2 sin π / 2 n π / 2 n C n = 1 2 sin π / 2 n π / 2 n
(9)
Figure 8
Figure 8 (pulseF.png)

Note:

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 1n 1 n . The triangle wave is continuous, but not differentiable and its Fourier series falls off like 1n2 1 n 2 .

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