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Notes on Fourier Series

Steven A. Tretter

October 30, 2013

Contents

1 The Real Form Fourier Series 3

2 The Complex Exponential Form of the Fourier Series 9

3 Fourier Series for Signals with Special Symmetries 13

3.1 Even Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Odd Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Half Wave Symmetry . . . . . . . . . . . . . . . . . . . . . . . 14

4 Fourier Series for Some Simple Operations on Periodic Sig-

nals15

4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 The Delay Theorem . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.5 Multiplication of Two Periodic Signals with the Same Period . 19

4.6 Fourier Coefficients for the Complex Conjugate of a Signal. . 20

4.7 Average Power and Parseval"s Theorem . . . . . . . . . . . . . 20

5 Minimum Mean-Square Error Approximation 22

6 Fourier Series and LTI Systems 23

7 Speed of Convergence25

1

8 Effect of Truncating a Fourier Series by a Coefficient Win-

dow26 2 Letx(t) be a periodic signal with periodT0andfundamental frequency

0= 2π/T0. Fourier showed that these signals can be represented by a

sum of scaled sines and cosines at multiples of the fundamental frequency. The series can also be expressed as sums of scaled complex exponentials at multiples of the fundamental frequency. A sinusoid at frequencynω0is called annth harmonic. This document presents the approach I have taken to Fourier series in my lectures for ENEE 322 Signal and SystemTheory. Unless stated otherwise, it will be assumed thatx(t) is a real, not complex, signal. However, periodic complex signals can also be represented by Fourier series.

1 The Real Form Fourier Series

as follows: x(t) =a02+∞? n=1a ncosnω0t+bnsinnω0t(1) This is called a trigonometric series. We will call it the real form of the

Fourier series.

To derive formulas for the Fourier coefficients, that is, thea?sandb?s, we need trigonometric identities for the products of cosines and sines. You should already know the following formulas for the cosine ofthe sum and difference of two angles. cos(a+b) = cosacosb-sinasinb(2) cos(a-b) = cosacosb+ sinasinb(3) Adding these two equations and dividing by two gives cosacosb=1

2cos(a+b) +12cos(a-b) (4)

Subtracting the second equation from the first and dividing by 2 gives sinasinb=1

2cos(a-b)-12cos(a+b) (5)

You should already know the following two trigonometric identities: sin(a+b) = sinacosb+ cosasinb(6) 3 sin(a-b) = sinacosb-cosasinb(7)

Adding these last two equations give

sinacosb=1

2sin(a+b) +12sin(a-b) (8)

To finda0/2 consider the integral ofx(t) over one complete periodT0. For some conveniently chosen starting timet0this is t 0+T0? t

0x(t)dt=T0a

0

2+∞?

n=1a nt 0+T0? t

0cosnω0tdt+bnt

0+T0? t

0sinnω0tdt(9)

Each integral in the sum is overncomplete periods of a sine or cosine and is zero. Therefore a0

2=1T0t

0+T0? t

0x(t)dt(10)

which is just the DC value ofx(t). To findanforn≥1 consider the following integral fork≥1: t 0+T0? t

0x(t)coskω0tdt=t

0+T0? t 0a 0

2coskω0tdt+∞?

n=1a nt 0+T0? t

0cosnω0tcoskω0tdt

+bnt 0+T0? t

0sinnω0tcoskω0tdt(11)

Using identity (4) gives

t 0+T0? t

0cosnω0tcoskω0tdt=1

2t 0+T0? t

0cos[(n-k)ω0t]dt+12t

0+T0? t

0cos[(n+k)ω0t]dt

(12) The first integral on the right-hand-side is over|n-k|complete periods of the cosine and is zero whenn?=kandT0/2 whenn=k. The second integral 4 is overn+kperiods and is always zero. Using identity (8) gives t 0+T0? t

0sinnω0tcoskω0tdt=1

2t 0+T0? t

0sin[(n+k)ω0t]dt+12t

0+T0? t

0sin[(n-k)ω0t]dt

(13) The two integrals on the right are zero even whenn=k. Therefore an=2T0t 0+T0? t

0x(t)cosnω0tdtforn≥1 (14)

To findbnforn≥1 consider the following integral fork≥1: t 0+T0? t

0x(t)sinkω0tdt=t

0+T0? t 0a 0

2sinkω0tdt+∞?

n=1a nt 0+T0? t

0cosnω0tsinkω0tdt

+bnt 0+T0? t

0sinnω0tsinkω0tdt(15)

Using identity (8) gives

t 0+T0? t

0cosnω0tsinkω0tdt=1

2t 0+T0? t

0sin[(k+n)ω0t]dt+12t

0+T0? t

0sin[(k-n)ω0t]dt

(16) The first integral on the right-hand-side is overk+ncomplete periods of the sine and is always zero. The second integral is over|k-n|periods and is always zero also. Using identity (5) gives t 0+T0? t

0sinnω0tsinkω0tdt=1

2t 0+T0? t

0cos[(n-k)ω0t]dt-12t

0+T0? t

0cos[(n+k)ω0t]dt

(17) Both integrals on the right-hand-side are zero except whenn=k. Then the first integral isT0/2. Therefore bn=2T0t 0+T0? t

0x(t)sinnω0tdtforn≥1 (18)

5 The dot product of the three dimensional vectorsA=axi+ayj+azk andB=bxi+byj+bzkisA·B=axbx+ayby+azbz. Give two functions x(t) andy(t), the integral t 0+T0? t 0x(t) y(t)dt is abstractly similar to the dot product. Two vectors are orthogonal if their dot product is zero. It was shown above that the integrals of cosnω0tcoskω0t and sinnω0tsinkω0tover an interval of lengthT0are always zero fork?=n. Also the integral of sinnω0tcoskω0tover an interval of lengthT0is zero for allnandkTherefore, these sines and cosines can be considered to be orthogonal basis vectors in an infinite dimensional vector space. The Fourier coefficients are the coordinates of the functionx(t) with respect to the basis vectors in this infinite dimensional space. Even though we have derived formulas for thean"s andbn"s this does not prove that the series converges tox(t) because the sines and cosines may not be a rich enough set of functions. Fortunately, it can be shown that a rather mild set of conditions know as the Dirichlet conditions guarantee the series converges point wise to the function except at discontinuities. All the real-world functions we encounter satisfy these conditions. See Oppenheim and Willsky

1for these conditions. Also, if the magnitude squared ofx(t)

integrated over one fundamental period is finite, it can be shown that the series converges tox(t) in the sense that the integral over one fundamental period of the squared magnitude of the error betweenx(t) and the series is converges to zero. This is known as convergence in the mean-square error sense. Even if a function is not periodic, the Fourier series will converge to the function over the interval of integration (t0,t0+T0) and will extend periodically outside this interval.

EXAMPLE 1 Symmetric Square Wave

Letx(t) be the symmetric square wave show by the dashed purple linesin Figure 1. The formula for one period of this square wave centered about the

1A.V. Oppenheim and A.S. Willsky,Signals and Systems, 2nd Edition, Prentice Hall,

1996, pp. 197-198.

6 origin is x(t) =?-Afor-T0/2< t <0

Afor 0< t < T0/2(19)

The average or DC value of this signal is

a 0

2=1T0T

0/2? -T0/2x(t)dt= 0 (20)

Forn≥1

a n=2 T0T 0/2? -T0/2x(t)cosnω0tdt= 0 forn≥1 (21) This is true because functionx(t) is odd and cosnω0is even, so the product x(t)cosnω0tis odd and the integral of this product symmetrically about the origin is zero. Using the facts thatω0T0= 2πandx(t) and sinnω0tare both odd so that their product is even, the coefficients of the sine terms are b n=2 T0T 0/2? -T0/2x(t)sinnω0tdt=4T0T 0/2? 0 x(t)sinnω0tdt=4T0T 0/2? 0

Asinnω0tdt

4A T0? -cosnω0tnω0? ?T 0/2 0 =2Anπ(1-cosnπ) =2Anπ[1-(-1)n] ?4A nπfornodd

0 forneven(22)

Using these coefficients, the Fourier series for the square wave can be written as x(t) =4A n=112n-1sin(2n-1)ω0t(23) Approximations tox(t) when the sum is truncated at theN= 2n-1 harmonic orn= (N+ 1)/2 forN= 1,3,21 and 41 are shown in Figure 1. Notice that there is a significant overshoot at a jump even asNbecomes larger. It can be shown that the jump remains at close to 9% of the height of the jump asNincreases. This is known as Gibbs" phenomenon. The 7 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t/T0x

1(t)/A

x(t)/A (a) N = 1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t/T0x

3(t)/A

x(t)/A (b) N = 3 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t/T0x

21(t)/A

x(t)/A (c) N = 21 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 -0.7-0.6-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t/T0x

41(t)/A

x(t)/A (d) N = 41 Figure 1: Some Truncated Fourier Series Approximations to a Square Wave 8 frequency of the ripples increases but the ripples move closer to the jump and decay more quickly away from the jump asNincreases.

We saw in class that

acosθ+bsinθ=⎷ a2+b2cos[θ-arctan(b/a)] (24) Therefore, the Fourier series can also be expressed as x(t) =a0

2+∞?

n=1?a2n+b2ncos[nω0t-arctan(bn/an)] (25) This form shows the amplitude and phase shift of each harmonic.

2 The Complex Exponential Form of the Fourier

Series

We will now see that the real form of the Fourier series can be converted into a more compact form that is a sum of scaled complex exponentials at multiples of the fundamental frequency. Remember that cosθ=ejθ+e-jθ

2and sinθ=ejθ-e-jθ2j(26)

Using these identities, the real form Fourier series becomes x(t) =a0

2+∞?

n=1a The series can be rearranged by collecting the terms involvingejnω0tand e -jnω0tresulting in x(t) =a0

2+∞?

n=1a n-jbn2ejnω0t+an+jbn2e-jnω0t(28)

Now define the new coefficientcnas

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