Periodic functions

APPENDIX A

Periodic functions

A.1. Periodic functions

A.1.1. A periodic function of time is one in which F(t) = F(t+T) = F(t+2T) = . . .

(A.l)

for all values of t. The time constant T is the period (Fig. A.la). Symmetry properties: Alternate function (Fig. A.lb):

Flt+T2\=-F(t). Even function (Fig. A.lc):

F(t) = F(-t).

Odd function (Fig. A.Id):

F(t)=-F(-t). The mean value of a periodic function is F=f

i (rTF(t)dt.

FIG.

(A.2)

A.l.

The effective value of a periodic function is a constant whose square is equal to the mean square of F(t):

n, = ~jTPdL 18*

(A3) 263

APPENDIX A. PERIODIC FUNCTIONS

[§A.l

A.1.2. Sinusoidal functions: F(i) = a sin — (t+t0)

t0 = constant.

(A.4)

a is the amplitude (Fig. A.2). One can also write F(t) = a sin 2jtj>(f+fo).

(A.5)

v = 1 / r is the frequency or number of periods per unit time. If time is in seconds, v is given in hertz. One can also write: F(t) = 0sina)(f+fo) F(f) = tf sin(co/-t-q9)

(A.6)

co — 2TC/T is the angular frequency. The angle cot+cp, a linear function of time is the phase. When f increases by an integral number of periods, the phase increases by an integral number times 2JT and represents the same trigonometric curve. The phase is expressed in radians, cp is the phase angle at the origin (for t = 0). One has (Fig. A.2): 0 2 0 = cplco.

FIG.

A.2.

One can write in place of (A.5) F(t) — a cos (cot -f v)

with

(A.7)

One has (Fig. A.2) OiO = y/co. The mean value of a sinusoidal function is zero. The effective value of a sinusoidal function is defined by Fe2ff

a2 CT = — J sin2cotdt

a2 = ^

(A.8)

Two sinusoidal functions with the same period Fi = fli sin (cor-{-(pi)

and

F 2 = a 2 sin (cot-\-cp2)

have a phase difference cp =

cpi—(p2.

(A.9)

For cp — 0, the functions are in phase and they pass through the same set of zero and maximal values. 264

§A.2]

THE USE OF COMPLEX VARIABLES

For

F2 = ci2 sin lcot+q>i±~^ ) = ±a2 cos (cot+cpi). (A.10)

and

One is maximal or minimal when the other is zero. All sinusoidal functions are in quadrature with their first derivatives and in opposition with their second derivatives. A.2. The use of complex variables

One simplifies calculations involving sinusoidal functions by replacing them with exponential functions using Euler's theorem: exp(jx) = c o s x + j sinx

{j = \/(— 1)}

(A.11)

The real part is even, the imaginary part odd. One replaces (A.6) by F = a exp j (wt+cp) = a exp (j
(A.12)

by introducing the complex amplitude a = a exp (j
(A.13

which contains the amplitude a and the phase tp at the same time. One has, with a* the conjugate to a: a2 = a-a* and using (A.ll) tan cp =

a sin w imaginary e J part of a = \ . a cos (p real part of a

(A.14) ,. , _ (A. 15)

The use of complex variables is made possible by the fact that the partial differential equations which one encounters are linear with constant coefficients. One knows that one gets a solution by adding together the particular solutions. The real part and the imaginary part of a complex solution must separately satisfy the equations and as a result form two distinct solutions. By replacing trigonometric functions with complex functions, one then simultaneously gets two solutions whose terms remain separated. Now a complex variable has no immediate physical meaning since the result of a measurement is always a real number. The result of calculations using imaginaries is often in the form (A.12). One finds the phase and amplitude using equations (A.14) and (A.15) with the result that the complex number no longer appears in the final result. It also happens that the result will be in the form

A+]B = C+)D

which implies A = C and B = D. 265

APPENDIX A. PERIODIC FUNCTIONS

[§A.3

A.3. The sum of sinusoidal functions with the same period

A.3.1. The sum of two sinusoidal functions Fi = tfi sin (cot+cpi) and

F 2 = 02 sin (cot+992)

is a sinusoidal function with the same period One must have

F = A sin (cot+&).

01 cos (pi sin co/-l-fli sin 991 cos a>f-f-02 cos 992 sin cot-{-a2 sin 992 cos cot = A c o s 0 sin cot-\-A s i n 0 cos eof. In this expression of the form C sin cot+C cos cof = 0, C and C must be zero (otherwise one has tan cot =—C'/C = const, and t = const.). Thus, 01 COS
a\ sin 991+02 sin 992 = ^4 sin 0 v42 = a\+d^+la&i

cos ((pL—
, a\ sin r991 + 02 sin rG92 tan (P = —. 01 COS 99i-h02 COS (p2

(A.16) (A.17) (A.18)

If Fi and F 2 are in phase, A = ai+a2. If Fi and F 2 are in opposition, A = 01 — a2. If Fi and F 2 are in quadrature A2 = a\+a\. All sinusoidal functions can be thought of as the sum of two sinusoidal functions in quadrature since F(t) = 0 sin (cof-f 99) = a cos (p sin cof+0 sin 99 cos cot.

(A.19)

A.3.2. FresneVs construction. The function Fi (which is not necessarily a vector) is represented by a vector OV± or length 01 and makes an angle 991 with the polar axis (Fig. A.3). The function F 2 is similarly represented by a vector OV2. The sum F is represented by the vector OV, the vector sum of OV\ and OV2. Its length is A and the angle xOV = 0 . Through projection on Ox and Oy, one again gets the two relationships (A.16). The phase angles are taken as positive in the usual trigonometric sense. The generalization to more than two functions is obvious.

FIG.

266

A.3.

§A.4]

FOURIER SERIES

A.3.3. The use of complex quantities. For a vector such as OV (Fig. A.3), one can take the corresponding complex number: V = x+j y = K(cos 99-hj sin 99) = V exp (j 99)

(A.20)

with V = V(*2+y2)

and

tan

y

.

(A.21)

If one takes functions to be added in the form (A. 12), the complex amplitude of the sum is equal to the sums of the complex amplitudes: (A.22)

A = Yai. i

Using (A.20) this equality is, indeed, equivalent to Fresnel's construction. Finally, one gets A2 using (A. 14). A.4. Fourier series

A.4.1. All periodic functions of period T can, if they satisfy certain analytic properties which are always fulfilled in the problems treated here (for example, being defined with a finite value, being continuous except for a finite number of values of t, . . . ) , be represented by a trigonometric series or a Fourier series (1807). F(t) = c 0 +

Z c„ cos \2TI-?■ -
(A.23)

The components of the sum have amplitudes cn and phases (pn. Their frequencies vn — n/T — nv0 are integral multiples of the fundamental frequency v0 = \/T. The components are called harmonics. The fundamental is therefore the first harmonic (n = 1) and that of index n the n-ih. harmonic. The constant interval between two successive harmonics is v0. One can also write /

ith

nt

tit

> tn

n=l \

a„ = cn cos (pn

_ -tu„bin

Ail

bn = c/; sin
Cn = VK + K)

_

\

(A.24) (A.25) (A.26)

The coefficients an and bn are defined by 2

an = Y J

r+r/2

J-T/2

">«= ~ 1

J-T12

F(t) cos 2TT - ~ dr

(A.27)

F

(A28)

Wsin

2jr

r"d/I

267

[§A.4

APPENDIX A. PEROIDIC FUNCTIONS

The constant component of the expansion, c0, represents the mean value of the function F(t) taken over one period. It is found from 1

Co

-r 1

r+r/2 -772 J-TI

(A.29)

W) dr.

Equations (A.27) and (A.28) are based on the equalities +r'2 772

i 1

2nnt Inn't J cos - — -cos - — dr = 0 J

J

2

+V . 2nnt . Inrit A sin ——sin —=- dt = 0 *

-772

•*

+™ . 2;rwr 2rcn'f J sin-—-cos — - d t = 0 l

•r/2

d

for « ^ n'. For w = n\ the integrals above are all \. Functions which behave in this way, that is, for which

r rb

Fn(x).Fn(x) dx = 0

ri)

(A.29')

(# = constant independent of «)

(A.29")

(n *

are called orthogonal in the domain a < JC < fc. When F„(x) -Fn(x) dx = #

the functions are normalized. Calculation of the Fourier coefficients c0, an, andfc„,constitutes harmonic analysis of the function F(t). There are special apparatus designed to do this procedure and they are called harmonic analysers. A.4.2. Spectra. If one plots the amplitudes of the terms in a Fourier expansion as a function of their frequency, one gets the frequency spectrum of the function F. These spectra generally give an incomplete knowledge of F since they give no information about the phase
xy "%> FIG.

268

A.4.

U^ FIG.

-•V A.5.

§A.5]

FOURIER TRANSFORMS AND INTEGRALS

If all the an are zero in (A.24), F is an odd function. If all the bn are zero, F is an even function. In these two cases, the frequency spectrum gives a complete knowledge of the function F(t). A.4.3. One can also write the Fourier series in the complex form 00

F(t)=

£

n = — 00

/

nt\

C„exp J 2 T T \

/

.

(A.30)

n here being a positive, negative, or zero integer. Taking into account the Euler relationships* one finds 2C„ = cn exp ( - j (p„) = an-] bn cn = 2\Cn\ Co =

Co.

The complex amplitude 2Cn is found from 1 C+T/2 / nt\ F(t) exp / - j 2n -f-\ dr. Cn = T J

(A.31)

If one characterizes the periodic function by its angular frequency a)0 = 2n\T rather than by its period, (A.30) becomes, with the use of (A.31), I

00

J

n= — oo

F(t) = j ^ £

exp(jfia>oO

r + T/2

F(t)exp(-jnoj0t)dt.

(A.32)

J—T/2

A.5. Fourier transforms and integrals

A.5.1. One can extend the trigonometric series expansion to functions which are not periodic. A non-rigorous but simple way of doing this consists of letting the period T in the expansion of a periodic function tend to infinity. Taking (A.32) and making the transformations coo — dco, ncoo -* eo. (A.33) T ->■ 00, co being the continuous angular frequency variable and dco its increment. The sum E is transformed into an integral and (A.32) becomes j

F(t) = ?~-

r + 00 J—oo

exp (j (at) dco

r + °°

F(t) exp ( - j cot) dt

(A.34)

V_oo

or F(t) = ~

[ + O°0(CD) exp (j wt) dco

(A.35)

by taking 0(a>) = r°° F(t) exp ( - j cor) dt.

(A.36) 269

[§A.5

APPENDIX A. PERIODIC FUNCTIONS

The equations (A.34) and (A.35) define a Fourier transformation relating the functions F(t) and 0{co) which are Fourier transforms of one another. These functions play roles whose symmetry one can accent by using with each a constant factor \J\/2TI. One has, however, a positive exponential and the other a negative. F(t) = ~

f+°°0(co) exp (j cot) dco,

(A.37)

&(co) = - ^

f + °°F(0 exp ( - j cor) dr.

(A.38)

The equations (A.35) or (A.37) represent the Fourier integral in its complex form. The function F(t) which is not periodic is represented by a sum of an infinite number of sinusoidal functions whose angular frequencies are infinitely close to one another. The complex amplitude of each of these is infinitely small and is given by dC = -0(co) 71

(A.39)

dco.

A.5.2. In the above, one treated F(x) as a mathematical function. However it was not clear what physical significance could be attributed to the negative harmonics. In fact, according to paragraph A.2, (A.37) can be given a real form. Write 1

F(t) = —^--

V2n ' Jo

[0(co) exp (j cot)+0(-co) exp ( - j cot)] dco.

(A.40)

The expression between the brackets is the sum of conjugate variables, equal to twice the real part, Re, thus, F(t) = 1/ — Re \

0(co) exp (j cot) dco.

One can also write (A.40) If0 F(t) = —r^r— V2n

0

1 C°° [0{co) + 0(~co)] cos cot dco + ——[0(co) - &(-to)] sin cot dco V2TT J0

J0

which is in the form

1 f°° F(t\ = = —7^— F(t) [A(co) cos cot 4- j B(co) sin cot] dco.

V2 ' n Jo

If F(t) is a real even function of t, one has F(t) = F(-t) being real. One has then F w

0(co)= 1 / - J 270

and 0(co) =0(-co\

(A.41) with 0

(A.42)

\ F(r)coscordr.

«A.6]

THE SUM OF TWO SINUSOIDAL FUNCTIONS

If F(t) is real and odd, F(t) = - F ( - / ) , and one has &(a>) = - 0 ( - c o ) a n d 0 is purely imaginary. (A.41) becomes

f

sin Mt dc ( o = - j ]/y —\ ° n J0 ^(^)

r2nrl # ( ) = j 1/ — J w

(A-43) ^ ( 0 sin w/ d/.

A.5.3. While the spectra of periodic functions are discontinuous spectra or line spectra (Figs. A.4 and A.5), the spectra of non-periodic functions are continuous spectra and the interval between the lines becomes infinitely small following (A.33). They form a continuous series. From (A.39) one has Af~*

0(co) = 7i—-.

(A.44)

The function 0(co) then measures the spectral density and not the amplitude as previously stated. The absolute value of 0(co) is often called the spectrum of the function. A.5.4. The importance of the Fourier integral in physics comes from the fact that a periodic function is only an abstraction, certainly very useful, but always giving an imperfect image of the natural phenomenon. Aperiodic function defined by (A.l) has neither beginning nor end and thus a periodic phenomenon which depends on the limited duration of an experiment cannot be described by such a function. It will improve as a description as the number of periods gets very large. If this number is less than one, no period will be manifested in the curve representing the function. It is not necessary for the variables which arise in a Fourier transformation to be t and OJ = ITC/T, as they have been in this treatment. In paragraph B.3 we will examine the wave case where the variables are the length of a wave train x and the wave vector a. In problems dealing with diffraction at infinity, when the light amplitude distribution on a wave surface depends only on the coordinate x as F(x), the diffracted amplitude A in a direction defined by a parameter | (inversely proportional to the wavelength) is given by a Fourier transform. Examples of this will be found in paragraphs 5.14, 7.12, 7.16, 11.9, and 14.8. A.6. The sum of two sinusoidal functions with closely adjacent periods Let

Fi = 0i sin coif

and

F 2 = tf2 sin co2f,

(A.45)

be the two functions. By writing F 2 = a2 sin (coi-h Aeo)f = a 2 sin (a>it +
[§A.6

APPENDIX A. PERIODIC FUNCTIONS

They are in opposition when y = {2K+ X)n and the amplitudes subtract. Thus the amplitude varies periodically. This is called modulation. Its period T is such that Ay = 2n, so that Aco-T = In or Av-T = 1. (A.46) The frequency, l/r,of the amplitude modulation is equal to the difference Av = | vi-v2\ of the sum of the frequencies of the functions. The difference between the maximum and minimum values of the resultant amplitude (expressed in percent of the maximum) is the modulation strength. In the special case where a\ — a2, one has / Aco\ (A.47) F = F1+F2 = 2a cos —^ sin |+

t -2-)'-

The amplitude oscillates between the values 0 and 2a. The modulation strength is 100%. This phenomenon is called beats (Fig. A.6).

f¥WVW\A: FIG.

A.6.

Reciprocally, if one periodically varies the amplitude of a sinusoidal vibration by any physical means, one can think of the modulated vibration as the sum of two sinusoidal .vibrations such as (A.45).

^lx

FIG.

272

A.7.

PROBLEMS

A.7. The function sin x/x Figure A.7 shows the form of this function which occurs in paragraphs 5.10,6.7,6.8, and 16.5. Problems A.3.A. Find the resulting amplitude from n sinusoidal functions of the same amplitude a whose initial phases are n/n, 2-r//i, 3rc//i, ... A.3.B. Same question with phases In/n, An In, 6ji/n, . . . A.3.C. Same question with initial phases , . . . A.3.D. Derive (A. 17) from (A.22). A.4.A. Study the following functions from the point of view of parity and alternation: sin x — 2 cos 2x\ sin x - 2 sin 3x; cos x+ cos 2x\ sin 5x4-2 sin Ix; cos x + cos 3 x - 2 cos 7A\ A.4.B. Give the Fourier expansions of the following functions: 3T 2 ' •*•

(a)

n T F = — from 0 to — ;

(b)

F = / from 0 to T, t - T from T to 2T...

(c)

F = / from 0 to — ,

from T to

and

from

T

T

~ to 0, from -- to T . . . (Fig. A.8).

(Fig. A.9).

- 1 + — from — to 3 - - . . . (Fig. A.10).

A.5. Find the spectra of the functions represented by the following curves: (a) a rectangle of height Fand base 2T (Fig. A . l l a ) ; (b) a triangle of height F a n d base 2T (Fig. A . l l b ) ; (c) Gauss' function in the form exp (-fi2t2 )(Fig. A . l i e ) ; and (d) a portion of a sinusoid containing n periods (Fig. A.lid).

i-QL

ii FIG.

:T

A.8.

a> 0

T

2T FIG.

FIG.

3T

A.9.

A. 10.

FIG.

A.ll.

273