Periodic Functions, Harmonics, and Why Fourier Series Exist

(Topic)

FS01: Periodic Functions, Harmonics, and Why Fourier Series Exist

Fourier series begin with a simple observation: many physical motions repeat.

A pendulum oscillates. A vibrating string repeats its motion. An alternating voltage cycles. A rotating shaft returns to the same orientation. A repeating pulse train returns after each pulse interval.

The first goal of Fourier analysis is therefore not to compute coefficients. It is to understand the mathematical structure of repetition.

This lecture develops that structure in small steps:

what it means for a function to be periodic,
the difference between a period and the fundamental period,
frequency and angular frequency,
amplitude and phase,
harmonics and harmonic number,
why integer multiples of one fundamental frequency fit together naturally,
why sums of harmonics remain periodic,
how simple harmonics begin to build complicated repeating waveforms.

The Fourier coefficient formulas are deliberately postponed. At the end of this article, the natural next question will be:

How can we measure how much of a particular harmonic is present in a function?

FS02 answers that question by introducing inner products of functions.

1 Periodic functions

A function f is periodic if there exists a positive number T such that

f(x + T) = f (x)

(1)

for every x for which both sides are defined.

The number T is called a period of the function.

Equation (1) says that if we move forward by T, the function value repeats exactly.

For a time-dependent function,

f(t + T ) = f(t),

(2)

so the system returns to the same function value after every interval T.

Figure 1. A periodic function repeats after a fixed horizontal displacement T. A nonperiodic function has no single positive displacement that reproduces the complete graph for all values of the independent variable.

1.1 The independent variable does not have to be time

Fourier series are often introduced with time-dependent oscillations, but the independent variable can be many things.

Examples include:

time t,
position x along a string,
angle 𝜃 around a circle,
position along a repeating crystal lattice,
spatial coordinate in a periodically repeated boundary condition.

For this reason, the notation

f(x + T) = f (x)

(3)

should be understood generically.

When the independent variable is time, T has units of time. When the independent variable is distance, the period has units of distance.

2 A period is not always the fundamental period

Suppose

f (t) = cos(ωt).

(4)

If T is one period, then

cos[ω(t + T)] = cos(ωt)

(5)

for all t.

Because cosine repeats when its argument increases by 2π,

ωT = 2 π

(6)

is the smallest positive choice.

Thus

2 π T = -ω-.

(7)

But if T is a period, then

2T, 3T, 4T, ...

(8)

are also periods because the function repeats after any whole number of cycles.

The fundamental period is the smallest positive period.

We will denote it by

T . 0

(9)

For

f (t) = cos(ωt),

(10)

the fundamental period is

T0 = 2π. ω

(11)

3 Frequency

Frequency counts cycles per unit time.

If one complete cycle requires time T₀, then the number of cycles completed in one unit of time is

1 f0 = T-. 0

(12)

The SI unit of frequency is the hertz:

1 Hz = 1 cycle∕s.

(13)

For example, if

T0 = 0.25 s,

(14)

then

f₀	=	(15)
	= 4 Hz.	(16)

The system completes four cycles each second.

4 Angular frequency

One complete cycle corresponds to an angular advance of

2π rad.

(17)

If the cycle takes time T₀, then the angular rate is

2π ω0 = --. T0

(18)

Using

f = 1-, 0 T0

(19)

we obtain

ω0 = 2πf0.

(20)

Therefore the three quantities are related by

f = -1-, ω = 2 πf , T = 2π-. 0 T0 0 0 0 ω0

(21)

Figure 2. A sinusoid can be viewed as the projection of uniform circular motion. One full revolution is one cycle, corresponding to an angular advance of 2π radians. This gives ω₀ = 2π∕T₀ = 2πf₀.

4.1 Why radians are especially natural

Radians make the phase of a sinusoid directly proportional to angle.

The argument

ω0t

(22)

is dimensionless because

1 [ω0] = ----- time

(23)

when radians are treated as dimensionless angular measure.

After one period,

ω₀(t + T₀)	= ω₀t + ω₀T₀	(24)
	= ω₀t + 2π.	(25)

Cosine and sine therefore return to the same values.

5 Amplitude

Consider

y(t) = A cos(ω0t).

(26)

The constant A is the amplitude.

For A > 0,

− A ≤ y (t) ≤ A.

(27)

The amplitude controls vertical scale, not the rate at which the waveform repeats.

Thus changing A does not change the period:

A cos[ω0 (t + T0)] = A cos(ω0t).

(28)

6 Phase

A more general sinusoid is

y(t) = A cos(ω t + ϕ ). 0

(29)

The constant ϕ is the phase angle.

It tells us where in its cycle the oscillation is located when t = 0.

At t = 0,

y (0 ) = A cos ϕ.

(30)

The phase can also be interpreted as a horizontal shift.

Factor ω₀ from the argument:

ω₀t + ϕ

= ω₀

(31)

Thus

[ ( ) ] A cos(ω t + ϕ ) = A cos ω t +-ϕ- . 0 0 ω0

(32)

Relative to A cos(ω₀t), the graph is shifted left by

-ϕ-. ω0

(33)

Equivalently, if we write

y(t) = A cos[ω (t − t )], 0 s

(34)

then the corresponding phase is

ϕ = − ω0ts.

(35)

Figure 3. Two sinusoids can have the same amplitude and frequency but different phase. A positive phase in Acos(ωt + ϕ) shifts the graph to the left by ϕ∕ω.

7 Combining sine and cosine at the same frequency

Fourier series use both sine and cosine terms. It is useful to see that a sine-plus-cosine pair at one frequency is still just one sinusoid with a different amplitude and phase.

Start with

y(t) = A cos(ωt) + B sin(ωt).

(36)

Suppose we want to rewrite this as

y (t) = C cos(ωt − δ).

(37)

Use the identity

cos(α − β) = cosα cosβ + sin α sin β.

(38)

Then

C cos(ωt − δ)	= C cos(ωt) cos δ	(39)
	+ C sin(ωt) sin δ.	(40)

Compare with Equation (36). The cosine coefficients must match:

A = C cosδ.

(41)

The sine coefficients must also match:

B = C sin δ.

(42)

Square both equations and add:

A² + B²	= C² cos ²δ + C² sin ²δ	(43)
	= C²(cos ²δ + sin ²δ)	(44)
	= C².	(45)

Therefore

√ -2-----2- C = A + B .

(46)

Divide Equation (42) by Equation (41):

B-= tanδ. A

(47)

Thus the phase may be written carefully as

δ = atan2 (B, A),

(48)

which places the angle in the correct quadrant.

This result will later let us move freely between sine-cosine coefficients and amplitude-phase descriptions.

8 What is a harmonic?

Suppose a periodic function has fundamental period

T0.

(49)

Its fundamental frequency is

1-- f0 = T0,

(50)

and its fundamental angular frequency is

2π- ω0 = T0.

(51)

The frequencies

fn = nf0, n = 1,2, 3,...

(52)

are called harmonics.

Equivalently,

ωn = n ω0.

(53)

The integer n is the harmonic number.

Thus:

n = 1 is the fundamental or first harmonic,
n = 2 is the second harmonic,
n = 3 is the third harmonic,
and so on.

The corresponding sinusoidal basis functions are

cos (n ω0t), sin(nω0t).

(54)

Figure 4. The first four harmonics. Higher harmonic number means more oscillations within the same fundamental period T₀. The nth harmonic completes exactly n cycles while the fundamental completes one.

9 The period of the nth harmonic

The nth harmonic is

yn(t) = cos(n ω0t).

(55)

Its angular frequency is

ωn = n ω0.

(56)

Therefore its smallest positive period is

T_n	=	(57)
	=	(58)
	=	(59)
	= .	(60)

So the second harmonic repeats twice as fast as the fundamental, the third harmonic three times as fast, and so on.

But an important subtlety is that every harmonic also repeats after the larger interval T₀.

For the nth harmonic,

cos[nω₀(t + T₀)]	= cos(nω₀t + nω₀T₀)	(61)
	= cos(nω₀t + 2πn)	(62)
	= cos(nω₀t).	(63)

Thus T₀ is a common period of every integer harmonic.

This fact is one of the central reasons Fourier series use integer multiples of a fundamental frequency.

10 Why integer harmonics line up after one fundamental period

Suppose a sinusoidal basis function has angular frequency ω and must fit a period T₀.

We require

cos[ω(t + T0)] = cos(ωt)

(64)

for every t.

The phase advance over one period must therefore be a whole number of turns:

ωT = 2πn, n ∈ ℤ. 0

(65)

Solve for ω:

2πn ω = ----. T0

(66)

But

ω = 2π. 0 T0

(67)

Hence

ω = nω0.

(68)

The harmonic frequencies are therefore not arbitrary. They are exactly the sinusoidal frequencies that complete an integer number of cycles in the interval T₀.

Figure 5. Integer harmonics all return to their starting phase after the common interval T₀. This is the geometric reason the Fourier basis is naturally indexed by integer harmonic number.

11 Superposition of harmonics remains periodic

Consider a finite harmonic sum

∑N SN (t) = A0 + [An cos(n ω0t) + Bn sin(nω0t )]. n=1

(69)

Evaluate it one fundamental period later:

S_N(t + T₀)	= A₀	(70)
	+ ∑ _n=1^NA_n cos[nω₀(t + T₀)]	(71)
	+ ∑ _n=1^NB_n sin[nω₀(t + T₀)].	(72)

Use

ω0T0 = 2π.

(73)

Then

nω0T0 = 2πn.

(74)

Therefore

cos(nω t + 2πn ) = cos(n ω t) 0 0

(75)

and

sin(n ω0t + 2 πn) = sin(nω0t).

(76)

Hence

SN (t + T0) = SN (t).

(77)

Any finite sum of integer harmonics is therefore periodic with common period T₀.

Its fundamental period can sometimes be smaller than T₀ if some low harmonics are absent, but T₀ is always a period.

12 The constant or DC component

The term

A0

(78)

in Equation (69) does not oscillate.

It is a constant offset.

In time-dependent signal language, it is often called the DC component.

Adding a constant does not change periodicity:

A0 = A0.

(79)

Geometrically, the constant component simply shifts the entire waveform upward or downward.

Later, the real Fourier series will conventionally write this constant term as

a0-. 2

(80)

FS05 will derive why this factor of 1∕2 appears.

13 How harmonics build a more complicated waveform

Consider

S(t)	= 1 + 2 cos(ω₀t)	(81)
	− 0.75 sin(2ω₀t)	(82)
	+ 0.40 cos(3ω₀t).	(83)

This contains:

a constant component 1,
a first-harmonic cosine of amplitude 2,
a second-harmonic sine of amplitude 0.75 with a negative sign,
a third-harmonic cosine of amplitude 0.40.

Each piece is simple. Their sum is less obviously sinusoidal.

Figure 6. Superposition of a constant term and the first three harmonics. The individual components are simple sinusoids, but their sum already produces a visibly more complicated repeating waveform.

This is the central Fourier idea in its simplest form:

complicated periodic shape can arise from simple periodic components,
the frequencies are organized as integer harmonics,
amplitudes and phases determine the detailed shape.

14 Commensurate frequencies

Integer harmonics are an especially simple example of commensurate frequencies.

Two angular frequencies ω₁ and ω₂ are commensurate if their ratio is rational:

ω1-= p- ω2 q

(84)

for integers p and q.

If the ratio is rational, a common period can exist.

For example, consider

f(t) = cos(6t) + sin(9t).

(85)

The frequencies are

6 rad∕s

(86)

and

9 rad ∕s.

(87)

Both are integer multiples of

3 rad ∕s.

(88)

Thus choose

ω0 = 3 rad ∕s.

(89)

Then

6 = 2ω0, 9 = 3ω0.

(90)

The two terms are the second and third harmonics of the common fundamental.

The common fundamental period is

T₀	=	(91)
	= s.	(92)

15 Incommensurate frequencies need not produce a periodic sum

Now consider

√-- g(t) = sin(2t) + sin ( 2 t).

(93)

Suppose, for contradiction, that this sum has a common period P inherited from both sinusoidal components.

The first component would require

2P = 2πm

(94)

for some integer m.

The second would require

√ -- 2P = 2πn

(95)

for some integer n.

Divide the second relation by the first:

√ -- 2 n ----= --. 2 m

(96)

But the left side is irrational while the right side is rational.

This is impossible.

Therefore the two component periods have no common integer multiple, and the sum does not repeat with one finite fundamental period.

This example explains why Fourier series for a function of period T₀ use a discrete set of commensurate frequencies

n ω0.

(97)

16 Preview of the real Fourier series

The real Fourier series has the form

∑∞ f(t) ∼ a0-+ [an cos(nω0t) + bnsin(nω0t)]. 2 n=1

(98)

At this stage, Equation (98) should be read structurally rather than computationally.

It says:

one constant component may be present,
the first harmonic may contain sine and cosine pieces,
the second harmonic may contain sine and cosine pieces,
the third harmonic may contain sine and cosine pieces,
the pattern continues to higher harmonic number.

We have not yet derived

an or bn.

(99)

That derivation must wait until we know how to measure the similarity between two functions.

17 Partial sums

An infinite series cannot be drawn by adding infinitely many terms one at a time.

Instead we use a partial sum.

Define

a0 ∑N SN (t) = --+ [an cos(n ω0t) + bn sin (n ω0t)]. 2 n=1

(100)

The integer N tells us the highest harmonic retained.

For example:

S₁ contains only the constant and first harmonic,
S₂ includes harmonics through n = 2,
S₁₀ includes harmonics through n = 10.

The sequence

S , S , S , ... 1 2 3

(101)

is what we actually study when asking whether a Fourier series converges.

18 Square-wave preview

A classic example is the odd square wave.

Later, FS07 will derive its coefficients from first principles. For now, accept the preview formula

∑N SN (t) = 4- 1-sin (nt ). π n=1 n n odd

(102)

The first approximation keeps only the first harmonic:

S1 (t) = -4sint. π

(103)

Including the third harmonic gives

( ) 4- 1- S3(t) = π sint + 3 sin 3t .

(104)

Including the fifth harmonic gives

4 ( 1 1 ) S5(t) = -- sin t +--sin3t + --sin 5t . π 3 5

(105)

As additional odd harmonics are added, the graph becomes increasingly square-like.

Figure 7. Preview of square-wave synthesis. Adding odd harmonics changes a smooth sinusoid into a waveform with increasingly flat plateaus and increasingly sharp transitions. FS07 will derive the coefficients rather than assume them.

The figure contains an important warning. Near the jump, the approximation develops overshoot. This does not disappear in the simple way one might first expect. The phenomenon is studied carefully in FS15 and FS16.

19 Why adding higher harmonics changes shape

Higher harmonics oscillate more rapidly.

The first harmonic controls large-scale variation over the entire period. The second harmonic can add structure on half-period scales. The tenth harmonic can add structure on scales about one-tenth of the period.

Thus harmonic number is connected to scale:

low harmonic number describes broad features,
high harmonic number describes finer features.

This is the beginning of the spectral viewpoint.

Later we will see a deeper connection between smoothness and the rate at which high-harmonic coefficients decrease.

20 Why Fourier series are plausible

We can now give a careful answer to the title question.

Fourier series are plausible for periodic functions because:

Sine and cosine are themselves periodic.
The integer harmonics all share the required common period T₀.
Linear combinations of those harmonics remain periodic.
Amplitudes and phases provide enough freedom to change the waveform shape.
Higher harmonics supply progressively finer variation within one period.

However, these facts do not yet prove that every periodic function equals its Fourier series at every point.

Different notions of convergence must be distinguished. Discontinuities require special care. Some functions are better behaved than others.

Those questions are postponed until FS14 through FS16.

At this stage, the correct conclusion is more modest:

integer harmonic sinusoids form a natural candidate basis for representing periodic functions.

The next several lectures develop the mathematics needed to make that statement precise.

21 Worked Example 1: period, frequency, and angular frequency

Suppose a repeating signal has period

T0 = 5.0 ms.

(106)

Convert to seconds:

T0 = 5.0 × 10−3 s.

(107)

The frequency is

f₀	=	(108)
	=	(109)
	= 200 Hz.	(110)

The angular frequency is

ω₀	= 2πf₀	(111)
	= 2π(200)	(112)
	= 400π rad∕s	(113)
	≈ 1256.64 rad∕s.	(114)

22 Worked Example 2: amplitude-phase conversion

Rewrite

y (t) = 3 cos(ωt) + 4 sin (ωt)

(115)

as one cosine.

The combined amplitude is

C	=	(116)
	= 5.	(117)

The phase is

δ	= atan2(4, 3)	(118)
	≈ 53.13^∘.	(119)

Therefore

y(t) = 5 cos(ωt − 53.13 ∘).

(120)

The original sine and cosine terms were not two distinct frequencies. They were two quadrature components of one sinusoid at the same frequency.

23 Worked Example 3: finding a common fundamental

Consider

f (t) = 2 cos(6t) − sin(9t).

(121)

The angular frequencies are

6 rad∕s and 9 rad ∕s.

(122)

Their greatest common angular frequency is

ω0 = 3 rad ∕s.

(123)

Then

6 = 2ω0

(124)

and

9 = 3ω0.

(125)

Thus the waveform contains the second and third harmonics of the fundamental.

Its fundamental period is

T₀	= s	(126)
	≈ 2.094 s.	(127)

24 Worked Example 4: identifying harmonic content

Consider

f(t)	= 2 + 3 cos(ω₀t)	(128)
	− sin(2ω₀t) + 0.5 cos(3ω₀t).	(129)

The components are:

DC component: 2,
first harmonic: cosine amplitude 3,
second harmonic: sine coefficient −1,
third harmonic: cosine amplitude 0.5,
all harmonics above n = 3: absent.

Every oscillating term has frequency equal to an integer multiple of ω₀, so the entire function has T₀ = 2π∕ω₀ as a period.

25 Worked Example 5: a smaller fundamental period can hide inside a chosen period

Consider

f(t) = cos(2 ω t) + 1sin(4ω t). 0 2 0

(130)

Both terms repeat after

T = 2π, 0 ω0

(131)

so T₀ is certainly a period.

But the first term has period

T0 ---, 2

(132)

and the second has period

T0-. 4

(133)

After T₀∕2:

cos	= cos(2ω₀t + 2π)	(134)
	= cos(2ω₀t),	(135)

and

sin	= sin(4ω₀t + 4π)	(136)
	= sin(4ω₀t).	(137)

Therefore T₀∕2 is a period of the whole function.

The chosen T₀ was a common period, but it was not the fundamental period.

This is why the phrase fundamental period must be used carefully.

26 Physical examples of harmonic decomposition

The harmonic viewpoint appears throughout physics.

Vibrating string

A string fixed at both ends supports normal modes with frequencies

fn = nf1.

(138)

An arbitrary initial shape can be decomposed into those spatial harmonics.

Acoustics

A musical tone may contain a fundamental plus overtones. The relative harmonic amplitudes help determine timbre.

Electrical signals

A periodic voltage can be decomposed into a DC term plus sinusoidal harmonics. A linear circuit can then be analyzed one frequency at a time.

Periodic forcing

A nonsinusoidal periodic force acting on an oscillator can be written as a sum of harmonic forces. Because the oscillator equation is linear, the individual harmonic responses can be superposed.

Partial differential equations

The heat and wave equations naturally produce sinusoidal eigenfunctions under common boundary conditions. Fourier series then become a method for matching arbitrary initial data.

These applications are developed later in the FS sequence rather than assumed here.

27 What FS01 has not yet answered

We now know why the harmonic frequencies

ωn = nω0

(139)

are natural for a period-T₀ problem.

But several major questions remain:

How do we determine the coefficient of one particular harmonic?
Why do unwanted harmonics disappear when extracting a coefficient?
Why are sine and cosine special relative to one another over a complete period?
Under what conditions does the infinite series converge to the original function?
What happens at a discontinuity?

The first three questions require the ideas of inner product and orthogonality.

Figure 8. FS01 establishes the frequency structure of the Fourier basis. FS02 and FS03 next supply the inner-product and orthogonality machinery needed to extract individual harmonic coefficients.

28 Self-check questions

A learner should be able to answer the following before moving to FS02.

What equation defines a periodic function?
What is the difference between a period and the fundamental period?
How are T₀, f₀, and ω₀ related?
What does phase change geometrically?
What is the nth harmonic?
Why does the nth harmonic complete exactly n cycles during one fundamental period?
Why is any finite sum of integer harmonics periodic with common period T₀?
What does the constant or DC component do to a waveform?
What is a partial sum?
Why can incommensurate frequencies fail to produce a periodic sum?
Why is a Fourier-series representation plausible but not yet proven by this lecture?

29 Practice exercises

Show directly that sin(5t) has fundamental period 2π∕5.
A signal has frequency 60 Hz. Find its period and angular frequency.
A cosine has angular frequency 20π rad∕s. Find its frequency and period.
Rewrite 5 cos(ωt) + 12 sin(ωt) as one cosine with a phase shift.
Identify the harmonic numbers present in
$f (t) = 1 − 2sin(3ω0t) + 4cos(7ω0t ).$ (140)
Determine a fundamental angular frequency for
$f (t) = cos(8t) + sin(12t).$ (141)

Then determine the fundamental period.
Show that
$cos(2t) + cos(3t)$ (142)

is periodic and find its fundamental period.
Explain why
$√ -- sin t + sin( 3 t)$ (143)

has no common period inherited from both component sinusoids.
Verify directly that the finite harmonic sum in Equation (69) satisfies S_N(t + T₀) = S_N(t).
Plot the square-wave partial sums S₁, S₃, and S₅ from Equation (102). Describe what changes and what does not.

30 Summary

A periodic function satisfies

f(t + T ) = f(t).

(144)

The smallest positive period is the fundamental period T₀.

Frequency and angular frequency are

1 f0 = ---, ω0 = 2πf0. T0

(145)

The integer harmonics are

ω = n ω . n 0

(146)

The nth harmonic completes n cycles during one fundamental period and therefore returns to its starting phase after T₀.

Consequently, finite sums of integer harmonics are periodic with common period T₀.

Amplitude determines scale. Phase determines horizontal position within the cycle. Higher harmonics add progressively finer structure to a waveform.

These facts explain why sine and cosine harmonics are natural building blocks for periodic functions.

The next article, FS02, asks how to measure the amount of one basis function inside another function. The answer begins by generalizing the vector dot product to an inner product of functions.

References

[1] PhysicsLibrary, FS00: Fourier Series Roadmap, Motivation, and Learning Path.

[2] R. V. Churchill and J. W. Brown, Fourier Series and Boundary Value Problems, McGraw-Hill.

[3] E. Kreyszig, Advanced Engineering Mathematics, Wiley.

[4] G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, Academic Press.

"Periodic Functions, Harmonics, and Why Fourier Series Exist" is owned by bloftin.
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