Equivalence of Real and Complex Fourier Forms

FS20: Equivalence of Real and Complex Fourier Forms

The real Fourier series and the complex Fourier series are not two different theories of periodic functions. They describe the same harmonic components using different coordinates. In FS04–FS06 we extracted coefficients of cosine and sine functions. FS18 rewrote those functions as oppositely rotating complex exponentials, and FS19 derived the complex coefficients by orthogonal projection. Now we will join those two developments into a reversible calculation.

A key conceptual question is this: if a real signal has one cosine-and-sine harmonic at frequency n, why does its complex spectrum have two coefficients, one at +n and one at −n? The answer is not additional physical energy or a new frequency. It is a different bookkeeping system for the very same real oscillation.

The central results we will earn for n ≥ 1 are

(1)

and

(2)

We will also show that C₀ = a₀∕2 and, for real-valued f, that C_−n = C_n.

1 Begin with the two formulas on the same interval

Let f be a real-valued, 2L-periodic function. Write

(3)

We use 𝜃 only as an abbreviation; the period in x remains 2L. The real Fourier form is

(4)

The complex Fourier form is

(5)

In the first sum, the index starts at 1, and there are two real coefficients for each positive harmonic. In the second sum, the index spans all integers and there is one complex coefficient for each signed harmonic. The complex sum also contains n = 0.

When making algebraic transformations, begin with finite sums. This avoids silently exchanging infinite sums or limits. Set

(6)

and

(7)

Our first task is to choose C_n so that Q_N(x) = R_N(x) identically, for every x, without taking N to infinity.

2 One cosine requires a positive and a negative frequency

Euler’s formula reads

(8)

Changing the sign of α gives

(9)

Add these equations. The imaginary terms cancel:

(10)

Subtract instead. The real parts cancel:

(11)

Therefore

(12)

and

(13)

These are identities, not approximations.

An important geometric point follows. The phasor e^in𝜃 rotates counterclockwise as 𝜃 increases for positive n, while e^−in𝜃 rotates in the opposite direction. Their sum has no imaginary component and produces a cosine. Their suitably scaled difference produces a sine.

3 Derive the conversion from real coefficients to complex coefficients

Take just one real harmonic, not the whole series:

(14)

Replace cosine and sine using Equations (12) and (13):

hn(x)	=
	+ .	(15)

The only algebraic subtlety is that

(16)

because i(−i) = 1. Therefore the sine part can be rewritten as

(17)

Now collect the multiplier of e^in𝜃 and the multiplier of e^−in𝜃:

hn(x)	= ein𝜃
	+ e−in𝜃.	(18)

Compare this equation with

(19)

The coefficients must be

(20)

and

(21)

What happens to the constant term?

At zero frequency, e^i0𝜃 = 1. There is only one zero-frequency mode, not a positive-negative pair. The constant term in Equation (6) is already a₀∕2. Thus

(22)

Do not apply Equation (20) at n = 0 while also inventing a separate C₋₀. That would count the DC component twice.

4 Derive the reverse conversion from complex to real

Now suppose the complex coefficients C_n and C_−n are known. Start with the two equations just obtained:

(23)

(24)

Add them:

(25)

so

(26)

Subtract the second equation from the first:

(27)

Divide by −2i, or multiply by i∕2:

(28)

Finally,

(29)

The two forms are therefore related by a reversible linear change of coordinates. We did not approximate anything in either direction.

Why real signals have conjugate pairs

For a real-valued function, the Fourier coefficients a_n and b_n are real numbers. Conjugating Equation (20) gives

(30)

Consequently,

(31)

and

(32)

The minus sign in the formula for b_n follows from our e^+in𝜃 synthesis convention. Other Fourier sign conventions can reverse it, so the safe approach is to derive the conversion once from Euler’s formula.

We can verify the real-valued nature of a conjugate pair directly:

Cnein𝜃 + C −ne−in𝜃	= C nein𝜃 + C nein𝜃
	= 2 Re .	(33)

Thus a conjugate pair is exactly the real harmonic a_n cos(n𝜃) + b_n sin(n𝜃).

5 Worked example 1: reconstruct a finite signal both ways

Reuse the FS19 waveform

(34)

From the real form we read

(35)

with every other nonzero-index coefficient equal to zero. Use our conversion formulas one harmonic at a time.

At zero frequency,

(36)

At the first harmonic, b₁ = 0, giving

(37)

At the second harmonic, a₂ = 0 and b₂ = −2, giving

(38)

Thus the complex form is the finite expression

(39)

Group the pair at n = ±1:

(40)

Group the pair at n = ±2:

(41)

We recover Equation (34) exactly.

As an independent check, use the reverse formulas. For n = 2,

(42)

and

(43)

Notice how the sine coefficient is stored as imaginary components with opposite signs in the complex spectrum.

6 Amplitude and phase: interpreting a complex coefficient

We often express one real harmonic as

(44)

Expand the cosine of a difference:

hn(x)	= An cos φn cos(n𝜃)
	+ An sin φn sin(n𝜃).	(45)

Compare with a_n cos(n𝜃) + b_n sin(n𝜃) to identify

(46)

Therefore

(47)

The phase φ_n is chosen with the correct quadrant, for example using the two-argument arctangent atan2(b_n,a_n).

Substitute Equation (46) into the complex conversion:

Cn	=
	= e−iφn .	(48)

Hence

(49)

The complex spectrum has two conjugate coefficients of magnitude A_n∕2, but the real sinusoid has amplitude A_n. It would be wrong to read |C_n| alone as the real amplitude without accounting for the negative-frequency partner.

A quick phase example

Take

(50)

The real coefficients are a₁ = and b₁ = 1. The complex coefficients are

(51)

Each has magnitude 1, while the recovered real sinusoid has amplitude 2. The negative argument of C₁ is consistent with our cos(x − φ) phase convention.

7 Worked example 2: the odd square wave

Recall from FS07 the odd square wave of amplitude A on [−π,π]:

(52)

It is odd, so a₀ = a_n = 0. Its sine coefficients are

(53)

For positive n, the complex coefficient is therefore

(54)

For odd positive n this reduces to

(55)

Its negative-frequency partner is

(56)

Both even-harmonic coefficients are zero; the conjugate pair at each odd harmonic recreates the sine term. Notice that |C_n| = 2A∕(πn) whereas the real sine amplitude is b_n = 4A∕(πn).

8 Worked example 3: an even rectangular pulse

Define a periodic centered pulse by

(57)

Because the pulse is even, all b_n = 0. Its constant term is

(58)

For n ≥ 1 the cosine coefficient is

an	= ∫ 0d cos dx
	= sin .	(59)

Thus the complex coefficients are

(60)

and

(61)

These coefficients are real and even in their index. This is the spectral signature of an even, real waveform about the chosen origin. The sign of a coefficient can still be negative, so a plot of coefficient magnitude alone loses information.

9 Time shifts change phase, not coefficient magnitude

For a periodic time signal, replace 𝜃 by ω₀t, where ω₀ = 2π∕T. Suppose

(62)

Shift the entire waveform to the right by t₀:

g(t)	= f(t − t0)
	= ∑ nCneinω0(t−t0)
	= ∑ neinω0t.	(63)

Therefore

(64)

Since |e^{−inω₀t₀}| = 1, the magnitude stays unchanged:

(65)

A delay changes the phases by −nω₀t₀; harmonics at higher n rotate farther.

To see the same transformation in real coefficients, set α = nω₀t₀ and expand a single shifted harmonic. One finds

(66)

(67)

These formulas are a rotation in the (a_n,b_n) coefficient plane. In the complex plane, the same rotation is expressed more economically by multiplication by one exponential.

A quick check

Start with f(t) = cos(ω₀t) and delay by T∕4. Then α = π∕2, and the shifted waveform is sin(ω₀t). The real formulas yield a₁^(g) = 0 and b ₁^(g) = 1. The complex formula yields C ₁^(g) = −i∕2, C₋₁^(g) = i∕2. Both descriptions agree.

10 Finite energy bookkeeping also agrees

FS12 established that orthogonal projection splits the squared norm among retained directions. For a real Fourier partial sum on [−L,L], orthogonality gives

(68)

Using the complex basis, whose squared norm is 2L, we obtain

(69)

But the finite sums are the same function. The right sides must agree. Indeed, for each positive n,

|Cn|2 + |C −n|2	= +
	= .	(70)

Multiplying by 2L gives precisely L(a_n² + b _n²). The DC contribution is 2L|C ₀|² = La ₀²∕2.

This identity is finite, so it does not yet require a theorem about the convergence of the infinite Fourier series. FS13 develops the infinite-series energy statement, Parseval’s theorem, under the appropriate completeness assumptions.

11 What if the function itself is complex-valued?

Our algebraic identities converting a_n,b_n and C_±n do not require f to be real, provided the real-form coefficients are allowed to be complex. However, the symmetry

(71)

does require the signal to be real-valued (in the usual almost-everywhere sense). For example,

(72)

has C₁ = 1 and C₋₁ = 0. This is perfectly valid for a complex signal; no conjugate partner is required.

Complex signals are useful mathematical objects, and in electronics the in-phase/quadrature signal is commonly represented as a complex envelope. The physical meaning of this complex notation depends on the system and should not be confused with the automatically real quantity Re{e^i𝜃}.

12 Physics example: periodic forcing of a damped oscillator

Consider the linear mass-spring-damper equation

(73)

Suppose the forcing is a real periodic signal of fundamental angular frequency ω₀. Write the complex expansions

(74)

For each exponential, differentiation multiplies by inω₀; differentiating twice multiplies by −(nω₀)². Consequently the equation for each harmonic becomes

(75)

For a stable forced response,

(76)

A numerical two-harmonic forcing

Let M = 1 kg, b = 2 N s∕m, k = 25 N∕m, and ω₀ = 3 rad∕s. Apply

(77)

The nonzero force coefficients are

(78)

At n = 1, the response is

Y 1	= =
	= .	(79)

At n = 2,

Y 2	= =
	= .	(80)

The negative-frequency responses are their conjugates because the forcing and the system coefficients are real. Convert the two positive-frequency coefficients back to the real basis:

(81)

(82)

Therefore the real steady-state displacement is

y(t)	= cos(3t) + sin(3t)
	− cos(6t) − sin(6t),	(83)

with displacement measured in meters. Direct substitution into Equation (73) reproduces the original force.

13 Choosing which representation to use

The real form is often the more intuitive choice when drawing a waveform, separating even and odd contributions, specifying a directly measured cosine or sine amplitude, or teaching elementary harmonic synthesis. The complex form is usually more compact when differentiating harmonics, handling shifts, keeping phase and amplitude together, or solving linear differential equations mode by mode.

Neither form is intrinsically more accurate. Their finite partial sums are identical functions when the coefficients are converted correctly. Any difference in an actual computation comes from truncation, rounding, or using different conventions, not from a mathematical difference between real and complex Fourier series.

Common convention traps

• Mixing C₀ and a₀: the constant term is C₀ = a₀∕2, not a₀.
• Ignoring negative frequencies: for a real sinusoid, two complex exponentials reconstruct one real harmonic.
• Forgetting the imaginary sign: with synthesis e^+in𝜃, b _n = −2 Im C_n for positive n.
• Reading |C_n| as the real amplitude: for n ≥ 1, the real amplitude is 2|C_n|.
• Treating e^in𝜃 as automatically real: a single complex exponential is generally complex; reality requires conjugate pairing.
• Confusing series with transforms: a periodic signal has discrete harmonic indices. FS26 develops the different continuous-frequency Fourier transform.

14 Self-checks

1. If f(x) = 5 + 4 cos 3x, find C₀,C₃,C₋₃ and all other nonzero coefficients. Answer: C₀ = 5, C_±3 = 2.
2. If C₂ = 1 − 2i and f is real, what is C₋₂ and what real harmonic do they represent? Answer: C₋₂ = 1 + 2i and 2 cos 2x + 4 sin 2x.
3. What is the amplitude of a real harmonic whose positive-frequency coefficient has modulus 0.4? Answer: 0.8.
4. Can C₋₁ = 0 and C₁ = 1 describe a real-valued waveform? Answer: not by themselves; they give e^ix, a complex waveform.
5. Why does cos x delayed by one quarter-period become sin x? Answer: the positive-frequency coefficient rotates by e^−iπ∕2 = −i; the real coefficient vector rotates from (a₁,b₁) = (1, 0) to (0, 1).

15 What FS20 has established

From FS02’s projection idea to FS19’s complex coefficient formula, we have arrived at a complete dictionary between real and complex representations. Euler’s formula pairs two counter-rotating exponentials into one real harmonic. Their complex coefficients store both the amplitude and phase, and their conjugate symmetry guarantees a real result. The finite projection energy is identical in the two descriptions, and linear physical systems can be solved harmonic by harmonic in complex form before returning to measurable real variables.

This prepares two important directions: FS21 develops Fourier modes as eigenfunctions in separation-of-variables problems, and FS26 explains how discrete harmonics become a continuous-frequency integral for nonperiodic signals.

References