Periodic Functions and Trigonometric Orthogonality, calculus lesson

A heartbeat, a vibrating string and the hum of a power line all repeat the same shape over and over. Functions that repeat are called periodic, and the simplest of them are sines and cosines. Before building other periodic functions out of these, you need one surprising fact. Over a full period, the product of two different harmonics always integrates to zero.

Key terms

Periodic functions

Definition A function $f$ defined on $\mathbb{R}$ is periodic with period $T > 0$ if

f(x + T) = f(x) \qquad \text{for every real } x.

If $T$ is a period, so is every positive integer multiple $nT$ . If $f$ is periodic with period $T$ and integrable over each bounded interval, its integral over any interval of length $T$ is the same,

\int_c^{c+T} f(x) \, dx = \int_0^T f(x) \, dx \qquad \text{for every real } c.

In plain words Shift the graph sideways by $T$ and you get the same graph.

Example The functions $\sin x$ and $\cos x$ have period $2\pi$ , and so they also have periods $4\pi$ , $6\pi$ and so on. The fractional part $x - \lfloor x \rfloor$ , a sawtooth climbing from $0$ toward $1$ on each interval $[n, n + 1)$ , has period $1$ .

Fundamental period

Definition The fundamental period of a periodic function is its smallest positive period, when one exists. Not every periodic function has one. A constant function has every $T > 0$ as a period, so there is no smallest. If $f$ is continuous, periodic and not constant, then a fundamental period exists, and every period is an integer multiple of it.

In plain words It is the length of one full repeat. Most functions you meet have one, but constants are the exception.

Example The function $\sin(3x)$ has fundamental period $\frac{2\pi}{3}$ , since

\sin\Bigl(3\Bigl(x + \frac{2\pi}{3}\Bigr)\Bigr) = \sin(3x + 2\pi) = \sin(3x)

and no smaller shift of $3x$ than $2\pi$ leaves the sine unchanged for every $x$ . The sum $\sin x + \sin(2x)$ has periods $2\pi$ and $\pi$ for its two parts, so $2\pi$ , the least common multiple, is a period of the sum. It is the fundamental period too. If $T$ is any period of $f(x) = \sin x + \sin(2x)$ , then $T$ is also a period of $f''(x) + 4f(x) = 3\sin x$ , and that forces $T$ to be a multiple of $2\pi$ . In general the least common multiple is a period of the sum but need not be the smallest one.

Angular frequency

Definition For a sinusoid $A \cos(\omega t + \varphi)$ or $A \sin(\omega t + \varphi)$ with $A \neq 0$ and $\omega > 0$ , whose amplitude is $\lvert A \rvert$ , the number $\omega$ is the angular frequency, measured in radians per unit of $t$ . The fundamental period and the ordinary frequency are

T = \frac{2\pi}{\omega}, \qquad \nu = \frac{1}{T} = \frac{\omega}{2\pi}.

In plain words Angular frequency counts how many radians the inside of the sine advances per unit of time. Divide by $2\pi$ to get full cycles per unit of time.

Example The signal $\sin(4\pi t)$ with $t$ in seconds has $\omega = 4\pi$ radians per second. Its period is

T = \frac{2\pi}{4\pi} = \frac{1}{2}

second, and its frequency is $\nu = 2$ cycles per second.

Harmonics

Definition Fix a base period $T > 0$ and put $\omega_0 = \frac{2\pi}{T}$ . For each integer $n \ge 1$ the $n$ th harmonic is the pair of functions

\cos(n \omega_0 t), \qquad \sin(n \omega_0 t),

with angular frequency $n \omega_0$ . Each harmonic has fundamental period $\frac{T}{n}$ , so $T$ is a period of every harmonic. The first harmonic is the fundamental.

In plain words Harmonics are the sine and cosine waves that fit into the base period a whole number of times. They are the building blocks for anything that repeats with period $T$ .

Example For $T = 2\pi$ we have $\omega_0 = 1$ , and the harmonics are $\cos(nx)$ and $\sin(nx)$ . The third harmonic $\sin(3x)$ completes three full cycles on $[0, 2\pi]$ , and its graph returns to its starting point at $x = 2\pi$ .

Inner product of functions

Definition For real functions $f$ and $g$ that are piecewise continuous on $[a, b]$ , their inner product on $[a, b]$ is

\langle f, g \rangle = \int_a^b f(x) \, g(x) \, dx.

It is symmetric, $\langle f, g \rangle = \langle g, f \rangle$ , linear in each argument, and satisfies $\langle f, f \rangle \ge 0$ . The norm of $f$ is $\lVert f \rVert = \sqrt{\langle f, f \rangle}$ . Two functions are orthogonal on $[a, b]$ if $\langle f, g \rangle = 0$ .

In plain words This is the dot product for functions. Multiply the two functions point by point and add up with an integral instead of a sum.

Example On $[0, 1]$ ,

\langle x, 1 \rangle = \int_0^1 x \, dx = \frac{1}{2}, \qquad \left\langle x, x - \frac{2}{3} \right\rangle = \int_0^1 \left(x^2 - \frac{2x}{3}\right) dx = \frac{1}{3} - \frac{1}{3} = 0.

So $x$ and $x - \frac{2}{3}$ are orthogonal on $[0, 1]$ . The norm of $x$ is

\lVert x \rVert = \sqrt{\int_0^1 x^2 \, dx} = \frac{1}{\sqrt{3}}.

Sine and cosine orthogonality

Definition For integers $m, n \ge 1$ ,

\int_{-\pi}^{\pi} \cos(mx) \cos(nx) \, dx = \begin{cases} 0, & m \neq n, \\ \pi, & m = n, \end{cases} \qquad \int_{-\pi}^{\pi} \sin(mx) \sin(nx) \, dx = \begin{cases} 0, & m \neq n, \\ \pi, & m = n, \end{cases}

\int_{-\pi}^{\pi} \sin(mx) \cos(nx) \, dx = 0, \qquad \int_{-\pi}^{\pi} \cos(nx) \, dx = \int_{-\pi}^{\pi} \sin(nx) \, dx = 0, \qquad \int_{-\pi}^{\pi} 1 \, dx = 2\pi.

So the functions $1, \cos x, \sin x, \cos 2x, \sin 2x, \ldots$ are pairwise orthogonal on $[-\pi, \pi]$ , and the same holds on any interval of length $2\pi$ . For period $2L$ the same statements hold for $\cos\frac{n\pi x}{L}$ and $\sin\frac{n\pi x}{L}$ on $[-L, L]$ , with $L$ in place of $\pi$ in the nonzero values and $2L$ for the constant.

In plain words Two different harmonics, or a sine and a cosine, cancel each other out over a full period. Only a harmonic paired with itself leaves something behind.

Example The product to sum identity

\cos A \cos B = \frac{1}{2}\bigl(\cos(A - B) + \cos(A + B)\bigr)

gives

\int_{-\pi}^{\pi} \cos(2x) \cos(3x) \, dx = \frac{1}{2} \int_{-\pi}^{\pi} \bigl(\cos x + \cos(5x)\bigr) dx = 0.

With $m = n = 3$ instead,

\int_{-\pi}^{\pi} \sin^2(3x) \, dx = \int_{-\pi}^{\pi} \frac{1 - \cos(6x)}{2} \, dx = \pi.

Check yourself

What is the fundamental period of $\cos\frac{\pi x}{3}$ ?

Show the answer

The angular frequency is $\omega = \frac{\pi}{3}$ , so

T = \frac{2\pi}{\pi/3} = 6.

What is the fundamental period of $\sin(2x) + \cos(3x)$ ?

Show the answer

It is $2\pi$ . The parts have fundamental periods $\pi$ and $\frac{2\pi}{3}$ , and $2\pi = 2 \cdot \pi = 3 \cdot \frac{2\pi}{3}$ , so $2\pi$ is a period. No smaller $T > 0$ works. A period $T$ of $f(x) = \sin(2x) + \cos(3x)$ is also a period of $f'' + 9f = 5\sin(2x)$ and of $f'' + 4f = -5\cos(3x)$ , so $T$ is a multiple of both $\pi$ and $\frac{2\pi}{3}$ , hence of $2\pi$ .

Compute $\int_{-\pi}^{\pi} \bigl(\sin x + 2\cos(3x)\bigr)^2 \, dx$ using orthogonality.

Show the answer

Expanding the square gives

\sin^2 x + 4 \sin x \cos(3x) + 4\cos^2(3x).

The middle term integrates to $0$ , and the others give $\pi + 4\pi = 5\pi$ .

Periodic Functions and Trigonometric Orthogonality

Key terms

Periodic functions

Fundamental period

Angular frequency

Harmonics

Inner product of functions

Sine and cosine orthogonality

Check yourself

How this lesson connects

Builds on

Used later in