Chebyshev polynomials

The formula

The first-kind Chebyshev polynomials Tn(x) are defined on the interval x in [-1, 1] by a cosine in disguise:

Tn(cos θ) = cos(n·θ)

Reading: substitute x = cos θ, and the n-th polynomial is just cos(nθ). Every oscillation property they have comes straight from that identity.

You almost never build them from the cosine, though. You use the three-term recurrence:

T0(x) = 1
T1(x) = x
T(n+1)(x) = 2·x·Tn(x) − T(n-1)(x)

Reading: each polynomial is 2x times the one below it, minus the one below that. Cheap, exact, and numerically stable. The first few come out as:

T2(x) = 2x² − 1
T3(x) = 4x³ − 3x
T4(x) = 8x⁴ − 8x² + 1

The zeros of Tn — the Chebyshev nodes — sit at

x_k = cos( (k − 1/2)·π / n )   for k = 1 … n

Reading: project equally spaced angles on a half-circle straight down onto the x-axis. The points bunch up near ±1 and spread out in the middle. That bunching is the whole trick.

The extrema of Tn land at x = cos(kπ/n), and at each one the polynomial hits exactly +1 or −1, alternating. That's the equiripple behavior in one line.

Where you meet it

Designing a Chebyshev Type I filter in a DSP review. The whole filter family is named after these polynomials. The squared magnitude response is |H(ω)|² = 1 / (1 + ε²·Tn²(ω/ωc)). Because Tn oscillates flat between ±1 inside the passband, the passband ripple is a constant 10·log10(1 + ε²) dB — you spread the error out evenly instead of letting it pile up. Pick ε for 1 dB ripple and you get ε ≈ 0.509; the passband magnitude then bounces between 1.0 and 0.891 and never worse. Trade that flat ripple for a steeper roll-off than a Butterworth of the same order.
Minimax / equiripple FIR design (Parks-McClellan / Remez). When an RF or DSP engineer runs the Remez exchange to design an FIR filter, the equiripple error they're chasing is Chebyshev's theorem in software. The optimal filter is the one whose error alternates to the same peak height across the band — same idea, applied to the frequency response instead of a raw polynomial.
Fitting a calibration curve or lookup table on the test stand. When you approximate a sensor's nonlinear response or a thruster's Cd-vs-Mach curve with a polynomial, sampling at Chebyshev nodes instead of evenly spaced points keeps the fit from oscillating wildly near the ends of the range (the Runge problem below). This is how function approximation in libraries and math coprocessors gets done under the hood.
A minimax argument on a review board. "Worst-case error" is minimax. Any time someone specifies a spec as "the peak deviation must be under X across the whole band," and someone else answers "then spread the error evenly," that second person is invoking Chebyshev whether they name him or not.

How it works

The magic is the substitution x = cos θ. On [-1, 1], Tn is a polynomial, but through the cosine it's a pure sinusoid in angle. A degree-n cosine hits its peak n+1 times as the angle sweeps 0 to π, and those peaks are all the same height, alternating sign. Verified numerically: T5 at its six interior/edge extrema reads +1, −1, +1, −1, +1, −1. Dead even. That's why they're called equiripple.

The property engineers actually cash in is the minimax one. Take the monic version, Tn(x)/2^(n-1) (monic = leading coefficient 1). Among all monic degree-n polynomials, that one has the smallest possible maximum absolute value on [-1, 1], and that minimum-of-the-maximum is exactly 2^(1-n). Checked to six digits for n = 1 through 5: the scaled peak lands on 1, 0.5, 0.25, 0.125, 0.0625 — precisely 2^(1-n) each time. No other polynomial of that degree keeps its worst-case value lower. That is the definition of a minimax solution.

Backing it is the Chebyshev equioscillation theorem: a degree-n polynomial is the best possible minimax approximation to a continuous function exactly when its error curve touches its maximum size at least n+2 times, alternating sign each touch. Flip that around and it's a working design rule — if your approximation error doesn't equiripple, you can always shave the peak down somewhere, so you're not optimal yet. When the ripples are all equal, you're done. That "am I done?" test is the practical payoff.

The gotcha that bites people is the Runge phenomenon. Fit a smooth-looking function like 1/(1 + 25x²) with a high-degree polynomial through equally spaced points and the fit goes berserk near the edges. Concrete numbers, degree 20: equally spaced nodes give a worst-case error of about 60 — the interpolant is off by sixty times the function's own scale. Switch to Chebyshev nodes and the same degree-20 fit lands within about 0.015. That's not a small tweak; it's the difference between garbage and a usable curve. The reason is that Chebyshev nodes crowd toward the interval ends, which is exactly where an equally spaced fit starves for information.

Two limits to respect. First, the whole story lives on [-1, 1]. Real problems have real units — a passband from 900 to 1000 MHz, a temperature range from -40 to 85 °C. You have to linearly map your interval onto [-1, 1] before the node formula means anything, and map back afterward. Forget the mapping and the nodes land in the wrong places. Second, outside [-1, 1] these polynomials don't oscillate at all — they shoot up like cosh(n·acosh x), growing explosively. That's not a bug; it's the same behavior that gives a Chebyshev filter its steep skirt just past the cutoff. Inside the band, flat ripple. Outside, a cliff.

History

Pafnuty Lvovich Chebyshev — born 16 May 1821 in Okatovo, in Russia's Kaluga region, died 8 December 1894 in St. Petersburg — did not set out to help anybody design a filter ^[1]^[2]. He came at it through machines. After an 1852 trip through Western Europe studying mechanisms, he got interested in linkages: the jointed bars, like the ones in a steam engine, that are supposed to convert rotary motion into a straight line but never quite do. The question was how to shape a linkage so its deviation from a true straight line was as small as possible across its whole travel ^[1]^[2].

That is a minimax problem in disguise — minimize the worst-case error over an interval — and in chasing it Chebyshev wrote the 1854 paper "Théorie des mécanismes connus sous le nom de parallélogrammes" ("Theory of mechanisms known as parallelograms"). The polynomials that carry his name today appeared in print for the first time in that paper on machine linkages ^[1]^[2]. The approximation theory grew out of his broader work on least-squares fitting, and he went on to apply the same machinery to interpolation and numerical quadrature ^[1].

The name gets spelled a dozen ways in the literature — Tchebycheff, Tschebyscheff, Chebyshov — which is why the polynomials are universally abbreviated Tn: the letter survives every transliteration. The equiripple property he uncovered sat mostly in the pure-math world until the 20th century, when filter designers realized a polynomial engineered to spread its error out flat was precisely the tool for spreading a filter's ripple out flat. A linkage problem from the age of steam became the backbone of RF and DSP filter tables.

Related tools

/tools/rc-filter
/tools/lc-resonance
/tools/q-factor-bandwidth
/tools/series-rlc-impedance
/tools/vswr-return-loss

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →