Wasteland Signal Mixer

Tune three harmonic amplitudes and a noise dial to hunt down a hidden ghost signal buried in the wasteland static.

Worked example — Wasteland Signal Mixer

RMS	0.73
Peak	1.26
Clarity	83%
Error	0.08
Static Noise	18%
Base Frequency	3x

These are the values the calculator shows when the page opens. Change any input above to recompute them.

Loading interactive simulation...

Any jagged wave is a stack of smooth ones 🖖

Any periodic wave, however jagged, can be built by stacking sine waves at integer multiples of a base frequency — a fundamental plus its harmonics. Fourier's insight also runs in reverse: a complex signal, whether it's a voice, a radio broadcast, or wasteland static, can be decomposed back into those same sine components. The clarity score above is really an inverse of the reconstruction error between your mix and the buried ghost wave - the same kind of math a real radio receiver's filters use to strip noise from a signal, or a JPEG uses to compress an image into a handful of dominant frequencies.

Same pitch, different shape 🖖

The base-frequency slider sets the pitch — how often the whole wave repeats — while the three amplitude sliders only reshape it. Because every harmonic is an exact integer multiple of the fundamental, stacking them enriches the wave's character without changing how fast it repeats. That is why a flute and a violin playing the same note sound unmistakably different yet share one pitch: same fundamental, different harmonic recipe. To lock the ghost you must find both its pitch and its recipe.

The pitch your ears invent 🖖

The ghost signal is anchored by its fundamental at frequency 3, but your ears do not actually need it. Strip out the fundamental and leave only the higher harmonics, and your brain still hears that original low pitch — a quirk called the missing fundamental. It is why a tiny phone speaker that cannot physically move enough air for deep bass can still make you hear it, rebuilding the fundamental from the spacing of the harmonics alone.

Problems solved in full

The 0.73 RMS from y = 0.90·sin(3x) + 0.45·sin(6x) + 0.25·sin(9x) 5 steps
The live equation reads y = 0.90·sin(3x) + 0.45·sin(6x) + 0.25·sin(9x), and the stats box scores it RMS 0.73. Derive that 0.73 from the amplitudes alone — and watch what never enters the calculation.
1. RMS means what it spells: square the wave, average that over a whole cycle, take the root. Squaring the sum leaves one square per harmonic and one cross product per pair, so the entire question is what the cross products do.
  
  $y_{rms} = \overline{y^{2}}, \overline{f} = \frac{1}{2 π} \int_{0}^{2 π} f d x$
2. They do nothing. sin(3x)·sin(6x) rewrites as half of cos(3x) minus half of cos(9x), and a whole cosine cycle averages to 0; the same collapse happens for every pair. In the tool's own arithmetic the cancellation is exact rather than approximate, because it sums 180 samples across one period and 3, 6 and 9 all divide 180.
  
  $\overline{sin m x sin n x} = 0 (m \neq = n), \overline{sin^{2} n x} = \frac{1}{2}$
3. The 3 squares survive, and each of them averages to 0.5. So the mean square is half of 0.81 + 0.2025 + 0.0625. Powers add; amplitudes do not.
  
  $\overline{y^{2}} = \frac{1}{2} (A_{1}^{2} + A_{2}^{2} + A_{3}^{2}) = \frac{1}{2} (0.81 + 0.2025 + 0.0625)$
4. Half of 1.075 is 0.5375, whose root is 0.7331 — the 0.73 in the box.
  
  $\overline{y^{2}} = 0.5375 ⟹ y_{rms} = 0.5375 = 0.7331 \to 0.73$
5. Read back what went in: 3 amplitudes, and nothing else. No phase, no frequency. The base frequency lever spans 1x to 8x and not one of those positions moves the RMS off 0.73. The peak is a different animal: |y| can never exceed 0.90 + 0.45 + 0.25 = 1.60, but how near it gets depends entirely on whether the 3 crests fall together, so the crest factor here ranges up to 2.18 where a lone sine is stuck at 1.414.
  
  $∣ y ∣ \leq A_{1} + A_{2} + A_{3} = 1.60, \frac{1.60}{0.7331} = 2.18 (2 = 1.414)$
Answer

RMS 0.73, out of 3 lever positions and nothing else. Split the 1.075 by term and the fundamental holds 75.3% of it, the second harmonic 18.8%, the third 5.8% — so 24.7% of what this transmitter radiates sits above the note it is named after. That ratio has a standard name, total harmonic distortion, √(A₂² + A₃²) / A₁ = 57.2%, and no box on the panel carries it. Nor does the panel separate the 2 limits these levers are up against. Average power follows the RMS, which step 5 just showed is blind to phase. Headroom follows the peak, which is not blind to it at all: the same 3 amplitudes cap the wave at 1.60, and where between 0.73 and 1.60 it actually sits is decided by phase alone. The RMS readout answers the power question and is silent on the other.

$THD = \frac{A _{2}^{2} + A _{3}^{2}}{A _{1}} = \frac{0.265}{0.90} = 57.2%$
Smallest mismatch the 3 amplitude levers can reach 5 steps
This is Static Hell: levers at 0.50, 0.60 and 0.40 with the base frequency at 5x, so the transmitted wave is y = 0.50·sin(5x) + 0.60·sin(10x) + 0.40·sin(15x). The ghost being hunted is built on 3x, with harmonics at 6x and 9x. Find the smallest mismatch the 3 amplitude levers can reach while the frequency stays where it is.
1. Call the mismatch d = y − g and square it. Averaging over a cycle splits d² into your mean square, the ghost's, and a cross term. Only the cross term can be negative, so only the cross term can cancel anything.
  
  $d = y - g, \overline{d^{2}} = \overline{y^{2}} - 2 \overline{y g} + \overline{g^{2}}$
2. Here it cannot. Every product inside it pairs a harmonic from {5, 10, 15} with one from {3, 6, 9}; the sets share no member; and a product of sines at different whole-number frequencies averages to 0 over the period. The cross term is not small. It is 0.
  
  ${5, 10, 15} \cap {3, 6, 9} = \emptyset ⟹ \overline{y g} = 0$
3. What is left is a sum of 2 non-negative pieces, and the levers reach only the first. Yours is half of 0.25 + 0.36 + 0.16, which is 0.385, and its root 0.6205 is the 0.62 the panel prints as RMS.
  
  $\overline{y^{2}} = \frac{1}{2} (0.25 + 0.36 + 0.16) = 0.385 ⟹ y_{rms} = 0.6205 \to 0.62$
4. So the mismatch obeys d_rms ≥ 0.62 whatever the ghost turns out to be — you need not know a single one of its amplitudes to say that. The Error box will not read 0.62, because it averages the absolute difference rather than the squared one; the bound is on a different average of the same wave.
  
  $\overline{d^{2}} = 0.385 + \overline{g^{2}} \geq 0.385 ⟹ d_{rms} \geq 0.62$
5. At 5x, then, every lever raised makes the mismatch worse, and the minimum sits with 0.50, 0.60 and 0.40 all driven to 0. At 3x the 2 harmonic sets coincide instead, the difference wave has amplitudes A_i − G_i, and those same 3 levers can drive the mismatch to exactly 0.
  
  $f = 3 : \overline{d^{2}} = \frac{1}{2} i \sum (A_{i} - G_{i})^{2} A_{i} \to G_{i} 0$
Answer

The best these levers can do at 5x is switch themselves off: transmitting nothing beats transmitting this. That makes the frequency control a gate, not an adjustment. 3 continuous sliders are worth precisely nothing until 1 integer is right, and no amount of care with them buys a fraction of the lock — which is a harsher rule than a tuning game usually implies. It also gives the RMS box a second job. Ordinarily it reports how loud you are; with the base frequency wrong it is a floor on how wrong you are, because all of your power lands in the error instead of cancelling any of the ghost's. Orthogonality is what makes the gate absolute: 2 waves built on disjoint sets of whole-number harmonics cannot cancel each other even slightly, however you scale them.

$A_{1}, A_{2}, A_{3} min \overline{d^{2}} = \overline{g^{2}}_{A_{1} = A_{2} = A_{3} = 0}$

References (2)

The decomposition the clarity score is built on: J. Fourier, Théorie analytique de la chaleur. Firmin Didot, Paris, 1822.
The same transform as used in image compression: N. Ahmed, T. Natarajan and K. R. Rao, "Discrete Cosine Transform." IEEE Transactions on Computers C-23(1), 90–93, 1974.

By the 42Math team · Updated 2026-07-29

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Pure Signal - Pure fundamental: y = sin(3x) — single harmonic, zero noise
Doof Warrior - Three-harmonic mix at freq=3 with moderate noise — default wasteland broadcast
Static Hell - High noise at freq=5 — clarity collapses below 10%
Ghost Match - Near-target mix — fine-tune to lock onto the ghost signal