Tensions in Practice: Shared nonlinear hardware in tension with spectral separation¶
Continuous signals · a specified quadratic response
Two separate inputs oscillate at 2 and 3 cycles per time unit. Each processing unit obeys the same ideal rule: output equals input plus its square. If the inputs are combined first, the square contains a cross-term that generates rates 1 and 5. If each is processed separately and the outputs are then added, that cross-term never appears. Both paths use the same final filter, which retains rates from 1 through 3.
Share a processing unit
Handle both inputs with one nonlinear element and fewer separate paths.
Keep the inputs from generating cross-products
Prevent their interaction from adding unwanted content inside the retained band.
Why these aims pull against each other
Processing a sum through a nonlinear curve differs from adding separately processed outputs. The shared path saves a processing unit but generates a difference-rate component inside the permitted band.
Choose an arrangement to see what changes and what remains difficult.
Finite illustrative comparisons. Text states carry the meaning; color is not a measured score or universal preference.
What this choice protects
What it costs
When it fits
Compare the arrangements
Combine first
Add the tones, apply y = x + x² once, and retain the band from 1 through 3.
| Signal path | New rates | |
|---|---|---|
| Input | Combine tones | None |
| Curve | One shared unit | 0, 1, 4, 5, 6 |
| Keep 1–3 | Same band filter | 1 remains |
- What it protects
- One nonlinear processing unit handles the combined signal.
- What it costs
- The new rate 1 survives the band filter; deleting that frequency would also remove any wanted content at rate 1.
- When it fits
- Hardware sharing matters and the application permits the added intermodulation, or intentionally uses it.
Illustration note: The ideal memoryless quadratic and unit-amplitude cosines are stipulated, not an amplifier measurement. Squaring their sum adds twice their product, producing sum and difference rates.
Process separately
Apply the same rule independently to each tone, then add outputs and use the same band filter.
| Signal path | New rates | |
|---|---|---|
| Input | Keep tones apart | None |
| Curve | Two separate units | 0, 4, 6 |
| Keep 1–3 | Combine, then filter | None remains |
- What it protects
- The rates 1 and 5 are never generated by cross-tone multiplication.
- What it costs
- Two processing units and separate input paths are required; each still generates its own harmonic before filtering.
- When it fits
- The component inputs are separately accessible and avoiding the cross-products justifies duplicated processing.
Illustration note: Squaring rate 2 produces DC and rate 4; squaring rate 3 produces DC and rate 6. DC means rate 0. This is not a general distortion cure for already mixed or internally multi-tone inputs.
What this illustration does—and does not—establish
Harmonic Distortion: Harmonics versus Intermodulation (In-Band Hazard) supplies in-band cross-products. The two paths are an editorial algebraic construction with explicitly different hardware counts.
- These are continuous unsampled signals; no aliasing or quantization is involved.
- The result is exact only for the specified quadratic response and ideal band filter.
- No power, monetary cost, device efficiency or real attenuation is quantified.
Source entries
Harmonic Distortion
Harmonic Distortion: Harmonics versus Intermodulation (In-Band Hazard) supplies the conflict examined here.
Harmonics versus Intermodulation (In-Band Hazard)
Single-tone harmonics land at integer multiples of the input, often *outside* the signal band and so filterable; but with multiple input frequencies the nonlinearity also generates intermodulation products at sums and differences, which can fall *inside* the band and cannot be filtered out without removing signal.