Convolution, Plancherel and Transforming a Derivative
Establish the three properties that make the transform useful: convolution becomes multiplication, energy is preserved, and differentiation becomes multiplication by the frequency.
What a learner can do afterwards
- Prove the convolution theorem and use it to compute a hard integral cheaply
- State Plancherel's theorem and interpret it as the transform being unitary
- Solve a linear constant-coefficient equation by transforming, dividing and inverting
1 · Read
The convolution theorem says the transform of a convolution is the product of the two transforms. A tangled sliding integral becomes plain multiplication, so a hard integral can collapse to a cheap product.
Plancherel says the transform preserves energy: the total size of a function equals the total size of its transform. In geometric terms the transform is unitary, more like a rotation than a stretch, so lengths and angles survive.
Differentiating turns into multiplying: the transform of the derivative f' is i times the frequency times the transform of f. Applied twice, the multiplier squares. To solve a linear equation with constant coefficients, transform both sides, divide by the multiplier, and invert back.
Use this order every time: transform both sides, collect the algebraic factor, divide, then invert. If the multiplier is zero somewhere, stop, since division there is not allowed.
Convolution becomes multiplication, energy is preserved, and derivatives become multipliers, which turns equations into algebra.
More to read
How to Calculate the Fourier Transform of a Function: 14 Steps
Fourier transforms trade signals for spectra. Decompose any waveform into sine ingredients, each with its own frequency and strength. Convolution in time becomes plain multiplication in frequency, which is why engineers transform before solving. Plancherel guarantees the energy counted either way matches exactly. Transform, multiply, invert back.
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2 · Watch
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14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.