Chirp Transform
z = ∫₋₄⁴ exp(−u²) · cos(x · u + y · u² + t) du
Open in the app The dials and keys named below are the app's.
What it draws
The height is a Fourier integral: z = ∫₋₄⁴ e^(−u²) · cos(x·u + y·u² + t) du. The Gaussian e^(−u²) is a pulse in u, and x is the frequency the integral tests it against. The term y·u² adds a phase growing with the square of u: a chirp, a tone whose pitch rises at a rate set by y. With the clock t the sheet is the real part of e^(it) times the Fourier transform of the chirped pulse, every row a spectrum.
Along y = 0 there is no chirp and the row is the transform of a plain Gaussian, another Gaussian, √π · e^(−x²/4): 1.772 at the centre and 0.652 at x = ±2. Every other row is that Gaussian stretched and twisted: √π · (1 + y²)^(−1/4) · e^(−x²/(4(1 + y²))) · cos(t + ½ · atan y − x²·y/(4(1 + y²))). The width grows as √(1 + y²), from 2 on the middle row to 6.32 at y = 3. The peak falls as the fourth root of 1 + y², to 1.00 at y = 3.
Why the spectrum spreads
A chirp adds no energy, it spreads it. The area under the square of a row's amplitude is the same for every y, π · √(2π) ≈ 7.87: the peak falls by exactly the factor the width grows. That is Parseval's theorem: a phase leaves the energy untouched. Radar uses it in reverse, compressing a long chirped echo to a spike. Far from the middle row the Gaussian barely matters against the chirp, and the row tends to a Fresnel integral, √(π/y) · cos(t + π/4 − x²/(4y)).
History
Pulse compression by a chirp was patented independently in the late 1940s, Robert Dicke's application of 1945 the first, then Sidney Darlington's and Sproule and Hughes's, and stayed classified. It reached the open literature in 1960, when Klauder, Price, Darlington and Albersheim published The Theory and Design of Chirp Radars.
Try
- Top: the ridge along x = 0 widens away from the middle row, and the first zero lines beside it come closest at y = ±1.
- Halve the pulse's width, integral(exp(−4·u²) · cos(x·u + y·u² + t), u, −4, 4): a shorter pulse has a broader spectrum, the middle row twice as wide with its peak down to √π/2 ≈ 0.886.
- Slide the pulse, integral(exp(−(u − 1)²) · cos(x·u + y·u² + t), u, −4, 4), then Play: a shift is a phase ramp in the spectrum. The middle row becomes √π · e^(−x²/4) · cos(x + t), waves running through the envelope.
- Make the phase cubic, integral(exp(−u²) · cos(x·u + y·u³ + t), u, −4, 4): the Airy integral. Each row ripples on one flank and slides smoothly to zero on the other.