convolved

Convolved is the past participle of the verb convolve, used to describe a function or signal that has undergone the convolution operation with another function or kernel. In mathematics, convolution is a integral or summation that combines two functions to produce a third function expressing how the shape of one is modified by the other.

In the continuous setting, the convolution of two functions f and g is defined by y(t) = ∫_{-∞}^{∞} f(τ) g(t−τ) dτ. In discrete time, the analogous operation for sequences x[n] and h[n] is y[n] = Σ_{k=-∞}^{∞} x[k] h[n−k], with finite sums when the sequences have finite support. Convolution is commutative (the result is the same if the order is swapped), associative, and distributive over addition. It has an identity element given by the Dirac delta function in the continuous case or the impulse sequence in the discrete case.

Convolved data often represents the response of a system to an input. In signal processing and communications, the output of a linear time-invariant system is the convolution of the input signal with the system’s impulse response, producing effects such as smoothing, blur, or filtering depending on the kernel. In image processing, convolving an image with a kernel applies filters for blurring, sharpening, or edge detection. The convolution operation is tightly connected to the Fourier transform: the Fourier transform of a convolution equals the product of the Fourier transforms, enabling efficient computation via the convolution theorem. Deconvolution seeks to reverse convolution in order to recover the original signal, but it can be ill-posed in the presence of noise.