convC

convC is a term used in some mathematical and engineering literatures to denote a generalized convolution operator associated with complex-valued signals or domains. The exact definition of convC can vary by context, but it commonly refers to a convolution that acts on functions taking values in the complex numbers, or a complex-valued convolution layer in neural networks. Because the name is not standardized, readers should consult the source for the precise convention used.

In its standard form, a continuous-time complex convolution of two square-integrable functions f and g from real numbers to complex numbers is defined by (f * g)(t) = ∫ f(τ) g(t − τ) dτ, assuming appropriate integrability and smoothing conditions. When f and g are real-valued, convC reduces to the real convolution. In practice, extra care is taken with conjugation in certain variants, such as cross-correlation.

Discrete-time convC follows the same principles with sums over integers: (f * g)[n] = Σ_k f[k] g[n − k]. In complex-valued contexts, it is common to represent the pair (f, g) as real and imaginary parts or to use specialized representations to preserve phase information. Efficient implementations use FFT-based convolution with padding to manage boundary effects.

Applications of convC appear in signal processing, communications, and computer vision, particularly where signals are naturally complex-valued or where phase information is critical. In neural networks, complex-valued convolution layers (a form of convC) enable learning with complex weights, which can improve handling of oscillatory or frequency-domain features. The operation also underpins the convolution theorem, linking time-domain filtering to frequency-domain multiplication.