This paper addresses the problem of estimating the parameters of constant-amplitude chirp signals that have single or multiple components and are embedded in noise. Chirp signals are widely employed in applications such as radar and telecommunications, and it is a key task in countermeasure techniques to estimate their parameters without prior information. Hence, a parameter estimation processor based on a complex-valued deep neural network (CV DNN) is proposed to perform this task efficiently. The CV DNN, which is designed for regression, consists of a function fitter and a predictor. The function fitter acts like a eigenfunction mapping: it maps the one-dimensional input into a two-dimensional feature map suitable for subsequent network learning. As a special feature extraction tool, the predictor extracts local features from the feature map and estimates parameters. Simulation results indicate that the CV DNN outperforms conventional processors. Moreover, it is more accurate than the Wigner-Hough transform while being several orders of magnitude faster, which will enable real-time signal processing with fewer computational resources. Furthermore, we demonstrate that the CV DNN shows strong robustness to changes in modulation parameters and the number of components of a chirp signal. This study shows the advantages of deep learning systems for signal parameter estimation.