Convolutional neural network (CNN) is one of the most widely-used successful architectures in the era of deep learning. However, the high-computational cost of CNN still hampers more universal uses to light devices. Fortunately, the Fourier transform on convolution gives an elegant and promising solution to dramatically reduce the computation cost. Recently, some studies devote to such a challenging problem and pursue the complete frequency computation without any switching between spatial domain and frequent domain. In this work, we revisit the Fourier transform theory to derive feed-forward and back-propagation frequency operations of typical network modules such as convolution, activation and pooling. Due to the calculation limitation of complex numbers on most computation tools, we especially extend the Fourier transform to the Laplace transform for CNN, which can run in the real domain with more relaxed constraints. This work more focus on a theoretical extension and discussion about frequency CNN, and lay some theoretical ground for real application.
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