  This paper introduces the concept of on-chip temporal optical computing,
based on dispersive Fourier transform and suitably designed modulation module,
to perform mathematical operations of interest, such as differentiation,
integration, or convolution in time domain. The desired mathematical operation
is performed as signal propagates through a fully reconfigurable on-chip
photonic signal processor. Although a few number of photonic temporal signal
processors have been introduced recently, they are usually bulky or they suffer
from limited reconfigurability which is of great importance to implement
large-scale general-purpose photonic signal processors. To address these
limitations, this paper demonstrates a fully reconfigurable photonic integrated
signal processing system. As the key point, the reconfigurability is achieved
by taking advantages of dispersive Fourier transformation, linearly chirp
modulation using four wave mixing, and applying the desired arbitrary transfer
function through a cascaded Mach-Zehnder modulator and phase modulator. Our
demonstration reveals an operation time of $200~ps$ with high resolution of
$300~fs$. To have an on-chip photonic signal processor, a broadband photonic
crystal waveguide with an extremely large group-velocity dispersion of $2.81
\times {10^{6}}~\frac{{{ps^2}}}{km}$ is utilized. Numerical simulations of the
proposed structure reveal a great potential for chip-scale fully reconfigurable
all-optical signal processing through a bandwidth of $400~GHz$.
