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We show analytically as well as experimentally a photonic technique for generation of frequency-tunable microwave signals. The technique involves propagating an array of uniformly spaced optical combs with identical comb profile into a length of dispersive single-mode fiber (SMF) prior to photodetection. A new scheme that involves a supercontinuum source and a Fourier-domain programmable optical processor as a spectral filter is proposed to generate the required array of optical combs. For an initial optical comb with a fixed repetition rate and a given length of standard SMF, our scheme shows that by choosing appropriate values of the number of optical combs, the number of comb lines in each optical comb, as well as the frequency spacing between the adjacent optical combs, we can generate a microwave signal whose frequency can be tuned to any integer multiple of the optical comb's repetition frequency that lies within the bandwidth of the photodetector. Based on the proposed scheme, using a 2 GHz supercontinuum source as an initial optical comb, we demonstrate the generation of stable microwave signals tunable to 8, 10, and 12 GHz.