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In this paper, a new class of circulant matrices built from the deterministic filter and the deterministic subsampling is introduced for convolution-based compressed sensing. The pseudo-random sequences are applied in the construction of compressed sensing matrices. By using Katz's and Bombieri's character sum estimation, we are able to design good deterministic compressed sensing matrices for sparse recovery. In the worst case, the sparsity bound in our construction is similar to that of binary compressed sensing matrices constructed by DeVore and partial Fourier matrices constructed by Xu and Xu. Moreover, in the average case, we show that our construction can reconstruct almost all k -sparse vectors with m ≥ O ( k log ⁡ N ) . [ABSTRACT FROM AUTHOR]