Modulated wideband converter (MWC) is a blind undersampling system for sparse multiband signal. By the available hardware and compressed sensing method, it can realize energy-efficient wideband spectrum sensing under sub-Nyquist sampling rate. In this paper, we propose an optimized circulant measurement matrix constructing method which is suitable for the MWC system, aiming at the problem that the random measurement matrix is hard to implement in hardware. Simulation results show that the MWC system can reconstruct the original signal with high probability using the proposed method. And its signal to noise ratio (SNR) and sampling channel number performance are improved 8dB and 4 channels respectively compared with traditional Toeplitz measurement matrix and circulant measurement matrix.