This paper concerns covariance matrix based spectrum sensing over frequency selective channels. The impact of frequency selective channels on the covariance matrix of received signals is analyzed in frequency domain and it is shown that the phases of channel spectra degrade the performance of covariance matrix based detectors. To overcome this problem, we propose a new detector which employs only the magnitude spectra of received signals and thereby achieves considerable performance gain. Theoretical performance, in terms of the false-alarm and detection probabilities of the proposed detector, is analyzed. Simulation results verify our theoretical analyses and demonstrate the superior performance of the proposed detector.