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The paper analyses the spectrum of a frequency domain multiplication-free adaptive filter (FDMAF) using a modified discrete Fourier transform (MDFT) pair. The proposed FDMAF employs the MDFT and differential pulse code modulation in an input part and the reference input vectors are used in an update equation for the adaptive filter coefficients. Also, it can be simply controlled by adjusting the amplitude of the MDFT pair output signals and can obtain an output signal with stable convergence characteristics and without accumulated error. The condition used to analyse the algorithm has zero-mean, wide-sense stationary and white Gaussian noise.