This paper proposes a technique for processing signals in which the relevant spectral feature consists of one or more narrow lines which have a nonzero Doppler shift with respect to the reference frequency. The proposed technique consists of applying a second demodulation process to the demodulated signal obtained as the receiver output. The purpose of such a demodulation is to remove the frequency shift between the observed spectral line and the center of the observed spectral band. This way, the selected spectral line is brought to baseband, allowing the study of its envelope as a slowly varying process. One efficient way to perform the second demodulation step is by a digital version of a quadrature demodulator. This idea is translated into an efficient algorithm, employing real multiplications and accumulators. An advantage of such an algorithm is that in order to observe two spectral lines in symmetrical positions with respect to the center frequency, common multiplication and accumulators can be used. A possible simplification of the algorithm is discussed in which sinusoids used for demodulation are replaced by square waves, so that multiplication reduces to a synchronous sign reversal. Possible distorsions due to this simplified algorithm are mentioned. Possible implementations of double demodulation in a digital signal processing system are discussed in the final part of the paper.