A normal QUAM receiver, would use a phase lock loop, and a VCO to generate a carrier, and a quadrature carrier, to synchronously demodulate the L+R and L-R audio signals. This scheme doesn't work with C-QUAM, because the QUAM envelope was modified in creating the C-QUAM signal. One way to restore the C-QUAM signal to the original QUAM form, so it can be processed by normal synchronous demodulators, is to correct the envelope with a "Cosine Corrector". The Vector math of this is too complicated to get into here, but it is basically an open loop process that depends on highly precise analog functions to generate the cosine of the C-QUAM signal phase, and then divide the C-QUAM signal by the cosine function, to regenerate the original QUAM signal. This process is described in the paper referenced below.
The method actually used in most analog C-QUAM receiver chips makes use of a high gain feedback loop which forces the output of the L+R synchronous demodulator to be equal to the output of a precision envelope detector, this is done as follows. First, since the envelope of the C-QUAM signal is a faithful copy of the L+R sum signal, an envelope detector is used to recover the L+R sum signal, or monophonic audio. The L+R sum signal from the envelope detector is then compared, in an error amplifier, with the L+R audio signal coming out of the QUAM synchronous demodulator. This error signal is then used to control a variable gain amplifier, or analog multiplier in the input to the QUAM synchronous quadrature demodulators, in such a way that the envelope is forced back to the normal QAUM form. The L-R output from the second synchronous quadrature demodulator, is then distortion free, and can be matrixed with the L+R signal from the envelope detector, to recreate the left and right audio signals at the output of the receiver.
To identify C-QUAM AM Stereo Broadcasts, 25 Hz tone is added the L-R stereo difference signal which modulates the quadrature carrier. The 25 Hz tone modulates the quadrature carrier with 5% modulation. Unlike the 19 kHz pilot tone used in FM Stereo, this 25 Hz tone does not play any part in the demodulation of the AM Stereo signal, it's only purpose is to identify AM Stereo Broadcasts, so the receiver can automatically switch into stereo mode, and light an "AM Stereo" light.
This is a somewhat simplified sketch of the operation of the CQUAM AM stereo system. More details on the CQUAM AM stereo system can be found in a paper by the inventors, Norman Parker, Francis Hilbert, and Yoshio Sakaie, titled "A Compatible Quadrature System For AM Stereo", which was published in the IEEE Transactions on Consumer Electronics, Vol. CE-23, No. 4, November 1977. This article discusses only the "Cosine Corrector" type of demodulator. The error feedback system outlined above, is described in the data sheet for the Motorola MC13020 CQUAM AM Stereo demodulator chip, "Advance Information Motorola CQUAM AM Stereo Decoder MC13020P" Motorola Data Sheet ADI-725, 1983.