In the context of QAM (Quadrature Amplitude Modulation), where both amplitude and phase are modulated, it's true that a standard Fast Fourier Transform (FFT) or Inverse Fast Fourier Transform (IFFT) operation alone does not explicitly preserve or provide phase shift information. However, the phase information is still present implicitly in the complex values obtained from the FFT.
To understand how QAM signals can be coded or deconstructed using FFT, we need to consider the relationship between the time domain and frequency domain representations of the signal.
In QAM, the modulated signal is typically represented as a complex-valued waveform, where the in-phase (I) and quadrature (Q) components carry both amplitude and phase information. The I and Q components are combined to form a complex-valued signal, which can be expressed as:
s(t) = I(t) + jQ(t),
where j is the imaginary unit.
When applying the FFT to this complex-valued signal, the resulting frequency domain representation will also be complex. Each complex value in the frequency domain corresponds to a specific frequency component and contains both magnitude and phase information. The magnitude represents the amplitude of the frequency component, while the phase encodes the phase shift.
To extract the phase information, one can compute the complex argument or phase angle of each complex value obtained from the FFT. This provides the phase shift information associated with each frequency component. The magnitude information can be obtained by calculating the absolute value or magnitude of the complex values.
So, while the FFT operation itself does not explicitly provide phase shift information, it is possible to recover the phase information from the complex values obtained through the FFT by calculating their phase angles. This allows for the coding or deconstruction of QAM signals using FFT transformation.
It's worth noting that proper demodulation or decoding of QAM signals typically involves additional steps beyond FFT, such as carrier recovery, synchronization, and symbol decoding, depending on the specific modulation scheme and system requirements.