User: receiving two signals on one antenna, interfering. They are the same signal but phase shifted. How do we extract the original signal?
Assistant: When you have two identical signals on a receiving antenna that are phase-shifted, they can interfere with each other in a way that complicates the process of recovering the original signal. However, there are methods for extracting the useful signal from the combined signals. Here’s a general approach to handle this situation:

### 1. **Understand the Interference:**
   - If the two signals are the same but phase-shifted (let’s say one is \( S(t) \) and the other is \( S(t + \phi) \), where \( \phi \) is the phase difference), the resulting signal can be expressed mathematically.
   - The combined signal can be written as:
     \[
     R(t) = S(t) + S(t + \phi)
     \]

### 2. **Use of Phase Information:**
   - Determine the phase difference \( \phi \). This can often be estimated using techniques such as cross-correlation, where you cross-correlate the received signal with a delayed version of itself.

### 3. **Signal Demodulation:**
   - If the signal is modulated (e.g., AM or FM), you need to demodulate the signal. This can be done using an appropriate demodulation technique once you understand the phase relationship.

### 4. **Signal Processing Techniques:**
   - **Coherent Detection:** If both signals are phase-coherent, you can use coherent detection techniques to extract the original signal. This involves mixing (multiplying) the received signal with a version of the signal at the expected frequency and phase.
   - **Adaptive Filtering:** You can implement adaptive filtering techniques to minimize the interference and enhance the signal. For example, you can use methods like Least Mean Squares (LMS) or Recursive Least Squares (RLS).

### 5. **Fourier Transform Techniques:**
   - Applying a Fourier Transform to your received signal can help you visualize the frequency components. Depending on the phase shift, you might be able to filter out one of the signals in the frequency domain.

### 6. **Vector Representation:**
   - Represent both signals as phasors in a complex plane. The original signal can be retrieved by vector addition of the individual signals represented as phasors.

### 7. **Use of Phase Shift Keying (if applicable):**
   - If the original signal uses phase shift keying, you can detect the phase shifts directly, which could facilitate the recovery of the original signal.

### 8. **Digital Signal Processing:**
   - If the signals can be digitized, using digital signal processing (DSP) techniques may enable you to apply filters, phase corrections, or even machine learning techniques to enhance the desired signal.

### Summary
Extracting the original signal from two phase-shifted versions of the same signal will depend heavily on your ability to measure or determine the phase difference and employ suitable signal processing techniques to separate the components. Depending on your specific context (i.e., type of modulation, level of phase difference, etc.), some techniques may be more effective than others.
User: tell me exactly how we would get the original signal (without demodulation) with the phase corrected. Also tell me how the SNR of the filtered signal will be with respect to the SNR of the original signals in isolation
Assistant: