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: To extract the original signal from two identical signals that are phase-shifted and interfering without demodulating, you can use coherent combining techniques. Below is a step-by-step process to obtain the original signal with the phase corrected, along with an analysis of how the Signal-to-Noise Ratio (SNR) changes.

### Steps to Extract the Original Signal

1. **Signal Model:**
   Assume the two signals can be represented as:
   - \( S_1(t) = A \cdot \cos(\omega t) \)
   - \( S_2(t) = A \cdot \cos(\omega t + \phi) \)

   where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase difference between the signals.

2. **Add the Signals:**
   The combined signal received can be expressed as:
   \[
   R(t) = S_1(t) + S_2(t) = A \cdot \cos(\omega t) + A \cdot \cos(\omega t + \phi)
   \]

3. **Use Trigonometric Identity:**
   Use the trigonometric identity for the sum of cosines:
   \[
   R(t) = A \cdot \left(\cos(\omega t) + \cos(\omega t + \phi)\right) = 2A \cdot \cos\left(\omega t + \frac{\phi}{2}\right) \cdot \cos\left(\frac{\phi}{2}\right)
   \]

   This means that the resulting signal can be expressed as a single cosine wave at an adjusted amplitude and phase.

4. **Correct the Phase:**
   To extract the original signal \( S_1(t) \), you need to phase-correct the resulting signal. If you know the phase shift \( \phi \), you can multiply the combined signal \( R(t) \) by a cosine wave with the appropriate phase to shift it back:
   \[
   S_{\text{extracted}}(t) = R(t) \cdot \cos\left(\frac{\phi}{2}\right)
   \]

5. **Implementation:**
   - Sample the combined signal \( R(t) \).
   - Estimate or input the known phase \( \phi \).
   - Apply the phase correction using the derived formula.

### Signal-to-Noise Ratio (SNR) Analysis

Assuming that both signals have an SNR of \( \text{SNR}_{\text{original}} \) (where noise is additive and white):

- **Combined Signal Noise:**
  When you combine the two signals, the noise from both channels can also interfere:
  \[
  N(t) = N_1(t) + N_2(t)
  \]
  Where \( N_1(t) \) and \( N_2(t) \) are the noise components associated with each signal. Under the assumption that the noise components are uncorrelated, the power of the combined noise will be the sum of the individual noise powers:
  \[
  P_N = P_{N_1} + P_{N_2} = 2P_N \text{ for equal noise powers}
  \]

- **Combined SNR:**
  The power of the combined signal \( P_R \) (assuming both signals are in phase after correction) will be:
  \[
  P_R = 2A^2
  \]
  Thus, the SNR of the combined signal is:
  \[
  \text{SNR}_{\text{combined}} = \frac{P_R}{P_N} = \frac{2A^2}{2P_{N}} = \frac{A^2}{P_N} = \text{SNR}_{\text{original}}
  \]

In conclusion, the SNR of the extracted signal after phase correction will remain the same as the SNR of the original signals when they are isolated, provided the assumptions hold true (e.g., noise is independent and identically distributed). 

### Final Summary
- **Extracting the Original Signal:** Use the coherent addition technique while correcting for the phase shift.
- **SNR Analysis**: The SNR of the extracted signal will equal the SNR of the original signals if the noise contributions are uncorrelated.
User: if you didnt do the phase correction, would you still be able to demodulate the frames?
Assistant: If you do not perform phase correction on the combined signal comprised of two identical phase-shifted signals, the ability to demodulate the frames would depend on a few factors, including the modulation type and the nature of the phase shift.

### Implications of Not Performing Phase Correction

1. **Signal Interference:**
   - If the two signals interfere constructively or destructively, it can significantly alter the amplitude and phase of the resultant signal \( R(t) \). This alteration could lead to distortions that make it difficult to demodulate the signal reliably.
   
2. **Demodulation Techniques:**
   - **Coherent Demodulation:** If you are using a coherent demodulation technique (e.g., coherent detection, phase shift keying), failing to correct the phase shift would generally result in the loss of essential phase information. The demodulator would not correctly interpret the signal variations, which could lead to incorrect symbol detection.
   - **Non-Coherent Demodulation:** On the other hand, if you are using non-coherent demodulation techniques (e.g., envelope detection for amplitude modulation), you might still be able to extract some information from the combined signal. Non-coherent techniques typically rely on amplitude rather than phase.

3. **Symbol Rate Considerations:**
   - If the frame rate or symbol rate is high enough, the continuous phase shift may not cause substantial misalignment over short intervals. In such cases, you might still recover some parts of the information, albeit potentially with degraded performance.

### Summary
1. **With Phase Correction**: You can demodulate the frames correctly using coherent techniques since the phase relationship will be accurate.
2. **Without Phase Correction**: 
   - You may still demodulate using non-coherent techniques, but with potentially reduced efficiency and accuracy.
   - For coherent techniques, you risk significant errors in symbol detection due to distortion introduced by the phase interference.

### Practical Recommendation
For optimal performance, especially with coherent modulation schemes, it is highly advisable to implement phase correction prior to demodulation. This will enhance the integrity of the signal and improve the overall error rate in detection.
User: what would change if the two signals are oppositely polarized circularly?
Assistant: