add_CFO: Simulating Carrier Frequency Offset in Wireless Communication Systems

Carrier Frequency Offset (CFO) is a common phenomenon in wireless communication systems, primarily caused by frequency mismatches between the local oscillators at the transmitter and receiver ends. Simulating CFO effects in signal processing helps test frequency offset compensation algorithms at the receiver side or analyze system performance. ### Fundamental Principle CFO introduces a time-varying phase rotation that can be represented by multiplying the signal with a complex exponential function. Assuming the input signal is s(t) and the carrier frequency offset is Δf, the CFO-affected signal s_CFO(t) can be expressed as: s_CFO(t) = s(t) · e^(j2πΔft) where Δf is the frequency offset value, typically measured in Hz. ### Implementation Approach Time Axis Generation: Generate a time point sequence t based on the signal's sampling rate, ensuring its length matches the input signal. In code implementation, this can be achieved using: t = np.arange(0, len(signal)/fs, 1/fs) where fs is the sampling frequency. Phase Rotation Calculation: Compute the phase offset 2πΔft for each time point. The algorithm should efficiently calculate this using vectorized operations for optimal performance. Complex Exponential Generation: Convert the phase offset into complex-form rotation factors e^(j2πΔft). This can be implemented using Euler's formula: rotation_factor = np.exp(1j * 2 * np.pi * delta_f * t). Signal Modulation: Perform point-wise multiplication between the input signal and the rotation factors to apply CFO. The core operation is: cfo_signal = original_signal * rotation_factor. ### Important Considerations Sampling Rate Impact: The frequency offset value Δf should be set reasonably, typically not exceeding a certain proportion of the signal bandwidth to avoid distortion. In implementation, validate that Δf < fs/2 to prevent aliasing. Normalized Time Processing: In discrete signal processing, the time axis t may require normalization based on the sampling interval. The implementation should ensure proper scaling of time indices for accurate phase calculations. Application Scenarios: Commonly used in communication system simulations, such as testing frequency offset robustness for modulation schemes like OFDM and QAM. The function should support batch processing for multiple signal frames in system-level simulations. This method effectively simulates frequency offset effects in real communication systems within simulation environments, facilitating algorithm development and performance analysis. The implementation should include error checking for input parameters and support both real and complex signal formats.