Fast Fourier Transform with Gaussian Noise Implementation

The mathematical expression x=2*sin(4*pi*t)+5*cos(8*pi*t) represents a composite signal containing two frequency components that can be analyzed using Fast Fourier Transform (FFT) to extract its spectral characteristics. In practical implementation, the signal is first discretized using appropriate time sampling (e.g., t=0:0.01:1 in MATLAB) to create a digital representation. The FFT algorithm is then applied using functions like fft() in MATLAB or numpy.fft.fft() in Python, which efficiently computes the discrete Fourier transform and returns complex-valued frequency coefficients. To simulate real-world scenarios where signals often contain noise, Gaussian noise can be added using random number generation functions such as randn() in MATLAB or np.random.randn() in Python. The noise amplitude is typically controlled by a scaling factor (e.g., noise_amplitude = 0.5) to achieve desired signal-to-noise ratio levels. After adding noise, FFT is performed again on the contaminated signal. Frequency analysis involves computing the magnitude spectrum using abs() function and plotting the results to visualize how noise affects the original frequency components. Key observations include potential spectral leakage, reduced peak amplitudes, and the appearance of noise floor across the frequency spectrum. This analytical approach is particularly valuable in signal processing applications such as audio processing (where it helps identify fundamental frequencies in noisy recordings), vibration analysis in mechanical systems, and digital communication systems for evaluating channel noise impacts. The comparison between clean and noisy signal spectra provides insights into noise robustness of signal processing algorithms and helps in designing effective filtering strategies.