FFT Analysis of Signal Spectrum

After sampling x(t) to obtain x(n), we need to analyze whether spectral aliasing will occur. For spectral analysis, we employ the FFT algorithm. The following are the requirements for the design debug report: 1. Working Principle Summary: The Fast Fourier Transform (FFT) algorithm efficiently computes the Discrete Fourier Transform (DFT) of a signal. It converts time-domain samples to frequency-domain components, enabling analysis of spectral characteristics and potential aliasing effects when the sampling rate violates the Nyquist criterion. 2. Design Approach: The implementation involves generating the signal x(t), sampling it to create discrete sequence x(n), and applying FFT with different window lengths (N=16 and N=1024). Key MATLAB functions include fft() for transformation, abs() for magnitude calculation, and fftshift() for proper frequency axis alignment. The code will handle signal generation, FFT computation, and visualization of both time-domain and frequency-domain representations. 3. Challenges and Solutions: Potential challenges include proper frequency axis scaling, handling complex FFT outputs, and understanding spectral resolution differences between N=16 and N=1024. Solutions involve using appropriate normalization, focusing on magnitude spectra, and comparing frequency bin widths between different FFT sizes. 4. Design, Debug Results and Analysis: The implementation will demonstrate how different FFT lengths affect frequency resolution. With N=16, coarse frequency resolution may show leakage effects, while N=1024 provides finer spectral details. The analysis will compare both results to identify aliasing artifacts and validate the sampling approach. 5. Program Code and Operational Steps: The MATLAB code will include: signal generation using trigonometric functions, sampling implementation, FFT computation with different N values, and plotting routines using subplot() for comparative visualization. Operation steps involve running the script and interpreting the resulting waveforms and spectra. To meet these requirements, the following steps are necessary: 1. Program to plot the waveform of signal x(t) using time-domain visualization techniques 2. Set N=16, perform FFT on x(n) and plot its magnitude-frequency characteristic curve with proper frequency axis labeling 3. Set N=1024, perform FFT on x(n) and plot its magnitude-frequency characteristic curve with enhanced frequency resolution 4. Analyze and compare the computational results from steps 2 and 3, focusing on spectral leakage, resolution, and potential aliasing indicators Please proceed with the design and debugging according to the above steps, and record relevant results and analysis.