Designing FIR Digital Filters Using Various Window Functions

In this document, we will demonstrate the design of FIR digital filters using various window functions. Specifically, we will implement linear phase FIR low-pass filters using both rectangular window and Hamming window approaches. The design specifications require a passband cutoff frequency and a unit impulse response h(n) with length N=21. The implementation includes plotting h(n) and its corresponding magnitude-frequency response characteristics to better analyze the filter's performance and features.

From a code implementation perspective, the design process involves several key steps: First, we define the filter order as N-1=20 and calculate the ideal low-pass filter coefficients. Then we apply the selected window functions (rectangular and Hamming) to truncate the ideal impulse response. The rectangular window provides a simple rectangular weighting function, while the Hamming window offers better side lobe attenuation with its specific cosine weighting coefficients: w(n) = 0.54 - 0.46*cos(2πn/(N-1)). Finally, we use FFT-based methods to compute and visualize the magnitude response, demonstrating the trade-offs between transition bandwidth and stopband attenuation characteristic of each window type.