Designing a Half-Band Filter for Signal Processing

In this task, we need to design a half-band filter and perform 2x interpolation on the signal for more refined filtering. First, we must determine the required filter specifications, such as bandwidth, center frequency, and stopband frequency. Using MATLAB's filter design functions (like fdesign.halfband or firpm), we can calculate critical design parameters including stopband attenuation and passband ripple. Next, we implement the half-band filter using efficient polyphase structures that exploit the symmetric nature of half-band filters, where approximately half the coefficients are zero - this reduces computational complexity by nearly 50%. For 2x interpolation, we employ an upsampling process that inserts zeros between original samples, doubling the sampling rate before applying the half-band filter. This filter efficiently removes imaging artifacts while preserving the original signal spectrum. The combination results in a smoother signal with minimal information loss. Finally, we perform signal decoding using appropriate reconstruction techniques to obtain the desired output. The entire process can be implemented using MATLAB's DSP System Toolbox with functions like upfirdn for efficient interpolation filtering.