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(b) Calculate the filter coefficients for a second ordercausalFIR lowpass filter withtheωcfrom part a.), designed using a rectangular window.(c) Using a frequency shifting property, write the coefficients of a second order causalFIR highpass filter (real coefficients) with the same cutoff frequency,ωc.Solution:
ECE 513 Exam #2Instructor: Dr. Cranos Williams9
ECE 513 Exam #2Instructor: Dr. Cranos Williams10Figure 4: Ideal Lowpass Filter Response.We can convert this to a high pass filter with the same cutoff frequencyωc=π2by shifting the response in frequency by +π(to the right) and-π(to the left).Figure 5: Ideal Highpass Filter Response.Using the requency shift property, this can be accomplished byˆh(n)=h(n)ejπn+e-jπn2=h(n)cos(πn)(26)Note that the exponentials are divided by 2. This is to account for overlappingspectra that occur at-πandπbecause the magnitude spectrum is repeating.Thus, a second order high pass filter with a cutoffωc=π2can be obtained fromh2(n) asˆh2(n)=h2(n)cos(πn)(27)