b Calculate the filter coefficients for a second order causal FIR lowpass

B calculate the filter coefficients for a second

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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:
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ECE 513 Exam #2 Instructor: Dr. Cranos Williams 9
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ECE 513 Exam #2 Instructor: Dr. Cranos Williams 10 Figure 4: Ideal Lowpass Filter Response. We can convert this to a high pass filter with the same cutoff frequency ω c = π 2 by 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 by ˆ h ( n ) = h ( n ) e jπn + e - jπn 2 = h ( n ) cos ( πn ) (26) Note that the exponentials are divided by 2. This is to account for overlapping spectra that occur at - π and π because the magnitude spectrum is repeating. Thus, a second order high pass filter with a cutoff ω c = π 2 can be obtained from h 2 ( n ) as ˆ h 2 ( n ) = h 2 ( n ) cos ( πn ) (27)
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ECE 513 Exam #2 Instructor: Dr. Cranos Williams 11 1
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