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What is the magnitude squared response of the normalized low pass Butterworth filter?(a) \(\frac{1}{1+Ω^{-2N}}\)(b) 1+Ω^-2N(c) 1+Ω^2N(d) \(\frac{1}{1+Ω^{2N}}\)I got this question in an online quiz.Asked question is from Butterworth Filters topic in chapter Digital Filters Design of Digital Signal Processing

Answer»

Right option is (d) \(\FRAC{1}{1+Ω^{2N}}\)

Explanation: We know that the MAGNITUDE response of a low PASS Butterworth filter of order N is given as

|H(jΩ)|=\(\frac{1}{\sqrt{1+(\frac{Ω}{Ω_C})^{2N}}}\)

For a normalized filter, ΩC =1

=> |H(jΩ)|=\(\frac{1}{\sqrt{1+(Ω)^{2N}}}\) => |H(jΩ)|^2=\(\frac{1}{1+Ω^{2N}}\)

Thus the magnitude SQUARED response of the normalized low pass Butterworth filter of order N is given by the equation,

|H(jΩ)|^2=\(\frac{1}{1+Ω^{2N}}\).



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