In 6.33, we considered a system commonly used to obtain a highpass filter from a lowpass

Chapter 6, Problem 6.34

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In 6.33, we considered a system commonly used to obtain a highpass filter from a lowpass filter and vice versa. In this problem, we explore the system further and, in particular, consider a potential difficulty if the phase of H(jw) is not properly chosen. (a) Referring to Figure P6.33, let us assume that H(jw) is real and as shown in Figure P6.34. Then 1 - o1 < H(jw) < 1 + o1, 0 ::; w ::; w 1, -02 < H(jw) < +02, w2 < w. Determine and sketch the resulting frequency response of the overall system of Figure P6.33. Does the resulting system correspond to an approximation to a highpass filter? (b) Now let H(jw) in Figure P6.33 be of the form (P6.34-1) where H1 (jw) is identical to Figure P6.34 and O(w) is an unspecified phase characteristic. With H(jw) in this more general form, does it still correspond to an approximation to a lowpass filter? (c) Without making any assumptions about O(w ), determine and sketch the tolerance limits on the magnitude of the frequency response of the overall system of Figure P6.33. (d) If H(jw) in Figure P6.33 is an approximation to a lowpass filter with unspecified phase characteristics, will the overall system in that figure necessarily correspond to an approximation to a highpass filter?