1. With reference to the bandreject filter transfer functions in Table 4.7, obtain equations for the transfer functions of: (a) An ideal bandpass filter. (b) A Gaussian bandpass filter. (c) A Butterworth bandpass filter. 2. Start with Eq. (5-46) and derive Eq. (5-48). 3. Assuming now that \\( x \\) and \\( y \\) are continuous quantities, show how you would solve Problems 5.19(b) and (c) using Eq. (5-61) directly. [Hint: Take a look at the solution to Problem 4.1(c).] 4. With reference to the alpha-trimmed filter defined in Eq. (5-31)]: (a) Explain why setting \\( d=0 \\) in the filter reduces it to an arithmetic mean filter. (b) Explain why setting \\( d=m n-1 \\) turns the filter into a median filter. 5. Repeat Problem 5.1 using a max filter. 6. Repeat Problem 5.1 using a contraharmonic mean filter with \\( Q=1 \\).