Fillers are a type of system. Thus, we can start with a system model of a desire

Question

Fillers are a type of system. Thus, we can start with a system model of a desired logic and then translate this logic (i.e.. its transfer function) into hardware for implementation. Historically, a filter has been constructed with electronic components (op-amps. resistors, capacitors, inductors), but. of course, the logic may be achieved via other means, such as a digital computer that has been programmed instead. This gives rise to HR methods, which are a class of filters with no immediate, direct continuous/analog equivalent. (a) Briefly explain what are two fundamental similarities and two fundamental differences between FIR and HR filters? (b) The passband and stopband ripple may be attenuated through window selection. Briefly, what is the relative relationship between the stopband ripple and the transition bandwidth and filter order? (c) Lathi (p. 763) notes that a rectangular window gives the smallest transition band. What order of a Hamming windowed Low-Pass Filler (LPF) would be needed to equal or belter the transition band of a 10^th order Rectangular Window LPF of the same cut-off frequency? (d) Bilinear Transformations and Equivalence One approach to realizing a digital filler is to transform the analog filler function (a complex function in the s-domain) to a digital one (a complex function in the z-domain). This may be accomplished via a bilinear transformation (or a conformal mapping) of the form: s = K (1 - z^-1/1 + z^-1) Where K is a design parameter. A common choice for this is K = 2/T_s, that is, Tustin's Rule. Using this approach, design an equivalent digital filter to a 4^th -order Chebyshev Type I LPF with a cutoff set at 1 rad/sec and with normalized or unit sampling. Please plot/sketch the Magnitude Response of both the Analog and Digital LPFs for Ohm = 0... pi rad/sec. (e) Extending on (d). design the best (highest signal/noise ratio) 100^th order non-recursive (FIR) you can for die 50-Hz notch filter application of Problem 2 in Problem Set 1. Please explain the design (i.e., justify which design strategy was used, if a Window is used, justify which window, etc.) in addition to the coefficients.

Explanation / Answer

(a) Differences:

1. Linear phase

2. Stable

3.Limited cycles

1. Nonlinear phase

2. Not stable

3. No limited cycles

Similarities:

1. FIR and IIR filter are digital filters.

(b) attenuation can be improved by decreasing the ripple.

FIR IIR

1. Linear phase

2. Stable

3.Limited cycles

1. Nonlinear phase

2. Not stable

3. No limited cycles

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