Question

Problem #1. Topics: Z Transform Find the Z transform of: x[n]=-(0.9 )n-2u-n+5] X(Z) Problem #2. Topics: Filter Design, Effective Time Constant Design a causal 2nd order, normalized, stable Peak Filter centered at fo 1000Hz. Use only two conjugate poles and two zeros at the origin. The system is to be sampled at Fs- 8000Hz. The duration of the transient should be as close as possible to teft 7.5 ms. The transient is assumed to end when the largest pole elevated to the effective discrete-time constant (neff) reach the value of ε-0.01 a) Find H(z), including R.О.С. H(Z2) b) Sketch the Pole-Zero plot of the filter, including the filter R.O.C and list the value of both poles. c) Find the approximate 3dB bandwidth of this filter in Hz and in radians/sample. d) Find the filter Frequency Response. H(w) e) Find the filter difference equation. y[n] f Sketch a Block Diagram realization of the filter. g) Sketch the Peak Filter magnitude frequency response. Include gain at ω:0, [H (O) I, ωο, lh(ω°) I, and ω-n, IH (n) 1, where the filter center frequency is called H(o)I h) Find the inverse filter including R.О.С. Hin,(z) i) Find the inverse filter impulse response in terms of delayed unit samples j) Draw a pole-zero plot for the inverse filter k) Find the inverse filter Frequency Response H(o) 1) Sketch the inverse Peak Filter magnitude frequency response. Include gain at Hin(a) m) Find the following products and explain the result. n) Draw a Block Diagram Realization of the Inverse Filter.

Answer 1

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