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EE 453: Homework 4Due in PDF on Canvas: 9 April 2021Fundamental Problems (20 pts)1. Consider the impulse responseh[n] = 0.25δ[n−1] + 0.5δ[n] + 0.25δ[n+ 1](a) Determine the magnitude respnose|H(ejω)|(b) What is the phase response of this filter? How can you obtain alinear-phasefilter from thish[n]?(c) Obtain a length threelinear-phase highpassfilter by suitably modifying the coefficients of thelinear-phase version ofh[n].2. Consider astable, causalIIR transfer function with squared-magnitude response given by|H(ejω)|2=9(1.09 + 0.6 cosω)(1.25−cosω))(1.36 + 1.2 cosω)(1.16 + 0.8 cosω)|H(ejω)|2=H(z)H(z−1)|z=ejωHINT:cosω7→12(z+z−1)(a) Determine astabletransfer functionH(z) such thatH(z)H(z−1)|z=ejωsatisfies the above squared-magnitude response(b) How manystable, distincttransfer functionsHi(z) are there such that:H1(z)H1(z−1)|z=ejω=H2(z)H2(z−1)|z=ejω=…=Hn(z)Hn(z−1)|z=ejω(c) Among the different transfer functionsHi(z), identify the minimum-phase, mixed-phase, and maximum-phase systems.(d) Plot the different pole-zero diagrams for each different transfer function, again identifying mini-mum/maximum/mixed phase(e) Calculate the all-pass filter which transforms the minimum-phase filter into the maximum-phasefilter3. Consider the IIR filter whose transfer function is given byH(z) =(0.2z−1+ 0.4z−2)(3−2.2z−1)(2−3z−1+ 4z−2)(1−0.7z−1)(a) Draw the (i) Direct Form I, (ii) Direct Form II, (iii) Direct Form II Transpose representations forH(z)(b) Draw two different cascade realizations ofH(z) using second-order sections(c) Implement the following Parallel Forms for this system function:•Parallel Form I:H(z) = 0.3143−0.02681−0.7z−1+−0.2875+0.7501z−11−1.5z−1+2z−2•Parallel Form II:H(z) =0.0188z−11−0.7z−1+0.3188z−1+0.5750z−21−1.5z−1+2z−24. For each of the following systems, the goal is to identify the associated minimum phase systems, so that|H(ejω)|=|Hmin(ejω)|.(a)H(z) =1−2z−11+13z−1(b)H(z) =(1+3z−1)(1−12z−1)z−1(1+13z−1)(c)H(z) =(1−3z−1)(1−14z−1)(1−34z−1)(1−43z−1)1
Advanced Problems (40 pts)5. Consider the IIR transfer functionH(z) =1 +316z−11 +38z−1−532z−1Suppose the fractional coefficients are quantized by rounding up to three fractional bits, so that [116→316)7→18, [316→516)7→28, etc.(a) Determine the Direct Form II system function forH(z) (Taking into account the quantization,Youdo not need to draw it)(b) Determine the Parallel system function forH(z) (Taking into account the quantization,You donot need to draw it)(c) Determine the Cascade system function forH(z) (Taking into account the quantization,You donot need to draw it)(d) Which representation in (a) – (c) had the least error due to quantization?6. Suppose that you want to deisgn a discrete-time lowpass Butterworth filter to meet the following speci-fications:Fp= 500 Hz,Fstop= 1000 Hz, with a minimum stopband attenuation of 50 dB.(a) UsingImpulse Invariance, determine the minimum filter order for the following sampling fre-quencies: 10 kHz, 5 kHz, 2.5 kHz(b) UsingBilinear Transformation, determine the minimum filter order for the following samplingfrequencies: 10 kHz, 5 kHz, 2.5 kHz7. (a) Prove thatimpulse invariancemaps thejΩ axis in thes-plane to the unit circle in thez-plane(b) Prove thatimpulse invariancepreserves stability(c) Prove that thebilinear transformationmaps thejΩ axis in thes-plane to the unit circle in thez-plane(d) Prove that thebilinear transformationpreserves stabilityPage 2
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