ECE565Midterm2PracticeProblems
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University of Massachusetts, Amherst**We aren't endorsed by this school
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ECE 565
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Electrical Engineering
Date
Jan 8, 2025
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2
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ECE 565 Digital Signal Processing - Midterm Exam 2Practice Problems1.Illuminati Confirmed: You are asked to design a FIR filter of length 4 with the target responseH(ejΩ) given below.(a) Provide the filter impulse responseh[n] and frequency responseH(ejΩ) obtained viawindowing.(b) Provide the filter impulse responseh[n] and frequency responseH) obtained viafrequency sampling. (The frequency selection is part of the design process!)(c) What would you change in the design specifications to improve the filters?2.You Can Count On Me: Theℓ0“norm” of a complex vector of lengthNis given by∥x∥0=limp→0+∥x∥p= number of nonzero entries ofx.Show that theℓ0“norm” is not a norm.3.Throwing Shade: The five shaded regions inR2below denote the “unit balls” for which the functionsfj:R2→Robey 0≤fj(x)≤1 forj∈ {a, b, c, d, e}– more formally,Bj={x∈R2: 0≤fj(x)≤1}.(1,3)(1,-3)(-1,3)(-1,-3)(1,0)(-1,0)(0,-1)(0,1)BaBb-1-111-1-111-1-111B1={x:kxk11}B2={x:kxk21}B1={x:kxk11}(a) Letx=32. Forp= 1,2,1, findr=kxkp, and sketchxandrBp(use di↵erent axesfor each of the three values ofp).(b) Consider the 5 shapes below.(1,3)(1,-3)(-1,3)(-1,-3)(1,0)(-1,0)(0,-1)(0,1)BaBb(1,-1)(-1,1)(-1,-1)(1,1)(-1,0)(0,-2)(1,0)(0,2)(2,-2)(1,1)(-1,-1)(-2,2)BcBdBeDetermine thejfor whichk·kBjis a valid norm. In the cases wherek·kBjisnota validnorm, explain why. The most convincing way to do this is to find vectors for which oneof the three properties of a valid norm are violated.5. LetAbe the 2⇥2 matrixA=1p244-1/21/2.Forx2R2, definekxkA=kAxk2.(a) Show thatk·kAis indeed a valid norm.(b) Sketch the unit ballBA={x:kxkA1}corresponding tok·kA.Feel free to useMATLAB.2Last updated 12:11, August 29, 2019(a) Determine which of these functionsfjcannotbe valid norms, and explain why.(b) For the remaining functions, what else do you need to know to be able to say that the function is a valid norm?4.Go Low: Consider a continuous-time low-pass filter with system functionH(s) =1s+a.(a) Suppose an IIR filter is obtained by applying impulse invariance toH(s) withTS= 2. What is the differenceequation for the resulting filter?(b) Suppose an IIR filter is obtained by applying the bilinear transformation toH(s) withT= 2. What is thedifference equation for the resulting filter?5.Aim High: You are asked to design a FIR filter of length 5 that approximates an idea high-pass filter with cutofffrequency ΩC=dπ10.(a) Provide the filter impulse responseh[n] and frequency responseH(ejΩ) obtained via windowing. Use a Hammingwindow.(b) Provide the filter impulse responseh[n] and frequency responseH(ejΩ) obtained via frequency sampling. (Thefrequency selection is part of the design process!)6.I Know Kung Fu: LetAbe the matrixA=aba−b, and for vectorsx∈R2, define∥x∥A=∥Ax∥2. Show that∥ · ∥Ais a valid norm.7.Low-Passed Out: You are asked to design a linear FIR filter of lengthM= 4 for which the magnitude response atΩ = 0, Ω =π/2, and Ω =πis specified as|Hr(ej0)|= 1,|Hr(ejπ/2)|=1√2,|Hr(ejπ)|= 0. Determine the impulseresponsehr[n] of this filter.1
8.Take Me Out To The Ball Game: LetBpbe the unit ball for theℓpnorm inRN. Fill in the right hand side belowwith an expression that depends only ony, using the properties of the inner product (including those proved inHomeworks). Show your work!(a) maxx∈B2⟨x, y⟩=.(b) maxx∈B∞⟨x, y⟩=.(c) ˆx= arg maxx∈B2⟨x, y⟩=.(d) ˆx= arg maxx∈B∞⟨x, y⟩=.Note that for (c) and (d) if there is more than one maximum, you can pick any vectorxthat achieves it.9.Existence is Pain: You are Mr. Meeseeks, and I ask you to help me define an inner product for the vector spaceRN.I come up with the bilinear function⟨x, y⟩M=∑Nn=1anx[n]y[n], whereais a real scalar. You must provide me asuitable value ofathat would make this map a valid inner product in order to disappear.(a) Using the definition of the inner product, provide a value ofathat would provide a valid inner product for⟨·,·⟩M.(b) As a service to future Mr. Meeseeks, what conditions does an arbitrary valueaneed to hold for the map aboveto be an inner product?10.Direction and Magnitude: In the vector spaceP3of all cubic polynomial signalsx(t) =a3t3+a2t2+a1t+a0withscalar setR, pointwise addition, and pointwise scalar multiplication. LetTbe the subset of such polynomials withR10x(t)dt= 0. Determine whetherTis a subspace ofP3.