presec1 - 2 θ = 1 and identity 2 cos(2 θ = cos 2 θ-sin 2...

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ECE 2200: Section I Problems (Week 2 / Spring 2009) 1. Describe a mixed-time-base system that has both continuous-time and discrete- time elements. Draw a block diagram. Designate the input and output signals. Provide example sketches of typical outputs caused by typical inputs. 2. Given two measurements of the signal y ( t i ) = cos(2 πft i + φ ) at times t 1 and t 2 , develop a method for computing f and φ . Impose restrictions on t 1 and t 2 such that a unique solution exists. Test numerically. 3. Consider relating the phase shift φ of a sinusoid to a time shift t 1 implicit in x ( t ) = A cos(2 πf 0 t + φ ) = A cos(2 πf 0 ( t - t 1 )) Assume that the period of x ( t ) is T 0 = 8 seconds. Are the following statements true or false? Explain your answer. (a) When t 1 = - 2 seconds, the value of the phase is φ = π 2 . (b) When t 1 = 3 seconds, the value of the phase is φ = 3 π 4 . (c) When t 1 = 7 seconds, the value of the phase is φ = π 4 . 4. Given identity 1 sin 2 θ + cos
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Unformatted text preview: 2 θ = 1 and identity 2 cos(2 θ ) = cos 2 θ-sin 2 θ in Table 2-2 from the text produce the identity sin 2 θ = 1 2 (1-cos(2 θ )) ConFrm numerically. 5. Consider an unforced, undamped second order di±erential equation, such as that describing the vibrating motion of a tuning fork or a weight hanging on the end of a perfect spring a d 2 dt 2 x ( t ) + bx ( t ) = 0 Consider the solution candidate x ( t ) = A cos( ω t ). Determine ω in terms of the di±erential equation coe²cients a and b . 1 6. The following measurements were made of the continuous-time signal x ( t ) = a cos( bπt-c ) + d for t > 0: (i) Frst maximum of 3.7 occurs at t = 0 . 4 seconds (ii) Frst minimum of-1 . 3 occurs at t = 0 . 6 seconds Determine a , b , c , and d . ConFrm numerically. 2...
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