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Unformatted text preview: 1 ECE-2025 Homework #3 Solutions Spring 2009 Problem 3.1*: (a) Concepts : DC-component, Average Value, Sinusoid, Period, Frequency. Approach: First, x(t) is separated into a DC-component and cosine component. The DC-component is the average value of the signal and can be determined by inspection. By applying the definition of a DC- component, it doesnt vary with time, so it must have a frequency of zero rad/sec or Hz. Next, subtract the DC- component from x(t) to extract the cosine component. To find the frequency of the cosine component select a full cycle of the signal and measure the duration on the time axis to obtain the period, T . The frequency is found from 1/T = f . Answer : The frequency of the DC-component is 0 Hz = 0 rad/sec. The frequency of the cosine component is 1/(1msec) = 1 kHz. (b) Concepts : DC-component, Sinusoid, Amplitude, Phase, Frequency. Approach: We know that we are looking for an equation of the form: ) 2 cos( ] [ ) ( + + = t f A component DC t x In part (a) we found the DC-component and the frequency, f . By inspecting the cosine component, we can determine its maxima and minima and therefore the amplitude, A . The phase, , can be determined by inspection and noting that a minimum occurs at t = 0 and is, therefore, 180-degrees out of phase. Alternatively, the formula T t m 2 = can be applied where t m is the time of the positive maximum. Answer : The maximum is 10 and the minimum is 10 of the cosine. Therefore, A = 10. By applying the time of positive maximum formula: = = = sec 1 sec) 5 . ( 2 2 m m T t m . Note that a phase of = is also correct. In summary, ) ) 1000 ( 2 cos( 10 10 ) ( + = t t x (c) Concepts : Spectrum, Complex Amplitude, Inverse Euler Formula Approach: To plot the spectrum we find the spectral components of x(t). The DC-component was found in part(a). To spectral components of the remaining cosine are found by applying the Inverse Euler Formula that provides a sum of complex conjugate pairs. Each of the complex numbers presents a complex amplitude and a frequency term. The spectral lines are plotted along a horizontal axis that varies with frequency. At each frequency a line is drawn with the height equal to the magnitude of the complex amplitude and labeled with the...
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