ECE111_2008SPRING_HW1__[0] - The Cooper Union Department of...

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The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Problem Set I: Signals & Spectra January 24, 2008 1. Let x ( t ) = e ° 3 t u ( t ) . Sketch x ( t ) , x ( t ° 1) and x (2 ° t ) by hand. 2. Let x be a discrete-time signal with support ° 1 ± n ± 3 with values given by f 4 ; 5 ; 3 ; 1 ; 2 g . Sketch x ( n ) , x ( n + 2) and x (1 ° n ) by hand. 3. The continous-time signal cos ° 4 ° 5 t ± and the discrete-time signal cos ° 4 ° 5 n ± are each periodic, but have di/erent periods! First, °nd the period for each signal (note that for a discrete-time signal to be periodic, the period MUST be an integer!). Next, sketch cos ° 4 ° 5 t ± in MATLAB for 0 ± t ± 4 T , where T is the period for this signal, and superimpose a stem plot of cos ° 4 ° 5 n ± over the same time span, and graphically con°rm the di/erent period of the discrete-time signal. 4. Use phasors (not trigonometric identities) to express the following in the form A cos ( !t + ° ) . Use MATLAB as a calculator (i.e., to convert between polar and rectangular form): 2 cos (50 ±t + 1 : 2) ° 3 cos (50 ±t + 0 : 8) 5. The impedance of a circuit at 10 MHz is 3 + j . Can this be modeled as a resistor in series with an inductor or capacitor? Find the resistance value and the inductance or capacitance value. 6. In this problem use the notation used in the class for exponential and trigonometric forms of Fourier series. A real signal over the interval 0 ± t ± 3 is characterized by Fourier coe¢ cients c m (complex exponential form). The following is known: c 0 = 2 , c 1 = 2 + j , c 2 = 3 ° 4 j . (a) There is enough information to determine some, but not all, of the other coe¢ - cients c m as well as (some but not all) the coe¢ cients a m ; b m . Determine these values. (b) What is the DC power? What is the power at the fundamental frequency? At the second harmonic? 7. Path-Loss Exponent for Wireless Communications The attenuation of a propagating wave in a wireless communication channel is often modeled with a path loss exponent, n , in that the power P r at the receiving antenna a distance r from the transmitter is given by: P r ( r ) = P 0 ² r 0 r ³ n
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