UNIVERSITY OF AKRON DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING 4400: 341 INTRODUCTION TO COMMUNICATION SYSTEMS - Spring 2015 SAMPLE FINAL EXAM TIME: 1 hour 30 minutes INSTRUCTIONS: (1) (2) (3) (4) (5) (6) (7) Write your name on each page Clearly show all your work, in steps, in the space assigned for each question. Answers without reasoning will not receive any credit. With some thinking, there are always short cuts to the answer. If you are taking too much time to answer a question it is advisable to leave it and return to it after you have attempted the rest of the problems. If you feel that you do not understand a question or it is ambiguous, please ask about it. The list of the needed equations and formulae is attached to the end of this booklet. Relax and Perform. GOOD LUCK Question Total Grade 1 20 2 20 3 30 4 10 Total 80 Your Score Name:_______________________________________ Problem 1 (20 points) −2t Consider the signal x(t) = e u(t) . a) Calculate its energy in time and frequency domains and verify the Parseval’s theorem for it. b) Calculate its autocorrelation and ESD functions. c) Calculate the essential bandwidth of this signal where the essential bandwidth must contain 95% of the signal energy. d) If we pass x(t) through a system whose impulse response is h(t) = δ (t − 4) , Find the output signal ESD and its energy. Name:_______________________________________ Problem 2 (20 points) a) The signal shown in the figure is supposed to be transmitted using DSB+C modulation. Calculate the required modulation index and maximum power efficiency of this modulation. b) We have a superheterodyne AM receiver with a cheap non-ideal local oscillator. In addition to fLO, this local oscillator always generates another frequency tone at fLO-10KHz. The intermediate frequency fIF is set to be at 200KHz. Plot the structure of the superheterodyne receiver. Assuming that the receiver is tuned to receive a channel at center frequency fc = 350KHz, calculate fLO. Calculate possible image frequencies. Name:_______________________________________ Problem 3 (30 points) An angle modulated signal with frequency fc = 150KHz is described by the equation, ϕ FM (t) = 10 cos(2π fc t + 0.01sin 2000π t) a) Find the power of the modulated signal and its frequency deviation. Estimate the bandwidth of the modulated signal. b) Design and sketch the block diagram of an Armstrong indirect FM modulator that generate a WBFM signal with carrier 96.3MHz and frequency deviation 20.48KHz using the above signal. Only frequency doublers are available as frequency multipliers. In addition, an oscillator with adjustable frequency from 13MHz to 14MHz is also available for mixing, along with bandpass filters of any specifications. c) To demodulate the WBFM signal generated in part (b), we use a slope detector as shown in the figure which is in fact a high pass filter. We know that the 3dB frequency of this filter is at 1/RC. Assuming R = 1KΩ, find a suitable value for the capacitor, C. Problem 4 (10 points) Consider the signal 𝑥 𝑡 = cos 10𝜋𝑡 + 2cos (30𝜋𝑡). a) Find out the Nyquist sampling rate for this signal. b) If we sample this signal at 25Hz, and then from the sampled signal try to reconstruct the original signal by passing in through a LPF with bandwidth 18Hz, find out the signal in time domain at the output of the LPF. Sketch the spectrum of this signal. Is it a perfect reconstruction? Explain. e ± jx = cos x ± j sin x sin 2 x = cos x = 1 (1 − cos 2x ) 2 cos 2 x = 1 T →∞ T Signal Power: Pg = lim Signal Energy: Eg = ∫ ∞ −∞ ∫ T /2 −T /2 e jx + e− jx 2 e jx − e− jx sin x = 2j 1 (1 + cos 2x ) 2 x(t)x * (t)dt = x(t)x * (t)dt = ∫ ∞ −∞ ∫ ∞ −∞ ∫a 2 dx 1 x = ( ) tan −1 ( ) 2 a a +x Sg ( f )df = Rg (0) Eg ( f )df = Rg (0) Fourier Transform and Inverse Fourier Transform: X( f ) = x(t) = ∞ ∫−∞ x(t)e− j 2πft dt ∞ ∫−∞ X( f )e j 2 πft df mmax − mmin 2A + mmax + mmin usefulpower Ps AM power efficiency: η = = totalpower Pc + Ps Δf FM and PM modulation index: β = where B is the message bandwidth B AM modulation index: µ = Δf = k f mp 2π • for FM and Δf = kp m p 2π for PM FM and PM bandwidth: BFM = 2B(β + 1) Nyquist Rate is 2B (B is the analog signal bandwidth). m 2p Quantization Noise Power: N = 3L2