EECE 340 Homework, Spring 2008

American University of Beirut Department of Electrical and Computer Engineering EECE 440 Signals and Systems Homework 3: Due July 18, 2006 Problem 1 Two signals m1(t) and m2(t), both band-limited to 500 rad/s, are to be transmitted simultaneously over a channel as shown below m1(t) + b X c m2(t) X a 2 cos20,000t 2 cos10,000t The modulated signal at point c is transmitted over a channel. Determine the bandwidth of the output signal at point c. Problem 2 The spectrum of the two band-limited signals s1(t) and s2(t) are shown below. S1(ω) S2(ω) -5 rad/s 5 rad/s -2 rad/s 2 rad/s Determine the bandwidth of the signal y(t) given by: y(t)=s1(t)+10s2(t)cos10t. Problem 3 The signal t cos T rect t )t( f ω0 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = is applied to the following system f(t) X Filter g(t) cos(2ω0t) The filter in the above figure is considered to be a unity gain ideal band-pass filter of mid-frequency (ω0) and bandwidth (2W) rad/s. Determine the output of this filter. Assume that rect(t/T) has a bandwidth of W rad/s. Problem 4 An electric filter system has the frequency response ω + ω ω = j 1 j H( ) What type of filter is this? Problem 5 Consider the signal s(t) given by: 20cos(220 )t 12cos(160 )t )t(s + = . This signal is present at the input of a unity gain ideal band-pass filter of mid- frequency 210 rad/s and bandwidth 25 rad/s. Determine the average power at the output of the filter. Problem 6 The spectrum of the signals f(t) and g(t) are shown below. Express g(t) as a function of f(t). F(ω) 1 ω -W W G(ω) 1 ω -ω0 -ω0 +W ω0 -W ω0 Problem 7 A sinusoidal signal of frequency 1 Hz is to be sampled periodically. Find the maximum allowable time interval between samples. Problem 8 The system shown below is used for scrambling audio signals. m(t) X LPF y(t) 0-15 KHz 2 cos(30,000π)t The output y(t) is the scrambled version of the input signal m(t). Let the spectrum of the signal m(t) be as shown below, write m(t) as a function of y(t) M(f) 1 f -15 Khz 15 KHz Problem 9 An amplitude modulated wave-form has the form [ ] s t t t t ( ) . cos( ) . cos( ) cos( ) = + + 10 1 05 200 05 400 2000 π π π a. Sketch the spectrum of s(t) b. Find the total power. c. Find the total side-band power d. What is the modulation index? Problem 10 An AM transmitter develops a carrier power output of 50 Watts across 1 Ohm resistive load. The carrier is modulated by a single tome with a modulation index of 0.8. Write the expression of the AM signal s(t) assume fm=5Khz and fc=1 Mhz. Problem 11 The envelope of the output of an AM modulator is shown below s(t) 12 8 4 t Determine the following: a. The modulation index. b. The carrier amplitude. Problem 12 An AM modulator operates with the message signal: m(t)=-6cos(20πt)- 2cos(60πt). The unmodulated carrier is given by: c(t)=100cos(200πt) and the modulation index is 0.5. a. Write the time-domain expression of the AM wave s(t). b. Write the time-domain expression of the USB of s(t) including carrier.