ME36500 Homework #11 Due: 12/4/2014

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Problem  #1  (30%)  You  wish  to  transmit  two  low-­‐frequency  pure-­‐tone  signals  using  amplitude  modulation  (AM).  The  first  signal  (𝑥!(𝑡))  is  a  10  Hz  sine  wave  with  a  4  volt  peak-­‐to-­‐peak  amplitude,  while  the  second  (𝑥!(𝑡))  is  a  15  Hz  sine  wave  with  a  6  volt  peak-­‐to-­‐peak  amplitude.  Remembering  that  the  carrier  frequency  should  be  much  greater  than  the  signal  frequency  when  implementing  AM,  you  use  a  1000  Hz  sine  wave  carrier  (𝑣!(𝑡))  in  conjunction  with  your  10  Hz  signal  and  a  1020  Hz  sine  wave  carrier  (𝑣!(𝑡))  in  conjunction  with  your  15  Hz  signal.  Each  carrier  signal  exhibits  peak-­‐to-­‐peak  amplitude  of  2  volts.  The  resulting  modulated  signal  is:  

𝑦 𝑡 = 𝑦! 𝑡 + 𝑦! 𝑡 = 𝑥! 𝑡 𝑣! 𝑡 + 𝑥! 𝑡 𝑣!(𝑡).  (A) Identify  appropriate  mathematical  expressions  for  𝑥! 𝑡 , 𝑥! 𝑡 , 𝑣! 𝑡  and  𝑣!(𝑡).  

(B) Write  down  an  expression  for  𝑦(𝑡),  using  appropriate  trig  identities  to  eliminate  all  trigonometric  products  (eliminate  terms  like  sin 𝛼 sin  (𝛽)).  

(C) Plot  the  magnitude  frequency  spectrum  (𝑀!  vs.  frequency  in  Hz).  

(D) With  regard  to  demodulation,  what’s  the  problem  with  the  above  scenario?  (E) Plot  the  magnitude  frequency  spectrum  (𝑀!  vs.  frequency  in  Hz)  if  the  frequency  of  

carrier  signal  𝑣!(𝑡)  is  doubled.  Why  is  this  arrangement  preferable  with  regard  to  demodulation?  

Problem  #2  (30%)  Assume  you  have  access  to  ideal  band-­‐pass  filters  and  can  perfectly  separate  the  modulated  signal  from  Problem  1  back  into  components  𝑦!(𝑡)  and  𝑦!(𝑡).  You  would  now  like  to  use  the  “multiplication  by  carrier  wave”  method  to  demodulate  the  signal.  

(A) Write  an  expression  for  signal  component  𝑦!(𝑡).  (B) Calculate  the  signal  𝑦!! 𝑡 = 𝑣! 𝑡 𝑦! 𝑡 ,  using  appropriate  trig  identities  to  eliminate  all  

terms  possessing  trigonometric  products  (don’t  leave  terms  like  sin 𝛼 sin  (𝛽)  or  cos 𝛼 sin  (𝛽)).  

(C) To  recover  the  original  signal  𝑥!(𝑡),  you  pass  𝑦!(𝑡)  through  an  ideal  low-­‐pass  filter  with  a  pass-­‐band  gain  of  2  and  a  cutoff  frequency  of  100  Hz.  Plot  the  resulting  magnitude  frequency  spectrum  (𝑀!  vs.  frequency  in  Hz).  

(D) Repeat  part  (C)  using  a  first-­‐order  (non-­‐ideal)  low-­‐pass  filter  with  a  pass-­‐band  gain  of  2  and  a  cutoff  frequency  of  100  Hz.    

(E) If  the  carrier  signal  peak-­‐to-­‐peak  amplitude  was  3  volts,  what  pass-­‐band  gain  would  be  appropriate  for  perfectly  recovering  the  input  signal  (assuming  an  ideal  low-­‐pass  filter  was  available)?  

Problem  #3  (40%)  A  coworker  is  attempting  to  transmit  a  10  Hz  test  signal  using  amplitude  modulation.  Unfortunately,  their  experimental  configuration  allows  the  test  signal  to  pick  up  some  60  Hz  noise  prior  to  modulation.  Thus,  the  input  to  the  modulator  is:  

ME36500 Homework #11 Due: 12/4/2014

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𝑥 𝑡 = 2 cos 2𝜋 ⋅ 10𝑡 +12 cos  (2𝜋 ⋅ 60𝑡)  

(A) If  modulation  occurs  with  a  carrier  wave  of  𝑣 𝑡 = sin 2𝜋 ⋅ 2000𝑡 ,  write  down  the  resulting  modulated  signal  equation,  and  plot  the  frequency  spectrum  (magnitude  only).  

(B) Calculate  the  signal-­‐to-­‐noise  (S/N)  ratio  of  the  modulated  signal.  

(C) To  improve  the  S/N  ratio,  your  coworker  wants  to  pass  the  modulated  signal  through  a  band-­‐pass  filter  prior  to  demodulation,  hoping  to  filter  out  spectral  components  associated  with  the  60  Hz  noise.  Do  you  think  this  will  be  effective?  Why?  

(D) Using  spare  parts,  your  coworker  starts  to  build  a  (non-­‐ideal)  band-­‐pass  filter,  comprised  of  a  first-­‐order  low-­‐pass  filter  (𝐾 = 1  and  𝜔!" = 2020  Hz)  and  a  first-­‐order  high-­‐pass  filter  (𝐾 = 1  and  𝜔!" = 1980  Hz).  Hoping  to  save  time,  you  sit  down  and  calculate  the  S/N  ratio  for  the  modulated  signal  after  it  passes  through  the  band-­‐pass  filter  your  coworker  is  attempting  to  construct.  What  is  your  answer,  and  how  does  it  compare  to  the  unfiltered  S/N  ratio  determined  in  part  (B)?  

(E) You  suggest  to  your  coworker  that  applying  a  low-­‐pass  filter  to  the  noisy  signal,  prior  to  modulation,  might  be  more  effective.  What  is  the  S/N  ratio  of  the  modulated  signal  if  𝑥(𝑡)  is  passed  through  a  first-­‐order  low-­‐pass  filter  (𝐾 = 1  and  𝜔!" = 25  Hz)  prior  to  modulation?  Why  is  this  approach  more  effective  than  the  band-­‐pass  filter  that  your  coworker  wanted  to  implement?