‫ﻦ ﻤﺎره س ﯿ ﻨﺎل و ﻢ‬
‫دا ه ﻨﺪ ﯽ ﮐﺎ ‪-‬دا ه ﯽ ﺮﻒ‬
‫م دوم ‪ ،‬ﺳﺎل ﯿ ﯽ ‪٨۶-٨٧‬‬
‫ر ﻮﻞ‪٨٧/١/١٨:‬‬
‫‪ .١‬ﻣﺴﺎﺋﻞ زﻳﺮ را از ﮐﺘﺎب اﭘﻨﻬﺎﱘ ﺣﻞ ﳕﺎﻳﻴﺪ‪:‬‬
‫أ‪3.8 .‬‬
‫ب‪3.13 .‬‬
‫ج‪3.22 .‬‬
‫د‪3.36 .‬‬
‫ﻩ‪3.44 .‬‬
‫‪ .٢‬ﺑﺎ اﺳﺘﻔﺎدﻩ از ﺗﻌﺮﻳﻒ ﺳﺮي ﻓﻮرﻳﻪ‪ ،‬ﺿﺮاﺋﺐ ﺳﺮي ﻓﻮرﻳﻪ ﺳﻴﮕﻨﺎل‬
‫هﺎي زﻳﺮ را ﺑﻴﺎﺑﻴﺪ‪.‬‬
‫أ‪x ( t ) = 2 sin( 3π t ) + cos( 4π t ) .‬‬
‫ب‪[δ ( t − 13 m ) + δ ( t + 23 m )] .‬‬
‫ج‪) .‬‬
‫‪π‬‬
‫‪3‬‬
‫‪n+‬‬
‫د‪n ) + 1 .‬‬
‫‪6π‬‬
‫‪17‬‬
‫‪10 π‬‬
‫‪19‬‬
‫∞‪+‬‬
‫‪m‬‬
‫)‪∑ ( − 1‬‬
‫= ) ‪x (t‬‬
‫∞‪m = −‬‬
‫(‪x [ n ] = cos‬‬
‫(‪n ) + cos‬‬
‫‪14 π‬‬
‫‪19‬‬
‫(‪x [ n ] = 2 sin‬‬
‫‪ .٣‬ﺑﺎ اﺳﺘﻔﺎدﻩ از ﺗﻌﺮﻳﻒ ﺳﺮي ﻓﻮرﻳﻪ‪ ،‬ﺳﻴﮕﻨﺎل هﺎي )‪ x(t‬ﻣﺘﻨﺎﻇﺮ‬
‫ﺑﺎ ﳕﺎﻳﺶ هﺎي ﺳﺮي ﻓﻮرﻳﻪ زﻳﺮ را ﺑﻴﺎﺑﻴﺪ‪.‬‬
‫أ‪a k = j δ [ k − 1] − j δ [ k + 1] + δ [ k − 3] + δ [ k + 3] .‬‬
‫‪,‬‬
‫‪ω 0 = 2π‬‬
‫ب‪.‬‬
‫‪k‬‬
‫)‬
‫‪1‬‬
‫‪3‬‬
‫‪a k = (−‬‬
‫‪,‬‬
‫‪= 1‬‬
‫‪0‬‬
‫‪ω‬‬
‫‪ .۴‬ﺑﺎ اﺳﺘﻔﺎدﻩ از ﺗﻌﺮﻳﻒ ﺳﺮي ﻓﻮرﻳﻪ‪ ،‬ﺳﻴﮕﻨﺎﳍﺎي ]‪ x[n‬ﻣﺘﻨﺎﻇﺮ ﺑﺎ‬
‫ﳕﺎﻳﺶ هﺎي ﺳﺮي ﻓﻮرﻳﻪ زﻳﺮ را ﺑﻴﺎﺑﻴﺪ‪.‬‬
‫أ‪.‬‬
‫)‪k‬‬
‫ب‪k ) .‬‬
‫ج‪.‬‬
‫‪8π‬‬
‫‪21‬‬
‫‪4π‬‬
‫‪19‬‬
‫(‪= cos‬‬
‫(‪k ) + 2 j sin‬‬
‫‪k‬‬
‫‪a‬‬
‫‪10 π‬‬
‫‪19‬‬
‫(‪a k = cos‬‬
‫]) ‪( − 1 ) m [ δ ( t − 2 m ) + δ ( t + 3 m‬‬
‫∞‪+‬‬
‫∑‬
‫∞‪m = −‬‬
‫= ‪ak‬‬