signal,part 1

‫ﻓﺼﻞ اول‬

‫ﻣﻌﺮﻓﻲ ﺳﻴﮕﻨﺎلهﺎي زﻣﺎن ﭘﻴﻮﺳﺘﻪ و زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫‪١‬‬

‫‪ -١-١‬ﺗﻌﺮﻳﻒ ﺳﻴﮕﻨﺎل‬ ‫ﺳﻴﮕﻨﺎل ﺗﺎﺑﻌﻲ اﺳﺖ آﻪ ﺣﺎوي اﻃﻼﻋﺎﺗﻲ درﺑﺎرﻩ رﻓﺘﺎر ﻓﻴﺰﻳﻜﻲ ﻳﻚ ﺳﻴﺴﺘﻢ اﺳﺖ‪.‬‬ ‫)‪ x(t‬ﺳﻴﮕﻨﺎل زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫]‪ x[n‬ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫‪ t ,n‬ﻣﺘﻐﻴﺮهﺎي ﻣﺴﺘﻘﻞ و ‪ x‬ﻣﺘﻐﻴﺮ واﺑﺴﺘﻪ ﻳﺎ ﺗﺎﺑﻊ ﻣﻲﺑﺎﺷﺪ‪.‬‬ ‫ﺳﻴﺴﺘﻢ ‪ :‬ﻣﺠﻤﻮﻋﻪاي از اﺟﺰاي ﮔﺮد ﺁﻣﺪﻩ در آﻨﺎر هﻢ‪.‬‬

‫‪ -٢-١‬ﻃﺒﻘﻪﺑﻨﺪي ﺳﻴﮕﻨﺎلهﺎ‬ ‫‪ -١-٢-١‬ﺳﻴﮕﻨﺎلهﺎي زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫‪ -٢-٢-١‬ﺳﻴﮕﻨﺎلهﺎي زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬ ‫) ‪x (t‬‬

‫] ‪x [n‬‬

‫‪n ∈z‬‬

‫‪t ∈ IR‬‬

‫ﺑﺪﻳﻬﻲ اﺳﺖ آﻪ ﺑﺎ ﻧﻤﻮﻧﻪﺑﺮداري از ﺳﻴﮕﻨﺎل زﻣﺎن ﭘﻴﻮﺳﺘﻪ ﻣﻲﺗﻮان ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ را ﺑﺪﺳﺖ ﺁورد‪.‬‬

‫‪ -٣-١‬ﺳﻴﮕﻨﺎلهﺎي زوج و ﺳﻴﮕﻨﺎلهﺎي ﻓﺮد‬ ‫) ‪ , Even x (t ) = x (−t‬ﻳﺎ زوج‬

‫) ‪x (t ) = −x (−t‬‬

‫‪, Odd‬ﻳﺎ ﻓﺮد‬

‫] ‪x [n ] = x [−n‬‬ ‫] ‪x [n ] = −x [−n‬‬

‫ﺗﺬآﺮ ‪ :١‬ﺳﻴﮕﻨﺎل ﻓﺮد ﮔﺴﺴﺘﻪ ﺑﺎﻻﺟﺒﺎر در ﻣﺒﺪا ﻣﺨﺘﺼﺎت ﻣﻘﺪار ﺻﻔﺮ دارد‪.‬‬ ‫ﺗﺬآﺮ ‪ :٢‬هﺮ ﺳﻴﮕﻨﺎل دﻟﺨﻮاﻩ را ﻣﻲﺗﻮان ﺑﻪ ﺻﻮرت ﻣﺠﻤﻮع دو ﺳﻴﮕﻨﺎل زوج و ﻓﺮد ﻧﻮﺷﺖ‪:‬‬

‫)) ‪x (t ) = Even (x (t )) + Odd (x (t‬‬ ‫‪1‬‬ ‫)) ‪(x (t ) + x (−t‬‬ ‫‪2‬‬

‫= )) ‪Even (x (t‬‬

‫‪1‬‬ ‫)) ‪(x (t ) − x (−t‬‬ ‫‪2‬‬

‫= )) ‪Odd (x (t‬‬

‫‪٢‬‬

‫ﻣﺜﺎل ‪(١‬‬

‫) ‪x (t ) = x (−t‬‬

‫‪t‬‬

‫) ‪x (t ) = −x (−t‬‬

‫‪T‬‬ ‫‪2‬‬

‫‪−1‬‬

‫‪t‬‬

‫‪−T‬‬ ‫‪2‬‬

‫‪−2‬‬

‫‪2‬‬

‫‪1‬‬

‫‪ -۴-١‬ﺳﻴﮕﻨﺎل ﻣﺘﻨﺎوب‬ ‫ﺳﻴﮕﻨﺎل ﻣﺘﻨﺎوب ﺑﻪ ﺳﻴﮕﻨﺎﻟﻲ ﮔﻔﺘﻪ ﻣﻲﺷﻮد آﻪ در ﺑﺎزﻩهﺎي زﻣﺎﻧﻲ ﻣﺸﺨﺺ ﻋﻴﻨﺎ ﺗﻜﺮار ﺷﺪﻩ ﺑﺎﺷﺪ‪.‬‬ ‫‪2π‬‬

‫‪T‬‬

‫‪k ∈Z‬‬

‫‪2π‬‬

‫‪N‬‬

‫‪k ∈Z‬‬

‫= ‪ω 0 = 2πf‬‬ ‫= ‪Ω 0 = 2πf‬‬

‫) ‪x (t ) = x (t + T ) = x (t + kT‬‬ ‫] ‪x [n ] = x [n + N ] = x [n + kN‬‬

‫ﺗﺬآﺮ ‪ :١‬ﺑﺪﻳﻬﻲ اﺳﺖ آﻪ اﮔﺮ ‪ N , T‬دورﻩ ﺗﻨﺎوب ﺑﺎﺷﻨﺪ ‪ ٢‬ﺑﺮاﺑﺮ و ‪ ٣‬ﺑﺮاﺑﺮ و ‪ ...‬ﺁن هﻢ دورﻩ ﺗﻨﺎوب اﺳﺖ‪.‬‬ ‫ﺗﺬآﺮ‪ :٢‬آﻮﭼﻜﺘﺮﻳﻦ دورﻩ ﺗﻨﺎوب دورﻩ ﺗﻨﺎوب اﺻﻠﻲ اﺳﺖ )‪ T‬در ﭘﻴﻮﺳﺘﻪ‪ N ،‬در ﮔﺴﺴﺘﻪ( و ﻓﺮآﺎﻧﺲ ﺗﻌﺮﻳﻒ ﺷﺪﻩ ﺑﺎ آﻮﭼﻜﺘﺮﻳﻦ‬ ‫دورﻩ ﺗﻨﺎوب‪ ،‬ﻓﺮآﺎﻧﺲ اﺻﻠﻲ اﺳﺖ‪ ω 0 ) .‬در ﭘﻴﻮﺳﺘﻪ‪ Ω 0 ،‬در ﮔﺴﺴﺘﻪ(‬ ‫ﺗﺬآﺮ‪ :٣‬دورﻩ ﺗﻨﺎوب ﺳﻴﮕﻨﺎل زﻣﺎن ﭘﻴﻮﺳﺘﻪ )‪ (T‬ﺑﺎﻳﺪ ﻋﺪد ﻣﺜﺒﺖ ﺑﺎﺷﺪ‪ ،‬درﺣﺎﻟﻴﻜﻪ دورﻩ ﺗﻨﺎوب ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ )‪ (N‬ﻋﻼوﻩ ﺑﺮ‬ ‫ﻣﺜﺒﺖ ﺑﻮدن ﺑﺎﻳﺴﺘﻲ ﺻﺤﻴﺢ ﻧﻴﺰ ﺑﺎﺷﺪ‪.‬‬ ‫ﻣﺜﺎل ‪(٢‬‬

‫) ‪x (t‬‬

‫] ‪x [n‬‬

‫‪11 n‬‬

‫‪4‬‬

‫‪3‬‬

‫‪−4‬‬

‫‪t‬‬

‫‪T‬‬

‫‪−T‬‬

‫دورﻩ ﺗﻨﺎوب اﺻﻠﻲ‬

‫دورﻩ ﺗﻨﺎوب اﺻﻠﻲ ‪N = 8‬‬

‫‪2T‬‬

‫‪ -۵-١‬ﺳﻴﮕﻨﺎلهﺎي اﻧﺮژي و ﺗﻮان‬ ‫∞‪+‬‬

‫] ‪E = ∑ x 2 [n‬‬ ‫∞‪−‬‬

‫‪(t ) dt‬‬

‫‪2‬‬

‫∞‪+‬‬

‫‪∫x‬‬

‫∞‪−‬‬

‫‪T‬‬ ‫‪2‬‬

‫‪2‬‬ ‫= ‪∫ x (t ) dt‬‬

‫‪−T‬‬ ‫‪2‬‬

‫‪٣‬‬

‫‪E = lim‬‬

‫∞→ ‪T‬‬

‫اﻧﺮژي‬

‫‪T‬‬

‫‪(t ) dt‬‬

‫‪2‬‬

‫‪2‬‬

‫‪∫x‬‬

‫‪−T‬‬ ‫‪2‬‬

‫‪1‬‬

‫‪T‬‬

‫‪T‬‬

‫= ‪(t ) dt‬‬

‫‪2‬‬

‫‪2‬‬

‫‪∫x‬‬

‫‪1‬‬

‫‪−T‬‬ ‫‪2‬‬

‫‪T‬‬

‫‪P av = lim‬‬

‫] ‪[n‬‬

‫∞→ ‪T‬‬

‫‪2‬‬

‫‪N −1‬‬

‫‪∑x‬‬ ‫‪n‬‬ ‫‪=0‬‬

‫‪1‬‬

‫‪N‬‬

‫= ‪P av‬‬

‫ﺗﻮان ﻣﺘﻮﺳﻂ‬

‫ﻳﻚ ﺳﻴﮕﻨﺎل ﺑﻪ ﻋﻨﻮان ﺳﻴﮕﻨﺎل اﻧﺮژي ﺷﻨﺎﺧﺘﻪ ﻣﻲﺷﻮد اﮔﺮ و ﺗﻨﻬﺎ اﮔﺮ ﻣﺤﺪود ﺑﺎﺷﺪ‪.‬‬

‫‪ -۶-١‬ﻋﻤﻠﻴﺎت روي ﻣﺘﻐﻴﺮ واﺑﺴﺘﻪ‬ ‫)‪(1‬‬

‫) ‪y (t ) = Ax (t‬‬

‫] ‪y [n ] = Ax [n‬‬

‫)‪(2‬‬

‫) ‪y (t ) = x 1 (t ) + x 2 (t‬‬

‫] ‪y [n ] = x 1 [n ] + x 2 [n‬‬

‫)‪(3‬‬

‫) ‪y (t ) = x 1 (t ) ⋅ x 2 (t‬‬

‫] ‪y [n ] = x 1 [n ] ⋅ x 2 [n‬‬

‫) ‪dx (t‬‬ ‫‪dt‬‬

‫)‪(4‬‬

‫= ) ‪y (t‬‬ ‫∞‪+‬‬

‫‪∫ x (λ) dλ‬‬

‫]‪y [n ] = x [n ] − x [n − 1‬‬ ‫‪n‬‬

‫] ‪∑ x [m‬‬

‫= ) ‪y (t‬‬

‫∞‪m = −‬‬

‫∞‪−‬‬

‫)‪(5‬‬

‫= ] ‪y [n‬‬

‫‪ -٧-١‬ﻋﻤﻠﻴﺎت روي ﻣﺘﻐﻴﺮ ﻣﺴﺘﻘﻞ‬ ‫‪ -١-٧-١‬ﺷﻴﻔﺖ زﻣﺎﻧﻲ‬ ‫) ‪y (t ) = x (t − t 0‬‬

‫] ‪y [n ] = x [n − n 0‬‬

‫ﺷﻴﻔﺖ ﺑﻪ راﺳﺖ‪ :‬اﮔﺮ ‪ t 0 > 0‬ﺑﺎﺷﺪ )‪ x(t‬را ﺑﻪ اﻧﺪازﻩ ‪ t 0‬ﺑﻪ ﺳﻤﺖ راﺳﺖ ﺷﻴﻔﺖ ﻣﻲدهﻴﻢ ﺗﺎ )‪ y(t‬ﺑﺪﺳﺖ ﺁﻳﺪ‪.‬‬ ‫ﺷﻴﻔﺖ ﺑﻪ ﭼﭗ‪ :‬اﮔﺮ ‪ t 0 < 0‬ﺑﺎﺷﺪ )‪ x(t‬را ﺑﻪ اﻧﺪازﻩ ‪ t 0‬ﺑﻪ ﺳﻤﺖ ﭼﭗ ﺷﻴﻔﺖ ﻣﻲدهﻴﻢ ﺗﺎ )‪ y(t‬ﺑﺪﺳﺖ ﺁﻳﺪ‪.‬‬ ‫ﺗﺬآﺮ ‪ :١‬ﭼﻨﺎﻧﭽﻪ ‪ t 0 > 0‬ﺑﺎﺷﺪ‪ ،‬ﺳﻴﮕﻨﺎل ) ‪ x (t − t 0‬از )‪ x(t‬ﻋﻘﺐﺗﺮ اﺳﺖ )ﺑﻪ ﻟﺤﺎظ زﻣﺎﻧﻲ( و ﭼﻨﺎﻧﭽﻪ ‪ t 0 < 0‬ﺑﺎﺷﺪ‪ ،‬ﺳﻴﮕﻨﺎل‬

‫) ‪x (t − t 0‬‬

‫از )‪ x(t‬ﺟﻠﻮﺗﺮ اﺳﺖ‪.‬‬

‫ﺗﺬآﺮ ‪ :٢‬در ﻣﻮرد ﺳﻴﮕﻨﺎلهﺎي زﻣﺎن ﮔﺴﺴﺘﻪ ﻧﻴﺰ ﺑﺴﺘﻪ ﺑﻪ اﻳﻨﻜﻪ ‪n 0‬‬ ‫اﻧﺪازﻩ ‪n 0‬‬

‫واﺣﺪ ﺷﻴﻔﺖ ﻣﻲدهﻴﻢ ﺗﺎ ]‪ y[n‬ﺑﺪﺳﺖ ﺁﻳﺪ‪.‬‬

‫‪۴‬‬

‫ﻣﺜﺒﺖ و ﻳﺎ ﻣﻨﻔﻲ ﺑﺎﺷﺪ ﺳﻴﮕﻨﺎل ]‪ x[n‬را ﺑﻪ راﺳﺖ و ﻳﺎ ﭼﭗ ﺑﻪ‬

‫ﻣﺜﺎل ‪(۴‬‬

‫ﻣﺜﺎل ‪(٣‬‬

‫) ‪x (t‬‬

‫] ‪x [n‬‬ ‫‪1‬‬

‫‪n‬‬

‫‪1 2‬‬

‫‪− 2 −1‬‬ ‫‪t‬‬

‫‪−1‬‬

‫‪1‬‬ ‫‪2‬‬

‫] ‪y 1 [n‬‬ ‫‪n‬‬

‫‪−1‬‬

‫‪−5‬‬

‫‪1‬‬ ‫‪2‬‬

‫) ‪y 1 (t‬‬

‫]‪y 1 [n ] = x [n + 3‬‬ ‫‪t‬‬

‫‪−1‬‬

‫‪0‬‬

‫] ‪y 2 [n‬‬ ‫‪n‬‬

‫‪1‬‬

‫‪3‬‬

‫‪−‬‬

‫) ‪y 2 (t‬‬

‫‪−1‬‬ ‫‪−1‬‬

‫]‪y 2 [n ] = x [n − 1‬‬ ‫‪t‬‬ ‫‪5‬‬ ‫‪2‬‬

‫‪3‬‬ ‫‪2‬‬

‫در ﺑﺴﻴﺎري از ﻣﺴﺎﺋﻞ ﺑﺎ اﺳﺘﻔﺎدﻩ از ﺷﻴﻔﺖ زﻣﺎﻧﻲ ﻣﻲﺗﻮان ﺳﻴﮕﻨﺎﻟﻲ را ﺑﻪ ﺻﻮرت زوج ﻳﺎ ﻓﺮد درﺁورد‪.‬‬

‫‪ -٢-٧-١‬وارونﺳﺎزي زﻣﺎﻧﻲ‬ ‫)‪(2‬‬

‫) ‪y 1 (t ) = x (t + 12‬‬

‫) ‪y (t ) = x (−t‬‬

‫] ‪y [n ] = x [−n‬‬

‫)‪ x(-t‬ﻳﺎ ]‪ x[-n‬اﻧﻌﻜﺎس )‪ x(t‬و ﻳﺎ ]‪ x[n‬ﻧﺴﺒﺖ ﺑﻪ ﻣﺤﻮر ﻗﺎﺋﻢ هﺴﺘﻨﺪ‪.‬‬

‫‪۵‬‬

‫)‪y 2 (t ) = x (t − 2‬‬

‫ﻣﺜﺎل ‪(۶‬‬

‫ﻣﺜﺎل ‪(۵‬‬

‫) ‪x (t‬‬

‫] ‪x [n‬‬ ‫‪t‬‬

‫‪1‬‬

‫‪n‬‬

‫‪−2‬‬

‫‪2‬‬

‫‪− T1‬‬

‫‪T2‬‬

‫) ‪y (t‬‬

‫‪−1‬‬

‫] ‪y [n‬‬

‫) ‪y (t ) = x (−t‬‬

‫‪t‬‬

‫‪T1‬‬

‫‪1‬‬

‫‪2‬‬

‫‪n‬‬

‫‪−2‬‬

‫‪−1‬‬

‫‪−T2‬‬

‫] ‪y [n ] = x [−n‬‬

‫ﻣﺜﺎل ‪(٧‬‬ ‫‪2‬‬

‫‪n‬‬

‫] ‪y [n‬‬

‫‪2‬‬

‫‪1‬‬ ‫‪−2‬‬

‫‪2‬‬

‫] ‪y [n ] = x [−n‬‬

‫‪n‬‬

‫] ‪x [n‬‬ ‫‪1‬‬ ‫‪−2‬‬

‫‪2‬‬

‫ﺑﺮاي ﺳﻴﮕﻨﺎﻟﻲ آﻪ ﻧﻪ زوج و ﻧﻪ ﻓﺮد اﺳﺖ وارون زﻣﺎﻧﻲ ﺁن ﻧﻪ ﻓﺮد و ﻧﻪ زوج اﺳﺖ‪ .‬اﻣﺎ ﺑﺮاي ﺳﻴﮕﻨﺎل زوج وارون زﻣﺎﻧﻲ زوج‬ ‫اﺳﺖ وﻟﻲ ﺑﺮاي ﺳﻴﮕﻨﺎل ﻓﺮد ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺧﺎﺻﻴﺖ ]‪ x[n]= -x[-n‬وارون زﻣﺎﻧﻲ ﺁن ]‪ -x[n‬اﺳﺖ‪.‬‬

‫‪ -٣-٧-١‬ﺗﻐﻴﻴﺮ ﻣﻘﻴﺎس زﻣﺎﻧﻲ‬

‫)‪(3‬‬

‫‪; a ∈ IR‬‬

‫] ‪ y [n ] = x [kn‬ﻳﺎ‬

‫) ‪y (t ) = x (at‬‬

‫‪;k ∈ Z‬‬

‫‪1‬‬

‫] ‪y [n ] = x [ n‬‬ ‫‪k‬‬

‫اﻟﻒ( اﮔﺮ ‪>1‬׀‪a‬׀ ﺑﺎﺷﺪ )‪ y(t‬ﻓﺸﺮدﻩ ﺷﺪﻩ ﺳﻴﮕﻨﺎل )‪ x(t‬ﺧﻮاهﺪ ﺑﻮد‪.‬‬ ‫ب( اﮔﺮ ‪<1‬׀‪a‬׀ ﺑﺎﺷﺪ )‪ y(t‬ﺑﺎز ﺷﺪﻩ ﺳﻴﮕﻨﺎل )‪ x(t‬ﺧﻮاهﺪ ﺑﻮد‪.‬‬ ‫ج( اﮔﺮ ‪ a < 0‬ﺑﺎﺷﺪ ﺑﺎﻳﺪ ﺑﻌﺪ از ﺗﻐﻴﻴﺮ ﻣﻘﻴﺎس ‪ ،‬وارون زﻣﺎﻧﻲ اﻧﺠﺎم داد‪.‬‬ ‫ﺗﺬآﺮ‪ :‬در زﻣﺎن ﭘﻴﻮﺳﺘﻪ ﻣﺎهﻴﺖ ﺳﻴﮕﻨﺎل ﻋﻮض ﻧﻤﻲﺷﻮد‪ ،‬اﻣﺎ در زﻣﺎن ﮔﺴﺴﺘﻪ ﻣﺎهﻴﺖ ﺳﻴﮕﻨﺎل ﺗﻐﻴﻴﺮ ﻣﻲآﻨﺪ و ﺳﻴﮕﻨﺎل ﺟﺪﻳﺪي‬ ‫ﺑﺪﺳﺖ ﻣﻲﺁﻳﺪ‪.‬‬ ‫ﻣﺜﺎل ‪(٨‬‬ ‫‪۶‬‬

‫) ‪y 1 (t‬‬ ‫‪t‬‬ ‫‪2‬‬

‫‪1‬‬

‫‪t‬‬ ‫‪2‬‬

‫‪1‬‬

‫‪2‬‬

‫) ‪y 2 (t‬‬

‫‪1‬‬

‫) ‪x (t‬‬

‫) ‪y 1 (t ) = x (2t‬‬

‫‪−1‬‬

‫‪t‬‬

‫‪1‬‬

‫‪1‬‬

‫‪−1‬‬

‫‪1‬‬

‫‪2‬‬

‫) ‪y 2 (t ) = x (−2t‬‬

‫‪−1‬‬

‫ﻣﺜﺎل ‪(٩‬‬

‫) ‪x (t‬‬ ‫‪1‬‬ ‫‪2‬‬

‫) ‪y (t ) = x (− t‬‬

‫‪t‬‬

‫‪t‬‬

‫‪T2‬‬

‫‪1‬‬ ‫‪2‬‬

‫) ‪y (t ) = v (−t‬‬

‫) ‪v (t ) = x ( t‬‬ ‫وارون زﻣﺎﻧﻲ‬

‫‪2T 2‬‬

‫‪− T1‬‬

‫‪− 2T 1‬‬

‫‪t‬‬

‫‪2T 1‬‬

‫‪− 2T 2‬‬

‫‪y 1 [n ] = x [2n ] , N = 2‬‬

‫ﻣﺜﺎل ‪(١٠‬‬

‫] ‪x [n‬‬ ‫‪2‬‬ ‫‪1‬‬

‫‪n‬‬

‫‪3‬‬

‫‪−3‬‬

‫ﺑﺮاي ﺣﻞ اﻳﻨﮕﻮﻧﻪ ﻣﺴﺎﺋﻞ ﺑﻪ روش زﻳﺮ ﻋﻤﻞ ﻣﻲآﻨﻴﻢ‪:‬‬ ‫ﺑﺎ ﺟﺎﻳﮕﺬاري ‪ n‬در ﻓﺮﻣﻮل ]‪ y[n‬ﻣﻘﺎدﻳﺮ ﺑﺪﺳﺖ ﺁﻣﺪﻩ ﺑﺮﺣﺴﺐ ]‪ x[n‬ﻣﻲﺑﺎﺷﻨﺪ‪.‬‬ ‫‪M‬‬

‫] ‪y 1 [n‬‬

‫‪y [−1] = x [−2] = 2‬‬ ‫‪y [1] = x [2] = 2‬‬

‫‪2‬‬

‫‪n‬‬

‫‪y [0] = x [0] = 2‬‬ ‫‪−1 1‬‬

‫دورﻩ ﺗﻨﺎوب ﺳﻴﮕﻨﺎل ﺟﺪﻳﺪ ] ‪ x [2n‬ﺑﺮاﺑﺮ ‪ N = 1‬اﺳﺖ‪.‬‬

‫]‪y [2] = x [4‬‬ ‫‪M‬‬

‫‪٧‬‬

‫ﺗﺬﮐﺮ‪ y[n]= x[kn] :‬ﻧﺴﺒﺖ ﺑﻪ ]‪ x[n‬ﻓﺸﺮدﻩ ﺷﺪﻩ‪ ،‬آﻪ ﺑﺮﺧﻲ از ﻣﻘﺎدﻳﺮ را از دﺳﺖ ﻣﻲدهﺪ‪.‬‬

‫] ‪y 2 [n ] = x [ 12 n‬‬

‫ﻣﺜﺎل ‪(١١‬‬ ‫‪M‬‬

‫‪y [−1] = 0‬‬

‫] ‪y 2 [n‬‬

‫‪y [0] = x [0] = 2‬‬ ‫‪1‬‬ ‫‪2‬‬

‫‪n‬‬

‫‪y [1] = x [ ] = 0‬‬

‫‪−4‬‬

‫‪4‬‬

‫‪y [2] = x [1] = 1‬‬ ‫‪M‬‬

‫دورﻩ ﺗﻨﺎوب ﺳﻴﮕﻨﺎل ﺟﺪﻳﺪ ﺑﺮاﺑﺮ ‪ N = 4‬اﺳﺖ‪.‬‬

‫]‪n‬‬

‫ﺗﺬﮐﺮ‪:‬‬

‫‪1‬‬

‫‪k‬‬

‫[ ‪y [n ] = x‬‬

‫ﻧﺴﺒﺖ ﺑﻪ ]‪ x[n‬ﺑﺎز ﺷﺪﻩ آﻪ در ﻧﺘﻴﺠﻪ ﺗﻌﺪادي ﺻﻔﺮ ﺑﻪ ]‪ y[n‬اﺿﺎﻓﻪ ﺧﻮاهﺪ ﺷﺪ‪ .‬ﺑﺪﻳﻦ ﺗﺮﺗﻴﺐ‬

‫ﻣﺎهﻴﺖ ﺳﻴﮕﻨﺎل ﮔﺴﺴﺘﻪ در اﺛﺮ ﺗﻐﻴﻴﺮ ﻣﻘﻴﺎس زﻣﺎﻧﻲ ﺗﻐﻴﻴﺮ ﻣﻲآﻨﺪ‪.‬‬ ‫ﺗﺬآﺮ‪ :‬اﮔﺮ ﺳﻴﮕﻨﺎل زﻣﺎن ﭘﻴﻮﺳﺘﻪ و ﻳﺎ زﻣﺎن ﮔﺴﺴﺘﻪ ﻣﺘﻨﺎوب ﺑﺎﺷﻨﺪ‪ ،‬در اﺛﺮ ﺗﻐﻴﻴﺮ ﻣﻘﻴﺎس زﻣﺎﻧﻲ در اﺛﺮ ﻓﺸﺮدﻩ ﺷﺪن‪ ،‬دورﻩ ﺗﻨﺎوب‬ ‫ﺳﻴﮕﻨﺎل ﺟﺪﻳﺪ آﻢ ﺷﺪﻩ و در اﺛﺮ ﺑﺎز ﺷﺪن‪ ،‬دورﻩ ﺗﻨﺎوب ﺳﻴﮕﻨﺎل ﺟﺪﻳﺪ اﻓﺰاﻳﺶ ﻣﻲﻳﺎﺑﺪ‪.‬‬

‫‪ -۴-٧-١‬رﺳﻢ ﺳﻴﮕﻨﺎل ﺑﻪ روش ﻣﻨﻈﻢ‬

‫)‪4‬‬

‫)‪4‬‬

‫‪1‬‬

‫‪1‬‬ ‫)] ‪(x [ n − n 0‬‬

‫‪k‬‬

‫)] ‪y [n ] = x [kn − n 0 ] (x [ n − n 0‬‬ ‫‪k‬‬ ‫] ‪x [n ] → v [n ] = x [n − n 0‬‬

‫) ‪x (t ) → v (t ) = x (t − b‬‬

‫) ‪v ( t ) → y ( t ) = v (at ) = x (at − b‬‬

‫] ‪v [n ] → y [n ] = v [kn ] = x [kn − n 0‬‬

‫ﻣﺜﺎل ‪(١٢‬‬

‫) ‪x (t‬‬

‫? = )‪y (t ) = x (−2t + 3‬‬

‫‪t‬‬

‫) ‪y (t ) = x (at − b‬‬

‫‪−1‬‬

‫‪1‬‬

‫‪٨‬‬

x (t + 3)

−4

−2

v (t ) = x (2t + 3)

t

y (t ) = v (−t )

t

− 2 −1

1

t

2

(١٣ ‫ﻣﺜﺎل‬

x [n ] 1 −2

2

n

y [n ] = x [3n − 2] = ?

y [n ] = v [3n ]

v [n ] = x [n − 2] 1

n

4

n

(١۴ ‫ﻣﺜﺎل‬ 0 1

x (t [ = 

t <3 t ≥3

x (t ) 1

t

3

y 1 (t ) = x (1 − t ) (‫اﻟﻒ‬

x (t + 1)

y 1 (t ) t

2 y 3 (t ) = x (1 − t ) + x (t − 2) (‫ج‬

−2

t

y 2 (t ) = x (3t ) (‫ب‬

٩

y 3 (t )

y 2 (t ) 1

−2

1 x [n ] =  0

1

t

5

t

1

‫( ﻣﻄﻠﻮب اﺳﺖ ﻗﺴﻤﺖ زوج و ﻓﺮد ﺳﻴﮕﻨﺎل زﻳﺮ؟‬١۵ ‫ﻣﺜﺎل‬

n = 0,1, 2, 3 o .w

‫ﻣﻴﺪاﻧﻴﻢ‬

x e = 12 (x [n ] + x [−n ])  1 x o = 2 (x [n ] − x [−n ])

x [n ]

1 0

x e [n ]

x o [n ]

1 1 2

−3

0

n

3

1 2

n

3

−1 2

١٠

0

4

n

‫‪ -٨-١‬ﻣﻌﺮﻓﻲ ﺳﻴﮕﻨﺎلهﺎي ﻣﻬﻢ‬ ‫دراﻳﻦ ﻗﺴﻤﺖ ﭼﻨﺪ ﺳﻴﮕﻨﺎل اﺳﺎﺳﻲ زﻣﺎن ﭘﻴﻮﺳﺘﻪ و زﻣﺎن ﮔﺴﺴﺘﻪ را ﻣﻌﺮﻓﻲ ﻣﻲآﻨﻴﻢ‪ .‬اﻳﻦ ﺳﻴﮕﻨﺎلهﺎ ﻧﻪ ﺗﻨﻬﺎ ﺑﻪ دﻓﻌﺎت ﭘﻴﺶ ﻣﻲﺁﻳﻨﺪ‬ ‫ﺑﻠﻜﻪ ﺗﻮﺳﻂ ﺁﻧﻬﺎ ﻣﻲﺗﻮان ﺳﻴﮕﻨﺎلهﺎي ﭘﻴﭽﻴﺪﻩاي را ﻓﺮﻣﻮﻟﻪ و ﺗﻮﻟﻴﺪ آﺮد‪ .‬ﻣﻬﻢﺗﺮﻳﻦ آﺎرﺑﺮد ﺁﻧﻬﺎ در ﺁزﻣﺎﻳﺸﮕﺎﻩ ﻣﺸﺨﺺ ﻣﻲﺷﻮد‪.‬‬

‫‪ -١-٨-١‬ﺳﻴﮕﻨﺎل ﻧﻤﺎﻳﻲ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫‪x (t ) = Be at‬‬

‫ب( ﻧﻤﺎﻳﻲ اﻓﺰاﻳﺸﻲ‬

‫اﻟﻒ( ﻧﻤﺎﻳﻲ آﺎهﺸﻲ‬

‫ﺑﺎ ﻓﺮض اﻳﻨﻜﻪ ‪ a , B ∈ IR‬ﺑﺎﺷﻨﺪ‪ ،‬ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﻣﺜﺒﺖ و ﻳﺎ ﻣﻨﻔﻲ ﺑﻮدن ‪ a‬ﺳﻴﮕﻨﺎل ﻧﻤﺎﻳﻲ ﺑﻪ ﻳﻜﻲ از دو ﻓﺮم اﻓﺰاﻳﺸﻲ و ﻳﺎ آﺎهﺸﻲ‬ ‫ﺧﻮاهﺪ ﺑﻮد‪.‬‬ ‫ﺗﺬآﺮ‪ :‬ﭼﻨﺎﻧﭽﻪ داﻣﻨﻪ ﺳﻴﮕﻨﺎل ﺧﺮوﺟﻲ ﺳﻴﺴﺘﻤﻲ ﺑﺎ اﻓﺰاﻳﺶ زﻣﺎن ﺑﻪ ﻃﻮر ﻧﺎﻣﺤﺪود زﻳﺎد ﺷﻮد‪ ،‬ﺳﻴﺴﺘﻢ ﺗﺤﺖ ﺑﺮرﺳﻲ ﺑﻪ ﻋﻨﻮان‬ ‫ﺳﻴﺴﺘﻢ ﻧﺎﭘﺎﻳﺪار ﺷﻨﺎﺧﺘﻪ ﻣﻲﺷﻮد‪.‬‬

‫‪١١‬‬

‫ﺳﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫) ‪x [n ] = B (r n‬‬

‫ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ ﻧﻤﺎﻳﻲ ﺑﺴﺘﻪ ﺑﻪ ﻣﻘﺎدﻳﺮ ﻣﺨﺘﻠﻒ ‪ r‬ﭼﻬﺎر ﺣﺎﻟﺖ ﻣﻲﺗﻮاﻧﺪ داﺷﺘﻪ ﺑﺎﺷﺪ‪.‬‬

‫اﻟﻒ( ﻧﻤﺎﻳﻲ اﻓﺰاﻳﺸﻲ‬

‫ب( ﻧﻤﺎﻳﻲ آﺎهﺸﻲ‬

‫ج( ﻧﻮﺳﺎﻧﻲ اﻓﺰاﻳﺸﻲ‬

‫د( ﻧﻮﺳﺎﻧﻲ آﺎهﺸﻲ‬

‫‪ -٢-٨-١‬ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ ) ‪ x (t ) = A cos(ωt + φ‬و‬

‫‪2π‬‬

‫‪T‬‬

‫= ‪ω = 2πf‬‬

‫ﺗﺬآﺮ‪ :‬ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ هﻤﻮارﻩ ﻣﺘﻨﺎوب و ﺑﺎ دورﻩ ﺗﻨﺎوب ‪ T‬اﺳﺖ‪.‬‬ ‫ﻣﺜﺎل ‪(١‬‬ ‫اﻟﻒ(‬

‫‪1‬‬ ‫‪6‬‬

‫) ‪x (t ) = cos( t‬‬

‫‪= 12π‬‬

‫‪2π‬‬ ‫‪1‬‬ ‫‪6‬‬

‫= ‪→T‬‬

‫‪١٢‬‬

‫‪8π‬‬ ‫)‪t‬‬ ‫‪31‬‬ ‫ب(‬ ‫‪31‬‬ ‫‪4‬‬

‫=‬

‫‪2π‬‬ ‫‪8π‬‬ ‫‪31‬‬

‫(‪x (t ) = cos‬‬ ‫= ‪→T‬‬ ‫‪2π‬‬

‫ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ ) ‪ x [n ] = A cos( Ωn + φ‬و‬

‫‪N‬‬

‫= ‪Ω = 2πf‬‬

‫ﺗﺬآﺮ‪ :‬ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ ﺳﻴﻨﻮﺳﻲ ﺑﻪ ﺷﺮﻃﻲ ﻣﺘﻨﺎوب اﺳﺖ آﻪ ﺑﺘﻮان‬

‫‪+‬‬

‫‪ N ∈ Z‬را ﺑﺪﺳﺖ ﺁورد ﺑﻪ ﻧﺤﻮي‬

‫آﻪ ] ‪ x [n ] = x [n + N‬ﮔﺮدد‪) .‬اﻳﻦ ﺳﻴﮕﻨﺎل ﺑﺮﺧﻼف ﺳﻴﮕﻨﺎل زﻣﺎن ﭘﻴﻮﺳﺘﻪ ﺳﻴﻨﻮﺳﻲ ﺑﻌﻀﺎ ﻣﺘﻨﺎوب ﻧﻴﺴﺖ‪(.‬‬ ‫‪+‬‬

‫‪2k π‬‬ ‫‪,N ∈ Z‬‬ ‫‪Ω‬‬

‫= ‪k ∈Z ⇒N‬‬

‫‪,‬‬

‫‪→ Ω N = 2kπ‬‬

‫) ‪A cos( Ωn + φ ) = A cos( Ωn + ΩN + φ‬‬

‫ﻣﺜﺎل ‪(٢‬‬ ‫اﻟﻒ(‬

‫)‪, (k=1‬‬

‫‪2π‬‬ ‫‪2kπ‬‬ ‫‪n ] ⇒ N = 2π = 12k = 12‬‬ ‫‪12‬‬ ‫‪12‬‬

‫[‪x [n ] = cos‬‬

‫ب(‬

‫)‪,( k=4‬‬

‫‪8π‬‬ ‫‪2kπ 31‬‬ ‫= ‪n ] ⇒ N = 8π‬‬ ‫‪k = 31‬‬ ‫‪31‬‬ ‫‪4‬‬ ‫‪31‬‬

‫[‪x [n ] = cos‬‬

‫ج(‬

‫‪= 12kπ‬‬ ‫ﻣﺘﻨﺎوب ﻧﻴﺴﺖ‬

‫‪2kπ‬‬ ‫‪1‬‬ ‫‪6‬‬

‫‪1‬‬ ‫‪6‬‬

‫= ‪x [n ] = cos[ n ] ⇒ N‬‬

‫‪ -٣-٨-١‬ﺳﻴﮕﻨﺎل ﻧﻤﺎﻳﻲ ﻣﺨﺘﻠﻂ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫‪ x (t ) = B e − jωt‬ﻳﺎ ‪x (t ) = Be jωt‬‬ ‫‪2π‬‬

‫‪jωt‬‬ ‫) ‪= B cos(ωt ) + jB sin(ωt‬‬ ‫‪, x (t ) = B e‬‬

‫‪ω‬‬

‫= ‪T‬‬

‫زﻣﺎن ﮔﺴﺴﺘﻪ ‪ x [n ] = B e − j Ωn‬ﻳﺎ ‪x [n ] = Be + j Ωn‬‬ ‫‪N ∈ Z + ,K ∈ Z‬‬

‫‪2kπ‬‬ ‫‪Ω ,‬‬

‫= ‪N‬‬

‫) ‪x [n ] = B e j Ωn = B cos( Ωn ) + jB sin( Ωn‬‬

‫‪+ j Ωn‬‬ ‫‪ x [n ] = Be‬ﻣﻲﺗﻮاﻧﺪ در زﻣﺎن ﻣﺘﻨﺎوب ﺑﺎ دورﻩ ﺗﻨﺎوب ‪ N‬ﺑﺎﺷﺪ‪.‬‬ ‫ﺗﺬآﺮ ‪ :١‬ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ ﻧﻤﺎﻳﻲ ﻣﺨﺘﻠﻂ‬

‫‪١٣‬‬

‫‪+ j Ωn‬‬ ‫‪ x [n ] = Be‬ﻋﻼوﻩ ﺑﺮ اﻳﻨﻜﻪ در زﻣﺎن ﻧﺴﺒﺖ ﺑﻪ ‪ n‬ﻣﻲﺗﻮاﻧﺪ ﻣﺘﻨﺎوب ﺑﺎﺷﺪ‪،‬‬ ‫ﺗﺬآﺮ ‪ :٢‬ﺳﻴﮕﻨﺎل زﻣﺎن ﮔﺴﺴﺘﻪ ﻧﻤﺎﻳﻲ ﻣﺨﺘﻠﻂ‬

‫در ﻓﺮآﺎﻧﺲ ﻧﻴﺰ ﻧﺴﺒﺖ ﺑﻪ ‪ Ω‬هﻤﻮارﻩ ﻣﺘﻨﺎوب اﺳﺖ‪.‬‬

‫‪e j Ωn = e j (Ω +2π )n = e j (Ω + 4π )n = e j (Ω +2mπ )n ,‬‬

‫‪m ∈z‬‬

‫ﺗﺬآﺮ ‪ :٣‬اﮔﺮ ﺳﻴﮕﻨﺎل ﺑﻪ ﺻﻮرت ﺣﺎﺻﻠﻀﺮب ﺑﻮد در ﺻﻮرت اﻣﻜﺎن ﺑﺎﻳﺪ ﺑﻪ ﺣﺎﺻﻞ ﺟﻤﻊ دو ﺳﻴﮕﻨﺎل ﺗﺒﺪﻳﻞ ﺷﻮد و دورﻩ ﺗﻨﺎوب‬ ‫هﺮ آﺪام را ﺟﺪاﮔﺎﻧﻪ ﺑﻪ دﺳﺖ ﺁورد‪ .‬ﺳﭙﺲ دورﻩ ﺗﻨﺎوب ﻣﺸﺘﺮك را ﺑﻪ دﺳﺖ ﻣﻲﺁورﻳﻢ‪.‬‬ ‫ﻣﺜﺎل ‪(٣‬‬ ‫اﻟﻒ(‬

‫‪x (t ) = e j 2t + e j 3t‬‬

‫‪2π‬‬ ‫‪2π 6π 8π‬‬ ‫= ‪= π ,2π ,3π ,4π ,... T 2‬‬ ‫=‬ ‫=‬ ‫‪= ...‬‬ ‫‪⇒ T = 2π‬‬ ‫‪2‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪3‬‬ ‫‪,‬‬

‫ب(‬

‫‪n‬‬

‫‪3π‬‬ ‫‪4‬‬

‫‪j‬‬

‫‪+e‬‬

‫‪n‬‬

‫‪2π‬‬ ‫‪3‬‬

‫‪j‬‬

‫= ‪T1‬‬

‫‪x [n ] = e‬‬

‫‪8k‬‬ ‫‪= 8 = 16 = 24 = ...‬‬ ‫‪3‬‬ ‫‪N = 24‬‬

‫=‬

‫‪2kπ‬‬ ‫‪3π‬‬ ‫‪4‬‬

‫‪= 3k = 3 = 6 = 9 = ... = 24‬‬

‫= ‪N2‬‬ ‫‪,‬‬

‫‪2kπ‬‬ ‫‪2π‬‬ ‫‪3‬‬

‫= ‪N1‬‬

‫‪ -٤-٨-١‬ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ ﻣﻴﺮاﺷﻮﻧﺪﻩ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫) ‪x (t ) = Be at cos(ωt + φ‬‬

‫ﺗﺬآﺮ‪ :‬ﺳﻴﮕﻨﺎل ﻧﻤﺎﻳﻲ ﻣﺘﻨﺎوب ﻧﺒﻮدﻩ و ﭘﺲ از ﺿﺮب ﺁن در هﺮ ﻋﺒﺎرﺗﻲ ﺑﺎﻋﺚ ﻣﻲﺷﻮد آﻪ ﺳﻴﮕﻨﺎل ﻧﻬﺎﻳﻲ ﻣﺘﻨﺎوب ﻧﺒﺎﺷﺪ‪.‬‬

‫اﻟﻒ( ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ ﻣﻴﺮاﺷﻮﻧﺪﻩ اﻓﺰاﻳﺸﻲ‬

‫‪١٤‬‬

‫ب( ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ ﻣﻴﺮاﺷﻮﻧﺪﻩ آﺎهﺸﻲ‬

‫زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫] ‪x [n ] = B (r n ) cos[Ωn + φ‬‬

‫اﻟﻒ( ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ ﻣﻴﺮاﺷﻮﻧﺪﻩ آﺎهﺸﻲ‬

‫ب( ﺳﻴﮕﻨﺎل ﺳﻴﻨﻮﺳﻲ ﻣﻴﺮاﺷﻮﻧﺪﻩ اﻓﺰاﻳﺸﻲ‬

‫‪١٥‬‬

‫‪-٩-١‬ﺗﻮاﺑﻊ وﻳﮋﻩ‬ ‫‪-١-٩-١‬ﺗﺎﺑﻊ ﭘﻠﻪ واﺣﺪ‬ ‫‪n ≥0‬‬ ‫‪n <0‬‬

‫‪1‬‬ ‫‪0‬‬

‫‪u [n ] = ‬‬

‫زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫‪t ≥0‬‬ ‫‪t <0‬‬

‫‪1‬‬ ‫‪0‬‬

‫‪u (t ) = ‬‬

‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫] ‪u [n‬‬

‫) ‪u (t‬‬

‫‪1‬‬

‫‪1‬‬

‫‪n‬‬

‫‪t‬‬

‫ﺗﺬآﺮ‪ :‬ﺗﻮاﺑﻌﻲ آﻪ ﻓﺮم هﻨﺪﺳﻲ دارﻧﺪ را ﻣﻲﺗﻮان ﺑﺮ ﺣﺴﺐ ﺗﺎﺑﻊ ﭘﻠﻪ ﺑﻴﺎن آﺮد‪:‬‬ ‫ﻣﺜﺎل ‪(٤‬‬

‫) ‪x 1 (t‬‬

‫) ‪x 1 (t ) = u (t + τ2 ) − u (t − τ2‬‬

‫‪t‬‬ ‫‪2‬‬

‫اﻟﻒ(‬

‫‪1‬‬

‫‪τ‬‬

‫‪2‬‬

‫) ‪x 2 (t‬‬ ‫‪1‬‬

‫)‪x 2 (t ) = u (t ) − u (t − 2‬‬

‫‪t‬‬

‫‪2‬‬

‫ب(‬

‫] ‪x [n‬‬

‫‪n‬‬

‫]‪x [n ] = u [n ] − u [n − 3‬‬ ‫ج(‬

‫]‪u [−n + 4‬‬

‫د(‬

‫]‪u [−n + 4‬‬ ‫‪n‬‬

‫ن(‬

‫‪2‬‬

‫]‪u [n + 4‬‬

‫‪n‬‬

‫‪4‬‬

‫]‪u [−n − 4‬‬ ‫‪١٦‬‬

‫‪−4‬‬

‫‪−τ‬‬

‫]‪u [−n − 4‬‬

‫]‪u [n − 4‬‬

‫‪n‬‬

‫‪n‬‬

‫‪−4‬‬

‫‪4‬‬

‫‪-٢-٩-١‬ﺗﺎﺑﻊ ﺿﺮﺑﻪ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫) ‪δ (t‬‬ ‫‪1‬‬

‫‪t =0‬‬ ‫‪t ≠0‬‬

‫‪t‬‬ ‫‪=0‬‬

‫‪−1‬‬

‫‪∫ δ (t )dt‬‬

‫∞‪−‬‬

‫‪,‬‬

‫ﻣﻘﺪاروﻳﮋﻩ ‪‬‬ ‫‪0‬‬

‫‪δ (t ) = ‬‬

‫‪0+‬‬

‫∞‪+‬‬

‫‪−‬‬

‫∞‪−‬‬

‫‪∫ δ (t )dt = ∫ δ (t )dt = 1‬‬ ‫‪0‬‬

‫راﺑﻄﻪ ﺑﻴﻦ ﺗﺎﺑﻊ ﭘﻠﻪ واﺣﺪ و ﺗﺎﺑﻊ ﺿﺮﺑﻪ واﺣﺪ‬

‫) ‪du (t‬‬ ‫‪dt‬‬ ‫‪t <0‬‬ ‫‪t ≥0‬‬

‫‪)dτ‬‬

‫= ) ‪δ (t‬‬ ‫‪0‬‬

‫‪∫ δ (λ)dλ =1‬‬

‫∞‪−‬‬

‫‪t‬‬

‫‪0‬‬

‫‪t‬‬

‫= ) ‪u (t‬‬

‫‪∫ δ (τ − t‬‬

‫∞‪−‬‬

‫‪t −t 0‬‬

‫= ‪∫ δ (λ)dλ‬‬

‫⇒‬

‫= ) ‪u (t − t 0‬‬

‫∞‪−‬‬

‫‪λ +t0 = τ‬‬ ‫ﺧﻮاص‪:‬‬‫‪ (١‬ﺑﻴﺎن ﺗﻮاﺑﻊ ﭼﻨﺪ ﺿﺎﺑﻄﻪاي ﺑﺎ ﻳﻚ ﺿﺎﺑﻄﻪ‪:‬‬

‫‪t ≥0‬‬ ‫) ‪o .w ⇒ y (t ) = x (t )u (t‬‬

‫) ‪x (t‬‬ ‫‪0‬‬

‫‪y (t ) = ‬‬

‫ﻣﺜﺎل ‪(٥‬‬

‫‪t ≥2‬‬ ‫‪o .w‬‬ ‫)‪⇒ y (t ) = x (t )u (t − 2‬‬

‫) ‪x (t‬‬ ‫‪0‬‬

‫‪y (t ) = ‬‬

‫‪ (٢‬ﺧﺎﺻﻴﺖ ﻏﺮﺑﺎﻟﻲ ﺗﺎﺑﻊ ﺿﺮﺑﻪ‬ ‫‪١٧‬‬

‫) ‪x (t ) ≠δ (t ) = x (0) δ (t‬‬ ‫) ‪x (t )δ (t − t 0 ) = x (t 0 )δ (t − t 0‬‬ ‫‪ (٣‬اﻧﺘﮕﺮال آﺎﻧﻮﻟﻮﺷﻦ‬ ‫∞‪+‬‬

‫) ‪x ( λ )δ (t − λ )dλ = x (t ) ∫ δ (t − λ)dλ = x (t‬‬ ‫∞‪−‬‬

‫∞‪+‬‬

‫∫‬

‫∞‪λ = −‬‬

‫‪ (٤‬زوج ﺑﻮدن ﺗﺎﺑﻊ ﺿﺮﺑﻪ‬

‫) ‪δ (t ) = δ (−t‬‬ ‫‪(*٥‬‬

‫) ‪δ (t‬‬

‫‪1‬‬

‫‪a‬‬

‫= ) ‪δ (at‬‬

‫زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫] ‪δ [n‬‬

‫‪n =0‬‬ ‫‪n ≠0‬‬

‫‪n‬‬

‫‪1‬‬ ‫‪0‬‬

‫‪δ [n ] = ‬‬

‫راﺑﻄﻪ ﺑﻴﻦ ﺗﺎﺑﻊ ﭘﻠﻪ واﺣﺪ و ﺗﺎﺑﻊ ﺿﺮﺑﻪ واﺣﺪ‬

‫‪u [n ] = δ [n ] + δ [n − 1] + ...‬‬ ‫‪n‬‬

‫∞‬

‫∞‪−‬‬

‫] ‪∑ δ [n − k ] =m∑n δ [m ] = m∑ δ [m‬‬ ‫‪k‬‬ ‫∞‪= −‬‬

‫]‪−k‬‬

‫=‬

‫∞‬

‫‪0‬‬

‫‪=0‬‬

‫‪n −n 0‬‬

‫‪∑ δ [k ] = k∑ δ [n − n‬‬ ‫‪k‬‬ ‫‪=0‬‬

‫=‬

‫∞‪= −‬‬

‫= ] ‪u [n − n 0‬‬

‫]‪δ [n ] = u [n ] − u [n − 1‬‬

‫]‪δ [n ] = u [−n ] − u [−n − 1‬‬

‫ﺗﺎﺑﻊ ﺿﺮﺑﻪ ﭼﻮن ﺗﺎﺑﻌﻲ زوج اﺳﺖ ﭘﺲ هﺮ دو راﺑﻄﻪ ﺑﺎﻻ ﻗﺎﺑﻞ ﻗﺒﻮل اﺳﺖ‪.‬‬ ‫ ﺧﻮاص‪:‬‬‫‪ (١‬ﺑﻴﺎن ﺗﻮاﺑﻊ ﭼﻨﺪ ﺿﺎﺑﻄﻪاي ﺑﺎ ﻳﻚ ﺿﺎﺑﻄﻪ‬

‫‪n ≤ −2‬‬ ‫]‪= x [n ]u [−n − 2‬‬ ‫‪o .w‬‬

‫] ‪x [n‬‬ ‫‪0‬‬

‫‪y [n ] = ‬‬

‫‪ (٢‬ﺧﺎﺻﻴﺖ ﻏﺮﺑﺎﻟﻲ ﺗﺎﺑﻊ ﺿﺮﺑﻪ‬

‫* اﻳﻦ راﺑﻄﻪ را اﺛﺒﺎت آﻨﻴﺪ‪.‬‬

‫‪١٨‬‬

‫] ‪y [n ] = x [n ] ≠δ [n ] = x [0]δ [n‬‬ ‫ﻣﺜﺎل ‪:6‬‬

‫]‪y [n ] = x [n + 2]δ [n − 4] = x [6]δ [n − 4‬‬ ‫‪(٣‬اﻧﺘﮕﺮال آﺎﻧﻮﻟﻮﺷﻦ‬ ‫∞‬

‫] ‪x [k ]δ [n − k ] = x [n ] ∑ δ [n − k ] = x [n‬‬ ‫∞‪k = −‬‬

‫∞‬

‫∑‬

‫∞‪k = −‬‬

‫‪ (4‬زوج ﺑﻮدن ﺗﺎﺑﻊ ﺿﺮﺑﻪ‬

‫] ‪δ [n ] = δ [−n‬‬

‫‪ -٣-٩-١‬ﺗﺎﺑﻊ ﺷﻴﺐ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫) ‪r (t‬‬ ‫‪t‬‬

‫‪t >0‬‬ ‫‪t <0‬‬

‫‪1‬‬

‫‪t‬‬ ‫‪r (t ) = ‬‬

‫‪0‬‬

‫راﺑﻄﻪ ﺑﻴﻦ ﺗﺎﺑﻊ ﺷﻴﺐ و ﭘﻠﻪ واﺣﺪ‬

‫) ‪dr (t‬‬ ‫‪dt‬‬

‫‪t‬‬

‫= ) ‪u (t‬‬

‫‪r (t ) = t u (t ) = ∫ u ( λ ) dλ‬‬ ‫∞‪−‬‬

‫زﻣﺎن ﮔﺴﺴﺘﻪ‬

‫] ‪r [n‬‬

‫‪n ≥0‬‬ ‫‪n <0‬‬

‫‪n‬‬

‫‪n‬‬ ‫‪r [n ] = ‬‬

‫‪0‬‬

‫راﺑﻄﻪ ﺑﻴﻦ ﺗﺎﺑﻊ ﺷﻴﺐ و ﭘﻠﻪ واﺣﺪ‬

‫? = ] ‪u [n‬‬

‫] ‪r [n ] = n u [n‬‬

‫‪ -٤-٩-١‬ﺗﺎﺑﻊ دوﺑﻠﺖ واﺣﺪ‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬

‫) ‪dδ (t‬‬ ‫‪dt‬‬

‫= ) ‪δ ′(t‬‬ ‫‪١٩‬‬

‫) ‪δ ′(t‬‬ ‫‪t‬‬ ‫ﺧﻮاص‪:‬‬

‫) ‪x (t )δ ′(t ) = x (0) δ ′(t ) − x ′(0)δ (t‬‬

‫ﺧﺎﺻﻴﺖ ‪(١‬‬ ‫اﺛﺒﺎت‪:‬‬

‫) ‪(x (t ) ≠f (t ))′ = x ′(t )f (t ) + x (t )f ′(t‬‬

‫) ‪if f (t ) = δ (t‬‬ ‫) ‪= x ′(t )δ (t ) + x (t )δ ′(t ) + x ′(0)δ (t ) + x (t )δ ′(t‬‬

‫↓‬ ‫‪(x (t )δ (t ))′‬‬ ‫=‬

‫‪(x (0) δ (t ))′‬‬ ‫=‬

‫) ‪x (0)δ ′(t ) = x ′(0)δ (t ) + x (t )δ ′(t ) ⇒ x (t )δ ′(t ) = x (0) δ ′(t ) − x ′(0)δ (t‬‬

‫) ‪− λ )dλ = x ′(t‬‬

‫ﺧﺎﺻﻴﺖ ‪(٢‬‬

‫∞‪+‬‬

‫‪∫ x (λ)δ ′(t‬‬

‫∞‪−‬‬

‫ﻣﺜﺎل ‪ (٧‬ﻣﻄﻠﻮب اﺳﺖ ﺑﻴﺎن )‪ f(t‬ﺑﺮ ﺣﺴﺐ ﺗﻮاﺑﻊ وﻳﮋﻩ ؟‬

‫) ‪f (t‬‬ ‫‪2‬‬

‫‪t‬‬

‫‪1‬‬ ‫‪1 2 3 4 5‬‬

‫از ﻣﻨﺘﻬﻲ اﻟﻴﻪ ﺳﻤﺖ ﭼﭗ ﺷﺮوع ﺑﻪ ﻧﻮﺷﺘﻦ ﻣﻲآﻨﻴﻢ‪ .‬هﺮ ﺟﺎ آﻪ ﺷﻜﻞ ﺗﻐﻴﻴﺮ آﻨﺪ ﻳﻌﻨﻲ ﺗﺎﺑﻊ ﻋﻮض ﺷﺪﻩ‪ .‬اوﻟﻴﻦ آﺎري آﻪ اﻧﺠﺎم‬ ‫ﻣﻲدهﻴﻢ ﺿﺮﻳﺐ زاوﻳﻪ آﻠﻴﻪ ﺧﻂهﺎ را ﺑﺪﺳﺖ ﻣﻲﺁورﻳﻢ ‪:‬‬ ‫در ﻓﺎﺻﻠﻪ ‪ 1 ≤ t ≤ 2‬ﺷﻜﻞ ﺗﻐﻴﻴﺮ آﺮدﻩ و داراي ﺷﻴﺐ ﺑﻪ ﺳﻤﺖ ﭘﺎﻳﻴﻦ اﺳﺖ‪ .‬در ﻓﺎﺻﻠﻪ ‪ 2 ≤ t ≤ 3‬ﻧﻴﺰ ﺷﻜﻞ دوﺑﺎرﻩ ﺗﻐﻴﻴﺮ‬ ‫ﻣﻲآﻨﺪ‪.‬‬

‫) ‪r (t‬‬

‫) ‪− r (t‬‬

‫‪⇒ m = −1‬‬

‫)‪(1,1‬‬ ‫)‪(2,0‬‬

‫‪⇒m =1‬‬

‫‪t‬‬

‫)‪(2,0‬‬ ‫)‪(3,1‬‬

‫)‪f (t ) = u (t ) − r (t − 1) + 2r (t − 2) − r (t − 3) + u (t − 4) − 2u (t − 5‬‬ ‫‪٢٠‬‬

‫‪t‬‬

f ′(t ) = δ (t ) − u (t − 1) + 2u (t − 2) − u (t − 3) + δ (t − 4) _ 2δ (t − 4) −t .‫ هﺮ ﻳﻚ از ﺗﻮاﺑﻊ زﻳﺮ را هﻤﺮاﻩ ﺑﺎ ﻣﺸﺘﻖ و اﻧﺘﮕﺮاﻟﺸﺎن رﺳﻢ آﻨﻴﺪ‬f (t ) = a e ‫ ( اﮔﺮ‬٨ ‫ﻣﺜﺎل‬

f 1 (t ) = f (t ) u (t ) (‫اﻟﻒ‬ ‫ﺣﻞ‬

f 1′(t ) = f ′(t ) u (t ) + f (t ) u ′(t ) = f ′(t ) u (t ) + f (0) δ (t ) + −a e −t u (t ) + a δ (t ) +∞

+∞

−t −t ∫ f (t ) dt = ∫ a e dt = −a e 0

0

+∞ 0

=a

f 2 (t ) = f (t − 1) u (t − 2) (‫ب‬ f 3 (t ) = f (t − 1) u (t − 1) (‫ج‬ f 4 (t ) = f (t − 1) u (t ) (‫د‬

٢١

‫ﺳﻴﺴﺘﻢ و ﺗﻌﺮﻳﻒ ﺁن ‪:‬‬ ‫ﭘﺮوﺳﻪاي آﻪ ﺑﺎﻋﺚ ﺗﻐﻴﻴﺮ و ﺗﺤﻮل در ﻳﻚ ﺳﻴﮕﻨﺎل ﻣﻲﺷﻮد ‪.‬‬ ‫ﻣﺠﻤﻮﻋﻪاي ﻣﻨﻈﻢ آﻪ ﺑﻪ آﻤﻚ ﻳﻜﺪﻳﮕﺮ هﺪف ﻣﺸﺨﺼﻲ را ﺑﺮﺁوردﻩ ﻣﻲﺳﺎزد‪.‬‬ ‫ﺑﻴﺎن رواﺑﻆ ﺑﻴﻦ ﺧﺮوﺟﻲ و ورودي در ﻳﻚ ﺳﻴﺴﺘﻢ ‪:‬‬ ‫زﻣﺎن ﭘﻴﻮﺳﺘﻪ‬ ‫‪ (١‬ﺑﺎ اﺳﺘﻔﺎدﻩ از ﻣﻌﺎدﻻت دﻳﻔﺮاﻧﺴﻴﻞ‬

‫‪ (٢‬ﺗﺎﺑﻊ ﺗﺒﺪﻳﻞ‬

‫زﻣﺎن ﮔﺴﺴﺘﻪ‬ ‫‪ (٢‬ﺗﺎﺑﻊ ﺗﺒﺪﻳﻞ‬

‫‪ (١‬ﻣﻌﺎدﻻت ﺗﻔﺎﺿﻠﻲ‬

‫‪ -١٠-١‬ﺗﻘﺴﻴﻢﺑﻨﺪي ﺳﻴﺴﺘﻢهﺎ‬ ‫‪ -١-١٠-١‬اﺗﺼﺎل ﺳﺮي‪:‬‬

‫ورودي‬

‫ﺧﺮوﺟﻲ‬ ‫‪ -٢-١٠-١‬اﺗﺼﺎل ﻣﻮازي‪:‬‬

‫ﺧﺮوﺟﻲ‬

‫ورودي‬

‫‪+‬‬

‫‪ -٣-١٠-١‬اﺗﺼﺎل ﻓﻴﺪﺑﻚ‪:‬‬ ‫ﺧﺮوﺟﻲ‬

‫‪+‬‬

‫ورودي‬

‫‪ -١١-١‬ﺧﻮاص ﺳﻴﺴﺘﻢهﺎ‬

‫‪٢٢‬‬

‫‪ -١-١١-١‬ﺣﺎﻓﻈﻪ‬ ‫ﺳﻴﺴﺘﻢ ﺑﺪون ﺣﺎﻓﻈﻪ اﺳﺖ اﮔﺮ ﺧﺮوﺟﻲ در هﺮ ﻟﺤﻈﻪ ﺑﻪ ورودي در هﻤﺎن ﻟﺤﻈﻪ ﺑﺴﺘﮕﻲ داﺷﺘﻪ ﺑﺎﺷﺪ‪ .‬ﺑﻪ ﻋﺒﺎرت دﻳﮕﺮ ﺳﻴﺴﺘﻢ ﺑﺎ‬ ‫ﺣﺎﻓﻈﻪ اﺳﺖ اﮔﺮ ﺧﺮوﺟﻲ ﺑﻪ ﻣﻘﺎدﻳﺮ ﮔﺬﺷﺘﻪ ورودي واﺑﺴﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫ﻣﺜﺎل ‪(١‬‬

‫‪2‬‬ ‫اﻟﻒ( ] ‪ y [n ] = 2x [n ] + x [n‬ﺑﺪون ﺣﺎﻓﻈﻪ‬

‫‪n‬‬

‫ب(‬

‫] ‪∑ x [k‬‬ ‫‪k‬‬

‫د(‬

‫) ‪v (t‬‬

‫‪1‬‬

‫‪R‬‬

‫= ) ‪i (t‬‬ ‫‪t‬‬

‫= ] ‪y [n‬‬

‫ن(‬

‫ﺑﺎ ﺣﺎﻓﻈﻪ‬

‫∞‪= −‬‬

‫‪1‬‬ ‫)]‪(x [n ] + x [n − 1] + x [n − 2‬‬ ‫‪3‬‬ ‫ج(‬

‫‪∫ v L (λ) dλ‬‬

‫) ‪di l (t‬‬ ‫و( ‪dt‬‬

‫= ] ‪y [n‬‬

‫∞‪−‬‬

‫‪1‬‬

‫‪L‬‬

‫ﺑﺪون ﺣﺎﻓﻈﻪ‬

‫= ) ‪i L (t‬‬

‫‪v L (t ) = L‬‬

‫ﺑﺎﺣﺎﻓﻈﻪ‬ ‫ﺑﺎ ﺣﺎﻓﻈﻪ‬

‫ﺑﺎ ﺣﺎﻓﻈﻪ‬

‫‪ -٢-١١-١‬ﭘﺎﻳﺪاري )‪(BIBO1‬‬ ‫ﺳﻴﺴﺘﻤﻲ ﭘﺎﻳﺪار اﺳﺖ آﻪ ﺑﻪ ازاي ورودي ﻣﺤﺪود ﺧﺮوﺟﻲ ﻣﺤﺪود ﺑﺪهﺪ‪.‬‬

‫‪x (t ) ≤ M x < ∞ , ∀t ⇒ y (t ) ≤ M y < ∞ , ∀t‬‬ ‫ﺗﺬآﺮ‪ :١‬اﮔﺮ ﺳﻴﺴﺘﻤﻲ ﺑﺎ ورودي ﻣﺤﺪود و ﺧﺮوﺟﻲ ﻧﺎﻣﺤﺪود داﺷﺘﻴﻢ اﻳﻦ ﺳﻴﺴﺘﻢ ﻧﺎﭘﺎﻳﺪار اﺳﺖ‪.‬‬ ‫ﺗﺬآﺮ ‪ :٢‬اﮔﺮ ﺳﻴﺴﺘﻤﻲ ﺑﺎ ورودي ﻧﺎﻣﺤﺪود و ﺧﺮوﺟﻲ ﻧﺎﻣﺤﺪود داﺷﺘﻴﻢ هﻴﭻ ﺻﺤﺒﺘﻲ ﻧﻤﻲﺗﻮان راﺟﻊ ﺑﻪ اﻳﻦ ﺳﻴﺴﺘﻢ آﺮد‪.‬‬

‫‪Bounded Input Bounded Output‬‬

‫‪٢٣‬‬

‫‪1‬‬

‫ﻣﺜﺎل ‪y (t ) = t x (t ) (٢‬‬ ‫) ‪ x (t ) = u (t‬را در ﻧﻈﺮ ﻣﯽﮔﻴﺮﻳﻢ‪:‬‬ ‫) ‪ y (t ) = t u (t ) = r (t‬و ) ‪ r (t‬ﺗﺎﺑﻌﻲ ﻧﺎﻣﺤﺪود اﺳﺖ ﭘﺲ ﺳﻴﺴﺘﻢ ﻧﺎﭘﺎﻳﺪار اﺳﺖ‪.‬‬

‫ﻣﺜﺎل ‪y (t ) = e x (t ) (٣‬‬ ‫∞ < ‪y (t ) = e x (t ) = e x (t ) ≤ M y‬‬ ‫) ‪x (t‬‬ ‫‪ e‬ﻋﺪد ﻣﻲﺷﻮد ﭘﺲ ﺑﺎز هﻢ ﻣﺤﺪود اﺳﺖ‪ .‬ﺑﻨﺎﺑﺮاﻳﻦ ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺮﻳﻒ رﻳﺎﺿﻲ اﻳﻦ ﺳﻴﺴﺘﻢ ﭘﺎﻳﺪار اﺳﺖ‪.‬‬ ‫ﭼﻮن‬

‫ﻣﺜﺎل ‪(٤‬‬

‫‪1‬‬ ‫)]‪(x [n ] + x [n − 1] + x [n − 2‬‬ ‫‪3‬‬

‫= ] ‪y [n‬‬

‫اﮔﺮ ] ‪ x [n‬در هﻤﻪ ﻟﺤﻈﺎت ﻣﺤﺪود ﺑﺎﺷﺪ ﺷﻴﻔﺖ ﻳﺎﻓﺘﻪهﺎي ﺁن هﻢ ﻣﺤﺪود اﺳﺖ ﭘﺲ ﺳﻴﺴﺘﻢ ﭘﺎﻳﺪار اﺳﺖ‬ ‫ﻣﺜﺎل ‪(٥‬‬ ‫ﻧﺎﭘﺎﻳﺪار ⇒ ∞ → ] ‪∞ : x [n ] = u [n ] → y [n‬‬ ‫→‬ ‫‪n‬‬ ‫‪n‬‬ ‫∞→‬

‫∞→‬

‫ﭘﺎﻳﺪار ⇒ ‪0‬‬ ‫→‬ ‫‪n‬‬ ‫∞→‬

‫‪r > 1 ⇒ r n‬‬ ‫‪‬‬ ‫‪x [n ]‬‬ ‫‪n‬‬ ‫‪r < 1 ⇒ r‬‬ ‫‪‬‬

‫‪n‬‬

‫‪y [n ] = r‬‬

‫ﻣﺜﺎل ‪(۶‬‬ ‫‪n‬‬

‫] ‪∑ u [k ] = r [n‬‬

‫∞‪k = −‬‬

‫‪n‬‬

‫= ] ‪∑ x [k ] → x [n ] = u [n ] → y [n‬‬

‫= ] ‪y [n‬‬

‫∞‪k = −‬‬

‫ﻧﺎﭘﺎﻳﺪار‬

‫‪ -٣-١١-١‬ﻋﻠﻴﺖ )‪(causality‬‬ ‫ﺳﻴﺴﺘﻤﻲ ﻋﻠﻲ اﺳﺖ آﻪ ﺧﺮوﺟﻲ در هﺮ ﻟﺤﻈﻪ ﺑﻪ ورودي در هﻤﺎن ﻟﺤﻈﻪ ﻳﺎ ﻟﺤﻈﺎت ﻗﺒﻞ ﺑﺴﺘﮕﻲ داﺷﺘﻪ ﺑﺎﺷﺪ ﺑﻪ ﻋﺒﺎرﺗﻲ ﺧﺮوﺟﻲ‬ ‫ﺳﻴﺴﺘﻢ ﺑﻪ ْﺁﻳﻨﺪﻩ ورودي ﺑﺴﺘﮕﻲ ﻧﺪارد‪.‬‬ ‫ﺗﺬآﺮ ‪ :١‬ﺳﻴﺴﺘﻢ ﺑﺪون ﺣﺎﻓﻈﻪ ﻣﻄﻤﺌﻨﺎ ﻋﻠﻲ اﺳﺖ‪.‬‬ ‫ﺗﺬآﺮ ‪ :٢‬ﺑﺮاي ﺳﻴﺴﺘﻢ ﻋﻠﻲ ﺷﺮط ﺳﻜﻮن ﺑﺮﻗﺮار اﺳﺖ‪.‬‬ ‫‪:‬ﺷﺮط ﺳﻜﻮن‬

‫‪t ≤ t 0 ⇒ y (t ) = 0; t ≤ t 0‬‬

‫;‪x (t ) = 0‬‬ ‫‪٢٤‬‬

‫‪t ≤ t0‬‬

‫;) ‪x 1 (t ) = x 2 (t‬‬

‫;) ‪t ≤ t 0 ⇒ y 1 (t ) = y 2 (t‬‬

‫ﻣﺜﺎل ‪(٧‬‬ ‫‪t‬‬

‫‪∫ i c (τ ) dτ‬‬

‫ﻋﻠﻲ‬

‫∞‪−‬‬

‫‪1‬‬

‫‪c‬‬

‫= ) ‪v c (t‬‬

‫ب(‬

‫)‪y (t ) = x (t + 1‬‬

‫ﻏﻴﺮﻋﻠﻲ‬

‫) ‪v c (t ) − v c (t − ∆t‬‬ ‫‪‬‬ ‫‪lim‬‬ ‫‪‬‬ ‫‪dv c  ∆t →0‬‬ ‫‪∆t‬‬ ‫‪=‬‬ ‫‪dt‬‬ ‫) ‪ lim v c (t + ∆t ) − v c (t‬‬ ‫‪∆t →0‬‬ ‫‪∆t‬‬

‫;‬

‫) ‪dv c (t‬‬ ‫‪dt‬‬

‫اﻟﻒ(‬

‫= ) ‪i c (t‬‬ ‫ج(‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ اﻳﻦ آﻪ ﺑﺮاي ﻣﺸﺘﻖ دو ﺗﻌﺮﻳﻒ وﺟﻮد دارد ﭼﻨﺎﻧﭽﻪ از دوﻣﻴﻦ ﺗﻌﺮﻳﻒ ﺑﺮاي ﺗﻌﻴﻴﻦ ﺟﺮﻳﺎن ﺧﺎزن اﺳﺘﻔﺎدﻩ ﻧﻤﺎﺋﻴﻢ ﺳﻴﺴﺘﻢ‬ ‫ﻣﺬآﻮر ﻏﻴﺮﻋﻠﻲ ﺧﻮاهﺪ ﺑﻮد‪.‬‬

‫)‪y (t ) = x (t ) cos(t + 1‬‬

‫د(‬

‫ﺧﺮوﺟﻲ ﻓﻘﻂ ﺑﻪ زﻣﺎن ﺣﺎل ورودي ﺑﺴﺘﮕﻲ دارد و ازﻃﺮﻓﻲ ﭼﻮن ﺳﻴﺴﺘﻢ ﺑﺪون ﺣﺎﻓﻈﻪ اﺳﺖ ﭘﺲ ﺣﺘﻤﺎ ﻋﻠﻲ اﺳﺖ‪.‬‬ ‫‪n‬‬

‫‪∑ x [k ] = x [n ] + x [n − 1] + ...‬‬

‫ﻋﻠﯽ‬

‫∞‪k = −‬‬

‫]‪n = −1 ⇒ y [−1] = x [1‬‬

‫ﻏﻴﺮ ﻋﻠﯽ‬

‫] ‪y [n ] = x [−n‬‬

‫ﻋﻠﯽ‬

‫‪1‬‬ ‫)]‪( x [n ] + x [n − 1] + x [n − 2‬‬ ‫‪3‬‬

‫= ] ‪y [n‬‬

‫ﻏﻴﺮ ﻋﻠﯽ‬

‫‪1‬‬ ‫)]‪(x [n + 1] + x [n ] + x [n − 1‬‬ ‫‪3‬‬

‫= ] ‪y [n‬‬

‫‪ -٤-١١-١‬ﻣﻌﻜﻮسﭘﺬﻳﺮي‬ ‫ﺳﻴﺴﺘﻤﻲ ﻣﻌﻜﻮسﭘﺬﻳﺮ اﺳﺖ آﻪ ﺑﻪ ازاي وروديهﺎي ﻣﺘﻤﺎﻳﺰ هﻤﻮارﻩ ﺧﺮوﺟﻲهﺎي ﻣﺘﻤﺎﻳﺮ داﺷﺘﻪ ﺑﺎﺷﺪ‪.‬‬ ‫اﮔﺮ ﺳﻴﺴﺘﻤﻲ ﻣﻌﻜﻮسﭘﺬﻳﺮ ﺑﺎﺷﺪ ﻣﻲﺗﻮان ﺑﻪ ﺻﻮرت زﻳﺮ ﻧﺸﺎن داد‪:‬‬

‫) ‪x (t‬‬

‫‪sys‬‬

‫‪sys −1‬‬

‫) ‪x (t‬‬

‫ﻣﺜﺎل‪:٤‬‬ ‫اﻟﻒ(‬

‫) ‪x (t‬‬

‫) ‪y (t ) = 2x (t‬‬

‫‪1‬‬ ‫‪2‬‬

‫×‬

‫= ] ‪y [n‬‬

‫ﻣﻌﻜﻮسﭘﺬﻳﺮ‬

‫‪×2‬‬

‫) ‪x (t‬‬

‫‪٢٥‬‬

‫ﻩ(‬ ‫ن(‬ ‫و(‬ ‫ی(‬

‫‪x (t ) = 1‬‬ ‫‪‬‬ ‫‪⇒ y (t ) = 1‬‬ ‫‪‬‬ ‫‪x (t ) = −1‬‬ ‫‪‬‬

‫ب(‬

‫‪t‬‬

‫‪1‬‬

‫‪∫ x ( λ) dλ‬‬

‫ج(‬

‫‪L‬‬

‫∞‪−‬‬

‫) ‪y (t ) = x (t‬‬ ‫‪2‬‬

‫ﻣﻌﻜﻮس ﻧﺎﭘﺬﻳﺮ‬

‫= ) ‪y (t‬‬ ‫ﺑﺎﻳﺪ ورودي ﺑﺪهﻴﻢ آﻪ ﺧﺮوﺟﻲ را از ﺣﺎﻟﺖ ﭘﺎﻳﺪار ﺧﺎرج ﻧﻜﻨﺪ‪.‬‬ ‫‪t‬‬

‫‪d‬‬ ‫‪dt‬‬

‫) ‪x (t‬‬

‫) ‪dx (t‬‬ ‫‪dt‬‬

‫ن(‬

‫‪∫ x ( λ)dλ‬‬

‫‪L‬‬

‫‪1‬‬

‫‪L‬‬

‫∫‬

‫∞‪−‬‬

‫) ‪x (t‬‬

‫= ) ‪y (t‬‬

‫ﭼﻮن ﺑﻪ ازاي هﻤﻪ ورديهﺎي ﺛﺎﺑﺖ ﺧﺮوﺟﻲ ﺻﻔﺮ ﻣﻲﺷﻮد ﭘﺲ ﻣﻌﻜﻮسﭘﺬﻳﺮ ﻧﻴﺴﺖ‪.‬‬ ‫‪n‬‬

‫] ‪∑ x [k‬‬ ‫‪k‬‬

‫و(‬

‫= ] ‪y [n‬‬

‫∞‪= −‬‬

‫‪n −1‬‬

‫] ‪⇒ y [n ] − y [n − 1] = x [n‬‬

‫] ‪x [n‬‬

‫‪−‬‬

‫] ‪y [n‬‬

‫∑‬

‫] ‪∑ x [k‬‬

‫= ]‪y [n − 1‬‬

‫∞‪k = −‬‬

‫‪sys‬‬

‫] ‪x [n‬‬

‫]‪Delay y [n − 1‬‬

‫‪ -٥-١١-١‬ﻧﺎﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن )‪Time invariant (TI‬‬ ‫اﮔﺮ رﻓﺘﺎر و ﻣﺸﺨﺼﻪهﺎي ﺳﻴﺴﺘﻢ در ﻃﻲ زﻣﺎن ﺛﺎﺑﺖ ﺑﺎﺷﺪ‪ .‬ﺑﻪ ﺁن ﺳﻴﺴﺘﻢ‪ ،‬ﻧﺎﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن ﮔﻔﺘﻪ ﻣﻲﺷﻮد‪.‬‬

‫‪sys‬‬

‫) ‪y (t‬‬ ‫‪sys‬‬

‫) ‪y (t − t 0‬‬

‫) ‪x (t‬‬

‫) ‪x (t − t 0‬‬

‫ﻣﺜﺎل ‪(٥‬‬ ‫‪t‬‬

‫اﻟﻒ(‬

‫‪∫ x (τ ) d τ‬‬

‫∞‪−‬‬

‫‪t‬‬

‫‪∫ x (τ )dτ‬‬

‫∞‪−‬‬

‫‪1‬‬

‫‪L‬‬

‫‪1‬‬

‫‪L‬‬

‫= ) ‪y (t‬‬

‫= ) ‪y (t‬‬

‫‪sys 1‬‬

‫) ‪x (t‬‬ ‫‪٢٦‬‬

‫ﺑﺎ ﺗﻮﺟﻪ ﺑﻪ ﺗﻌﺮﻳﻒ اﮔﺮ ﺷﻴﻔﺖ ﻳﺎﻓﺘﻪ ورودي را ﺑﻪ ﺳﻴﺴﺘﻢ اﻋﻤﺎل آﻨﻴﻢ ﺧﺮوﺟﻲ هﻢ ﺑﻪ هﻤﺎن اﻧﺪازﻩ ﺷﻴﻔﺖ ﭘﻴﺪا آﻨﺪ‪ ،‬ﺳﻴﺴﺘﻢ ﻣﻮرد ﻧﻈﺮ‬ ‫ﻧﺎﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن ﺧﻮاهﺪ ﺑﻮد‪.‬‬ ‫‪t‬‬

‫‪1‬‬

‫‪∫ z (τ )dτ‬‬

‫‪L‬‬

‫∞‪−‬‬

‫= ﺧﺮوﺟﻲ‬ ‫‪t −t 0‬‬

‫‪∫ x (τ ) dτ‬‬

‫∞‪−‬‬

‫‪1‬‬

‫‪L‬‬

‫‪sys 1‬‬

‫=‬

‫‪t −t 0 = λ‬‬

‫) ‪z (t‬‬ ‫‪t‬‬

‫‪∫ x (t‬‬

‫‪− t 0 ) dτ‬‬

‫∞‪−‬‬

‫‪1‬‬

‫‪L‬‬

‫‪t‬‬

‫=‬

‫‪∫ z (τ ) dτ‬‬

‫∞‪−‬‬

‫‪1‬‬

‫‪L‬‬

‫= ﺧﺮوﺟﻲ‬

‫)‪−t0‬‬ ‫ﻧﺎﻣﺘﻐﻴﺮﺑﺎزﻣﺎن‬

‫ب(‬

‫‪1‬‬ ‫) ‪x (t‬‬ ‫) ‪R (t‬‬

‫= ) ‪y (t‬‬

‫‪1‬‬ ‫) ‪x (t‬‬ ‫) ‪R (t‬‬ ‫‪1‬‬ ‫) ‪z (t‬‬ ‫) ‪R (t‬‬

‫= ﺧﺮوﺟﻲ‬

‫) ‪x (t‬‬

‫‪sys 2‬‬

‫) ‪z (t‬‬

‫‪sys 2‬‬

‫?‬ ‫‪1‬‬ ‫) ‪x (t − t 0 ) = y (t − t 0‬‬ ‫) ‪R (t‬‬

‫= ﺧﺮوﺟﻲ ⇒ ) ‪z (t ) = x (t − t 0‬‬

‫‪1‬‬ ‫) ‪x (t − t 0‬‬ ‫) ‪R (t − t 0‬‬

‫ﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن‬

‫?‬

‫=‬

‫) ‪y (t ) = t x (t‬‬

‫) ‪y (t ) = t x (t‬‬

‫‪sys 3‬‬

‫) ‪ = t z (t‬ﺧﺮوﺟﻲ‬

‫) ‪x (t‬‬ ‫) ‪z (t‬‬

‫‪sys 3‬‬

‫?‬

‫) ‪ = t x (t − t 0 ) = y (t − t 0‬ﺧﺮوﺟﻲ ⇒ ) ‪z (t ) = x (t − t 0‬‬ ‫?‬

‫) ‪=(t − t 0 )x (t − t 0‬‬ ‫د(‬

‫‪t −t 0‬‬

‫‪∫ x (λ) dλ = y (t‬‬

‫∞‪−‬‬

‫= ) ‪y (t‬‬

‫ج(‬

‫?‬

‫⇒ ) ‪z (t ) = x (t − t 0‬‬

‫ﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن‬

‫) ‪y (t ) = x (at‬‬

‫) ‪y (t ) = x (at‬‬

‫) ‪ = z (at‬ﺧﺮوﺟﻲ‬

‫‪sys 4‬‬ ‫‪sys 4‬‬

‫) ‪x (t‬‬ ‫) ‪z (t‬‬ ‫‪٢٧‬‬

‫‪1‬‬

‫‪L‬‬

‫= ﺧﺮوﺟﻲ‬

‫?‬

‫) ‪ = x (at − t 0 ) = y (t − t 0‬ﺧﺮوﺟﻲ ⇒ ) ‪z (t ) = x (t − t 0‬‬ ‫?‬

‫)) ‪= x (a (t − t 0‬‬

‫ن(‬

‫ﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن‬

‫] ‪y [n ] = r n x [n‬‬

‫] ‪y [n ] = r n x [n‬‬

‫] ‪x [n‬‬

‫‪sys 5‬‬

‫] ‪ = r n z [n‬ﺧﺮوﺟﻲ‬

‫] ‪z [n‬‬

‫‪sys 5‬‬

‫?‬

‫] ‪ = r n x [n − n 0 ] = y [n − n 0‬ﺧﺮوﺟﻲ ⇒ ] ‪z [n ] = x [n − n 0‬‬ ‫?‬

‫] ‪= r n −n 0 x [n − n 0‬‬

‫ﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن‬

‫] ‪y [n ] = nx [n‬‬

‫ﻩ(‬

‫] ‪y [n ] = n x [n‬‬

‫] ‪x [n‬‬

‫‪sys 6‬‬

‫] ‪ = n z [n‬ﺧﺮوﺟﻲ‬

‫] ‪z [n‬‬

‫‪sys 6‬‬

‫?‬

‫] ‪ = n x [n − n 0 ] = y [n − n 0‬ﺧﺮوﺟﻲ ⇒ ] ‪z [n ] = x [n − n 0‬‬ ‫?‬

‫] ‪=(n − n 0 )x [n − n 0‬‬

‫ﻣﺘﻐﻴﺮ ﺑﺎ زﻣﺎن‬

‫‪ -٦-١١-١‬ﺧﻄﻲ ﺑﻮدن‬ ‫ﺳﻴﺴﺘﻢ ﺧﻄﻲ ﺑﻪ ﺳﻴﺴﺘﻤﻲ ﮔﻔﺘﻪ ﻣﻲﺷﻮد آﻪ اﺻﻞ ﺟﻤﻊ ﺁﺛﺎر ﺑﺮاي ﺁن ﺻﺪق آﻨﺪ‪.‬‬ ‫ﺷﺮط ﺧﻄﻲ ﺑﻮدن‪:‬‬

‫) ‪y (t‬‬ ‫ﺟﻤﻊ ﺁﺛﺎر ;‬

‫‪sys‬‬

‫) ‪y 1 (t ) + y 2 (t‬‬ ‫هﻤﮕﻨﻲ;‬

‫) ‪ay (t‬‬

‫) ‪1) x 1 (t ) + x 2 (t‬‬ ‫‪⇒ ‬‬ ‫) ‪2) ax (t‬‬ ‫‪sys‬‬ ‫‪‬‬

‫) ‪y 1 (t‬‬

‫) ‪y 2 (t‬‬ ‫ﺗﺬآﺮ‪ :‬در ﺑﺮﻗﺮاري هﻤﮕﻨﻲ ﺿﺮﻳﺐ ﺛﺎﺑﺖ ‪ a‬هﺮ ﻋﺪدي ﻣﻲﺗﻮاﻧﺪ ﺑﺎﺷﺪ‪ ،‬ﺣﺘﻲ ﻋﺪد ﻣﺨﺘﻠﻂ‪.‬‬ ‫ﻣﺜﺎل ‪(٦‬‬ ‫‪٢٨‬‬

‫‪sys‬‬

‫‪sys‬‬

‫‪sys‬‬

‫) ‪ x (t‬‬ ‫‪‬‬ ‫‪‬‬ ‫) ‪x (t‬‬ ‫‪ 1‬‬ ‫‪‬‬ ‫‪‬‬ ‫) ‪x (t‬‬ ‫‪ 2‬‬

y (t ) = x 2 (t )

‫ﻏﻴﺮﺧﻄﻲ‬

(x 1 + x 2 )

sys

ax (t )

sys

(x 1 + x 2 ) 2 ≠ x 12 + x 22 y (t ) = x (t )x (t − 1)

‫ﻏﻴﺮﺧﻄﻲ‬

(‫اﻟﻒ‬

(‫ب‬

(ax (t ))(ax (t − 1)) = a 2 x (t )x (t − 1) ≠ ax (t )x (t − 1)

٣) y (t ) = sin( x (t ))

.‫ ﻏﻴﺮﺧﻄﻲ هﺴﺘﻨﺪ‬sin, cos ‫ﻏﻴﺮﺧﻄﻲ اﺳﺖ ﭼﻮن ﺗﻮاﺑﻊ‬

1 0

y (t ) = 

x τ0 x <0

(‫ج‬

x (t ) = 1 → y (t ) = 1 .‫ ﺧﺎﺻﻴﺖ هﻤﮕﻨﻲ ﻧﻘﺾ ﺷﺪﻩ ﭘﺲ ﻏﻴﺮﺧﻄﻲ اﺳﺖ‬a x (t ) = a → y (t ) = 1

y [n ] = nx [n ] x 1 [n ] + x 2 [n ] → n (x 1 [n ] + x 2 [n ]) = n x 1 [n ] + n x 2 [n ] = y 1 [n ] + y 2 [n ] a x [n ] → n (a x [n ]) = a (n x [n ]) = a y [n ] y [n ] = Real (x [n ])

‫ﻏﻴﺮﺧﻄﻲ‬

(‫د‬

‫ﺧﻄﻲ‬

(‫ﻩ‬

x [n ] = r + js → y [n ] = r a x [n ] = a (r + js ) ⇒ if a = j 2 ⇒ −2S ≠ ay [n ] y [n ] = 2x [n ] + 3

x 1 [n ] + x 2 [n ] ⇒ 2(x 1 [n ] + x 2 [n ]) + 3 ≠ (2x 1 [n ] + 3) + (2x 2 [n ] + 3) ≠ y 1 [n ] + y 2 [n ]

٢٩

(‫ن‬ ‫ﻏﻴﺮﺧﻄﻲ‬

‫ﻧﻤﻮﻧﻪ ﻣﺴﺎﺋﻞ ﻓﺼﻞ‬ ‫‪ -١‬ﺗﻌﻴﻴﻦ آﻨﻴﺪ آﺪام ﻳﻚ از ﺳﻴﮕﻨﺎل هﺎي زﻳﺮ ﻣﺘﻨﺎوﺑﻨﺪ؟‬ ‫اﻟﻒ(‬

‫] ‪x 2 [n ] = u [n ] + u [−n‬‬

‫ﻳﻚ راﻩ ﺗﺸﺨﻴﺺ ﻣﺘﻨﺎوب ﺑﻮدن رﺳﻢ ﺳﻴﮕﻨﺎل اﺳﺖ‪.‬‬

‫] ‪x 2 [n‬‬ ‫‪n‬‬

‫‪1‬‬

‫‪2‬‬

‫وﺟﻮد ﻳﻚ ﻧﺎﭘﻴﻮﺳﺘﮕﻲ در ﻧﻘﻄﻪ ﺻﻔﺮ ﺑﺎﻋﺚ ﻣﻲﺷﻮد آﻪ ﻧﺎﻣﺘﻨﺎوب ﺷﻮد‪.‬‬

‫ب(‬

‫] ‪x 3 [n ] = ∑ δ [n − 4k ] − δ [n − 1 − 4k‬‬

‫در هﺮ دورﻩ ﺗﻨﺎوب ‪ ٢ ، N = 4‬ﺿﺮﺑﻪ را ﺷﺎﻣﻞ ﺷﺪﻩ اﺳﺖ‪.‬‬

‫] ‪x 3 [n‬‬ ‫‪1‬‬

‫‪n‬‬

‫‪5‬‬

‫‪−3‬‬

‫‪4‬‬

‫‪−1‬‬

‫ج(‬

‫‪2π‬‬ ‫‪n‬‬ ‫‪7‬‬

‫‪⇒ N = 35‬‬

‫‪−j‬‬

‫‪−e‬‬

‫‪4π‬‬ ‫‪n‬‬ ‫‪7‬‬

‫‪j‬‬

‫‪−4‬‬

‫‪x [n ] = 1 + e‬‬

‫‪2kπ‬‬ ‫)‪= 5k = 5 , (k = 1‬‬ ‫‪2π‬‬ ‫‪5‬‬

‫= ‪N2‬‬

‫‪,‬‬

‫‪2kπ‬‬ ‫)‪= 7k = 7 (k = 1‬‬ ‫‪4π‬‬ ‫‪7‬‬

‫‪ -٢‬ﺗﻌﻴﻴﻦ آﻨﻴﺪ آﺪام ﻳﻚ از ﺳﻴﮕﻨﺎلهﺎي زﻳﺮ ﻣﻌﻜﻮسﭘﺬﻳﺮ اﺳﺖ؟‬ ‫اﻟﻒ(‬

‫]‪y [n ] = x [n ] x [n − 2‬‬

‫]‪x [n ] = Aδ [n ] , y [n ] = A 2δ [n ] δ [n − 2] = 0 , x [n ] = δ [n − 3‬‬ ‫‪⇒ y [n ] = δ [n − 3] δ [n − 5] = 0‬‬ ‫ﺣﺎﻓﻈﻪدار اﺳﺖ وﻟﻲ ﻣﻌﻜﻮسﭘﺬﻳﺮ ﻧﻴﺴﺖ‪.‬‬

‫ب(‬

‫‪n τ0‬‬ ‫‪n ≤ −1‬‬

‫]‪x [n + 1‬‬ ‫] ‪x [n‬‬

‫‪y [n ] = ‬‬

‫‪= x [n + 1] u [n ] + x [n ] u [−n − 1] ,‬‬

‫‪٣٠‬‬

‫= ‪N1‬‬

‫] ‪if x [n ] = u [n ] ⇒ y [n ] = u [n + 1] u [n ] + u [n ] u [−n − 1] = u [n‬‬ ‫‪n1τ44‬‬ ‫‪n4‬‬ ‫‪0 n1τ44‬‬ ‫‪02‬‬ ‫‪, n4≤4‬‬ ‫‪−12, 4‬‬ ‫‪τ3‬‬ ‫‪−‬‬ ‫‪31‬‬ ‫] ‪=u [n‬‬

‫‪=0‬‬

‫‪if x [n ] = u [n − 1] ⇒ y [n ] = u1[4‬‬ ‫‪n2‬‬ ‫] ‪] u [n ] + u [n − 1] u [−n − 1] = u [n‬‬ ‫‪43 14442444‬‬ ‫‪3‬‬ ‫‪n1τ4‬‬ ‫‪1,n ≤ −1‬‬ ‫‪243‬‬ ‫‪=0‬‬

‫‪n1τ402‬‬ ‫‪,n τ 0‬‬ ‫‪43‬‬ ‫] ‪u [n‬‬

‫ﻣﻌﻜﻮسﭘﺬﻳﺮ ﻧﻴﺴﺖ‪.‬‬ ‫ﺗﻤﺮﻳﻦ( ﺁﻳﺎ اﻳﻦ ﺳﻴﺴﺘﻢ ﻣﻌﻜﻮسﭘﺬﻳﺮ اﺳﺖ ﻳﺎ ﻧﻪ؟ اﮔﺮ هﺴﺖ ﻣﻌﻜﻮﺳﺶ را ﺑﻪ دﺳﺖ ﺁورﻳﺪ‪.‬‬

‫‪n τ1‬‬ ‫‪n =0‬‬ ‫‪n ≤ −1‬‬

‫]‪x [n − 1‬‬ ‫‪‬‬ ‫‪y [n ] = 0‬‬ ‫] ‪x [n‬‬ ‫‪‬‬

‫‪ -١٢-١‬ﺧﻼﺻﻪ‬ ‫ﺳﻴﮕﻨﺎل ﺗﺎﺑﻌﻲ اﺳﺖ آﻪ ﺣﺎوي اﻃﻼﻋﺎﺗﻲ درﺑﺎرﻩ رﻓﺘﺎر ﻓﻴﺰﻳﻜﻲ ﻳﻚ ﺳﻴﺴﺘﻢ اﺳﺖ‪.‬‬ ‫هﺮ ﺳﻴﮕﻨﺎل دﻟﺨﻮاﻩ را ﻣﻲﺗﻮان ﺑﻪ ﺻﻮرت ﻣﺠﻤﻮع دو ﺳﻴﮕﻨﺎل زوج و ﻓﺮد ﻧﻮﺷﺖ‪.‬‬ ‫در ﺑﺴﻴﺎري از ﻣﺴﺎﺋﻞ ﺑﺎ اﺳﺘﻔﺎدﻩ از ﺷﻴﻔﺖ زﻣﺎﻧﻲ ﻣﻲﺗﻮان ﺳﻴﮕﻨﺎل را ﺑﻪ ﺻﻮرت زوج ﻳﺎ ﻓﺮد درﺁورد‪.‬‬ ‫آﻮﭼﻜﺘﺮﻳﻦ دورﻩ ﺗﻨﺎوب ﺳﻴﮕﻨﺎل دورﻩ ﺗﻨﺎوب اﺻﻠﻲ اﺳﺖ و ﻓﺮآﺎﻧﺲ ﻣﺘﻨﺎﺳﺐ ﺑﺎ دورﻩ ﺗﻨﺎوب اﺻﻠﻲ ﻓﺮآﺎﻧﺲ اﺻﻠﻲ اﺳﺖ‪.‬‬ ‫ﺗﻮاﺑﻌﻲ آﻪ ﻓﺮم هﻨﺪﺳﻲ دارﻧﺪ را ﻣﻲﺗﻮان ﺑﺮ ﺣﺴﺐ ﺗﻮاﺑﻊ وﻳﮋﻩ ﺑﻴﺎن آﺮد‪.‬‬ ‫ﺗﻮاﺑﻊ ﺿﺮﺑﻪ واﺣﺪ‪ ،‬ﭘﻠﻪ واﺣﺪ و دوﺑﻠﺖ واﺣﺪ ﺧﻮاص ﻣﻨﺤﺼﺮ ﺑﻪﻓﺮدي دارﻧﺪ‪.‬‬ ‫اﮔﺮ ﺳﻴﺴﺘﻤﻲ ﺑﺎ ورودي ﻣﺤﺪود و ﺧﺮوﺟﻲ ﻧﺎﻣﺤﺪود داﺷﺘﻴﻢ اﻳﻦ ﺳﻴﺴﺘﻢ ﻧﺎﭘﺎﻳﺪار اﺳﺖ‪.‬‬ ‫ﺳﻴﺴﺘﻢ ﺑﺪون ﺣﺎﻓﻈﻪ ﻣﻄﻤﺌﻨﺎ ﻋﻠﻲ اﺳﺖ‪.‬‬

‫‪٣١‬‬