X t 2p sin t t pt t 2 15 1 st 05 0 05 1 15 2 0 2 4 6

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Unformatted text preview: chigan) Fall 2012 September 6, 2012 10 / 130 Lecture Notes 5 Bandwidth Spectrum for Rectangular Pulses 0 −5 −10 SX(f) (dB) −15 −20 −25 −30 −35 −40 −8 −6 −4 −2 0 2 4 6 8 frequency EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 11 / 130 Lecture Notes 5 Bandwidth Example 2: Half Cosine/Sine Pulses Consider a pulse that consists of a half sinusoid. √ x (t ) = 2P sin(π t /T )pT (t ) 2 1.5 1 s(t) 0.5 0 −0.5 −1 −1.5 −2 0 2 4 6 8 10 time (s) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 12 / 130 Lecture Notes 5 Bandwidth Spectrum for Half Cosine Pulses The half cosine pulse is x (t ) = transform is given by X (f ) = Z∞ √ −∞ = = = = = = EECS 455 (Univ. of Michigan) 2P sin(π t /T )pT (t ) The Fourier 2P sin(π t /T )pT (t )e Z∞ √ 2P −∞ = √ 2j √ ZT 2P 2j 0 √ ZT 2P (e (e j π t /T j π t /T −e −e −j 2π ft −j π t / T −j π t / T j π t /T (1−2fT ) (e 2j 0 " j π t /T (1−2fT ) √ e 2P 2j j π (1 − 2fT )/T " √ 2P ej π (1−2fT ) − 1 −e + e dt ) pT ( t ) e )e −j 2π ft −j 2π ft dt dt −j π t /T (1+2fT ) )dt # −j π t /T (1+2fT ) T j π (1 + 2fT )/T e−j π (1+2fT ) − 1 0 # + 2j j π (1 − 2fT )/T j π (1 + 2fT )/T " # √ −j 2π fT −1 2PT −e −e−j 2π fT − 1 + 2j j π (1 − 2fT ) j π (1 + 2fT ) # " √ 2PT 1 1 −j 2π fT (1 + e ) + 2π (1 + 2fT ) (1 − 2fT ) Fall 2012 September 6, 2012 13 / 130 Lecture Notes 5 Bandwidth Spectrum for Half Cosine Pulses X (f ) = = = = = |X ( f ) | 2 SY (f ) = = = EECS 455 (Univ. of Michigan) √ 2PT 2π √ 2PT (1 + e (1 + e −j 2π fT −j 2π fT ) " ) " 2 ( 1 − 4f 2 T 2 ) 2 ( 1 − 4f 2 T 2 ) # # 2π # " √ 2 2PT −j 2π fT /2 j 2π fT /2 −j 2π fT /2 e (e +e ) 2π ( 1 − 4f 2 T 2 ) " # √ 2 2PT −j 2π fT /2 e (2 cos(2π fT /2)) 2T 2) 2π ( 1 − 4f # " √ 1 2 2PT −j 2π fT /2 e (cos(2π fT /2)) π ( 1 − 4f 2 T 2 ) " # 8PT 2 cos2 (2π fT /2) (π 2 ) 1 T ( 1 − 4f 2 T 2 ) 2 |X ( f ) | 8PT (π 2 ) " 2 cos2 (2π fT /2) ( 1 − 4f 2 T 2 ) 2 # Fall 2012 September 6, 2012 14 / 130 Lecture Notes 5 Bandwidth Summary for Half Cosine Pulses x (t ) = X (f ) = √ 2P sin(π t /T )pT (t ) 8PT 2 cos2 (π fT ) π 2 (1−4f 2 T 2 )2 The energy of the pulse is E = x 2 (t )dt = |X (f )|2 df = PT The power spectral density of the pulse is SY (f ) = The power SY (f )df = P 8PT cos2 (π fT ) . π 2 (1−4f 2 T 2 )2 Notice that the spectrum falls of as 1/f 4 rather than 1/f 2 . EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 15 / 130 Lecture Notes 5 Bandwidth Spectrum for Sine Pulses Power Spectral Density 1 Rectangular Pulse 0.8 0.6 0.4 Sinusoidal Pulse 0.2 0 −3 −2 −1 0 1 2 3 frequency (Hz) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 16 / 130 Lecture Notes 5 Bandwidth Spectrum for Sine Pulses 10 Power Spectral Density (dB) 5 0 Sinusoidal Pulse −5 −10 −15 Rectangular Pulse −20 −25 −30 −35 −40 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 frequency (Hz) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 17 / 130 Lecture Notes 5 Bandwidth Spectrum for Sine Pulses 10 Power Spectral Density 0 −10 Rectangular Pulse Sinusoidal Pulse −20 −30 −40 −50 −10 EECS 455 (Univ. of Michigan) −5 0 frequency (Hz) Fall 2012 5 10 September 6, 2012 18 / 130 Lecture Notes 5 Bandwidth Example 3: Square-Root Raised-Cosine Pulses x (t ) = X (f ) = √ PT sin(π (1 − α)t /T ) + 4αt /T cos(π (1 + α)t /T ) . π [1 − (4αt /T )2 ]t /T √ PT , T P 2 [1 |X (f )| = PT 2 SY (f ) = PT 2 2 EECS 455 (Univ. of Michigan) 2 − sin(π T (|f | − PT 2 , [1 − sin(π T (|f | − 0, PT , [1 − sin(π T (|f | − 0, 1 2T 1 2T 1 2T )/α)], 1−α 2T )/α)], − 0 ≤ |f | ≤ 12Tα 1−α 1+α 2T ≤ |f | ≤ 2T otherwise. )/α)], Fall 2012 1−α 2T + ≤ 12Tα 0 ≤ |f | ≤ ≤ |f | − 0 ≤ |f | ≤ 12Tα + ≤ |f | ≤ 12Tα otherwise. 1−α 2T September 6, 2012 19 / 130 Lecture Notes 5 Bandwidth Square-Root Raised-Cosine Pulses 1.4 α=0.7 1.2 1 x(t) 0.8 0.6 α=0.35 0.4 0.2 0 α=0.1 −0.2 −0.4 0 2 4 6 8 10 time (s) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 20 / 130 Lecture Notes 5 Bandwidth Spectrum for Square-Root Raised-Cosine Pulses 1.2 1 α=0.35 Y |S (f)| 0.8 0.6 0.4 α=0.1 0.2 0 −1.5 EECS 455 (Univ. of Michigan) −1 −0.5 0 0.5 frequency (Hz) Fall 2012 α=0.7 1 1.5 September 6, 2012 21 / 130 Lecture Notes 5 Bandwidth Spectrum for Square-Root Raised-Cosine Pulses 5 0 −5 −10 α=0.35 |SY(f)| (dB) −15 −20 −25 α=0.1 −30 α=0.7 −35 −40 −45 −50 −1.5 EECS 455 (Univ. of Michigan) −1 −0.5 0 0.5 frequency (Hz) Fall 2012 1 1.5 September 6, 2012 22 / 130 Lecture Notes 5 Band...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.

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