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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online