Xs t ys t 2 sin2 fc t ys t xs t 2 sin2

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 5 Up and Down Conversion Up and Down Conversion Re[Xc (f )], Im[Xc (f )] −W EECS 455 (Univ. of Michigan) f W Fall 2012 September 6, 2012 35 / 130 Lecture Notes 5 Up and Down Conversion Up and Down Conversion Im[Xc (f )] −W f W EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 36 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of yc (t ) Re[Yc (f )], Im[Yc (f )] EECS 455 (Univ. of Michigan) f fc −fc Fall 2012 September 6, 2012 37 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of yc (t ) Im[Yc (f )] −f c − W fc − W −f c + W −fc EECS 455 (Univ. of Michigan) fc + W f fc Fall 2012 September 6, 2012 38 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of ys (t ) √ Now consider multiplication by − 2 sin(2π fc t ). xs (t ) × ys (t ) √ − 2 sin(2π fc t ) √ ys (t ) = −xs (t ) 2 sin(2π fc t ) √ j2 Ys (f ) = [Xs (f − fc ) − Xs (f + fc )] 2 EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 39 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of ys (t ) √ Thus multiplication by − 2 sin(2π fc t ) shifts the spectrum up and down also except that the real part becomes the imaginary part and the imaginary part is inverted and becomes the real part in addition to a reduction by 1/2. Re[Xs (f )], Im[Xs (f )] −W EECS 455 (Univ. of Michigan) f W Fall 2012 September 6, 2012 40 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of ys (t ) Im[Xs (f )] −W f W EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 41 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of yc (t ) Re[Ys (f )], Im[Ys (f )] −fc − W fc + W −ffc + W −c EECS 455 (Univ. of Michigan) fc − W Fall 2012 f fc September 6, 2012 42 / 130 Lecture Notes 5 Up and Down Conversion Spectrum of yc (t ) Im[Ys (f )] −fc − Wfc + W − EECS 455 (Univ. of Michigan) f fc fc − Wfc + W −fc Fall 2012 September 6, 2012 43 / 130 Lecture Notes 5 Up and Down Conversion IQ Modulation Now consider adding these two functions together. √ 2 cos(2π fc t ) xc (t ) × yc (t ) + xs (t ) × y (t ) ys (t ) √ − 2 sin(2π fc t ) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 44 / 130 Lecture Notes 5 Up and Down Conversion IQ Modulation y (t ) = yc (t ) + ys (t ) √ √ = xc (t ) 2 cos(2π fc t ) − xs (t ) 2 sin(2π fc t ) = xe (t ) cos(2π fc t + θ (t )) The signal xe (t ) is called the envelope and θ (t ) is called the phase. Y (f ) = Yc (f ) + Ys (f ) xs (t ) θ (t ) = tan−1 [ xc (t ) √ 2 2 2(xc (t ) + xs (t ))1/2 xe (t ) = √ y (t ) = Re[(xc (t ) + jxs (t )) 2ej 2πfc t ] The signal xc (t ) + jxs (t ) is called the lowpass complex equivalent of the signal x (t ). EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 45 / 130 Lecture Notes 5 Up and Down Conversion IEEE 802.11 This is from the IEEE 802.11 (WiFi) standard (page 555)   $3F:7?3F;53> 5A@H7@F;A@E ;@ F:7 E;9@3> 67E5D;BF;A@E 4HE TRANSMITTED SIGNALS WILL BE DESCRIBED IN A COMPLEX BASEBAND SIGNAL NOTATION 4HE ACTUAL TRANSMITTED SIGNAL IS RELATED TO THE COMPLEX BASEBAND SIGNAL BY THE FOLLOWING RELATION   D ( ( ) F  (7 [ D F EXP J π 8 5 F ] WHERE (7  85 REPRESENTS THE REAL PART OF A COMPLEX VARIABLE DENOTES THE CARRIER CENTER FREQUENCY EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 46 / 130 Lecture Notes 5 Up and Down Conversion GSM from 3GPP This is from the 3GPP Standard for GSM [3GPP TS 05.04 Release 1999 V8.4.0 (2001-11)] 3.6 Modulation The modulated RF carrier during the useful part of the burst is therefore: where Es is the energy per modulating symbol, f0 is the centre frequency and 0 is a random phase and is constant during one burst. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 47 / 130 Lecture Notes 5 Up and Down Conversion IQ Modulation Re[Y (f )] EECS 455 (Univ. of Michigan) f fc − W f +W fcc −fc −f + W −fc − W c Fall 2012 September 6, 2012 48 / 130 Lecture Notes 5 Up and Down Conversion IQ Modulation Im[Y (f )] −fc − W fc − W −ffc + W −c EECS 455 (Univ. of Michigan) f fc + Wfc Fall 2012 September 6, 2012 49 / 130 Lecture Notes 5 Up and Down Conversion Signal Decomposition The signals xc (t ) and xs (t ) can be recovered from y (t ) by mixing down to baseband and filtering out the double frequency terms. Note that we need the exact phase of the local oscillators to do this perfectly. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 50 / 130 Lecture Notes 5 Up and Down Conversion Signal Decomposition √ 2 cos(2π fc t ) × zc (t ) GLP (f ) xc (t ) y (t ) × zs (t ) GLP (f ) xs (t ) √ − 2 sin(2π fc t ) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 51 / 130 Lecture Notes 5 Up and Down Conversion Signal Decomposition GLP (f ) is an ideal low pass filter with transfer function GLP (f ) = 1 |f | ≤ W and GLP (f ) = 0 otherwise. GLP (f ) −W EECS 455 (Univ. of Michigan) f W Fall 2012 September 6, 2012 52 / 130 Lecture Notes 5 Up and Down Conversion Signal Decomposition Consider the spectrum of zc (t ). This is given by √ 2 Zc (f ) = [Y (f − fc ) + Y (f + fc )]. 2 Similarly the spectrum of zs (t ) is √ j2 [Y (f − fc ) − Y (f + fc )]. Zs (f ) = 2 EECS 455 (Univ. of M...
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