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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 (200111)] 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...
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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.
 Fall '08
 Stark

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