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Unformatted text preview: Complex ContinuousTime Filter Tutorial Jackson Harvey 2 Outline Mixing and Image Problem Complex Signals and Quadrature Downconversion Mismatch and Image Rejection Ratio Real and Complex Filter Theory CMOS IF Filter Circuits (Real and Complex) z Passive RC Filters z Active RC Filters z MOSFETC Filters z Transconductors z GmC Filters z Resistively Terminated LC Ladder Filters z Gyrators and GyratorC Filters z Published Examples 3 Frequency Downconversion (Mixing) Data is translated from baseband to RF for transmission Rx must translate from RF back to baseband for processing Multiplication in time domain >> convolution in frequency domain >> frequency translation Both sum and difference frequencies are generated [ ] ) cos( ) cos( 2 1 ) cos( ) cos( LO C LO C LO C + + = 4 Image Problem Frequencies on both sides of LO frequency are converted to IF z One side has desired signal, other side has image z After downconversion, image and signal cannot be separated z Image appears as a cochannel interferer at IF z Image is often larger than signal z For low IF, image cannot be filtered before downconversion Distance between signal and image is 2X IF frequency 5 Solution 1: Homodyne RX LO frequency = RF frequency, IF frequency is DC Two copies of signal end up at IF (frequency inverted) z Signal must be symmetric about LO (no quadrature data) z Carrier frequency offset leads to selfinterference Many problems: DC offsets, LO pulling, 1/f noise, LO radiation, evenorder distortion 6 Solution 2: Heterodyne RX Image frequency is filtered prior to downconversion For good filtering, image must be far from signal z High IF frequency (f signal f image  = 2 * f IF ) z Multiple downconversion steps required Cascade of filters and mixers z High power, large area, many discrete components 7 Complex Signals Complex Number z Composed of real part and imaginary part Real Signal z Can be decomposed into sinusoids z Specified by frequency, phase, and amplitude z Symmetric about DC (frequency is unsigned) Complex Signal z Composed of real signal and imaginary signal z Any two real signals can make up an imaginary signal z No longer symmetric about DC Need signed frequency Gives phase relation of Re{X} and Im{X} } Im{ } Re{ X j X X + = )} ( Im{ )} ( Re{ ) ( t X j t X t X + = 8 Euler Relationship Defines a complex sinusoid Positive frequency only ( ) 2 cos t j t j o o o e e t + = ( ) = 2 sin t j t j o o o e e j t t j e t j t ) sin( ) cos( = + 9 Complex Signal Identities Complex exponential Swap Re and Im Invert Re Invert Im Re and Im equal t j e t j t ) sin( ) cos( = + = + 2 ) cos( ) sin( t j e t j t t j e t j t ) sin( ) cos( = t j e t j t ) sin( ) cos( = + [ ]...
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 Fall '04
 Harjani

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