6 003 Signals and Systems Modulation May 6 2010 Course VI Underground Guide Please give us and future students feedback on 6 003 by participating in the Course VI Underground Guide Survey https sixweb mit edu student evaluate 6 003 s2010 Communications Systems Signals are not always well matched to the media through which we wish to transmit them signal applications audio telephone radio phonograph CD cell phone MP3 video television cinema HDTV DVD internet coax twisted pair cable TV DSL optical fiber E M Amplitude Modulation Amplitude modulation can be used to match audio frequencies to radio frequencies It allows parallel transmission of multiple channels z1 t x1 t cos w1t z2 t x2 t cos wct cos w2t z3 t x3 t cos w3t z t LPF y t Superheterodyne Receiver Edwin Howard Armstrong invented the superheterodyne receiver which made broadcast AM practical Edwin Howard Armstrong also invented and patented the regenerative positive feedback circuit for amplifying radio signals while he was a junior at Columbia University He also invented wide band FM Amplitude Phase and Frequency Modulation There are many ways to embed a message in a carrier Here are three Amplitude Modulation AM y1 t x t cos c t Phase Modulation PM y2 t cos c t kx t Rt y3 t cos c t k x d Frequency Modulation FM Frequency Modulation In FM the signal modulates the instantaneous carrier frequency Z t x d y3 t cos c t k z t d i t c t c kx t dt Frequency Modulation Compare AM to FM for x t cos m t AM y1 t cos m t 1 1 cos c t t FM y3 t cos c t m sin m t t Advantages of FM constant power no need to transmit carrier unless DC important bandwidth Frequency Modulation Early investigators thought that narrowband FM could have arbitrarily narrow bandwidth allowing more channels than AM Wrong Z t x d y3 t cos c t k Z t Z t x d x d sin c t sin k cos c t cos k If k 0 then Z t cos k x d 1 Z t Z t sin k x d k x d Z t y3 t cos c t sin c t k x d Bandwidth of narrowband FM is the same as that of AM integration does not change bandwidth Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m m sin m t m 0 t m cos m sin m t 1 0 1 t increasing m Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 0 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 1 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 2 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 5 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 10 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 20 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 30 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c t sin m sin m t x t is periodic in T 2 therefore cos m sin m t is periodic in T m cos m sin m t 1 t 0 1 m 40 ak 0 10 20 30 40 50 60 k Phase Frequency Modulation Find the Fourier transform of a PM signal x t sin m t y t cos c t mx t cos c t m sin m t cos c t cos m sin m t sin c …
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