6.003: Signals and Systems Modulation December 6, 2011 1

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 Modulation can improve match based on frequency . 2

Amplitude Modulation Amplitude modulation can be used to match audio frequencies to radio frequencies. It allows parallel transmission of multiple channels. ③ ✶ ✭t✮ ① ✶ ✭t✮ ❝♦s ✇ ✶ t ③ ✷ ✭t✮ ③✭t✮ ① ✷ ✭t✮ ▲P❋ ②✭t✮ ❝♦s ✇ ❝ t ❝♦s ✇ ✷ t ③ ✸ ✭t✮ ① ✸ ✭t✮ ❝♦s ✇ ✸ t 3

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 wideband FM. 4

Amplitude, Phase, and Frequency Modulation There are many ways to embed a “message” in a carrier. � � Amplitude Modulation (AM) + carrier: y 1 ( t ) = x ( t ) + C cos( ω c t ) Phase Modulation (PM): y 2 ( t ) = cos( ω c t + kx ( t )) � � t � Frequency Modulation (FM): y 3 ( t ) = cos ω c t + k −∞ x ( τ ) dτ PM: signal modulates instantaneous phase of the carrier. y 2 ( t ) = cos( ω c t + kx ( t )) FM: signal modulates instantaneous frequency of carrier. t � � y 3 ( t ) = cos ω c t + k x ( τ ) dτ ' −∞ v " φ ( t ) d ω i ( t ) = ω c + φ ( t ) = ω c + kx ( t ) dt 5

Frequency Modulation Compare AM to FM for x ( t ) = cos( ω m t ) . � � AM: y 1 ( t ) = x ( t ) + C cos( ω c t ) = (cos( ω m t ) + 1 . 1) cos( ω c t ) t � � � t � k FM: y 3 ( t ) = cos ω c t + k −∞ x ( τ ) dτ = cos( ω c t + sin( ω m t )) ω m t Advantages of FM: • constant power • no need to transmit carrier (unless DC important) • bandwidth? 6

Frequency Modulation Early investigators thought that narrowband FM could have arbitrar- ily narrow bandwidth, allowing more channels than AM. t � � � � y 3 ( t ) = cos ω c t + k x ( τ ) dτ −∞ ' v " φ ( t ) d ω i ( t ) = ω c + φ ( t ) = ω c + kx ( t ) dt Small k → small bandwidth. Right? 7

Frequency Modulation Early investigators thought that narrowband FM could have arbitrar- ily narrow bandwidth, allowing more channels than AM. Wrong! � � t y 3 ( t ) = cos ω c t + k x ( τ ) dτ −∞ � � � � t t = cos( ω c t ) × cos k x ( τ ) dτ − sin( ω c t ) × sin k x ( τ ) dτ −∞ −∞ If k → 0 then � � t cos k x ( τ ) dτ → 1 −∞ � � t t sin k x ( τ ) dτ → k x ( τ ) dτ −∞ −∞ � � t y 3 ( t ) ≈ cos( ω c t ) − sin( ω c t ) × k x ( τ ) dτ −∞ Bandwidth of narrowband FM is the same as that of AM! (integration does not change the highest frequency in the signal) 8

Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y ( t ) = cos( ω c t + mx ( t )) = cos( ω c t + m sin( ω m t )) = cos( ω c t ) cos( m sin( ω m t ))) − sin( ω c t ) sin( m sin( ω m t ))) x ( t ) is periodic in T = 2 π , therefore cos( m sin( ω m t )) is periodic in T . ω m 1 sin( ω m t ) 1 0 t − 1 cos(1 sin( ω m t )) 1 0 t − 1 9

Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y ( t ) = cos( ω c t + mx ( t )) = cos( ω c t + m sin( ω m t )) = cos( ω c t ) cos( m sin( ω m t ))) − sin( ω c t ) sin( m sin( ω m t ))) x ( t ) is periodic in T = 2 π , therefore cos( m sin( ω m t )) is periodic in T . ω m m sin( ω m t ) m 0 t − m cos( m sin( ω m t )) 1 0 t increasing m − 1 21

Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y ( t ) = cos( ω c t + mx ( t )) = cos( ω c t + m sin( ω m t )) = cos( ω c t ) cos( m sin( ω m t ))) − sin( ω c t ) sin( m sin( ω m t ))) x ( t ) is periodic in T = 2 π ω m , therefore cos( m sin( ω m t )) is periodic in T . cos( m sin( ω m t )) 1 0 t − 1 m = 0 | a k | k 0 10 20 30 40 50 60 22

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