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. z1(t) x1(t) cos w1t z2(t) z(t) x2(t) LPF y(t) cos w2t cos wct z3(t) x3(t) cos w3t 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 in­ vented wide-band FM. 4 Amplitude, Phase, and Frequency Modulation There are many ways to embed a “message” in a carrier. Amplitude Modulation (AM) + carrier: y1(t) = x(t) + C cos(ωct) Phase Modulation (PM): y2(t) = cos(ωct + kx(t)) t Frequency Modulation (FM): y3(t) = cos ωct + k −∞ x(τ )dτ PM: signal modulates instantaneous phase of the carrier. y2(t) = cos(ωct + kx(t)) FM: signal modulates instantaneous frequency of carrier. t y3(t) = cos ωct + k x(τ)dτ −∞ ' v " φ(t) d ωi(t) = ωc + φ(t) = ωc + kx(t) dt 5 Frequency Modulation Compare AM to FM for x(t) = cos(ωmt). AM: y1(t) = x(t) + C cos(ωct) = (cos(ωmt) + 1.1) cos(ωct) t R t k FM: y (t) = cos ω t + k x(τ )dτ = cos(ω t + sin(ω t)) 3 c −∞ c ωm 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. Z t y3(t) = cos ωct + k x(τ)dτ −∞ ' v " φ(t) d ωi(t) = ωc + φ(t) = ωc + 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! 0 Z t y3(t) = cos ωct + k x(τ)dτ −∞ 0 Z t 0 Z t = cos(ωct) × cos k x(τ)dτ − sin(ωct) × sin k x(τ )dτ −∞ −∞ If then k → 0 0 Z t cos k x(τ)dτ → 1 −∞ 0 Z t Z t sin k x(τ )dτ → k x(τ)dτ −∞ −∞ 0 Z t y3(t) ≈ cos(ωct) − sin(ωct) × 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(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 1 sin(ωmt) 1 0 t −1 cos(1 sin(ωmt)) 1 0 t −1 9 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 2 sin(ωmt) 2 0 t −2 cos(2 sin(ωmt)) 1 0 t −1 10 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 3 sin(ωmt) 3 0 t −3 cos(3 sin(ωmt)) 1 0 t −1 11 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 4 sin(ωmt) 4 0 t −4 cos(4 sin(ωmt)) 1 0 t −1 12 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 5 sin(ωmt) 5 0 t −5 cos(5 sin(ωmt)) 1 0 t −1 13 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 6 sin(ωmt) 6 0 t −6 cos(6 sin(ωmt)) 1 0 t −1 14 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 7 sin(ωmt) 7 0 t −7 cos(7 sin(ωmt)) 1 0 t −1 15 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 8 sin(ωmt) 8 0 t −8 cos(8 sin(ωmt)) 1 0 t −1 16 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 9 sin(ωmt) 9 0 t −9 cos(9 sin(ωmt)) 1 0 t −1 17 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 10 sin(ωmt) 10 0 t −10 cos(10 sin(ωmt)) 1 0 t −1 18 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 20 sin(ωmt) 20 0 t −20 cos(20 sin(ωmt)) 1 0 t −1 19 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m 50 sin(ωmt) 50 0 t −50 cos(50 sin(ωmt)) 1 0 t −1 20 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m m sin(ωmt) m 0 t −m cos(m sin(ωmt)) 1 0 t increasing m −1 21 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m cos(m sin(ωmt)) 1 0 t −1 m = 0 |ak| k 0 10 20 30 40 50 60 22 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m cos(m sin(ωmt)) 1 0 t −1 m = 1 |ak| k 0 10 20 30 40 50 60 23 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal. y(t) = cos(ωct + mx(t)) = cos(ωct + m sin(ωmt)) = cos(ωct) cos(m sin(ωmt))) − sin(ωct) sin(m sin(ωmt))) 2π x(t) is periodic in T = , therefore cos(m sin(ω t)) is periodic in T . ωm m cos(m sin(ωmt)) 1 0 t −1 m = 2 |ak| k 0 10 20 30 40 50 60 24 Phase/Frequency Modulation Find the Fourier transform of a PM/FM signal.