6.003: Signals and Systems Modulation May 6, 200
Communications Systems Signals are not always well matched to the media through which we wish to transmit them. signal audio video internet applications telephone, radio, phonograph, CD, cell phone, MP3 television, cinema, HDTV, DVD 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. x (t) z (t) cos t x 2 (t) z 2 (t) z(t) LPF y(t) cos 2t cos ct x 3 (t) z 3 (t) cos 3t
Superheterodyne Receiver Edwin Howard Armstrong invented the superheterodyne receiver, which made broadcast AM practical. Edwin Howard Armstrong also invented and patented the regenerative (positive feedbac) 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): y (t) = x(t) cos(ω c t) Phase Modulation (PM): Frequency Modulation (FM): y 2 (t) = cos(ω c t + x(t)) ( ) t y 3 (t) = cos ω c t + x(τ)dτ
Frequency Modulation In FM, the signal modulates the instantaneous carrier frequency. ( t ) y 3 (t) = cos ω c t + x(τ)dτ }{{} φ(t) ω i (t) = ω c + d φ(t) = ω c + x(t) dt
Frequency Modulation Compare AM to FM for x(t) = cos(ω m t). AM: y (t) = (cos(ω m t)+.) cos(ω c t) t FM: y 3 (t) = cos(ω c t + m sin(ω 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! ( t ) y 3 (t) = cos ω c t + x(τ )dτ ( t ) ( t ) = cos(ω c t) cos x(τ )dτ sin(ω c t) sin x(τ )dτ If hen ( t ) cos x(τ)dτ ( t ) t sin x(τ)dτ x(τ )dτ ( t ) y 3 (t) cos(ω c t) sin(ω c t) x(τ )dτ Bandwidth of narrowband FM is the same as that of AM! (integration does not change bandwidth)
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. m sin(ω m t) m 0 t m cos(m sin(ω m t)) 0 t increasing m
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 0 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 2 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 5 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 0 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 20 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 30 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 40 a 0 0
x(t) is periodic in T = ω m, therefore cos(m sin(ω mt)) is periodic in T. cos(m sin(ω m t)) m = 50 a 0 0
Fourier transform of first part. = cos(ω } c t) cos(m sin(ω {{ m t))) sin(ω } c t) sin(m sin(ω m t))) y a (t) Y a (jω) m = 50 ω c ω c 50ω m ω
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. m sin(ω m t) m 0 m sin(m sin(ω m t)) 0 t increasing m t increasing m
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 0 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 2 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 5 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 0 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 20 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 30 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 40 b 0 0
x(t) is periodic in T = ω m, therefore sin(m sin(ω mt)) is periodic in T. sin(m sin(ω m t)) m = 50 b 0 0
Fourier transform of second part. = cos(ω c t) cos(m sin(ω m t))) sin(ω } c t) sin(m sin(ω {{ m t))) } y b (t) Y b (jω) m = 50 ω c ω c 50ω m ω
Fourier transform. }{{}}{{} y a (t) y b (t) Y (jω) m = 50 ω c ω c 50ω m ω
Frequency Modulation Wideband FM is useful because it is robust to noise. AM: y (t) = (cos(ω m t)+.) cos(ω c t) t FM: y 3 (t) = cos(ω c t + m sin(ω m t)) t FM generates a very redundant signal, which is resilient to additive noise.
Summary Modulation is useful for matching signals to media. Examples: commercial radio (AM and FM) Close with unconventional application of modulation in microscopy.
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