6.3: Signals and Sysems Modulaion December 6, 2
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Communicaions Sysems Signals are no always well mached o he media hrough which we wish o ransmi hem. signal audio video inerne applicaions elephone, radio, phonograph, CD, cell phone, MP3 elevision, cinema, HDTV, DVD coax, wised pair, cable TV, DSL, opical fiber, E/M Modulaion can improve mach based on frequency.
Ampliude Modulaion Ampliude modulaion can be used o mach audio frequencies o radio frequencies. I allows parallel ransmission of muliple channels. x () z () cos w x 2 () z 2 () z() LPF y() cos w 2 cos w c x 3 () z 3 () cos w 3
Superheerodyne Receiver Edwin Howard Armsrong invened he superheerodyne receiver, which made broadcas AM pracical. Edwin Howard Armsrong also invened and paened he regeneraive (posiive feedback) circui for amplifying radio signals (while he was a junior a Columbia Universiy). He also invened wide-band FM.
Ampliude, Phase, and Frequency Modulaion There are many ways o embed a message in a carrier. Ampliude Modulaion (AM) + carrier: y () = ( x() + C ) cos(ω c ) Phase Modulaion (PM): Frequency Modulaion (FM): y 2 () = cos(ω c + kx()) ( y 3 () = cos ω c + k ) x(τ)dτ PM: signal modulaes insananeous phase of he carrier. y 2 () = cos(ω c + kx()) FM: signal modulaes insananeous frequency of carrier. ( ) y 3 () = cos ω c + k x(τ)dτ } {{ } φ() ω i () = ω c + d d φ() = ω c + kx()
Frequency Modulaion Compare AM o FM for x() = cos( ). AM: y () = ( x() + C ) cos(ω c ) = (cos( ) +.) cos(ω c ) FM: y 3 () = cos ( ω c + k x(τ)dτ) = cos(ω c + k sin( )) Advanages of FM: consan power no need o ransmi carrier (unless DC imporan) bandwidh?
Frequency Modulaion Early invesigaors hough ha narrowband FM could have arbirarily narrow bandwidh, allowing more channels han AM. ( ) y 3 () = cos ω c + k x(τ)dτ } {{ } φ() ω i () = ω c + d d φ() = ω c + kx() Small k small bandwidh. Righ?
Frequency Modulaion Early invesigaors hough ha narrowband FM could have arbirarily narrow bandwidh, allowing more channels han AM. Wrong! ( y 3 () = cos ω c + k ) x(τ)dτ ( = cos(ω c ) cos k If k ( hen ) cos k x(τ)dτ ( ) sin k x(τ)dτ k x(τ)dτ ( y 3 () cos(ω c ) sin(ω c ) k ) ( ) x(τ)dτ sin(ω c ) sin k x(τ)dτ ) x(τ)dτ Bandwidh of narrowband FM is he same as ha of AM! (inegraion does no change he highes frequency in he signal)
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. sin( ) cos( sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 2 sin( ) 2 2 cos(2 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 3 sin( ) 3 3 cos(3 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 4 sin( ) 4 4 cos(4 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 5 sin( ) 5 5 cos(5 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 6 sin( ) 6 6 cos(6 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 7 sin( ) 7 7 cos(7 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 8 sin( ) 8 8 cos(8 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 9 sin( ) 9 9 cos(9 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. sin( ) cos( sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 2 sin( ) 2 2 cos(2 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. 5 sin( ) 5 5 cos(5 sin( ))
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. m sin( ) m m cos(m sin( )) increasing m
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 2 a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 5 a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 2 a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 3 a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 4 a k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore cos(m sin( )) is periodic in T. cos(m sin( )) m = 5 a k k 2 3 4 5 6
Phase/Frequency Modulaion Fourier ransform of firs par. x() = sin( ) y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω }{{} c ) sin(m sin( ))) ya() Y a (jω) m = 5 ω c ω c ωm ω
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. m sin( ) m m sin(m sin( )) increasing m increasing m
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 2 b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 5 b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 2 b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 3 b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 4 b k k 2 3 4 5 6
Phase/Frequency Modulaion Find he Fourier ransform of a PM/FM signal. y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω c ) sin(m sin( ))) x() is periodic in T = 2π, herefore sin(m sin( )) is periodic in T. sin(m sin( )) m = 5 b k k 2 3 4 5 6
Phase/Frequency Modulaion Fourier ransform of second par. x() = sin( ) y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω }{{} c ) sin(m sin( ))) }{{} ya() y b () Y b (jω) m = 5 ω c ω c ωm ω
Phase/Frequency Modulaion Fourier ransform. x() = sin( ) y() = cos(ω c + mx()) = cos(ω c + m sin( )) = cos(ω c ) cos(m sin( ))) sin(ω }{{} c ) sin(m sin( ))) }{{} ya() y b () Y (jω) m = 5 ω c ω c ωm ω
Frequency Modulaion Wideband FM is useful because i is robus o noise. AM: y () = (cos( ) +.) cos(ω c ) FM: y 3 () = cos(ω c + m sin( )) FM generaes a redundan signal ha is resilien o addiive noise.
Summary Modulaion is useful for maching signals o media. Examples: commercial radio (AM and FM) Close wih unconvenional applicaion of modulaion in microscopy.
6.3 Microscopy Dennis M. Freeman Sanley S. Hong Jekwan Ryu Michael S. Mermelsein Berhold K. P. Horn
6.3 Model of a Microscope microscope Microscope = low-pass filer
Phase-Modulaed Microscopy microscope