102_1_Lecture12 - UCLA Spring 2009-2010 Systems and Signals...

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UCLA Spring 2009-2010 Systems and Signals Lecture 12: Moduation and Demodulation May 10, 2010 EE102:Systems and Signals; Spr 09-10, Lee 1
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Modulation The modulation theorem tells us how to place a message signal m ( t ) on a carrier cos( ω c t ) . How do we recover m ( t ) from the modulated signal? What are the problems? Are there better receivers? Can I use other modulation methods to make the receiver easier? EE102:Systems and Signals; Spr 09-10, Lee 2
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Double-Sideband, Suppressed Carrier (DSB-SC) Modulation This is a complicated way of saying ”multiply by a cosine.” We have a message (baseband) signal, and cosine cos( ω c t ) at a frequency ω c . The modulated message signal and its Fourier transform are m ( t ) cos( ω c t ) 1 2 [ M ( j ( ω + ω c )) + M ( j ( ω - ω c ))] where m ( t ) M ( j ω ) . The block diagram is × m ( t ) cos ( ! c t ) m ( t ) cos ( ! c t ) This signal and its spectrum are illustrated on the next page: EE102:Systems and Signals; Spr 09-10, Lee 3
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m ( t ) - m ( t ) m ( t ) cos ( ! c t ) m ( t ) t t ! c - ! c ! ! M ( j ! ) 1 2 M ( j ( ! - ! c )) 1 2 M ( j ( ! + ! c )) EE102:Systems and Signals; Spr 09-10, Lee 4
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We can think of modulation as frequency domain convolution. F [ m ( t ) cos( ω c t )] = 1 2 π [ M ( j ω ) * ( πδ ( ω + ω c ) + πδ ( ω - ω c ))] ! ! c - ! c ! ! c - ! c ! = * 1 2 ! M ( j ! ) 1 2 M ( j ( ! + ! c )) 1 2 M ( j ( ! - ! c )) !" ( # - # c ) !" ( # + # c ) EE102:Systems and Signals; Spr 09-10, Lee 5
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To demodulate this signal, consider what happens if we multiply again by cos( ω c t ) . Again, we can think of this as a convolution in the frequency domain: ! c - ! c ! ! c - ! c ! = ! c - ! c ! 2 ! c - 2 ! c 0 0 0 Lowpass Filter * 1 2 ! 1 2 M ( j ( ! + ! c )) 1 2 M ( j ( ! - ! c )) 1 4 M ( j ( ! - 2 ! c )) 1 4 M ( j ( ! + 2 ! c )) 1 2 M ( j ! ) !" ( # - # c ) !" ( # + # c ) EE102:Systems and Signals; Spr 09-10, Lee 6
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After the convolution there is a replica of the spectrum centered at ω = 0 , which we can extract with a lowpass filter. The modulated signal spectrum is F [ m ( t ) cos( ω c t )] = 1 2 M ( j ( ω + ω c )) + 1 2 M ( j ( ω - ω c )) Multiplying this by cos( ω c t ) corresponds to convolving in frequency, F m ( t ) cos 2 ( ω c t ) = 1 2 π 1 2 M ( j ( ω + ω c )) + 1 2 M ( j ( ω - ω c )) * [ πδ ( ω + ω c ) + πδ ( ω - ω c )] = 1 4 [ M ( j ( ω + ω c )) + M ( j ( ω - ω c ))] * [ δ ( ω + ω c ) + δ ( ω - ω c )] = 1 4 M ( j ( ω + 2 ω c )) + 1 2 M ( j ω ) + 1 4 M ( j ( ω - 2 ω c ))
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