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slides_digital - ECE-501 Phil Schniter April 10, 2008...

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ECE-501 Phil Schniter April 10, 2008 Digital Communication (Ch. 6,7,10): Transmission consists of 1. pulse shaping: ˜ m ( t )= n a [ n ] g ( t - nT ) , 2. modulation: s ( t ) = Re { ˜ m ( t ) e j 2 π f c t } . Reception consists of 1. demodulation: ˜ v ( t LPF { 2 r ( t ) e - j 2 π f c t } , 2. ±ltering: y ( t ) = ˜ v ( t ) * q ( t ) , 3. sampling: y [ m ]= y ( mT ) . ˜ m ( t ) s ( t ) r ( t v ( t ) y ( t ) a [ n ] y [ m ] nT t = mT e j 2 π f c t 2 e - j 2 π f c t g ( t ) LPF q ( t ) Re × × channel digital modulation digital demodulation analog QAM mod analog QAM demod Building on analog QAM mod/demod components, digital mod adds pulse shaping & demod adds ±ltering/sampling. Simplifying via the complex-baseband equivalent channel: ˜ m ( t ) ˜ v ( t ) y ( t ) a [ n ] y [ m ] nT t = mT g ( t ) q ( t ) ˜ h ( t ) ˜ w ( t ) + 38 ECE-501 Phil Schniter April 10, 2008 Transmitter pulse shaping is used to convert the symbol sequence { a [ n ] } into the continuous message ˜ m ( t ) : ˜ m ( t ± n a [ n ] g ( t - nT ) “baseband message” T = “symbol period” Thus, ˜ m ( t ) can be seen to be a superposition of scaled and time-shifted copies of the pulse waveform g ( t ) . Example, if the symbol sequence ² a [0] ,a [1] [2] [3] [4] ³ equals [1 , 3 , - 1 , 1 , 3] , then the square pulse g ( t ) shown below left yields the message ˜ m ( t ) shown below right. ˜ m ( t n a [ n ] g ( t - nT ) a [ n ] g ( t - nT ) for n =0 , ..., 4 t t t t t t t 1 1 - 1 3 3 g ( t ): g ( t - T g ( t - 2 T g ( t - 3 T g ( t - 4 T T 2 T 3 T 4 T 5 T 6 T T 2 T 3 T 4 T 5 T 6 T 39
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ECE-501 Phil Schniter April 10, 2008 Receiver fltering (via q ( t ) ) has two goals: 1. noise suppression (i.e., SNR improvement), 2. inter-symbol interFerence (ISI) prevention. Noise suppression was brie±y discussed on slide 13 and will soon be revisited in more detail. Next we describe ISI. Realize that, in the ideal digital comm system, the n th output y [ n ] would simply equal the n th input a [ n ] . But in practice, y [ n ] can be corrupted by interFerence From the other symbols { a [ m ] } m ± = n , known as “inter-symbol interFerence,” and noise. ISI-prevention For the noiseless trivial channel : Consider the idealized system ˜ m ( t ) y ( t ) a [ n ] y [ m ] nT t = mT g ( t ) q ( t ) y ( t )= ± q ( τ m ( t - τ ) d τ For ˜ m ( t ² n a [ n ] g ( t - nT ) = ² n a [ n ] ± q ( τ ) g ( t - nT - τ ) d τ = ² n a [ n ] p ( t - nT ) For p ( t g ( t ) * q ( t ) . 40 ECE-501 Phil Schniter April 10, 2008 Thus, the idealized system can be re-drawn as y ( t ) a [ n ] y [ m ] nT t = mT p ( t ) where y [ m ]= y ( mT ² n a [ n ] p ( mT - nT ² n a [ n ] p ( ( m - n ) T ) . To make y [ m a [ m ] (i.e., prevent ISI), we need p ( t ) 1 t 0 T - T 2 T - 2 T 3 T - 3 T p ( mT 1 m =0 0 m ± which is known as the “Nyquist Criterion.” This criterion can be simply stated as p ( mT δ [ m ] using δ [ m 1 m 0 m ± “discrete-time impulse,” or “Kronecker delta.” a [ n ² m = -∞ a [ m ] δ [ n - m ] “siFting property.” 41
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ECE-501 Phil Schniter April 10, 2008
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slides_digital - ECE-501 Phil Schniter April 10, 2008...

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