12. Nyquist Sampling, Pulse-Amplitude Modulation, and Time- Division Multiplexing

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1 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac 2. Nyqui Sapling, Pule-Apliude Modulaion, and Tie- Diviion Muliplexing Many analogue counicaion ye are ill in wide ue oday. Thee include AM, FM, and PM ye. I analogue ignal are o be ranied digially, hey have o be convered o dicree aple. The converion o an analogue ignal ino a dicree-ie apled ignal i accoplihed by apling he analogue ignal a regular ie inerval T. T i called he apling period and = / T i known a he apling rae. Deiniion []. A ignal ( i called a band-liied ignal i M( = or > Hz (2. where i he highe-requency pecral coponen o (. Conider a periodic recangular waveor ( o period T = /, uni apliude, and pule widh. The rigonoeric Fourier erie o ( i ( = T + 2 ( in 2πn /2 T n = 2πn /2 co 2πn (2.2 c = T + 2 c T n co 2πn (2.3 n = = d + 2d n = in co 2πn (2.4 where c =, c n = Fourier erie i in, and d = T. The correponding exponenial ( = d n = in e j 2πn (2.5 I we uliply ( by (, we obain c ( = ( ( = ( (d + 2d n = = d( ( + 2 n = in in co 2πn co 2πn (2.6 c ( coni o he coponen ( and an ininie nuber o DSB ignal a 2.

2 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac apling requencie, 2, 3, The Fourier ranor o c ( i S c ( = dm( + d n = in [M( - n + M( + n ] (2.7 Figure 2. how he waveor and pecra aociaed wih ignal apling. Figure 2. Waveor and pecra aociaed wih ignal apling. Figure 2.2 how wha happen i = 2 and < 2. Figure 2.2 Signal pecra or (a = 2 and (b < 2. Since he bandwidh o ( i, we ee ha he pecra do no overlap i > 2 and he pecru aociaed wih he ignal ( can be eparaed ro oher uing a low-pa iler wih a cuo requency o. When < 2, he pecra overlap. Since he requency conen in hee region o overlap add, he ignal i diored. The diorion i called aliaing and i i no longer poible o recover ( ro i aple value by low-pa ilering. Sapling Theore [2] Le (nt be he aple value o ( where n i an ineger. The apling heore ae ha he ignal ( can be reconruced ro (nt wih no diorion i he apling requency > 2. The iniu apling rae 2 i called he Nyqui apling rae. Proo. Since M( i a non-periodic bandliied uncion, we can ake a new uncion which i periodic a requency bu no overlapping in he requency-doain. Le he periodic requency uncion be Mp (, a hown in Figure 2.3. M( i a band-liied verion o M p (, Figure 2.3 M( repreened a a periodic requency uncion. The exponenial Fourier erie o M p ( i M p ( = c n e jn (2.8 n = where > 2 and 2.2

3 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac = 2π/ (2.9 The coeicien c n are given by c n = /2 M p (e -jn d = /2 /2 M p ( e -jn(2π/ d (2. /2 However, he Fourier ranor ell u ha ( = F - [M(] = M( e j2π d = /2 M p ( e j2π d /2 (2. Coparing equaion (2. and (2., we ee ha, i = n, we obain c n = ( n (2.2 Thi ay ha we can obain each c n ro he aple value o ( a ie = n. Once c n i known, we can obain M p ( ro equaion (2.8, and once M p ( i known, we can obain ( ro equaion (2.. Subiuing c n ino equaion (2.8, we ge M p ( = n = ( n e jn (2.3 Subiuing hi expreion or M p ( ino equaion (2., we ge /2 ( = F - [M(] = [ ( n /2 n = e jn ] e j2π d /2 = n = ( n e jn e j2π d /2 = n = ( n in[ π ( n ] π ( n 2.3

4 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac = n = ( n in [π ( n ] π ( n Weighing acor (2.4 Equaion (2.4 how ha each aple i uliplied by a weighing acor. Signal Reconrucion [2, 3] The proce o reconrucing an analogue ignal ( ro i aple i known a inerpolaion. How do we reconruc ( ro i aple ( n? Conider he aple ignal o ( hown in Figure 2.4. Figure 2.4 Saple o (. Le M n ( be he Fourier ranor o he n-h aple ( n. I << /, ( can be aued o be conan over he apling ie and /2 M n ( = ( n e -j2π d ( n /2 e -j2π(n/ /2 d /2 M n ( ( n e -j2π(n/ (2.5 Fro our knowledge o baic PAM heory, we can recover he analogue ignal uing a low-pa iler wih a cuo requency o /2 (>. Aue ha we have an ideal low-pa iler whoe raner uncion i H( = K e -j2π d, where K i a conan and d i a ie delay. Wihou lo o generaliy, we e he iler gain K = and he iler delay d =. Le g n ( be he iler oupu repone o he n-h inpu aple ( n. The Fourier ranor o g n ( i G n ( = H(M n ( = M n ( and 2.4

5 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac g n ( /2 = G n ( e j2π d = G n ( e j2π d /2 = ( n in[ π ( n ] π ( n (2.6 For a linear ideal iler, he iler oupu repone o all inpu aple i ju he u o he iler oupu o each inpu aple, or g( = g n ( n = = n = ( n in[ π ( n ] π ( n (2.7 = ( (2.8 Equaion (2.7 yield value o g( beween aple a a weighed u o all aple value. g( i no only deined a he apling inan, bu i i proporional o ( a all inan o ie. Thi i hown in Figure 2.5. Figure 2.5 Filer repone o inpu aple. Pracical Sapling Frequency and Pule-Apliude Modulaion Table 2. Pracical apling requency value or audio and broadca ignal Signal > 2 Acual apling requency Audio 3.3 khz > 6.6 khz 8 khz Muic 2 khz > 4 khz 44. khz TV 4 MHz > 8 MHz The apling heore i very iporan becaue i allow u o replace an analogue ignal by a dicree aple and reconruc he analogue ignal ro i aple value. I open door o any new echnique o counicaing analogue ignal by aple. A ye raniing aple value o he analogue ignal i called a pule-apliude odulaion (PAM ye and i hown in Figure 2.6. Figure 2.6 PAM ye. 2.5

6 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac Tie Diviion Muliplexing (TDM One o he baic proble in counicaion engineering i he deign o a pule counicaion ye which allow ignal ro any uer o be ranied iulaneouly over a ingle counicaion channel. We ee ro he apling proce ha, wih << T, here i a ie gap beween wo conecuive aple in a ingle-uer PAM ye. Suppoe ha we have everal dieren ignal o he ae or dieren bandwidh. I we aple he ignal in a equenial anner, we can pu he aple in he ie gap. All hee ignal aple can now be ranied along a ingle counicaion channel. A he receiving end, he ignal can be eparaed and recovered. We now have a ie-uliplexed ye. Such a uliplexing echnique i called ie diviion uliplexing (TDM. I peri he iulaneou raniion o everal ignal on a ie-hared bai. Figure 2.7 how he ranier, he receiver, and he pecru o a 5-uer TDM PAM ye wih = 8 aple/. Figure 2.7 TDM ye. Reerence [] H. P. Hu, analog and Digial Counicaion, McGraw-Hill, 993. [2] M. Schwarz, Inoraion Traniion, Modulaion, and Noie, 4/e, McGraw-Hill, 99. [3] M. S. Roden, Analog and Digial Counicaion Sye, 3/e, Prenice Hall,

7 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac ( M ( ( -T T 2 2 ( c Envelope Envelope c n T - 2 T M( S ( c 2 T Guard band Figure 2. Waveor and pecra aociaed wih ignal apling. 2.7

8 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac Envelope S c ( T M( - (a 2 =2 Envelope S ( c T M( - 2 Region o overlap < 2 (b Figure 2.2 Signal pecra or (a = 2 and (b < 2. M ( p M ( Figure 2.3 M( ued o ake a periodic requency uncion. 2.8

9 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac ( ( Saple o ( T <<T Figure 2.4 Saple o (. g g n ( n ( ( ( Reconruced ignal g n +( ( n -2 n - n n + n +2 Figure 2.5 Filer repone o inpu ignal aple. ( c ( Sapler PAM ignal Tranier // LPF ~ ~ Recovered ignal Receiver Figure 2.6 PAM ye. 2.9

10 Nyqui Sapling, Pule-Apliude Modulaion, and Tie Diviion Muliplexing on Mac ( ( 2 ( 5 = 8 aple/ c ( Copoie PAM ignal // LPF ~ ~ ~ ~ ~ ~ ^ ( ^ ( 2 ^ ( 5 Source Deinaion (a Tranier and receiver c ( 25 µ 25 µ (b Waveor o TDM ignal Figure 2.7 TDM ye. 2.