Digital Encoding And Decoding

Transcription

1 Digial Encoding And Decoding Dr. George W Benhien Augus 13, 2007 Revised March 30, george@gbenhien.ne

2 1 Inroducion Many elecronic communicaion devices oday process and ransfer informaion digially. Examples are cable elevisions, cell phones, cable/dsl modems, and wireless rouers. Digial informaion is specified by a sequence of zeroes and ones such as Each zero or one is called a binary digi or bi. Digial informaion is usually embedded in an elecric or opical signal. This aricle describes a number of mehods for encoding digial informaion in an elecric or opical signal and laer exracing his informaion. I have divided hese mehods ino wo ypes. The firs ype (someimes referred o as baseband) involves he encoding of digial informaion in digial signals consising of a sequence of recangular pulses. These mehods are used primarily in local area Eherne neworks. The second ype ofen referred o as broadband) involves modulaion of one or more sinusoidal carrier waves. These mehods are used in cable/dsl broadband neworks, saellie communicaion, cell phones, digial TV, and wireless neworks. Some of he specific echniques ha will be discussed are Mancheser encoding, 4B/5B block encoding, MLT-3 encoding, Frequency Shif Keying (FSK), Quadraure Phase Shif Keying (QPSK), Quadraure Ampliude Modulaion (QAM), Orhogonal Frequency Division Muliplexing (OFDM), Frequency Hopping Spread Specrum (FHSS), and Direc-Sequence Spread Specrum (DSSS). In digial encoding ime is broken up ino a succession of equal ime inervals of lengh T. In each ime inerval eiher one or a group of bis of he given sequence are encoded in he signal. T is called he symbol period and he group of bis encoded during a ime inerval T is called a symbol. The rae a which symbols are encoded is called he baud rae. The rae a which bis are encoded is called he bi rae. 2 Encoding Digial Informaion in Digial Signals A digial signal consiss of a sequence of equal widh recangular pulses. The iming of he pulses is conrolled by an inernal clock whose oupu is an alernaing sequence of high and low valued recangular pulses. A high/low pair is called a clock cycle. 2.1 NRZ encoding The simples digial signal represening a bi sequence uses jus wo volage levels and represens a 1 by he higher volage and a zero by he lower volage. This ype of encoding is called NRZ (Non-Reurn o Zero). An example of an NRZ encoded signal is shown in Figure 1. Alhough simple, his mehod for encoding digial informaion has some serious drawbacks and is seldom used. Firs, i is difficul o keep he clocks of he source and receiver synchronized if here happen o be long sequences of ones or zeros. The receiver uses ransiions in level o deermine clock cycle boundaries. Second, i is impossible o disinguish beween a long sequence of zeros and he absence of a signal. Third, a long series of zeros or ones causes he average signal value, 1

3 Clock NRZ Daa bis Figure 1: Example of NRZ encoding. which is used o disinguish beween high and low values, o drif. Thus, for many reasons, i is desirable o have frequen ransiions beween he high and low values. 2.2 NRZI encoding Anoher simple encoding mehod, called NRZI (Non-Reurn o Zero Invered), changes level for a 1 bi and says a he same level for a 0 bi. An example of an NRZI encoded signal is shown in Figure 2. Clock NRZI Daa bis Figure 2: Example of NRZI encoding. This mehod ges rid of he problems associaed wih long srings of ones, bu does nohing abou long srings of zeros. We will see laer ha he NRZI mehod can be effecively used in combinaion wih oher mehods ha don produce long srings of zeros. 2

4 2.3 Mancheser encoding In Mancheser encoding 0 and 1 bis are represened in a clock cycle by he signals shown in Figure Figure 3: Mancheser encoding of 0 and 1 bis. Here he signal ransiion occurs in he middle of he cycle. An example of a Mancheser encoded sequence is shown in Figure 4. Clock Mancheser Daa bis Figure 4: An example of a Mancheser encoded signal. This encoding mehod is used in 10 Mbs (Megabis per second) 10BaseT Eherne neworks. Mancheser encoding solves he problems menioned previously in connecion wih NRZ encoding. However, since he signal alernaes level every clock cycle, Mancheser encoding has a broader frequency specrum han NRZ. For 100 Mbs and higher Eherne neworks, he specrum of a Mancheser encoded signal exends pas he high frequency limi for unshielded wised pair Eherne cables. Thus, oher encoding schemes are used for high speed Eherne B/5B encoding 4B/5B is a block encoding scheme designed o break up long srings of ones and zeros wihou increasing he frequency bandwidh. In his scheme he bi sequence is broken up ino four bi blocks. Each block of four bis is replaced wih a five bi block according o Table 1. 3

5 Table 1: Four bi o five bi conversion able. Four bi daa Five bi code Four bi daa Five bi code The five bi codes were seleced so ha here is no more han one leading zero and no more han wo railing zeros. Thus, when he codes are srung ogeher, here can be no more han hree consecuive zeros. The sring of bis afer he replacemen are ransmied using NRZI. As we saw before, NRZI effecively handles srings of ones. An example of a 4B/5B encoded signal is shown in Figure 5. Clock 4B/5B NRZI Daa bis Figure 5: An example of 4B/5B encoding. 4B/5B encoding followed by NRZI is used in 100BaseTX (Fas Eherne) neworks in conjuncion wih he mulilevel encoding scheme MLT-3 described in he nex secion. Since only half of he five bi codes are used in Table 1, he remaining codes can be used for oher purposes. 2.5 MLT-3 encoding MLT (Muli Level Threshold) encoding is used o decrease he high frequency conen of he signal. MLT-3 uses hree levels denoed by -1, 0, and 1. The process cycles hrough he four values 4

6 -1, 0,+1, 0. I moves o he nex of he four saes in a cyclical manner o ransmi a 1 bi, and says in he same sae o ransmi a 0 bi. An example of MLT-3 encoding is shown in Figure 6. The fases an MLT-3 signal can go hrough a complee cycle is four clock cycles. Thus, he high frequency limi of an MLT-3 signal will be abou one-fourh ha of a Mancheser encoded signal. In Fas Eherne (100 Mbps) MLT-3 is applied o he signal generaed by 4B/5B and NRZI. Higher order block codes such as 8B/10B and higher order muli-level mehods such as MLT-5 are used in higher speed Eherne neworks. Clock MLT-3 Daa bis Figure 6: An example of MLT-3 encoding. 3 Encoding Digial Informaion by Modulaing Analog Signals Anoher class of mehods encodes digial daa by modulaing one or more sinusoidal carrier waves. The process of backing ou he digial informaion from he modulaed wave is called demodulaion. A device for performing modulaion and demodulaion is called a modem (modulaordemodulaor). The hree basic mehods for modulaion are ampliude modulaion, frequency modulaion, and phase modulaion. Someimes a combinaion of one or more of hese basic mehods is used. We will someimes refer o a modulaion/demodulaion echnique as coheren or incoheren. A coheren modulaion echnique is one in which he phase of he signal is conrolled, and an incoheren modulaion echnique is one in which he phase of he signal is no conrolled. A coheren demodulaion scheme is one in which use is made of he phase of he signal. An incoheren demodulaion scheme makes no use of he phase of he signal. Anoher facor ha is imporan in all modulaion schemes is he rae a which he frequency specrum of he modulaed signal decays as frequency increases. I urns ou ha he rae of decay depends on he smoohness of he modulaed signal. Smooher ime funcions have frequency specra ha approach zero faser as frequency increases. Since ransmied signals are ofen resriced o a cerain frequency band, slow decay of he frequency specrum can lead o leakage of he signal ino oher bands. Generally modulaed signals wih jump disconinuiies decay as 5

7 one over he frequency, coninuous modulaed signals decay as one over he frequency squared, modulaed signals wih a coninuous derivaive decay as one over he frequency cubed, and so on. I is paricularly difficul o obain smoohness of he modulaed signal a he ransiions beween symbol periods. 3.1 Ampliude modulaion In ampliude modulaion he ampliude of he carrier wave is allowed o vary from inerval o inerval in order o specify he digial informaion. If only wo differen ampliudes are used, hen one ampliude would correspond o a 0 bi and he oher would correspond o a 1 bi. If four ampliudes are used, hen we could encode wo bis in each inerval. The four ampliudes would correspond o he bi paerns 00, 01, 10, and 11. Similarly, we can define higher order encoding schemes where 8 ampliudes could encode 3 bis per inerval, 16 ampliudes could encode 4 bis per inerval, ec. An ampliude modulaed signal could be wrien as follows x./ D a./ sin.2f c / (1) where a./ is consan for each symbol period and f c is he frequency of he carrier wave. Ampliude modulaion is more sensiive o noise han he oher echniques we will discuss. I is seldom used by iself, bu i is used in conjuncion wih oher modulaion echniques. An example of an ampliude modulaed signal involving wo ampliudes is shown in Figure Ampliude Time Figure 7: Ampliude modulaed signal corresponding o he bi sequence (101110). Ampliude modulaed signals are usually no coninuous a he swiching imes nt, n D 1;2;:::. An ampliude modulaed sine wave as in equaion (1) will be coninuous if he carrier wave is zero 6

8 a he edges of each symbol period. This will be rue if he carrier frequency f c is seleced so ha here are an inegral number of half-cycles in a symbol period, i.e., if f c D p for some ineger p. 3.2 Frequency modulaion In frequency modulaion he frequency of he signal is allowed o vary from inerval o inerval. The mos common ype of frequency modulaion for digial encoding is Frequency Shif Keying (FSK) in which only wo frequencies are used. In FSK one frequency corresponds o a zero bi and he oher frequency corresponds o a one bi. An FSK signal can be wrien as follows x./ D A cos 2 f c C m./f (2) where A is a fixed ampliude, f c is a cener frequency, f is a frequency incremen, and m./ is a digial funcion ha is eiher plus or minus one on each symbol inerval. Here he wo frequencies are f c f and f c C f. An example of an FSK signal is shown in Figure Ampliude Time Figure 8: Frequency modulaed signal corresponding o he bi sequence (101110). FSK signals are usually no smooh a he boundary beween inervals. In he nex secion we will see how FSK signals can be made coninuous by adding a variable phase angle. The usual mehod for demodulaing an FSK signal is o pass he signal hrough wo band pass filers cenered a he wo ransmission frequencies. The one having he larges oupu over a symbol period corresponds o he frequency used in ha symbol period. 7

9 3.3 Coninuous Phase FSK Modulaion (CPFSK) A coninuous phase FSK signal has he form x./ D A cos 2f c C 2f Z 0 m./ d : (3) where m is a digial message signal ha is consan on each ime inerval of widh T. The argumen of he cosine is coninuous and hence x./ is coninuous. If is in he inerval nt.nc 1/T, hen Z 0 Xn 1 m./ d D T m k C. nt /m n kd0 D m n C T Xn 1 m k nt m n kd0 where m k is he value of m./ in he inerval kt;.k C 1/T and n D 2f D m n C n =.2f / (4) T Subsiuing equaion (4) ino equaion (3), we ge Xn 1 m k nt m n : kd0 x./ D A cos.2f c C 2m n f C n / for nt.n C 1/T : (5) Thus, x./ is a coninuous FSK signal wih an added phase erm in each inerval. In a laer secion we will look a Minimum Shif Keying (MSK) which is an imporan subclass of CPFSK signals. 3.4 Phase modulaion Phase modulaion uses signals of he form x./ D A cos 2f c C./ (6) where./can akeon one of afiniese of values in each symbol period. The signal in equaion (6) can be wrien in he alernae form x./ D I./cos.2f c / C Q./ sin.2f c / (7) where I./ D A cos./ and Q./ D A sin./. A wo-dimensional plo of he possible pairs.i; Q/ is called a consellaion diagram. 8

10 Phase modulaed signals are usually demodulaed using a coheren scheme. Muliplying he signal x./ in equaion (7)bycos.2f c / and using rigonomeric formulas for double angles, we ge x./cos.2f c / D I./cos 2.2f c / C Q./sin.2f c /cos.2f c / D 1 I./Œ1 C cos.4f 2 c/ C 1 Q./ sin.4f 2 c/: (8) Passing x./cos.2f c / hrough a low-pass filer gives 1 I./. Muliplying he signal x./ by 2 sin.2f c / and using rigonomeric formulas for double angles, we ge x./sin.2f c / D I./sin.2f c /cos.2f c / C Q./ sin 2.2f c / D 1 I./sin.4f 2 c/ C 1 Q./Œ1 cos.4f 2 c/ : (9) Passing x./sin.2f c / hrough a low-pass filer gives 1 Q./. The closes poin o.i; Q/ in he 2 consellaion diagram is used o obain he symbol. One of he mos popular forms of phase modulaion is Quadraure Phase Shif Keying (QPSK). In QPSK he phase./ akes on one of he four values =4, 3=4, 5=4, or7=4 in each symbol period. The signals corresponding o hese four phase angles are A cos.2f c / sin.2f c / = p 2, A cos.2f c / sin.2f c / = p 2, A cos.2f c / C sin.2f c / = p 2,andA cos.2f c / C sin.2f c / = p 2.IfweleA D p 2, hen he pair.i; Q/ akes on he values.1; 1/,. 1; 1/,. 1; 1/,and.1; 1/. Each signal x./ corresponds o a unique pair.i; Q/. Figure 9 shows he poins corresponding o he pairs.i; Q/ for QPSK along wih he associaed symbols. Q I Figure 9: Consellaion diagram for QPSK modulaion. The poins in his consellaion diagram are ofen associaed wih poins in he complex plane. For phase shif keying he poins in he consellaion diagram all lie on a circle. QPSK can be easily generaed using equaion (7). Le d./ be a digial signal represening he bi sequence o be encoded. Choose d./ o be -1 on an inerval for a zero bi and +1 on an inerval for a one bi. Le d 0 ;d 1 :::: be he successive values of d./. Defined I./ o be a digial signal wih 9

11 symbol period represening he even values d 0 ;d 2 ;:::. Define d Q./ o be a digial signal wih symbol period represening he odd values d 1 ;d 3 ;:::. Figure 10 shows he signals d, d I,andd Q corresponding o he bi sequence The QPSK signal is obained by leing I./ D Ad I./= p 2 and Q./ D Ad Q./= p 2. d() d 0 d 1 d 5 d 6 d 7 d 2 d 3 d 4 d I () d 0 d 6 d 2 d 4 d Q () d 1 d 5 d 7 d 3 0 T 3T 4T 5T 6T 7T 8T Figure 10: Digial signals used in QPSK modulaion. The form of QPSK we have consruced has he same bi rae as he original, bu he symbol period is doubled. Doubling he symbol period has he effec of narrowing he frequency bandwidh of he signal. We could have kep he same symbol period for he QPSK signal and he bi rae would hen have doubled. An alernae form of QPSK uses he four phases 0, =2,, and3=2. QPSK is widely used in cable/dsl modems along wih he Quadraure Ampliude Mehods o be discussed laer. Higher order phase shif keying mehods can be consruced. Figure 11 shows phases ha could be used o represen 3 bis per symbol. This modulaion scheme is called 8-PSK. Some sysems use a modificaion of phase shif keying called Differenial Phase Shif Keying (DPSK). Insead of using phase angles relaive o a fixed sandard, DPSK uses phase angles relaive o he phase in he preceding symbol period. DPSK is simpler o implemen han ordinary PSK, bu has larger demodulaion errors. 10

12 Q I Figure 11: Consellaion diagram for 8-PSK modulaion. 3.5 Minimum Shif Keying (MSK) The goal of minimum shif keying is o obain a smooher signal and hus a faser decaying frequency specrum. There are several ways o approach MSK. We will consider MSK as a modificaion of QPSK. In QPSK we modulaed he carrier waves cos 2f c and sin 2f c by he digial signals d I./ and d Q./. Suppose ha we replace he square pulses in d I and d Q by half cycle sinusoidal pulses as in x./ D d I./ cos cos.2f c / C d Q./ sin sin.2f c / (10) where d I./ and d Q./ are he same digial signals used previously. The funcions d I and d Q are coninuous a odd muliples of T,andsin is zero a even muliples of T. Therefore, he second erm on he righ-hand-side of equaion (10) is coninuous. Since cos vanishes a odd muliples of T, he signal x./ will be coninuous if we can modify d I./ so ha i is coninuous a even muliples of T. This is easily done. We only need o shif d I./ o he lef by T as is illusraed in Figure 12. Wih his modificaion x./ is called an MSK signal. We will now show ha he MSK signal x./ is in fac a coninuous phase FSK signal. 11

13 d() d 0 d 1 d 5 d 6 d 7 d 2 d 3 d 4 d I () d 0 d 6 d 2 d 4 d Q () d 1 d 5 d 7 d 3 -T 0 T 3T 4T 5T 6T 7T 8T Figure 12: Shifed digial signal d I Since d I and d Q are eiher plus or minus one on each inerval, applying rigonomeric ideniies for sums and differences of angles o equaion (10)gives x./ DCcos.2f c / when d I./ DC1and d Q./ DC1 (11a) D cos.2f c / when d I./ D 1and d Q./ D 1 (11b) DCcos.2f c C / when d I./ DC1and d Q./ D 1 (11c) D cos.2f c C / when d I./ D 1and d Q./ DC1: (11d) These equaions can be wrien more concisely as x./ D d I./ cos 2f c d I./d Q./ D cos 2f c d I./d Q./ 2 4T C./ : (12) where./ D 1 d I./ =2. Thus, x./ is an FSK signal wih f D 1=4T and a phase on each inerval of 0 or. The name MSK is applied o his signal since f is he minimum frequency incremen ha will allow he signals corresponding o f c C f and f c f o be orhogonal over a symbol period. 12

14 Le us now look a he derivaive of x./. Differeniaing equaion (10), we obain Px./ D d I./ sin cos.2f c / 2f c cos sin.2f c / C d Q./ cos sin.2f c / C 2f c sin cos.2f c / h D d I./ i C d Q./2f c sin cos.2f c / h C d I./2f c C d Q./ i cos sin.2f c /: (13) Since sin is zero for an even muliple of T and cos is zero for an odd muliple of T, Px./ will be zero a all muliples of T if f c is chosen so ha cos.2f c / D 0 for an odd muliple of T sin.2f c / D 0 for an even muliple of T. These condiions will hold if f c is an odd muliple of f D 1=4T.Iff c is chosen in his way, hen he MSK signal will no only be coninuous, bu will have a coninuous derivaive. We menioned earlier ha he f used in MSK was he smalles incremen ha allowed he signals corresponding o f c C f and f c f o be orhogonal over every symbol period. Le us look now a he condiions necessary for orhogonaliy. Using rigonomeric ideniies for he sum and difference of wo angles we obain Z.nC1/T nt cos 2.f c C f / cos 2.f c f / d D 1 2 D 1 2 Z.nC1/T nt sin.4fc / 4f c C cos.4fc / C cos.4f / d sin.4f / 4f.nC1/T nt : (14) For orhogonaliy we need his inegral o vanish over each symbol period. I can be seen from equaion (14) ha he inegral will vanish if f c and f saisfy he condiions f c D p for some ineger p (15a) 4T f D q for some ineger q<p: (15b) 4T The smalles f saisfying he condiion (15b) is he one used in MSK. The process described for he MSK signal is no he only way o obain a smooh FSK signal. Consider an FSK signal x./ of he form x./ D r./cos 2f c C s./2f (16) where r./ and s./ are 1 on each symbol inerval. The only way we can hope o mach slopes a he boundary beween each pair of inervals is for he cosine erm in equaion (16) o have zero 13

15 slope a each boundary poin nt, n D 1;2;:::. The value of he cosine a he zero slope poins is 1. If we can mach he zero slopes, hen he funcion r./ can be chosen so as o mach he values a his boundary. Therefore, we wan he condiions cos 2.f c C f /nt D 1 cos 2.f c f /nt D 1 (17a) (17b) o hold for all n. These condiions will hold if f c and f saisfy he following relaions 2.f c C f /T D p for some ineger p (18a) 2.f c f /T D q for some ineger q<p: (18b) By adding and subracing equaions (18a)and(18b) we obain f c D p C q (19) 4T f D p q 4T : (20) Figure 13 shows a smooh FSK signal corresponding o he parameers p D 8, q D 4, andt D 1. MSK corresponds o he cases where p D q C 1. comparing he condiions (19) and(20) wih he condiions (15a) and(15b) we see ha he wo frequency signals will be orhogonal on every symbol period. The orhogonaliy can be used o separae he wo frequencies in he demodulaion process Ampliude Time Figure 13: Smooh FSK signal corresponding o he bi sequence (101110). Le us now look a he demodulaion of an MSK signal. An MSK signal can be demodulaed using a coheren scheme. Muliplying equaion (10) bycos.2f c / and using rigonomeric ideniies 14

16 for double angles, we ge x./cos.2f c / D 1 d 2 I./ cos Œ1 C cos.4f c / C 1 2 d Q./ sin sin.4f c /: (21) Passing x./cos.2f c / hrough a low pass filer, we obain he signal x c./ defined by x c./ D 1 d 2 I./ cos : (22) If we now inegrae x c./ over an inerval Œ.2k 1/T;.2k C 1/T, we obain Z.2kC1/T.2k 1/T x c./ d D 1 d 2 I.2kT / ˇˇˇˇ.2kC1/T sin.2k 1/T D 2d I.2kT / T. 1/k : (23) From his resul we can obain d I.2kT /. Similarly, muliplicaion of equaion (10)bysin.2f c / yields x./sin.2f c / D 1 d 2 I./ cos sin.4f c / C 1 2 d Q./ sin Œ1 C cos.4f c / : (24) Passing x./sin.2f c / hrough a low pass filer, we obain he signal x s./ defined by x s./ D 1 d 2 Q./ sin : (25) If we now inegrae x s./ over an inerval Œ2kT;.2k C 2/T, we obain Z.2kC2/T.2kC2/T x s./ d D 1 d 2 Q.2k C 1/T 2kT cos T ˇˇˇˇ D 2d Q.2k C 1/T 2kT. 1/k : (26) From his resul we can obain d Q.2k C 1/T. The original digial sequence can be reconsruced from he d I and d Q values. There is a modificaion of MSK called Gaussian Minimum Shif Keying (GMSK) ha is used in a number of sysems. GMSK follows he same process as MSK excep ha he modulaing digial signals are smoohed wih a Gaussian filer. The process is picured in Figure 14. The smoohing decreases he bandwidh and he iner-channel inerference, bu i increases he iner-symbol inerference. The demodulaion of a GMSK signal is similar o ha used for an MSK signal. GMSK is used in a number of cellular devices as well as Blueooh devices for shor range conneciviy. 15

17 () d I Gaussian Filer π Cos 2 T Cos2πf c d() x() d Q () Gaussian Filer π Sin 2 T Sin2πf c Figure 14: Generaion of a GMSK signal 3.6 Quadraure Ampliude Modulaion (QAM) Ampliude and phase modulaion are ofen combined. In Quadraure Ampliude Modulaion (QAM) he signal has he form s./ D I./cos.2f c / C Q./ sin.2f c / (27) where I./ and Q./ are consan on each symbol period. The signal given in equaion (27) can be wrien in he alernae form s./ D a./ cos 2f c./ (28) where a./ D p I 2./ C Q 2./, cos D I./=a./, andsin D Q./=a./. In his form we see ha QAM can be considered as an ampliude and phase modulaion scheme. Phase modulaion schemes such as QPSK are special cases of QAM. The I -Q represenaion in equaion (27) is usually preferred over he ampliude-phase represenaion since he signal is easier o generae in his form. The I and Q values corresponding o differen symbols are usually represened by a recangular grid of poins in he I -Q plane. The QAM scheme based on he diagram in Figure 15 consiss of 16 poins and is called 16-QAM. This modulaion scheme encodes 4 bis per symbol. 16

18 Q I Figure 15: Consellaion diagram for 16-QAM modulaion. QAM signals can be demodulaed using a coheren scheme like ha used for phase shif keying. Muliplying he signal s./ by cos.2f c / and using rigonomeric formulas for double angles, we ge s./ cos.2f c / D I./cos 2.2f c / C Q./ sin.2f c /cos.2f c / D 1 I./Œ1 C cos.4f 2 c/ C 1 Q./ sin.4f 2 c/: (29) Passing s./ cos.2f c / hrough a low pass filer, we ge 1 I./. Similarly, muliplicaion of he 2 signal by sin.2f c / gives s./ sin.2f c / D I./cos.2f c /sin.2f c / C Q./ sin 2.2f c / D 1 I./sin.4f 2 c/ C 1 Q./Œ1 cos.4f 2 c/ : (30) Passing s./ sin.2f c / hrough a low pass filer, we ge 1 Q./. The symbol of he closes poin o 2.I; Q/ in he consellaion diagram is aken as he decoded symbol. The modulaion schemes QPSK, 16-QAM, 64-QAM (8 8 grid), and 256-QAM (16 16 grid) are widely used in cable/dsl modems. The higher order QAM modulaion schemes provide higher bi raes, bu require higher signal-o-noise raios in order o work correcly. Ofen he order of he modulaion scheme is chosen adapively depending on he qualiy of he ransmission channel. 3.7 Orhogonal Frequency Division Muliplexing (OFDM) Orhogonal Frequency Division Muliplexing (OFDM) is probably he mos complicaed of he mehods described in his paper, bu i is widely used in wireless devices. OFDM is based on 17

19 he Discree Fourier Transform (DFT). For a sequence of complex values x 0 ;x 1 ;:::;x N 1 he Discree Fourier Transform X 0 ;X 1 ;:::;X N 1 of his sequence is defined by I can be shown ha X n D x m D 1 N NX 1 md0 NX 1 nd0 x m e i2mn=n : (31) X n e i2mn=n : (32) This relaion is called he inverse DFT. The DFT and inverse DFT can be compued rapidly using Fas Fourier Transform (FFT) algorihms or devices. The sequence fx m g is usually considered o be in he ime domain, and he sequence fx n g is usually considered o be in he frequency domain. If T is he symbol period, we can wrie equaions (31)and(32) as follows x. m / D 1 N X.f n / D NX 1 nd0 NX 1 md0 X.f n /e i2f n m D x m (33a) x. m /e i2f n m D X n (33b) where m D m, f n D nf, D T=N,andf D 1=T. Thus, he x. m / values can be considered as sampled values of he ime funcion x./ given by x./ D 1 N NX 1 nd0 X.f n /e i2f n : (34) I can be shown ha he funcions fe i2f n g are orhogonal over each symbol period, i.e., Z.kC1/T kt e i2f m e i2f n d D 0 m n: (35) In he OFDM mehod he symbols are represened by complex values ha are hen used for he frequency componens X n in equaion (32). The consellaion diagrams inroduced in connecion wih he PSK and QAM modulaion schemes can be looked upon as defining mappings beween symbols and complex values. Using one of he consellaion diagrams we assign he complex number corresponding o he firs symbol o X 0, he complex number corresponding o he second symbol o X 1, and so on unil he complex number corresponding o he N -h symbol is assigned o X N 1. We hen obain a sequence of ime values fx m g by means of he inverse DFT defined in equaion (32). This sequence is in general complex. The real and imaginary pars of his sequence can be convered o coninuous funcions of ime on he inerval T using a digial-oanalog converer (DAC). The real and imaginary funcions of ime on each symbol period are ofen used o modulae carrier signals cos.2f c /and sin.2f c /respecively. A diagram of he process isshowninfigure16. 18

20 X 0 real DAC cos (2π f c ) Bi sequence Serial o parallel X 1... FFT -1 s() X N-2 X N-1 imag DAC sin (2πf c ) Figure 16: Diagram showing he seps in Orhogonal Frequency Division Muliplexing. We can hink of he frequency componens X n as represening N parallel channels. The firs N symbols produce a ime signal on he firs symbol period, he nex N symbols produce a ime signal onhesecond symbolperiod,... Thus,he firssep inheofdmprocessamouns o convering a linear sequence of bis ino N parallel sequences. Since his parallelism allows a large number of bis o be handled a each sep, he symbol period used is ofen aken o be larger han is used in oher mehods. To demodulae an OFDM signal we firs shif he signal down o baseband using he same echnique used for QPSK. The signal s./ is given by s./ D u./ cos.2f c / C v./ sin.2f c / (36) where u./ and v./ are he real and imaginary pars of he complex signal x./. Muliplying s./ by cos.2f c /, we obain s./ cos.2f c / D u./ cos 2.2f c / C v./ sin.2f c /cos.2f c / D 1 u./œ1 C cos.22f 2 c/ C 1 v./ sin.22f 2 c/: (37) Thus, passing s./ cos.2f c / hrough a low-pass filer gives us 1 u./. Similarly, 2 s./ sin.2f c / D 1 u./ sin.22f 2 c/ C 1 v./œ1 cos.22f 2 c/ : (38) Thus, passing s./ sin.2f c / hrough a low-pass filer gives us 1 v./. Havingu./ and v./, we 2 can form x./ D u./ C iv./. Sampling x./ a he imes 0;T=N;:::;.N 1/T =N, we obain x 0 ;x 1 ;:::;x N 1. Using he DFT defined by equaion (31), we can compue X 0 ;X 1 ;:::;X N 1. The mapping represened by he consellaion diagram can hen be used o decode he symbols. Finally, he symbols can be srung ogeher o obain he ransmied bi sequence. OFDM is widely used in wireless communicaion devices. For example, wireless rouers conforming o he specificaions IEEE a, IEEE g, and he proposed IEEE n make use of OFDM. Some of he advanages of he OFDM mehod are ha i makes very efficien use of he frequency specrum, i has very lile inerference beween symbols, and i has greaer resisance o mulipah disorion han mos oher mehods. 19

21 3.8 Frequency Hopping Spread Specrum (FHSS) Frequency Hopping Spread Specrum (FHSS) is a echnique used in many wireless devices in order o spread he ransmied informaion over a wider frequency band. There are several reasons why his spreading is desirable. (1) Spread-specrum signals are highly resisan o narrow-band noise and jamming. (2) Spread-specrum signals are difficul o inercep. They end o look like background noise. (3) Spread-specrum signals can share a frequency band wih many ypes of convenional ransmissions wih minimal inerference. In FHSS he carrier frequency hops over a predeermined bu random-like sequence of frequencies. The paern of frequency hops mus be known by boh he ransmier and he receiver. If he period of he frequency hopping is shorer han he symbol period (fas hopping), hen here is a buil in redundancy since each symbol will occur in more han one carrier. This redundancy is useful in reducing he effecs of narrow band noise and jamming. The main disadvanage o fas hopping is ha coheren deecion is difficul and seldom used. If he period of he frequency hopping is greaer han he symbol period (slow hopping), hen coheren deecion schemes are feasible and he implemenaion is easier. The disadvanage of slow hopping is ha narrow-band noise can desroy one or more bis of informaion making error correcing codes almos a necessiy. Any of he echniques discussed previously can be used o modulae he carrier signals. FHSS allows muliple devices wih differen hopping codes o operae simulaneously. Since he hopping paern is random like, his echnique offers some securiy agains eavesdropping. Frequency Hopping Spread Specrum (FHSS) is used in a number of wireless devices including Blueooh devices. 3.9 Direc-Sequence Spread Specrum (DSSS) Direc Sequence Spread Specrum (DSSS) is anoher echnique for spreading he digial informaion over a wider frequency band. Suppose he bi sequence is encoded in a signal m./ consising of a sequence of square pulses having period T and ampliudes 1. Le d./ be anoher signal consising of 1 ampliude square pulses bu having a much smaller period T=N for some large N. Typically N is en or more. The predeermined sequence of 1 values in d./ is chosen o have a random-like paern. The signal s./ is obained by muliplying m./ and d./, i.e., s./ D m./d./: (39) The signal d./ mus be known by boh he ransmier and receiver. Ofen he paern of plus and minus ones in d./ is generaed by an algorihm based on some shared seed. The signal s./ has a noise-like appearance and hus provides some proecion agains eavesdropping. Figure 17 shows an example of he various signals ha make up a DSSS signal. 20

22 1 m() -1 1 d() -1 1 s() -1 Figure 17: Example of Direc Sequence Spread Specrum. Since d./ only akes on he values 1, muliplying d by iself produces a funcion ha is idenically one. Therefore, muliplicaion of equaion (39) byd./ gives m./ D s./d./. Thus, decoding of his process is very simple. There is an inheren redundancy in DSSS ha is useful in overcoming narrow band noise and jamming. Since he resuling signal in DSSS is also digial, his mehod can also be combined wih any of he modulaion echniques discussed previously. DSSS is used in a number of wireless applicaions. For example, wireless rouers conforming o he specificaion IEEE b use DSSS and wireless rouers conforming o he specificaion IEEE g use DSSS and OFDM Code Division Muliple Access (CDMA) Analog cell phones divide he frequency band up ino a number of smaller sub bands called channels and assign each user a differen channel. Digial cell phones commonly use a echnique called Code Division Muliple Access (CDMA) ha allows each user o use he whole frequency band. CDMA is a form of DSSS in which each user is assigned a differen pseudo random signal o muliply he original digial signal. The pseudo random signals assigned o differen users are chosen so ha he correlaion beween hem is small. The DSSS signals are used o modulae a high frequency carrier. QPSK is ofen used for his modulaion. 21