Principles of Digital Data Transmission Chapter 7

Principles of Digital Data Transmission Chapter 7 Lectured by Dr. Yun Q. Shi Dept. of Electrical & Computer Engr. New Jersey Institute of Technology [email protected] Text used for the course: , 4th Edition, Lathi and Ding, Oxford

Introduction • A significant portion of communication in 1990s was in analog. • It has been replaced rapidly by digital communication. • Now most of the communications become digital, with analog communication playing a minor role. • This chapter addresses the framework (several major aspects) of digital data transmission • All we learnt in this course so far is utilized here. • Hence, this chapter is naturally a conclusion of this course.

Dr. Shi

Lathi & Ding-Digital Commun

2

1

Digital Communication Systems: Source • Digital Communication Systems: Figure 7.1 Source encoder, Baseband modulation (Line coding), Digital carrier modulation, Multiplexer, Channel, Regenerative repeater, Receiver (detection).

• Source:

The input

– A sequence of digits – We discuss mainly: The binary case (two symbols) – Later in this chapter: The M-ary case (M symbols) more general case Dr. Shi


3

Figure 7.1 Fundamental building blocks of digital communication systems

Figure 7.1 Fundamental building blocks of digital communication systems

Dr. Shi


4

2

Line Coder (Transmission Coder) • The output of a multiplexer is coded into electrical pulses or waveforms for the purpose of transmission over the channel. • This process is called a line coding or transmission coding • Many possible ways of assigning waveforms (pulses) to the digital data. 1  

• On-off: Dr. Shi

↔ a 0 ↔

pulse p (t) no pulse


5

Line Coder … • Polar:  1 ↔ ↔ 0

a a

pulse pulse

p (t) − p (t)

• Bipolar (pseudoternary or alternate mark inversion (AMI))

1 ↔  0 ↔

a pulse p (t) or - p (t) depending on whether the previous 1 is encoded by -p (t) or p (t) no pulse

In short, pulses representing consecutive 1’s alternate in sign. • All the above three could use half-width pulses. It is possible to select other width. • Figure 7.1 (parts a, b and c). Dr. Shi


6

3

Line Codes

Dr. Shi


7

Line Coder … • Full width pulses are often used in some applications. – i.e., the pulse amplitude is held to a constant value throughout the pulse interval (it does not have a chance to go to zero before the next pulse begins). – These schemes are called non return-to-zero (NRZ) in contrast to return-to-zero (RZ) – Figure 7.2 shows • An On-off NRZ signal • A Polar NRZ signal in d and e parts of the figure Dr. Shi


8

4

Multiplexer • Usually, the capacity of a practical channel >> the data rate of individual sources. • To utilize this capacity effectively, we combine several sources through a digital multiplexer using the process of interleaving. • One way: A channel is time-shared by several messages simultaneously. Dr. Shi


9

Regenerative Repeater • Used at regularly spaced intervals along a digital transmission line to: 1) detect the incoming digital signal and 2) regenerate new clean pulses for further transmission along the line. • This process periodically eliminates, and thereby combats the accumulation of noise and signal distortion along the transmission path. • The periodic timing information (the clock signal at R b Hz) is required to sample the incoming signal at a repeater. • R b (rate): pulses/sec Dr. Shi


10

5

Regenerative Repeater… • The clock signal can be extracted from the received signal. – e.g., the polar signal when rectified ⇒ a clock signal at Rb Hz • The on-off signal = A periodic signal at Rb + a polar signal (Figure 7.) • When the periodic signal is applied to a resonant circuit tuned to Rb Hz, the output, a sinusoid of Rb Hz, can be used for timing. Dr. Shi


11

On-off Signal Decomposition

Dr. Shi


12

6

Regenerative Repeater… rectified

• The bipolar signal an on-off signal ⇒ the clock signal can also be extracted. • The timing signal (the output of the resonant circuit) is sensitive to the incoming pattern, sometimes. – e.g. in on-off, or   0 bipolar 

↔

no pulse

• If there are too many zeros in a sequence, will have problem. – no signal at the input of the resonant circuit for a while – sinusoids output of the resonant circuit starts decaying

• Polar scheme has no such a problem. Dr. Shi


13

Transparent Line Code • A line code in which the bit pattern does not affect the accuracy of the timing information is said to be a transparent line code. – The polar scheme is transparent. – The on-off and bipolar are not transparent (nontransparent).

Dr. Shi


14

7

Desired Properties of Line Codes 1.

Transmission bandwidth: as small as possible

2.

Power efficiency: for a specific bandwidth and detection error rate, transmitted power should be as small as possible.

3.

Error detection and correction capability: as strong as possible

4.

Favorable power spectral density: desirable to have zero PSD at (dc), called DC null, because of ac coupling requires this. If at dc, ω = 0 , PSD ≠ 0 , then dc wanders in the pulse stream, which is not desired.

5.

Adequate timing content: should be possible to extract the clock signal (timing information).

6.

Transparency Dr. Shi


15

PSD of Various Line Codes • Consider a general PAM signal, as shown in Figure 7.4 (b): Tb = 1 t = kTb Rb Rb – the k th pulse in the pulse train y (t ) = ak p (t ) • p (t): the basic pulse • P (ω) [P(f)]: Fourier spectrum of p (t) • a k: arbitrary and random

– The on-off, polar, bipolar line codes are all special cases of the general pulse train y(t), a k = 0, +1, -1 Dr. Shi


16

8

A Random PAM signal

All 3 line codes can be represented this way

New approach to determine PSD of y(t)

Dr. Shi


17

PSD of Various Line Codes … • How to determine the PSD? – Figure 7.4 – An attractive general approach Rx (τ ), S x (ω ) need to be studied once Then for different p(t), ⇒ different PSD – x(t): an impulse train – xˆ (t ) : a rectangular pulse train ( ε ⋅ hk = a k ) when ε → 0, xˆ (t ) → x(t ) (Figure 7.5) Dr. Shi


18

9

Dr. Shi


19

PSD of Various Line Codes … R0 = ak2

⇒ time average of a k2

R1 = ak .ak +1 ,L L, Rn = ak .ak +n

1 RX (τ ) = Tb S X (ω ) =

1 Tb

∞

∑ R δ (τ − nT )

n = −∞

n

b

∞

∑ Rn e− jnωTb =

n = −∞

1 Tb

∞   [ ] R + 2 R cos( n ω T ) ∑ 0 n b   n =1  

S y (ω ) = P (ω ) S X (ω ) 2

Different line codes Dr. Shi

⇒ different P(ω ) ⇒ different S y (ω ) Lathi & Ding-Digital Commun

20

10

Polar Signaling 1 ↔ p(t) 0 ↔ -p(t)

} ⋅ ak = ±1, equally

R0 = ak2 = 1

R1 = ak .ak +1 = 0 Rn = 0, n ≥ 1

likely, ak2 = 1

Q ak .ak +1 = 1 or − 1 equally likely Similar reasoning

S y (ω ) = P (ω ) .S X (ω ) 2 1 = P(ω ) . Tb 2

Dr. Shi


21

Polar Signaling… • To be specific, assume p(t) is a rectangular pulse of width T b (a half-width rectangular 2 pulse)    t  p (t ) = re c t    T b   2  T  ω Tb  P ( ω ) = b s in c   2  4  T  ω Tb  S y ( ω ) = b s in c 2   4  4  Dr. Shi


1st zero − cros sin g :

ω⋅Tb 4

= π ⇒ω =

or ⇒ f =

4π Tb

2 = 2Rb Tb 22

11

PSD of polar signal (half-width rectangle)

Dr. Shi


23

Polar Signaling … • Comment 1:

• • •

•

The essential bandwidth of the signal (main lobe) = 2 Rb For a full-width pulse, ⇒ Rb “Polar Signaling is not bandwidth efficient.” Comment 2: “No error-detection or error-correction capability.” Comment 3: “Non-zero PSD at dc( ω = 0)” ⇒ This will rule out the use of ac coupling in transmission. Comment 4: “Most efficient scheme from the power requirement viewpoint” for a given power, the detection-error probability for a polar scheme is the smallest possible. Comment 5: “Transparent” Dr. Shi


24

12

Achieving DC Null in PSD by Pulse Shaping Q P (ω ) =

∫ ∴ P (0) = ∫

∞

−∞

∞

−∞

p ( t ) e − j ω t dt p ( t )dt

⇒ If the area under p(t) = 0, P(0) = 0 .

Dr. Shi


25

Manchester (split-phase, twinned-binary) Signal

Dr. Shi


26

13

On-off Signaling • 1 ↔ p(t) 0 ↔ no pulse R

0

R

n

S

X

=

1 2 1 = 4

=

(ω ) =

∞

1 1 + 2Tb 4Tb

1 1 + 4Tb 4Tb

Dr. Shi

n ≥ 1

for all

∞

∑

e

∑

e

− jn ω T b

n = −∞ n ≠0

− jn ω T b

n = −∞


27

On-off Signaling S X (ω ) =

1 2π + 4Tb 4 T b2

∞

∑

n = −∞



δ  ω − 

2π n Tb

  

P (ω )   2π ∞ 2π n    S y (ω ) = 1 + δ  ω −  ∑ 4Tb  Tb n = −∞  Tb   For a half-width rectangular pulse, 2

S y (ω ) =

Tb 2π  ω Tb   sinc 2   1 + T 16 4    b

∞

∑

n = −∞



δ ω − 

2π n    Tb  

Note: 1. For some derivation, refer to the next four pages. 2. PSD is shown in Figure 7.8. 3. PSD consists of both a discrete part and a continuous part. Dr. Shi


28

14

Some Derivation (Lathi’s book, pp. 58-59, Example 2.12) • A unit impulse train: FS

g (t) =

∞

∑

n = −∞

D ne

jn ω b t

δ T (t ) = CombT (t ) = g (t ) 0

= =

0

1 Tb

∞

∑

e

n = −∞

1 2 + Tb Tb

jn ω b t ∞

∑

n = −∞

cos( n ω b t )

1 ∞ ∑ δ ( f − nf 0 ) T 0 n = −∞ 2π ∞ 2π G ( ω ) = FT { g ( t )} = ) δ (ω − n ∑ T 0 n = −∞ T0 G ( f ) = FT { g ( t )} =

≅ ω 0 COMB Dr. Shi

ω0

(ω )


29

Example 2.12

Dr. Shi


30

15

Some Derivation… • Fact 1 (Lathi’s book, p.83, Eq. (3.20a, 3.20b): 1↔ δ(f

)

e j 2πf bt ↔ δ ( f − f b )

• Fact 2:

[e

jω bt

]

↔ δ (ω − ωb )

Also,

(t ) ↔ 1 δ (t − nT b ) ↔ δ

Dr. Shi

e

− jn ω T

b


31

Some Derivation … • Fact 3:

(

)

∞ ∑ δ t − nT b n1=4−∞ 42443

FS =

∞ jnω t 1 2π b ω = ∑ e b T n = −∞ T b 1b4 44442444443

b FT 644744 8 ∞ − jnω t b ∑e n = −∞ Dr. Shi

b FT =

ω

(

∞ ∑ δ ω − nω b b n = −∞


) 32

16

PSD of An On-off Signal

Dr. Shi


33

On-off Signaling • This is reasonable since, as shown in Fig. 7.2, an on-off signal can be expressed as the sum of a polar and a periodic component • Comment 1: For a given transmitted power, it is less immune to noise interference. – Q Noise immunity ∝ difference of amplitudes representing binary 0 and 1. – If a pulse of amplitude 1 or –1 has energy E, then a pulse of amplitude 2 has energy (2)2E = 4E.

Dr. Shi


34

17

On-off Signaling … • For polar:Q

1 Tb

∴ • For on-off:

digits are transmitted per second polar signal power  1  2E  = power = 4 E  2 T  b  Tb

1 E = E   =  Tb  Tb

Power is now as large as twice of above.

• Comment 2: Not transparent Dr. Shi


35

Bipolar Signaling (Pseudoternary or Alternate Mark Inverted (AMI)) A. 0 ↔ no pulse 1 ↔ p(t) or –p(t) depending on whether the previous 1 was transmitted by –p(t) or p(t) B. [p(t), 0 , -p(t)]: In reality, it is ternary signaling. C. Merit: a dc null in PSD demerit: not transparent Dr. Shi


36

18

Bipolar Signaling… 1 N N  1 ( ± 1) 2 + (0)  =  N →∞ N 2 2  2 k 1 N 3N 1  R (1) = lim ( − 1) + (0) = −  N →∞ N 4 4 4  111, 101, 110, 011, 010, 001, 000 ak ak + n R (2)

D. R ( 0 ) = Nlim →∞

R ( 2 ) = lim

N →∞

R n = lim

N →∞

1 N

1 N

∑a

2 k

= lim

1 N N 5N  (1) + ( − 1) + (0)  = 0  N 8 8 8 

∑a

k

a k + n = 0, n > 1

k

Dr. Shi


37

Bipolar Signaling… E.

S y (ω ) = P(ω ) S X (ω ) 2 P (ω )  ∞  R n e − jnω Tb  = ∑  Tb  n = −∞  2

P (ω ) = Tb

2

P (ω ) = Tb P (ω ) = 2 ⋅ Tb Dr. Shi

2

∞   R + 2 R n cos nω Tb  ∑  0 n =1   2

(7.10b)

(7.10c)

1 1   + 2 ⋅ ( − ) cos ω T b  4 2 

P (ω ) (1 − co s ω Tb ) = Tb

2


 ω Tb  sin 2    2  38

19

Bipolar Sampling … ⇒

As ω=0 (dc) regardless of P(ω) A dc null (desirable for ac coupling)

S y (ω ) = 0

sin 2 (

ω Tb 2

)= 0

at

ω =

⇒ b a n d w id th = R b H z ,

2π 1 ,∴ f = = Rb Tb Tb reg a rd less

For a half-width rectangular pulse,

Dr. Shi

⇒ Zero:

of

P (ω )

  t Tb , P (t ) = rect  2  T b  2   ω Tb P (ω ) = s in c  2  4  T  b

          


S y (ω ) =

39

Tb  ωT   ωT  sinc 2  b  sin 2  b  4  4   2 

ɷ1Tb=4π 4π ω1 = Tb f1 = 2 Rb

ɷ2Tb=2π 2π ω2 = Tb f 2 = Rb

⇒

1 Rb =    Tb  essential ⋅ bandwidth

Half that of polar or on-off signaling. Twice that of theoretical minimum bandwidth (channel b/w). Table 3.1:

Dr. Shi

  ω  t   ωτ  rect   ↔ τ sin c   = τ sin c  2  τ   2    τ  Lathi & Ding-Digital Commun

40

20

Dr. Shi


41

Merits of Bipolar Signaling 1.Spectrum has a dc dull. 2.Bandwidth is not excessive. 3.It has single-error-detection capability. Since, a single detection error ⇒ a violation of the alternation pulse rule.

Dr. Shi


42

21

Demerits of Bipolar Signaling 1. Requires twice as much power as a polar signaling. Distinction between A, -A, 0 vs. Distinction between A/2, -A/2 2. Not transparent Dr. Shi


43

Pulse Shaping S y (ω ) = P (ω ) S X (ω ) 2

=

P (ω ) Tb

2

( R0 + 2

∞

∑R

n

cos nωTb )

[Slide 19]

n =1

• Different signaling (line coding, SX (ω) ) ⇒ diff.S y (ω ) • Different pulse shaping (P(ω)) ⇒ diff. S y (ω ) An additional factor. • Intersymbol Interference (ISI) (refer to the next slide) – Time-limited pulses (i.e., truncation in time domain) not band-limited in frequency domain – Not time-limited, causes problem  band-limited signal (i.e., truncation in frequency domain)

Dr. Shi


44

22

Intersymbol Interference (ISI) in Detection B. Sklar’s Digital Communications 1988

Dr. Shi


Back to slide 44

45

Phase Shaping…

Intersymbol Interference (ISI) • Whether we begin with time-limited pulses or band-limited pulses, it appears that ISI cannot be avoided. An inherent problem in the finite bandwidth transmission. • Fortunately, there is a way to get away: be able to detect pulse amplitudes correctly. Dr. Shi


46

23

Nyquist Criterion for Zero ISI • The first method proposed by Nyquist. 1, t = 0    1  p (t ) =  0, t = ± nTb Tb =  Rb 

(7.22)

T b : separation between successive transmitted pulses

• Example 6.1 (p.256) p(t ) = sinc(2π Bt ),

B=

1 1 = Rb 2Tb 2

Since 2πBTb=π, Tb is the 1st zero crossing.

• Minimum bandwidth pulse that satisfies Nyquist Criterion Fig. 7.10 (b and c) Dr. Shi


47

Minimum Bandwidth Pulse (Nyquist Criterion)

Dr. Shi


48

24

Nyquist Criterion for Zero ISI… 1 p (t ) = sinc (π Rb t ) =  0

P (ω ) =

 ω 1 rect  Rb  2π R b

Table 3.1:

t=0 t = ± nTb

  

 1   Tb =  Rb  

bandwidth =

Rb 2

FT

 ω  Sinc (Wt ) ↔ rect   π  2W 

W

Dr. Shi


49

Problems of This Method •

Impractical 1. − ∞ < t < ∞ 2. It decays too slowly at a rate 1/t. ⇒ Any small timing problem (deviation) ⇒ ISI

•

Solution: Find a pulse satisfying Nyquist criterion (7.22), but decays faster than 1/t.

•

Nyquist: Such a pulse requires a bandwidth k⋅ Dr. Shi

Rb , 2

1≤ k ≤ 2


50

25

p (t ) ↔ P (ω )

• Let

– The bandwidth of P(ω) is in (Rb /2, Rb) – p(t) satisfies Nyquist criterion Equation (7.22)

• Sampling p(t) every Tb seconds by multiplying p(t) by an impulse train δ T (t ). b

⇒ b ⇒

p (t )δ

p (t) =

T

b

(t) = δ (t)

FT 1 T b

∞

∑

P (ω

− n ω

b

P (ω

− n ω

n = −∞

Equation G (ω ) =

) = 1

( 6 .4 ) 1 T s

∞

∑

n = −∞

s

)

Dr. Shi


51

Dr. Shi


52

26

Derivation

• |P(ω)| is odd symmetric in y1-o-y2 system. Dr. Shi


53

Derivation… ω

b • The bandwidth of P(ω) is 2 + ω x – ω x : the excess bandwidth

–

γ

roll − offfactor

=

=

excess bandwidth theoretica l min imum bandwidth

ωx ω = 2⋅ x ωb ωb

0 ≤ r ≤1

2 R

rR

R

• The bandwidth of P(ω) is BT = 2b + 2b = ( 1 + γ ) 2b • P(ω), thus derived, is called a Vestigial Spectrum. Dr. Shi


54

27

Realizability • A physically realizable system, h(t), must be causal, i.e., h(t) = 0 , for t < 0. (The n.c. & s.c.) • In the frequency domain, the n.c. & s.c. is known as Paley-Wiener criterion ∞

ln H (ω )

−∞

1+ω 2

∫

dω < ∞

• Note that, for a physically realizable system, H(ω) may be zero at some discrete frequencies. But, it cannot be zero over any finite band • So, ideal filters are clearly unrealizable. Dr. Shi


55

• From this point of view, the vestigial spectrum P(ω) is unrealizable. • However, since the vestigial roll-off characteristic is gradual, it can be more closely approximated by a practical filter. • One family of spectra that satisfies the Nyquist criterion is   ωb    1 / 2    P (ω ) =  0   1    

Dr. Shi

)  π (ω −  2   1 − sin   2ω x         


ω− ω > ω <

ωb 2

ωb 2

ωb 2

< ωx +ωx −ωx

56

28

Realizability … ωx = 0 ω ωx = b 4

ωx =

ωb 2

 (r = 0)   ( r = 0 .5 )  ( r = 1)   

Shown in the figure on next slide

• Comment 1: Increasing ω x (or r) improves p(t), i.e., more gradual cutoff reduces the oscillatory nature of p(t), p(t) decays more rapidly. Dr. Shi


57

Pulses satisfying Nyquist Criterion

Dr. Shi


58

29

Realizability… • Comment 2: As

γ

= 1, i.e., ω x

=

ωb 2

 ω  ω  1 rect   P (ω ) = 1 + cos 2 2 Rb   4πRb 

 ω   ω  rect   = cos 2   4 Rb   4πRb 

• This characteristic is known as: The raised-cosine characteristic The full-cosine roll-off characteristic Dr. Shi


59

Realizability… p (t ) = Rb

A. B. C.

D.

co s(π R b t ) S in c (π R b t ) 1 − 4 R 2bt 2

Bandwidth is Rb (γ =1) p(0) = Rb p(t) = 0 at all the signaling instants & at points midway between all the signaling instants p(t) decays rapidly as 1/t3 ⇒ relatively insensitive to derivations of Rb , sampling rate, timing jitter and so on. Dr. Shi


60

30

Realizability… E. Closely realizable F. Can also be used as a duobinary pulse. G. H c (ω ) : channel transfer factor Pi ( k ) : transmitted pulse Pi (ω ) ⋅ H c (ω ) = P (ω )

Received pulse at the detector input should be P(ω) [p(t)]. Dr. Shi


61

Signaling with Controlled ISI: Partial Response Signals • The second method proposed by Nyquist to overcome ISI. n = 0,1 1 • Duobinary pulse: p(nTb ) = 0 for all other n

Dr. Shi


62

31

Signaling with Controlled ISI… • Use polar signaling 1 ↔ p (t ) 0 ↔ − p (t )

• The received signal is sampled at t = nTb p(t) = 0 for all n except n= 0,1. • Clearly, such a pulse causes zero ISI with all the pulses except the succeeding pulses. Dr. Shi


63

Signaling with Controlled ISI… • Consider two such successive pulses located at 0 and Tb. – If both pulses are positive, the sample value at t = Tb ⇒ 2. – If both pulses are negative, ⇒ -2 – If pulses are of opposite polarity, ⇒ 0

• Decision Rule: If the sample value at t = Tb is • Positive ⇒ 1, 1 • Negative ⇒ 0, 0 • Zero ⇒ 0, 1 or 1, 0

Dr. Shi


64

32

Signaling with Controlled ISI… • Table 7.1

Transmit ed Seq. Sample of x(Tb) Detected seq.

Dr. Shi

1 1 0 1 1 0 0 0 1 0 1 1 1 1 2 0 0 2 0 -2 -2 0 0 0 2 2 1 1 0 1 1 0 0 0 1 0 1 1 1


65

Signaling with Controlled ISI: Duobinary Pulses • If the pulse bandwidth is restricted to be it can be shown that the p(t) must be: p (t) =

Rb 2

sin( π R b t ) π R b t (1 − R b t )

 ω 2 P (ω ) = cos  Rb  2 Rb

  rect 

 ω   2π R b

 −  e 

j

ω 2 Rb

ωb

• Note: The raised-cosine pulse with ω = 2 , ( r = 1), satisfies the condition (7.3b) to be signaling with controlled ISI – refer to Figure 7.13 (slides 57). x

Dr. Shi


66

33

Signaling with Controlled ISI Duobinary Pulses…

Dr. Shi


67

Use of Differential Coding • For the controlled ISI method, 0-valued sample ⇒ 0 to 1 or 1 to 0 transition. • An error may be propagated. • Differential coding helps. • In differential coding, “1” is transmitted by a pulse identical to that used for the previous bit. “0” is transmitted by a pulse negative to that used for the previous bit. • Useful in systems that have no sense of absolute polarity. Fig 7.17 Dr. Shi


68

34

Differential Code

Dr. Shi


69

Scrambling • Purpose: A scrambler tends to make the data more random by – Removing long strings of 1’s or 0’s – Removing periodic data strings. – Used for preventing unauthorized access to the data.

• Example. (Structure) Scrambler: S ⊕ D 3T ⊕ D 5T = T D nT : ⊕:

T delayed by n units Modulo 2 Sum

1⊕1 = 0 0⊕0=0 Dr. Shi

→ Incomplete version

1⊕ 0 = 1 0 ⊕1 = 1


70

35

Dr. Shi


71

Scrambling… • Descrambler:

Incomplete version

R = T ⊕ (D

3

⊕ D 5 )T

= [1 ⊕ ( D 3 ⊕ D 5 )] T = (1 ⊕ F ) T ∆

F = D

Dr. Shi

3

⊕ D5


72

36

Example 7.2 • The data stream 101010100000111 is fed to the scrambler in Fig. 7.19a. Find the scrambler output T, assuming the initial content of the registers to be zero. – From Fig. 7.19a we observer that initially T = S, and the sequence S enters the register and is returned as through the feedback path. This ( D 3 ⊕ D 5 ) S = FS new sequence FS again enters the register and is returned as F 2 S , and so on. Hence, (7.41) T = S ⊕ FS ⊕ F 2 S ⊕ F 3 S ⊕ ... = (1 ⊕ F ⊕ F 2 ⊕ F 3 ⊕ ...) S

Complete version

Dr. Shi


73

Example 7.2 … • Recognizing that We have

(

)(

F = D3 ⊕ D5

)

F 2 = D 3 ⊕ D 5 D 3 ⊕ D 5 = D 6 ⊕ D10 ⊕ D 8 ⊕ D8

• Because modulo-2 addition of any sequence with itself is zero, D 8 ⊕ D 8 = 0 and F 2 = D 6 ⊕ D10

• Similarly,

(

)(

)

F 3 = D 6 ⊕ D10 D 3 ⊕ D 5 = D 9 ⊕ D11 ⊕ D13 ⊕ D15 Dr. Shi


74

37

Example 7.2 … …… and so on …… Hence, T = (1⊕ D3 ⊕ D5 ⊕ D6 ⊕ D9 ⊕ D10 ⊕ D11 ⊕ D12 ⊕ D13 ⊕ D15 ⊕...)S

Because DnS is simply the sequence S delayed by n bits, various terms in the preceding equation correspond to the following sequences:

Dr. Shi


75

Example 7.2 … S = 1010101000 00111 D 3 S = 0001010101 00000111 D 5 S = 0000010101 0100000111 D 6 S = 0000001010 1010000011 1 D 9 S = 0000000001 0101010000 0111 D 10 S = 0000000000 1010101000 00111 D 11 S = 0000000000 0101010100 000111 D 12 S = 0000000000 0010101010 0000111 D 13 S = 0000000000 0001010101 00000111 D 15 S = 0000000000 000001010101000 00111 T = 1011100011 01001 Dr. Shi


76

38

Example 7.2 … • Note that the input sequence contains the periodic sequence 10101010…, as well as a long string of 0’s. • The scrambler output effectively removes the periodic component as well as the long strings of 0’s. • The input sequence has 15 digits. The scrambler output up to the 15th digit only is shown, because all the output digits beyond 15 depend on the input digits beyond 15, which are not given. • We can verify that the descrambler output is indeed S when this sequence T is applied at its input. • Homework: Learn to be able to determine where no terms needs to be considered. Dr. Shi


77

Regenerative Repeater •

Three functions: 1. Reshaping incoming pulses using an equalizer. 2. Extracting timing information Required to sample incoming pulses at optimum instants. 3. Making decision based on the pulse samples

Dr. Shi


78

39

Regenerative Repeater…

Dr. Shi


79

Preamplifier and Equalizer • A pulse train is attenuated and distorted by transmission medium, say, dispersion caused by an attenuation of high-frequency components. • Restoration of high frequency components ⇒ increase of channel noise • Fortunately, digital signals are more robust. – Considerable pulse dispersion can be tolerated

• Main concern: Pulse dispersion ⇒ ISI ⇒ increase error probability in detection Dr. Shi


80

40

Zero-Forcing Equalizer • Detection decision is based solely on sample values. ⇒ No need to eliminate ISI for all t. All that is needed is to eliminate or minimize ISI at their respective sampling instants only. • This could be done by using the transversal-filter equalizer, which forces the equalizer output pulse to have zero values at the sampling (decisionmaking ) instant. (refer to the diagram in next slide) Dr. Shi


81

Dr. Shi


82

41

Zero-Forcing Equalizer … c0 =1   ck = 0 ∀k ≠ 0

• Let

tap setting

⇒ P0 (t ) = Pr (t − NTb ) P0 (t ) = Pr (t )

If ignore delay.

• Fig 7.22 (b) indicates a problem : a1, a-1, a2, a-2, …are not negligible due to dispersion. • Now, want to force a1= a-1= a2= a-2 =…= 0 • Consider ck’s assume other values (other tap setting). Dr. Shi


83

Zero-Forcing Equalizer … • Consider ck’s assume other values (other tap setting). N

p0 ( t ) =

∑c

n pr ( t

− nTb )

n =− N

N

p0 ( kTb ) =

∑c

n pr [(k

− n )Tb ],

k = 0, ± 1, ± 2 ,...

n =− N

• For simplicity of notation: p0 ( k ) =

∞

∑c

n =−∞

n

pr ( k − n)

• Nyquist Criterion:

Dr. Shi

p0 ( k ) = 0

∀k ≠ 0

p0 ( k ) = 1

for k = 0 Lathi & Ding-Digital Commun

84

42

Zero-Forcing Equalizer … •

⇒ A set of infinitely many simultaneous equations:

2N+1: cn’s Impossible to solve this set of equations. • If, however, we specify the values of p0(k) only at 2N+1 points: k =0 1 p0 ( k ) =  0 k = ±1,±2,...,± N

then a unique solution exists. Dr. Shi


85

Zero-Forcing Equalizer … • Meaning: Zero ISI at the sampling instants of N preceding and N succeeding pulses. • Since pulse amplitude decays rapidly, ISI beyond the Nth pulse is not significant for N>2 in general.

Dr. Shi


86

43

Zero-Forcing Equalizer … • Page 7.37

Think about when n = 2, 1, 0, -1, -2; an example in next slide Dr. Shi


87

Zero-Forcing Equalizer … • Page 7.37

Dr. Shi


88

44

Eye Diagram • A convenient way to study ISI on an oscilloscope • Figure

Dr. Shi


89

Timing Extraction • • •

The received signal needs to be sampled at precise instants. Timing is necessary. Three ways for synchronization: 1.

2. 3.

Dr. Shi

Derivation from a primary or a secondary standard (a master timing source exists, both transmitter and receiver follow the master). Transmitting a separate synchronizing signal (pilot clock) Self synchronization (timing information is extracted from the received signal itself). Lathi & Ding-Digital Commun

90

45

Timing Extraction … • Way 1:

Suitable for large volumes of data high speed comm. Systems. High cost. • Way 2: Part of channel capacity is used to transmit timing information. Suitable for a large available capacity. • Way 3: Very efficient. • Examples: – On-off signaling (decomposition: Fig. 7.2) – Bipolar signaling rectification on-off signaling Dr. Shi


91

Timing Jitter • Small random deviation of the incoming pulses from their ideal locations. • Always present, even in the most sophisticated communication systems. • Jitter reduction is necessary about every 200 miles in a long digital link to keep the maximum jitter within reasonable limits. • Figure 7.23 Dr. Shi


92

46

Figure 7.23

Dr. Shi


93

Detection –Error Probability • Received signal = desired + AWGN Example: Polar signaling and transmission (Fig. 7.24, the next slide) Ap : Peak signal value • Polar: Instead of ± Ap , received signal = ± Ap + n • Because of symmetry, the detection threshold = 0. – If sample value > 0, ⇒ “1” – If sample value < 0, ⇒ “0” – Error in detection may take place

Dr. Shi


94

47

Figure 7.24

Dr. Shi


95

Error Probability for Polar Signal • With respect to Figure 7.24 P (∈ / 0 ) = probability that n > Ap P (∈ / 1) = probability that n < -Ap

p (n) =

n: AWGN ⇒ P(∈ 0) = =

Dr. Shi

1 2π

1

∞ Ap

∫σ

n

e

1

σ n 2π −

1 2 ( x) 2

∫

∞ Ap

2π σ n e

1 n −  2 σn

  

e

1 n −  2 σ n

  

2

2

dn

x=

n

σn

 Ap  dx = Q    σn  Lathi & Ding-Digital Commun

96

48

Error Probability for Polar Signal… 1 Q( y ) = 2π

where • Similarly,

∫

∞

y

e

−

x2 2

dx

 Ap  P(∈ 1) = Q  σn 

 − Ap to − ∞     Symmetric 

• Summary:

P (∈) = P (∈,0) + P (∈,1)

Here, assume P(0)= P(1)=1/2.

= P (∈ 0) P(0) + P (∈ 1) P(1) =

A 1 [P(∈ 0) + P(∈ 1)] = Q  p 2 σn

• Q(x):

   Complementary error function: erfc (x) 1 Q ( x) ≅ x 2π

Dr. Shi

x2

0 .7  −  1 − 2 e 2 , x > 2 x   Lathi & Ding-Digital Commun

97

Error Probability for Polar Signal… • If

Ap

σn

= k , P(∈ 0) = P(∈ 1) = P(∈) = Q(k )

k Q(k)=P(ε)

1 0.1587

e.g. 3.16 ×10−5 ⇒ one of wrongly

Dr. Shi

2

3

4

5

6

0.0227 0.00135 3.16 ×10−5 2.87 ×10 −7 9 . 9 × 10 − 9

3.16 × 105

pulses will be possibly detected


98

49

Error Probability for On-off Signals • In this case, we have to distinguish between Ap and 0. 1 Ap • Threshold: 2 • Error Probabilities:  A  1 P(∈ 0) = prob. of

P(∈ 1) = prob. of P(∈) =

n>

p  Ap = Q σ 2  n  Ap  −1  n< Ap = Q 2  2σ n 

 A  1 [P(∈ 0) + P(∈ 1)] = Q p  2  2σ n 

2

Also assume 0 and 1 equally probable

P(∈) > P(ε ) (in the case of polar signaling) Dr. Shi


99

Dr. Shi


100

50

Error Probability for Bipolar Signals 1 ↔ Ap or − Ap 0 ↔ 0v

⇒

 A

A 

p p If the detected sample value: ∈  − 2 , 2  ↔ 0  

Ap    P(∈ 0) = prob. n > 2   Ap  A     + prob. n < − p  = prob. n > 2  2   

o.w. ↔ 1

Ap   A    = 2Q p  = 2 prob. n > 2    2σ n  Dr. Shi


101

Error Prob. for Bipolar Signals… Ap    when P (∈ 1) = prob . n < − positive pulse used 2   Ap    when negative or prob . n > pulse used 2    Ap   = Q   2σ n  1 P (∈) = P (∈ 0 ) P ( 0 ) + P (∈ 1) P (1) = [P (∈ 0 ) + P (∈ 1) ] 2  Ap   > P (ε ) = 1 . 5Q  for on − off signaling  2σ n  Dr. Shi


102

51

Detection Error Probability…  Ap     Q  σn    A   on-off is in middle, Q  p    2σ n    A  bipolar is the worst, 1.5Q  p    2σ n  

• Summary, polar is the best,

Assumption: “0” and “1” Equally likely

• Another factor: P(∈) decreases exponentially with the signal power. Dr. Shi


103

Comparison among three line codes • To obtain:

P(∈) = 0.286 × 10−6 Ap

• We need:

σn Ap

σn Ap

σn

=5

for polar case

= 10

Q Q(5) = 0.286 × 10−6

 Ap   Q P (∈) = Q 2 σ  n for bipolar case Q P (∈) = 1.5Q Ap   2σ   n −6 = 0.286 × 10

for on-off case

= 10 . 16

c Dr. Shi


Ap 2σ n

= 5.08 104

52

Comparison among three line codes…  Ap 

• For the same error rate, required SNR  σ n  , Polar < On-off < Bipolar

• For the same

A  SNR  σ p   n

(from previous example)

, the caused error rate

Polar < On-off < Bipolar

(from next example)

• Polar is most efficient in terms of SNR vs. error rate • Between, on-off and bipolar: Only 16% improvement of on-off over bipolar • Performance of bipolar case ≈ performance of on-off case in terms of error rate. Dr. Shi


105

Example a) Polar binary pulses are received with peak amplitude Ap = 1 mV. The channel noise rms amplitude is 192.3 µV. Threshold detection is used, and 1 and 0 are equally likely. b) Find the error probability for (i) the polar case, and the on-off case (ii) the bipolar case if pulses of the same shape as in part (a) are used, but their amplitudes are adjusted so that the transmitted power is the same as in part (a) Dr. Shi


106

53

Example … a) For the polar case

Ap

σn

=

10 −3 = 5 .2 192.3(10 −6 )

From Table 8.2 (4th ed), we find

P (∈) = Q ( 5 . 2 ) = 0 . 9964 × 10 − 7

b) Because half the bits are transmitted by nopulse, there are, on the average, only half as many pulses in the on-off case (compared to the polar). Now, doubling the pulse energy is accomplished by multiplying the pulse by 2 (in order to keep the power dissipation same) Dr. Shi


107

Example… Thus, for on-off Ap is 2 times the Ap in the polar case.

•Therefore, from Equation (7.53)  A  P(∈) = Q p  = Q (3.68) = 1.66 × 10 − 4  2σ n 

•As seen earlier, for a given power, the Ap for both the on-off and the bipolar cases are identical. Hence, from equation (7.54)  Ap P (∈) = 1.5Q   2σ n Dr. Shi

  = 1.749 × 10 −4 


108

54

M-ary Communication • Digital communications use only a finite number of symbols • Information transmitted by each symbol increases with M.

IM = log2 M bits

IM

: information transmitted by an M-ary symbol

• Transmitted power increases as M2 , i.e., to increase the rate of communication by a factor of log2 M , the power required increases as M2. (see an example below) Dr. Shi


109

M-ary Communication… • Most of the terrestrial digital telephone network: Binary • The subscriber loop portion of the integrated services digital network (ISDN) uses the quarternary code 2BIQ shown in Figure. 7.28

Dr. Shi


110

55

Dr. Shi


111

Pulse Shaping in Multi-amplitude Case • Nyquist Criterion  can be used for M-ary case  Controlled ISI  • Figure 7.28: One possible M-ary Scheme • Another Scheme: Use M orthogonal pulses: ϕ1 (t ), ϕ 2 (t ),..., ϕ M (t ) • Definition: Tb c i = j ϕ i ( t )ϕ j ( t ) =  0 0 i ≠ j

∫

Dr. Shi


112

56

Pulse Shaping in Multi-amplitude Case • The figure in next slide: One example in M orthogonal pulses:  2π ⋅ k ⋅ t , 0 < t < Tb k = 1, 2,..., M sin Tb ϕk (t ) =   

0, otherwise

• In the set, pulse frequency: k

1 1 2 M : , ,..., Tb Tb Tb Tb

M times that of the binary scheme

(see an example below)

Dr. Shi


113

Dr. Shi


114

57

Pulse Shaping in Multi-amplitude Case • In general, it can be shown that the bandwidth of an orthogonal M-ary scheme is M times that of the binary scheme – In an M-ary orthogonal scheme, the rate of communication is increased by a factor of log 2 M at the cost of an increase in transmission bandwidth by a factor of M. Dr. Shi


115

Digital Carrier Systems • So far: Baseband digital systems – Signals are transmitted directly without any shift in frequency. – Suitable for transmission over wires, cables, optical fibers.

• Baseband signals cannot be transmitted over a radio link or satellites. – Since it needs impractically large size for antennas – Modulation (Shifting signal spectrum to higher frequencies is needed) Dr. Shi


116

58

Digital Carrier Systems… • A spectrum shift to higher frequencies is also required when transmitting several messages simultaneously by sharing the large bandwidth of the transmission medium, • FDM: Frequency-division Multiplexing. (FDMA: Frequency-division Multiplexing Access) Dr. Shi


117

Several Types of Modulation • Amplitude-Shift Keying (ASK), also known as onoff keying (OOK) m(t ) cos(ω c t ) : m(t ) : on-off baseband signal (modulating signal)

cos(ω c t )

Dr. Shi

: carrier


118

59

Modulation Types… • Phase-Shift Keying (PSK) m(t ) : polar Signal 1 ↔ p(t) cos(ωct),0 ↔ − p(t) cos(ωct) = p(t) cos(π −ωct) = p(t) cos(ωct −π )

Dr. Shi


119

Modulation Types… • Frequency-Shift Keying (FSK) – Frequency is modulation by the base band signal.

1 ↔ ω c1 ,0 ↔ ω c0 – Information bit resides in the carrier frequency.

Dr. Shi


120

60

PSD of ASK, PSK & FSK

Dr. Shi


121

FSK • FSK signal may be viewed as a sum of two interleaved ASK signals, one with a modulating frequency ω c , the other, ω c 0

1

– Spectrum of FSK = Sum of the two ASK – Bandwidth of FSK is higher than that of ASK or PSK

Dr. Shi


122

61

Demodulation A. Review of Analog Demodulation • Baseband signal and bandpass signal The end of the 2nd part of the slides of Ch. 2 and Ch. 3

• Double-sideband Suppressed Carrier (DSB-SC) Modulation m(t) ↔ M(ω)

message signal

1 cos(ωct) ↔ [δ (ω +ωc ) +δ (ω −ωc )] carrier 2 1 m(t)cos(ωct) ↔ [ M(ω +ωc ) + M(ω −ωc )] modulated signal 2 Dr. Shi


123

Figure 4.1 (a), (b), (c) DSB-SC Modulation

124

62

DSB-SC Demodulation • Demodulation: Synchronization Detection (Coherent Detection) • Figure 4.1 (e) and (d)

  

Using a carrier of exactly the same frequency & pulse

Dr. Shi

125

DSB-SC Demodulation …

[m(t ) cos(ωct )]cos(ωct ) = e(t ) = 1 [m(t ) + m(t ) cos(2ωct )] 2

1 1 M (ω ) + [M (ω + 2ωc ) + M (ω − 2ωc )] 2 4 1 Aftera LPF ⇒ M (ω ) 2

E (ω ) =

1 m(t ) 2

Dr. Shi

126

63

Figure 4.1 (d), (e) DSB-SC demodulation

127

AM Signal • Motivation: – Requirements of a carrier in frequency and phase synchronism with carrier at the transmitter is too sophisticated and quite costly – Sending the carrier ⇒ a. eases the situation b. requires more power in transmission c. suitable for broadcasting Dr. Shi

128

64

Amplitude Modulation (AM) ϕ AM (t ) = A cos(ω c t ) + m(t ) cos(ω c t ) = [A + m(t )]cos(ω c t ) ϕ AM (t ) ⇔

1 [M (ω + ω c ) + M (ω − ω c )] 2

+ π ⋅ A[δ (ω + ω c ) + δ (ω − ω c )]

Dr. Shi

129

• AM Signal and its Envelope

130

65

• Demodulation of AM signals – Rectifier Detection:

Figure 4.11 Output:

– Envelope Detector:

1

π

m(t )

Figure 4.12 Output: A + m(t )

Dr. Shi

131

132

66

133

B. Back to Digital Communication System: Demodulation •

Demodulation of Digital-Modulated Signals is similar to Demodulation of Analog-Modulated Signals

•

Demodulation of ASK Synchronous (Coherent) Detection Non-coherent Detection: Say, envelope detection

Dr. Shi


134

67

Demodulation of PSK • Cannot be demodulated non-coherently (envelope detection). – Since, envelope is the same for 0 and 1

• Can be demodulated coherently. • PSK may be demodulated non-coherently if use: Differential coherent PSK (DPSK)

Dr. Shi


135

Differential Encoding • Differential coding: – “1”  encoded by the same pulse used to encode the previous data bit no transition – “0”  encoded by the negative of pulse used to encode the previous data bit transition

• Figure 7.30 (a) • Modulated signal consists of pulses ± A cos(ω ct ) with a possible sign ambiguity Dr. Shi


136

68

Differential Encoding • Figure

Dr. Shi


137

Differential Encoding… • Tb : one-bit interval • If the received pulse is identical to the previous pulse 1, A2 Y (t ) = A2 cos 2 (ω c t ) = [1 + cos(2ω ct )] 2

z (t ) =

2

A 2

• If 0 is transmitted

Y (t ) = − A2 cos 2 (ω c t ) = − z (t ) = −

Dr. Shi

A2 [1 + cos(2ω ct )] 2

A2 2 Lathi & Ding-Digital Commun

138

69

Demodulation of FSK • FSK can be viewed as two interleaved ASK signals with carrier frequency ω c and ω c , respectively. • Hence, FSK can be detected by non-coherent detection (envelope) or coherent detection technique 0

Dr. Shi


1

139

Demodulation of FSK

Dr. Shi


140

70

Demodulation of FSK • In (a), non-coherent detection (envelope), ωc

– H0(ω)

0

Turned to – H1(ω)

respectively ωc

1

• Comparison: above or bottom Whose output is large 0 or 1

Dr. Shi


141

71

Principles of Digital Data Transmission Chapter 7

Recommend Documents