SQNR with and without Companding

1

SamplingPAM- Pulse Amplitude Modulation

(continued)

EELE445-14Lecture 17

SQNR with and without

Companding

EELE445-13 Lecture 17

(continuation of lecture 16)

2

SQNR - summary

powernoiseonquantizatiM

VPnq 2

2max

3=

2max

2max

2 433V

PV

PMPPSQNR x

nx

nq

x ×===

• M is the number of quantization levels• n is the number of bits• Vmax is ½ the A/D input range

SQNRdB

12max

2max

≤

≤

VP

VP

x

x The SQNR decreases asThe input dynamic rangeincreases

8.46log10| 2max

10 ++⎟⎟⎠

⎞⎜⎜⎝

⎛≅ n

VPSQNR x

dB

range quantizer scale full the in bits of number the is

quantizer the of range peak to peakthe2

is

n

V

•

•1

max

3

Summary of SQNR|dB for linear, μ−law, A-law

( )[ ]

[ ]

quantizer the of level design peak the is power signal input the is

26c)-(3 )compandinglaw -(

26b)-(3 )compandinglaw -(

26a)-(3 )quantizing (uniform

25)-(3

max

2

max

ln1log2077.4

1lnlog2077.4

log2077.4

02.6

VxP

AA

xV

nSQNR

rmsx

rms

dB

=

+−≅

+−≅

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

+=

α

μμα

α

α

6

Exam 1Wednesday February 26

4

7

Line Codes:Baseband Digital Signals

ELE445-14Lecture 17

Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0

PCM Signal Transmission

PCMSignal

5

PCM Waveforms

Shaped Polar NRZ

Polar NRZ

Unipolar NRZ

Digital Signaling – Signal Vectors

okN

k k Tttwtw <<= ∑ =0)()(

1ϕ

Where wk is the digital data and φk(t) is the set of basis waveforms

Example:

( )( )( ) multilevelstateforw

bipolarorpolarforwUnipolarforw

codesNRZforTT

codesRZforTTTtt

so

so

o

475,.25,.25.,75.1,1

1,0

2)(

−−∈−∈

∈=

=⎟⎟⎠

⎞⎜⎜⎝

⎛Π=ϕ

6


Figure 3–11 Representation for a 3-bit binary digital signal.

Vector lengths are in units of E

Vector Signaling

∑=

=N

kkk twtw

1

)()( φ∫

0

0

T

∫0

0

T∫=

bT

kk ttww0

* )()( φ

∫=bT

ttww0

1*

1 )()( φ

The φk are orthogonal basis functions, wk are weightsthat correspond to the digital data.

7


Figure 3–13 Binary-to-multilevel signal conversion.

( ) multilevelstateforwcodesNRZforTT

codesRZforTTTtt

so

so

o

43,1,1,3

2)(

−−∈=

=⎟⎟⎠

⎞⎜⎜⎝

⎛Π=ϕ


Figure 3–12 Binary signaling (computed).

8


Figure 3–14 L = 4-level signaling (computed).

Desired Properties of Line Codes• Self-synchronization:

There is enough timing information built into the code so that bit synchronizers can be designed to extract the clock signal. A long series of 1’s or 0’s should not cause a problem.

• Low probability of bit error:Receiver can be designed that will recover the binary data with a low probability of bit error when the input data signal is corrupted by noise or ISI (intersymbol interference)

• Good Spectral Properties:If the channel is ac coupled, the PSD of the line code signal should be negligible at frequencies near zero. The signal bandwidth needs to be sufficiently small compared to the channel bandwidth, so that ISI will not be a problem.

9

Desired Properties of Line Codes

•Transmission bandwidth:As small as possible

•Error detection capability:It should be possible to implement this feature easillybythe addition of channel encoders and decoders, or the feature should be incorporated into the line code.

• Transparency:The data protocol and line code are designed so that every possible sequence of data is faithfully and transparently received. (The sequence of bits does not matter.)


Figure 3–15 Binary signaling formats.

Good: simpleBad: DC

Good: Lowest BER

NRZ: non-return to zero has smallest occupied bandwidthbut must constrain consecutive 0’s or 1’s

10


Figure 3–15 Binary signaling formats.

Bipolar

20

Line Codes:Power Spectral Density (PSD)

ELE445-14Lecture 18

11

PSD of Line Codes

iikn

l

in

k

kfTj

sy

PaakR

ekRTfF

fP s

)()(

)()(

)(

1

22

+=

∞

−∞=

∑

∑

=

= π

Binary Signals: Ts =TbMultilevel Signals: Ts = l Tb

Pi is the probability of anan+k occurring

for random data

R(k) example for a deterministic data file

010-11Bipolar A = 0, (1,-1)-11-111Polar A= -1,101011Unipolar A= 0,1

01011Data

bitsofnumbertheisNwhere

aaN

kR kn

N

Nkn )(1)( +

−=∑=

12

R(k) example Polar , k=0

11111(an)2

1(1/Ν)Σ(an)2

-11-111an+0

-11-111an

01011Data

)(1)( kn

N

Nknaa

NkR +

−=∑=


-1-1-11-1ana+1

-0.6(1/Ν)Σ ana+1

1-111-1an+1

-11-111an

01011Data

Shift Right 1

See Mathcad file bpsk_psd.xmcd and linecodepsd.xmcd

13


k

R(k)

1

-0.6

PSD of Line Codes

14

PSD of Line Codes

PSD of Line Codes

Check the website for Matlab and mathcad files to plot the psd of line codes

15


Figure 3–16 PSD for line codes (positive frequencies shown).

39b)-(3⎥⎦

⎤⎢⎣

⎡+⎟⎟

⎠

⎞⎜⎜⎝

⎛= )(11sin

4)(

22

fTfT

fTTAfPbb

bbNRZunipolar δ

ππ



41)-(32

2 sin)( ⎟⎟⎠

⎞⎜⎜⎝

⎛=

b

bbNRZpolar fT

fTTAfPπ

π

16



43)-(3⎥⎦

⎤⎢⎣

⎡−+⎟⎟

⎠

⎞⎜⎜⎝

⎛= ∑

∞=

−∞=

n

n bbb

bbRZunipolar T

nfTfT

fTTAfP )(112/

2/sin16

)(22

δπ

π



( ) 45)-(3bb

bbRZbipolar fT

fTfTTAfP π

ππ 2cos(1

2/2/sin

8)(

22

−⎟⎟⎠

⎞⎜⎜⎝

⎛=

Bipolar

17



( ) 46c)-(32/sin2/

2/sin)( 22

2b

b

bbNRZManchester fT

fTfTTAfP π

ππ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

34

Eye DiagramsClock Recovery

Regenerative RepeaterSpectral Efficiency


18

35

Exam in Class



Figure 3–18 Distorted polar NRZ waveform and corresponding eye pattern.

EYE Diagrams

19

EYE Diagrams

Eye diagram of a 20 Gbps data stream The time scale is set at 10 picoseconds/division.

20


Figure 3–19 Regenerative repeater for unipolar NRZ signaling.

•The ability to use a regenerative repeater is one of the major advantages of a digital binary system over an analog system


Figure 3–20 Square-law bit synchronizer for polar NRZ signaling.

21


Figure 3–20 Square-law bit synchronizer for polar NRZ signaling.


Figure 3–21 Early–late bit synchronizer for polar NRZ signaling.

22


Figure 3–22 Binary-to-multilevel polar NRZ signal conversion.


Figure 3–22 Binary-to-multilevel polar NRZ signal conversion.

23

Spectral Efficiency Spectral Efficiency

Definition: The spectral efficiency of a digital signal is given by the number of bits per second that can be supported by each hertz of bandwidth.

where R is the data rate in bits per second and B is the bandwidth in Hz

( ) HzbitsinBR /sec/=η

Spectral Efficiency Limits

• Shannon’s Law is the best we can do•We are a long way from this•Multiple bits/symbol get us closer (l- multilevel PCM)• η is in (bits/sec)/Hz

DAtheinusedbitsofnumbertheisHzsbit

signalingmultilevelforNS

BC

/)573(/)/(

:

)563(1log2max

−=

−⎟⎠⎞

⎜⎝⎛ +==

llη

η

24

Spectral Efficiency

lR/lMultilevel polar NRZ

1/22RManchester NRZ

1RBipolar RZ

1/22RUnipolar RZ

1RPolar NRZ

1RUnipolar NRZ

η=R/B(Hz)

Spectral EfficiencyFirst null BandwidthCode Type

Table 3-6 SPECTRAL EFFICIENCIES OF LINE CODES

SQNR with and without Companding

Documents