SQNR with and without Companding
Transcript of SQNR with and without Companding
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SamplingPAM- Pulse Amplitude Modulation
(continued)
EELE445-14Lecture 17
SQNR with and without
Companding
EELE445-13 Lecture 17
(continuation of lecture 16)
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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
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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
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Exam 1Wednesday February 26
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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
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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
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
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.
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–13 Binary-to-multilevel signal conversion.
( ) multilevelstateforwcodesNRZforTT
codesRZforTTTtt
so
so
o
43,1,1,3
2)(
−−∈=
=⎟⎟⎠
⎞⎜⎜⎝
⎛Π=ϕ
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–12 Binary signaling (computed).
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
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.
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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.)
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
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
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–15 Binary signaling formats.
Bipolar
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Line Codes:Power Spectral Density (PSD)
ELE445-14Lecture 18
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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)( +
−=∑=
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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 +
−=∑=
R(k) example Polar , k=1
-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
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R(k) example Polar , k=1
k
R(k)
1
-0.6
PSD of Line Codes
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PSD of Line Codes
PSD of Line Codes
Check the website for Matlab and mathcad files to plot the psd of line codes
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–16 PSD for line codes (positive frequencies shown).
39b)-(3⎥⎦
⎤⎢⎣
⎡+⎟⎟
⎠
⎞⎜⎜⎝
⎛= )(11sin
4)(
22
fTfT
fTTAfPbb
bbNRZunipolar δ
ππ
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–16 PSD for line codes (positive frequencies shown).
41)-(32
2 sin)( ⎟⎟⎠
⎞⎜⎜⎝
⎛=
b
bbNRZpolar fT
fTTAfPπ
π
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–16 PSD for line codes (positive frequencies shown).
43)-(3⎥⎦
⎤⎢⎣
⎡−+⎟⎟
⎠
⎞⎜⎜⎝
⎛= ∑
∞=
−∞=
n
n bbb
bbRZunipolar T
nfTfT
fTTAfP )(112/
2/sin16
)(22
δπ
π
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–16 PSD for line codes (positive frequencies shown).
( ) 45)-(3bb
bbRZbipolar fT
fTfTTAfP π
ππ 2cos(1
2/2/sin
8)(
22
−⎟⎟⎠
⎞⎜⎜⎝
⎛=
Bipolar
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–16 PSD for line codes (positive frequencies shown).
( ) 46c)-(32/sin2/
2/sin)( 22
2b
b
bbNRZManchester fT
fTfTTAfP π
ππ
⎟⎟⎠
⎞⎜⎜⎝
⎛=
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Eye DiagramsClock Recovery
Regenerative RepeaterSpectral Efficiency
EELE445-14 Lecture 19
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Exam in Class
EELE445-14 Lecture 20
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–18 Distorted polar NRZ waveform and corresponding eye pattern.
EYE Diagrams
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EYE Diagrams
Eye diagram of a 20 Gbps data stream The time scale is set at 10 picoseconds/division.
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
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
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–20 Square-law bit synchronizer for polar NRZ signaling.
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–20 Square-law bit synchronizer for polar NRZ signaling.
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–21 Early–late bit synchronizer for polar NRZ signaling.
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Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–22 Binary-to-multilevel polar NRZ signal conversion.
Couch, Digital and Analog Communication Systems, Seventh Edition ©2007 Pearson Education, Inc. All rights reserved. 0-13-142492-0
Figure 3–22 Binary-to-multilevel polar NRZ signal conversion.
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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
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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