Upload
abraham-lanki
View
240
Download
1
Embed Size (px)
Citation preview
8/3/2019 Baseband transmission2003
1/26
Baseband Transmission and Line
Coding
Presented by:
T.F.Lai
Member of IEEE, IET & WWF
Electrical Project Consultant (http://electricalprojecttestimonial.blogspot.com/)
8/3/2019 Baseband transmission2003
2/26
Outline of the Presentation
Introduction
Baseband Centre Point Detection
Baseband binary error rates in Gaussian noise
Multilevel baseband signaling
Error accumulation over multiple hops
Unipolar signaling
Polar signaling
Dipolar signaling
8/3/2019 Baseband transmission2003
3/26
Introduction
This chapter addresses two issues central to baseband signal transmission,
digital signal detection and line coding.
The analysis of detection processes is important since this leads to the
principle objective measure of digital communications system quality, i.e.
the probability of symbols being detected in error, as quantified byprobability error.
Appropriate coding of the transmitted symbol pulse shape can minimize
the probability of error by ensuring that the spectral characteristic of the
digital signal are well matched to the transmission channel.
The line code must also permit the receiver to extract accurate PCM bitand word timing signals directly from the received data, for proper
detection and interpretation of the digital pulses.
8/3/2019 Baseband transmission2003
4/26
Baseband centre point detection
The detection of digital signals involves two processes:
Reduction of each received voltage pulse (i.e. symbol) to a single
numerical value.
Comparison of this value with a reference voltage (or, for multisymbol
signaling, a set of reference voltage) to determine which symbol was
transmitted.
In the case of symbols represented by different voltage levels the simplest
way of achieving 1 is to sample the received signal plus noise; 2 is then
implemented using one or more comparators.
In the case of equiprobable, binarym symbols (zero and one) represented
by two voltage levels (e.g. 0V and 3V) intuition tells us that a sensible
strategy would be to set the reference, Vref(Figure 6.1(b)), mid-way
between the two voltage levels (i.e. at 1.5V).
8/3/2019 Baseband transmission2003
5/26
Baseband centre point detection
Decisions would then be made at the centre of each symbol period on the
basis of whether the instantaneous voltage (signal plus noise) is above or
below this reference.
Sampling the instantaneous signal plus noise voltage somewhere near the
middle of the symbol period is called centre point detection, Figure 6.1(a).
The noise present during detection is often either Gaussian or
approximately Gaussian.
8/3/2019 Baseband transmission2003
6/26
Baseband Binary Error Rates in
Gaussian Noise Figure 6.2(a) shows the pdf of a binary information signal which can take
on voltage levels V0 and V1 only.
Figure 6.2(b) shows the pdf of a zero mean Gaussian noise process, Vn(t),
with RMS value W V.
Since the process has zero mean the RMS value and standard deviationare identical.
Figure 6.2(c) shows the pdf of the sum of the signal and noise voltage.
Figure 6.2(c) is perhaps no surprise we can think of the quasi-DC
symbol voltage biasing the mean value of the noise to V0 and V1 .
Figure 6.2(c) is thus the convolution of 6.2(a) and (b).
For equiprobable symbols the optimum decision level is set at (V0 + V1)/2.
8/3/2019 Baseband transmission2003
7/26
Baseband Binary Error Rates in
Gaussian Noise
8/3/2019 Baseband transmission2003
8/26
Baseband Binary Error Rates in
Gaussian Noise Given that the symbol 0 is transmitted (i.e. voltage level V0) then the
probability, Pe1, that the signal plus noise will be above the threshold at
the decision instant is given by twice the shaded area under the curve
p0(Vn) in Figure 6.2(c):
------------------- (1)
Using the change of variable x = (Vn V0)/2W this becomes:
------------------------ (2)
n2
)VV(
2/)VV(
1e dve2
1P
2
20n
10
W
g
TW!
g
W
T!
22/)VV(
2x
1e
01
dxe1
P
8/3/2019 Baseband transmission2003
9/26
Baseband Binary Error Rates in
Gaussian Noise The incomplete integration in equation 2 cannot be evaluated analytically
but can be recast as a complementary error function, erfc(x), defined by:
----------------------- (3)
Thus equation 2 becomes:
----------------------- (4)
Alternatively, since erfc(z) and the error function, erf(z), are related by:
erfc(z) = 1 erf(z) -------------------------------------- (5)
g
(
T!z
2x
dxe2
)z(erfc
W
! 22
VV
erfc2
1
P01
1e
8/3/2019 Baseband transmission2003
10/26
Baseband Binary Error Rates in
Gaussian Noise Then Pe1 can also be written as:
------------------------- (6)
The advantage ofusing error (or complementary error) function in the
expression for Pe1 is that this function has been extensively tabulated.
If the digital symbol one is transmitted then the probability Pe0, that the
signal plus noise will be below the threshold at the decision instant is:
--------------- (7)
-
W
!
22
VVerfc1
2
1P 011e
n2
)VV(2/)VV(
0e dve2
1P
2
2
1n10
W
g TW!
8/3/2019 Baseband transmission2003
11/26
Baseband Binary Error Rates in
Gaussian Noise It is clear from the symmetry of this problem that Pe0 is identical to both
Pe1 and the probability of error, Pe, irrespective of whether a one or zero
was transmitted.
Noting that the probability of error depends on only symbol voltage
difference and not absolute voltage levels, Pe can be rewritten in terms of(V = V1 V0:
--------------------------- (8)
Equation (8) is valid for both unipolar signaling and polar signaling.
-
W
(!
22
Verfc1
2
1Pe
8/3/2019 Baseband transmission2003
12/26
Baseband Binary Error Rates in
Gaussian Noise(Figure 6.3)
8/3/2019 Baseband transmission2003
13/26
Baseband Binary Error Rates in
Gaussian Noise Whilst the x-axis of Figure 6.3 is clearly related to signal to noise ratio it is
not identical to it for all pulse levels and shapes.
This is partly because it involves the signal only at the sampling instant
(whereas a conventional SNRuses a time averaged signal power) and
partly because the use of(V neglects any transmitted DC component. For NRZ, unipoar, rectangular pulse signaling, Figure 6.4, the normalised
peak signal power is Speak = (V2 and the average signal power is S = (V2/2.
The normalised Gaussian noise power is N = W2.
We therefore have:
------------------------ (9)
2/12/1
peak N
S2
N
SV
!
!
W
(
8/3/2019 Baseband transmission2003
14/26
Baseband Binary Error Rates in
Gaussian Noise Where (S/N)peak indicates peak signal power divided by average noise
power.
Substituting for (V/W in equation (8) we therefore have:
---------------------- (10)
Or
----------------------------(11)
-
!
2/1
peak
eNS
221erf1
21P
-
!
2/1
e N
S
2
1
erf12
1
P
8/3/2019 Baseband transmission2003
15/26
Baseband Binary Error Rates in
Gaussian Noise (Figure 6.4/6.5)
8/3/2019 Baseband transmission2003
16/26
Baseband Binary Error Rates in
Gaussian Noise For NRZ, polar, rectangular pulse signaling with the same voltage spacing
as in the unipolar, Figure 6.5, the peak and average signal powers are
identical, Speak = S = ((V/2)2 and we can therefore write:
------------------------ (11)
Substitute into equation (8) we now have:
------------------------- (12)
2/12/1
peak N
S2N
S2
V
!
!W
(
-
!
2/1
e N
S
2
1
erf12
1
P
8/3/2019 Baseband transmission2003
17/26
Baseband Binary Error Rates in
Gaussian Noise Whilst probabilities of symbol and bit error are the quantities usually
calculated, it is the symbol error rate (SER) or bit error rate (BER) which is
usually measured.
These are simply the number of symbol or bit errors occurring per unit
time measured over a convenient period. The symbol error rate is clearly related to the probability of symbol error
by:
SER = PeRs
Where Rs is the symbol rate in symbol/s or baud.
Bit error rate is related to the probability of bit error by:
BER = PbRsn = PbRb
8/3/2019 Baseband transmission2003
18/26
Multilevel baseband signaling
Figure 6.6 shows a schematic diagram of a multilevel or multisymbol
signal.
If the number of equally spaced, allowed levels is M then the symbol plus
noise pdfs look like those in Figure 6.7 (drawn for M=4).
The probability of symbol error for the M-2 inner symbols (i.e. any but thesymbols represented by the lowest or highest voltage level) is now twice
that in the binary case.
This is because the symbol can be in error if the signal plus noise voltage is
too high or too low.
8/3/2019 Baseband transmission2003
19/26
Multilevel baseband signaling P
eM|
inner symbol= 2P
e-------------------------------------------- (13)
The symbol error for the two outer levels is identical to that for the binarycase,
PeM|outer symbol = Pe -----------------------------------------------(14)
Once again assuming equiprobable symbols, the average probability ofsymbol error is:
-------- (15)
Substitute for Pe from equation 8,
-------------------- (16)
eeeeM PM
)1M(2P
M
2P2
M
2MP !!
-
W(!
22Verf1
M1MPeM
8/3/2019 Baseband transmission2003
20/26
Error accumulation over multiple hops
All signal transmission media (e.g. cables, waveguides, optical fibres)
attenuate signals to a greater or lesser extent.
This is even the case for the space through which radio waves travel.
For long communication paths attenuation might be so severe that the
sensitivity of normal receiving equipment would be inadequate to detectthe signal.
In such cases the signal is boosted in amplitude at regular intervals along
the transmitter receiver path.
The equipment which boost the signal is called repeater and the path
between adjacent repeaters is called a hop. A repeater along with its associated, preceding, hop is called a section.
Thus long distance communication is usually achieved using multiple hops.
8/3/2019 Baseband transmission2003
21/26
Error accumulation over multiple hops
Repeaters used on multi-hop links can be divided into essentially two
types.
These are amplifying repeaters and regenerative repeaters.
For analogue communication linear amplifiers are required.
For digital communications either type of repeater could be used butnormally regenerative repeaters are employed.
Figure 6.8 shows a schematic of an m-hop link.
8/3/2019 Baseband transmission2003
22/26
Error accumulation over multiple hops
If a binary signal with voltage levels s(/2 is transmitted then the voltagereceived at the input of the first amplifying repeater is sE(V/2 + n1(t)whereE is the linear voltage attenuation factor (or voltage gain factor < 1)
and n1(t) is the random noise (with standard deviation, or RMS value, W)
added du
ring the first hop. In a well designed system the voltage gain, GV, of the repeater will be just
adequate to compensate for the attenuation over the first hop, GV = 1/E.
At the output of the first repeater the signal is restored to its original level
(s(V/2) but the noise signal is also amplified to a level GVn1(t) and willnow have an RMS value of G
VW.
The probability of error for m-hop link is therefore given by:
-
W
(!
m22
Verf1
2
1|P hopsme
8/3/2019 Baseband transmission2003
23/26
Unipolar signaling
Unipolar signaling (also called on-off keying) refers to a line code in which
one binary symbol (denoting a digital 0, for example) is represented by the
absence of a pulse and other binary symbol (denoting a digital 1) is
represented by the presence of a pulse.
There are two common variations on unipolar signaling, namely non-return to zero and return to zero.
In the former case the duration X of the mark pulse to the duration T0 of
the symbol slot.
In the latter case X is less than T0.
Both NRZ and RZunipolar signals have a non-zero average DC level.
Transmission of these signals over links with either transformer or
capacitor coupled AC repeaters results in the removal of this line and the
consequent conversion of the signals to a polar format.
8/3/2019 Baseband transmission2003
24/26
Unipolar signaling
Furthermore, since the continuous part of the RZ and NRZ signal spectrum
is non-zero at 0Hz then AC coupling results in distortion of the transmitted
pulse shapes.
If the AC coupled lines behave as high pass RC filters (which is typically the
case) then the distortion takes the form of an exponential decay of thesignal amplitude after each transition.
This effect, referred to as signal droop, is illustrated in Figure 6.13:
8/3/2019 Baseband transmission2003
25/26
Polar signaling
In polar signaling systems a binary one is represented by a pulse g1(t) and
a binary zero by the opposite (or antipodal) pulse g0(t) = -g1(t).
Polar signaling has a significant power advantage over unipolar signaling.
Fundamentally this is because the pulses in a unipolar scheme are only
orthobonal whilst the pulses in polar scheme are antipodal. Another way of explaining the difference in performance is to observe that
the average or DC level transmitted with unipolar signals contains no
information and is therefore wasted power.
Polar binary signaling also has the advantage that, providing the symbols
are equiprobable, the decision threshold is 0V. This means that no automatic gain control is required in the receiver.
8/3/2019 Baseband transmission2003
26/26
DiPolar signaling
Dipolar is designed to produce a spectral null at 0Hz.
This makes it especially well suited to AC coupled transmission lines.
The symbol interval, T0, is split into positive and negative pulses each of
width T0/2, Figure 6.11.