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EE303: Communication Systems
Professor A. ManikasChair of Communications and Array Processing
Imperial College London
An Overview of Fundamentals: Digital Modulators and Line Codes
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 1 / 54
Table of Contents1 Introduction2 First Modelling of Digital Modulators
Examples of Binary Modulators
3 Second Modelling of Digital ModulatorsSignal ConstellationDistance Between two M-ary signalsConstellation Diagram of Main Modems4ASK and QPSK Block StructuresQPSK: CommentsExamples of QPSK-Rx’s Constellation Diagram
4 Performance Evaluation Criteria5 Line Codes (Wireline Digital Communications)
IntroductionMain Types of Line Codes and their PSDPopular Line CodesExamples of Using Line Codes:Example-1: Connecting Systems with RS232Example-2: PSTN and HDB3 Line Codes
PSD(f) of "line-code" signalsExample: Autocorr. & PSD(f) of a Bipolar Line Code
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 2 / 54
Introduction
IntroductionGeneral Block Structure of a Digital Communication System
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 3 / 54
Introduction
Let us focus on the "discrete channel"
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 4 / 54
Introduction
With reference the previous figures:
at point T : s(t) waveform.I The digital modulator
F takes γcs -bits at a time at some uniform rate rcs = 1Tcs
andF transmits one of M = 2γcs distinct waveforms s1(t), . . . , sM (t)i.e. we have an M -ary communication system.
F If γcs = 1 we have one bit at a time{0 7−→ s11 7−→ s2
i.e. a binary comm. system
I A new waveform (corresponding to a new γcs -bit-sequence) istransmitted every Tcs seconds
at point T̂ : noisy waveform r(t) = βs(t) + n(t).
I The transmitted waveform s(t), affected by the channel, is received at point T̂
at point B̂2 : a binary sequence.I based on the received signal r(t) the digital demodulator has to decidewhich of the M waveforms si (t) has been transmitted in any giventime interval Tcs
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 5 / 54
Introduction
If M = 2⇒ Binary Digital Modulator ⇒ Binary Comm. SystemIf M > 2⇒ M-ary Digital Modulator ⇒ M-ary Comm. System
B2 B̂2
F =
Pr(D1|H1), Pr(D1|H2), . . . , Pr(D1|HM )Pr(D2|H1), Pr(D2|H2), . . . , Pr(D2|HM )
...... . . . ,
...Pr(DK |H1), Pr(DK |H2), . . . , Pr(DK |HM )
. (1)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 6 / 54
Introduction
Binary Comm Systems: use M = 2 possible waveforms
{Tcs←−→s0(t),
Tcs←−→s1(t)};Tcs = Tb (2)
M-ary Comm Systems: use M possible waveforms
{s1(t)←−→Tcs
, s2(t)←−→Tcs
, . . . , sM (t)←−→Tcs
}; Tcs = γcsTb (3)
The M signals (or channel symbols) are characterized by their energyEi
Ei =∫ Tcs
0s2i (t) · dt; ∀i (4)
Furthermore their similarity (or dissimilarity) is characterized by theircross-correlation
ρij =1√EiEj
∫ Tcs
0si (t) · s∗j (t) · dt (5)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 7 / 54
Introduction
B2
B̂2
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 8 / 54
First Modelling of Digital Modulators
First Modelling of Digital Modulators
Note: Transforming a sequence of 0’s and 1’s to a sequence of ±1’so/p = 1− 2×i/p (6)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 9 / 54
First Modelling of Digital Modulators
Examples of Binary Modulators
A binary digital modulator maps 0’s and 1’s onto two analogue
symbol-waveforms s0(t) and s1(t), that is{0 7→ so (t)1 7→ s1(t)
0 ≤ t ≤ TcsThe three basic binary digital modulators
I ASK (Amplitude Shift-Keyed) 0 7→ so (t) = A0 · cos(2πFc t)1 7→ s1(t) = A1 · cos(2πFc t)for 0 ≤ t ≤ Tcs
I PSK (Phase Shift-Keyed) 0 7→ so (t) = Ac · cos(2πFc t)1 7→ s1(t) = Ac · cos(2πFc t + 180◦)for 0 ≤ t ≤ Tcs
I FSK (Frequency Shift-Keyed) 0 7→ so (t) = Ac · cos(2πF0t)1 7→ s1(t) = Ac · cos(2πF1t)for 0 ≤ t ≤ Tcs
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 10 / 54
First Modelling of Digital Modulators
The above binary digital modulators using complex representation:
I ASK (Amplitude Shift-Keyed)
0 7→ so (t) = A0 · exp(j2πFc t)1 7→ s1(t) = A1 · exp(j2πFc t)for 0 ≤ t ≤ Tcs
I PSK (Phase Shift-Keyed)0 7→ so (t) =
=+Ac︷ ︸︸ ︷Ac exp(j0◦) · exp(j2πFc t)
1 7→ s1(t) = Ac exp(j180◦)︸ ︷︷ ︸
=−Ac
· exp(j2πFc t)
for 0 ≤ t ≤ Tcs
I FSK (Frequency Shift-Keyed)
0 7→ so (t) = Ac · exp(j2πF0t)1 7→ s1(t) = Ac · exp(j2πF1t)for 0 ≤ t ≤ Tcs
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 11 / 54
First Modelling of Digital Modulators
Communication Systems Classification:1 Coherent (if demodulator is coherent, i.e. it uses a copy of the carrier)2 Non-coherent (if demodulator is non-coherent, e.g. it does not use thecarrier, e.g. envelope detector)
Note: Optimum demodulators are coherent.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 12 / 54
Second Modelling of Digital Modulators Signal Constellation
Second Modelling of Digital ModulatorsSignal Constellation
We may represent the signal si (t) by a point w si in a D-dimensionalEuclidean space with D ≤ M.The set of points (vectors) specified by the columns of the matrix
W =[w s1 , w s2 , . . . , w sM
](7)
is known as “signal constellation”.
Ei = wHsi w si =∥∥w si∥∥2 (8)
ρij =1√EiEj
wHsi w sj =wHsi w sj
‖w si ‖∥∥∥w sj ∥∥∥ (9)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 13 / 54
Second Modelling of Digital Modulators Distance Between two M -ary signals
Distance Between two M-ary Signals
The distance between two signals si (t) and sj (t) is the Euclideandistance between their associate vectors w si and w sj
i.e. dij =∥∥∥w si − w sj∥∥∥
=√(w si − w sj )H (w si − w sj )
=√wHsi w si + w
Hsj w sj − 2wHsi w sj
=√Ei + Ej − 2ρij
√EiEj
⇒ d2ij = Ei + Ej − 2ρij√EiEj
It is clear from the above that the Euclidean distance dij associatedwith two signals si (t) and sj (t) indicates, like the cross-correlationcoeffi cient, the similarity or dissimilarity of the signals.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 14 / 54
Second Modelling of Digital Modulators Constellation Diagram of Main Modems
Constellation Diagram of Main ModemsConsider an M-ary System having the following signals
{s1(t), s2(t), . . . , sM (t)} with 0 ≤ t ≤ Tcs
M-ary ASKI channel symbols: si (t) = Ai · cos(2πFc t) whereAi =given= 2i − 1−M (say)
I dimensionality of signal space = D = 1I if M = 4 then
s t1( )0100 11 10s t2( ) s t3( ) s t4( )
Es1 Es2 Es3 Es4Origin
I if M = 8 then
s t1( )000
Es1
001s t2( )
Es2
011s t3( )
Es3
010s t4( )
Es4Origin
s t5( )110
Es5
s t6( )
Es6
101s t7( )
Es7
100s t8( )
Es8
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 15 / 54
Second Modelling of Digital Modulators Constellation Diagram of Main Modems
M-ary PSK
I channel symbols: si (t) = A. cos(2πFc t + 2π
M . (i − 1) + φ)
for i = 1, 2, ..,I dimensionality of signal-space = D = 2
M = 4 & φ = 0◦ M = 4 & φ = 45◦ M = 8 & φ = 0◦
s t1( )
01
0011
s t2( )
s t3( )
Es1
Es2
Es3 Origin
10s t4( )
Es4
s t1( )00Es1
01s t2( )
Es2
11s t3( )
Es3
Origin
10s t4( )
Es4
450Origin
s t1( )000 Es1
001s t2( )
Es2
011s t3( )
Es3
010s t4( )
Es4
s t5( )110Es5
s t6( )
Es6
101s t7( )
Es7
100s t8( )
Es8
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 16 / 54
Second Modelling of Digital Modulators Constellation Diagram of Main Modems
M-ary FSK:very diffi cult to be represented using constellation diagram
with
fi − fj = n 12Tcs
fi + fj = m 12Tcs
wheren,m = integers
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 17 / 54
Second Modelling of Digital Modulators Constellation Diagram of Main Modems
M-ary Amplitude & Phase - M-ary QAM:1 channel symbols: si (t) = Ai · cos(2πFc t + ϕi + φ) i = 1, 2, . . . ,M
or snm(t) = An · cos(2πFc t + ϕm + φ)
n = 1, 2, . . . ,M1m = 1, 2, . . . ,M2M = M1 ×M2
2 dimensionality of signal space = D = 2PAM-PSK QAM
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 18 / 54
Second Modelling of Digital Modulators 4ASK and QPSK Block Structures
4ASK and QPSK Block StructuresAn ASK (M=4) Communication System
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 19 / 54
Second Modelling of Digital Modulators 4ASK and QPSK Block Structures
A QPSK Communication System
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 20 / 54
Second Modelling of Digital Modulators QPSK: Comments
QPSK: CommentsA QPSK modulator has four “channel-symbols”which are describedby the following equation:
si (t) = A cos(2πFc t +2π
M(i − 1) + φ) for i = 1, 2, 3, 4 (10)
withM = 4 and 0 ≤ t ≤ Tcs
and the modulation process is described by the so called“constellation diagram”.
The previous figure (page 20) shows the constellation for φ = 45◦.
From this figure it is clear that the constellation diagram shows themapping of binary digits to QPSK channel symbols (constellationpoints) as well as the square root of the energy
√Es of the channel
symbols. The diagram also indicates a Gray code mapping frombinary digits to channel symbols (constellation points).
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 21 / 54
Second Modelling of Digital Modulators QPSK: Comments
Overall, using complex number representation, it is clear from theconstellation diagram that
00 7→ s1(t) =√Es exp(j45◦)︸ ︷︷ ︸
=m1
exp(j2πFc t)
01 7→ s2(t) =√Es exp(j135◦)︸ ︷︷ ︸
=m2
exp(j2πFc t)
11 7→ s3(t) =√Es exp(j225◦)︸ ︷︷ ︸
=m3
exp(j2πFc t)
10 7→ s4(t) =√Es exp(j315◦)︸ ︷︷ ︸
=m4
exp(j2πFc t)
(11)
In other words,
si (t) = mi exp(j2πFc t) for i = 1, 2, 3 and 4 (12)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 22 / 54
Second Modelling of Digital Modulators QPSK: Comments
It is important to point out that the four symbols m1,m2,m3 and m4are known as baseband QPSK “channel symbols” and are usedby the “QPSK constellation symbol mapping”block shown in theprevious figure.
For instance, if the bit-pair at the input of the QPSK modulator is“01” then the output is the baseband channel symbol m2.
With reference to the figure given in page 20, the term exp(j2πFc t)(see Equations 11 and 12) is shown at the Transmitter as theup-conversion from baseband to bandpass.
In a similar fashion the down-conversion from baseband to bandpassis shown at the receiver’s front-end using the complex conjugate ofthe transmitter’s carrier, i.e. using exp(−j2πFc t).
Thus, overall, we have exp(j2πFc t) exp(−j2πFc t) = 1.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 23 / 54
Second Modelling of Digital Modulators QPSK: Comments
Based on the above discussion it is clear that the presence of thecarrier does not affect the analysis of the system.
Therefore, it is common practice to ignore the carrier when analyzingcommunication systems, by working on the baseband.
For the rest of this explanatory note the carrier term will be ignoredfrom both Tx and Rx.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 24 / 54
Second Modelling of Digital Modulators QPSK: Comments
Summarising
a QPSK modulator/demodulator is represented by its constellationdiagram and the QPSK symbol mapper transforms the binarysequence to a sequence of QPSK complex channel symbols mi ,forming the baseband QPSK message signal m(t) of bandwidth Bi.e.
m(t) = ∑n
m1,m2,m3,m4︷︸︸︷a[n] · c(t − n · Tcs ); nTcs ≤ t < (n+ 1) · Tcs
where c(t) = rect
{tTcs
}{a[n]} = sequ. of independent data symbols (mi )B = rcs
2 =1
2Tcs
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 25 / 54
Second Modelling of Digital Modulators QPSK: Comments
Thus with reference to the binary sequence of bits “001001”(message), by looking at the QPSK constellation diagram it is clearthat we have the following mapping
00 m1 = A exp(j45◦)10 m4 = A exp(j315◦)01 m2 = A exp(j135◦)
whereA =√Es
Note that, for a binary system
m(t) ≡∑n
±1︷︸︸︷a[n] · c(t − n · Tcs ); nTcs ≤ t < (n+ 1) · Tcs
where c(t) =rect
{tTcs
}{a[n]} = sequ. of independent data bits (±1s)B = rcs
2 =1
2Tcs
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 26 / 54
Second Modelling of Digital Modulators Examples of QPSK-Rx’s Constellation Diagram
Examples of Plots of QPSK- Receiver’s ConstellationDiagram
1.5 1 0.5 0 0.5 1 1.51.5
1
0.5
0
0.5
1
1.5
Re
Im
8 6 4 2 0 2 4 6 8
x 106
8
6
4
2
0
2
4
6
8x 106
ReIm
"good" "bad"
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 27 / 54
Performance Evaluation Criteria
Performance Evaluation Criteria
In general the quality of a digital communication system is expressedin terms of the accuracy with which the binary digits delivered at theoutput of the detector represent the binary digits that were fed intothe digital modulator.
It is generally taken that it is the fraction of the binary digits thatare delivered back in error that is a measure of the quality of thecommunication system.
This fraction, or rate, is referred to as the bit error probability p e ,or, Bit-Error-Rate BER.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 28 / 54
Performance Evaluation Criteria
The performance of M-ary communication systems is evaluated bymeans of the average probability of symbol error p e,cs , which, forM > 2, is different than the average probability of bit error (orBit-Error-Rate BER), pe .
That is{pe ,cs 6= pe for M > 2pe ,cs = pe for M = 2 (i.e. Binary Communication Systems)
However because we transmit binary data, the probability of bit errorρe is a more natural parameter for performance evaluation than ρe ,cs .
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 29 / 54
Performance Evaluation Criteria
Although, these two probabilities are relatedi.e.
pe = f{pe ,cs}their relationship depends on the encoding approach which isemployed by the digital modulator for mapping binary digits to M-arysignals (channel symbols)
whereγcs = log2M
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 30 / 54
Line Codes (Wireline Digital Communications) Introduction
Introduction: Lines Codes (Wireline Digital Comms)Line codes are used for data transmission in a metallic wire, twistedpair, cable.Line coding is baseband signal transmission having:
I a Tx (Digital Modulator1 without carrier)I a cable (communication channel)I a Rx (Digital Demodulator1 without carrier).
Appropriate coding of the transmitted symbol pulse shape canminimise the probability of error by ensuring that the spectralcharacteristics of the digital signal are well matched to thetransmission channel.The line code must also permit the receiver to extract accurate PCMbit and word timing signals directly from the received data, for properdetection and interpretation of the digital pulses.for more info see Chapter 6 in [Ian Glover and Peter Grant, "DigitalCommunications", 3rd edition,Pearson Education Limited, 2010]
1Commonly used names: Tx="Line Encoder" and Rx="Line Decoder".Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 31 / 54
Line Codes (Wireline Digital Communications) Main Types of Line Codes and their PSD
Main Types of Line Codes and their PSD
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 32 / 54
Line Codes (Wireline Digital Communications) Main Types of Line Codes and their PSD
Main Types of Line Codes and their PSD (cont.)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 33 / 54
Line Codes (Wireline Digital Communications) Main Types of Line Codes and their PSD
Main Types of Line Codes and their PSD (cont.)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 34 / 54
Line Codes (Wireline Digital Communications) Main Types of Line Codes and their PSD
Main Types of Line Codes and their PSD (cont.)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 35 / 54
Line Codes (Wireline Digital Communications) Popular Line Codes
Popular Line Codes
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 36 / 54
Line Codes (Wireline Digital Communications) Popular Line Codes
Popular Line Codes (cont.)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 37 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-1: Connecting Systems
Example (1)
3rd system: H3(f )
H3(f ) =1
1+ j2πfN.B.: This is a Low Pass Filter
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 38 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-1: Connecting Systems (cont.)
Example (1)
3rd system: H3(f )
H3(f ) =A(f )
1+ A(f ).B(f )NB: This is a Control System
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 39 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-1: Connecting Systems (cont.)
H(f ) = H1(f ).H2(f ).H3(f )
h(t) = h1(t) ? h2(t) ? h3(t)
the dot-symbol denotes "multiplication" and ? denotes "convolution".
These days we have "system-on-chip" (SOC)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 40 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-1: Connecting SystemsConnecting Systems - Far Apart
Wire Connection? No (due to effects equivalent to "Friction")
Wireline Comm Connection? Yes
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 41 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-1: Connecting Systems (cont.)Connecting Systems with RS232 Data Port
It establishes a two-way (i.e. "full-duplex") data communicationchannel using a single cable of length up to 15m. It uses:
I the pin 2 to Tx dataI the pin 3 to Rx data
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 42 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Transmitted signal corresponding to the letter "A", using datatransmission over a cable (RS232 data interface/port)
N.B.:one byte of asynchronous transmission
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 43 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-1: Connecting Systems (cont.)
Note: The RS232 is a wireline communication system connecting"smart" devices, e.g. Computers, CPU, PLC (Programmable LogicControllers), etc.:
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 44 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-2: PSTN and HDB3 Line Codes
Example :
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 45 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-2: PSTN and HDB3 Line Codes (cont.)
HDB3 is used in Europe (and many other countries) in PublicSwitched Telephone Networks (PSTN) according to the 2nd CCITTrecommendation (32 channel PCM)
it is used to transmit a frame of duration 1Fs= Ts = 125µs,
containing data from:I 30 users (one 8 bit word per user),I one Frame Alignment 8-bit word, andI one signalling info 8-bit word
using an HDB3 line encoder at the Tx and HDB3 line decoder in theRx.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 46 / 54
Line Codes (Wireline Digital Communications) Examples of Using Line Codes:
Example-2: PSTN and HDB3 Line Codes (cont.)Single-Channel Path of 2nd CCITT rec. (32-channels PCM)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 47 / 54
Line Codes (Wireline Digital Communications) PSD(f) of "line-code" signals
PSD(f) of Line Code SignalsSignals - Definition:
s(t) = ∑na[n]c(t − nTcs ); nTcs < t < (n+ 1)Tcs (13)
whereI c(t) is an energy signal of duration Tcs . This implies that c(t − nTcs )is defined in the interval nTcs < t < (n+ 1)Tcs
I {a[n]} is a binary sequenceAutocorrelation function of a binary sequence {a[n]}
Raa[k ] = E {a[n]a[n+ k ]} (14)
=I
∑i=1(a[n]a[n+ k ])i th-pair .Pr(i
th-pair) (15)
I where I denotes the total number of "pairs"
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 48 / 54
Line Codes (Wireline Digital Communications) PSD(f) of "line-code" signals
PSDs (f ) =|FT(c (t))|2
Tcs·
Raa[0] ++∞∑
k=−∞k 6=0
Raa[k ] exp(−j2πfkTcs )
(16)
Note that if Raa[k ] ={E{
a2n}for k = 0
E {an} · E {an+k} for k 6= 0
i.e. Raa[k ] ={
µ2a + σ2a for k = 0 where µa = mean and σa = stdµ2a for k 6= 0
then
(16)⇒PSDs (f ) = σ2a|FT(c(t))|2
Tcs︸ ︷︷ ︸Continuous Spectrum
+µ2aT 2cs· comb 1
Tcs(|FT(c(t))|2)︸ ︷︷ ︸
Discrete Spectrum
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 49 / 54
Line Codes (Wireline Digital Communications) PSD(f) of "line-code" signals
EXAMPLE. . . FOR YOU. . .if an = 0, 1 with Pr {an = 0} = Pr {an = 1} = 1
2 find the spectrumof “unipolar RZ” line-code signal.
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 50 / 54
Line Codes (Wireline Digital Communications) Example: Autocorr. & PSD(f) of a Bipolar Line Code
Example: Autocorr. & PSD(f) of a Bipolar Line Code
1 Show that for a bipolar line code the autocorrelation function of thecode sequence {a[n]} is as follows:
Raa[k ] =
1/2 if k = 0−1/4 if k = 10 if k ≥ 2
(17)
2 If Pr(0) = Pr(±1) = 0.5 and c(t) =rect{
tTcs
}, derive an expression
for the power spectral density PSD(f ) for the bipolar line codewaveform
m(t) =∞
∑n=−∞
a[n] · c(t − nTcs ) (18)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 51 / 54
Line Codes (Wireline Digital Communications) Example: Autocorr. & PSD(f) of a Bipolar Line Code
Solution: Autocorrelation R[0], R[1]
Raa[0] = E{
a[n]2}=
{0× 0 = 0→ 1
2(±1)× (±1) = +1→ 1
2
}= 0× 1
2+ 1× 1
2
=12
(19)
Raa[1] = E {a[n]a[n+ 1]} =
1st 2nd 1st×2rd Pr
0, 0, = 0 1/4
0, ±1, = 0 1/4
±1, 0, = 0 1/4
±1, ∓1, = −1 1/4
= 0× 1
4+ 0× 1
4+ 0× 1
4+ (−1)× 1
4
= −14
(20)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 52 / 54
Line Codes (Wireline Digital Communications) Example: Autocorr. & PSD(f) of a Bipolar Line Code
Solution: Autocorrelation R[k],k>1
Raa[k ]with k>1
=
1st 2nd 3rd 1st×3rd Pr
0, 0, 0 = 0 1/8
0, 0, ±1 = 0 1/8
0, ±1, 0 = 0 1/8
0, ±1, ∓1 = 0 1/8
±1, 0, 0 = 0 1/8
±1, 0, ∓1 = −1 1/8
±1, ∓1, 0 = 0 1/8
±1, ∓1, ±1 = +1 1/8
= (ignoring the 2nd column and multiplying 1st with 3rd we have)
= 0× 68+ 1× 1
8+ (−1)× 1
8= 0 (21)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 53 / 54
Line Codes (Wireline Digital Communications) Example: Autocorr. & PSD(f) of a Bipolar Line Code
Solution: PSD(f)
PSD(f ) =
∣∣∣FT(rect{ tTcs
})∣∣∣2
Tcs{R [0] + 2R [k ] cos(2πkTcs )}
=T 2cs sinc
2(fTcs )Tcs
{12− 2× 1
4cos(2πTcs )
}
= Tcs sinc2(fTcs ){12− 12cos(2πTcs )
}
= Tcs sinc2(fTcs ) · sin2(2πTcs2
)(22)
Prof. A. Manikas (Imperial College) EE303: Digital Modulators and Line Codes v.18 54 / 54