1. CHAPTER 6 PASS-BAND DATA TRANSMISSION
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Outline
6.1. Introduction
6.2. Pass-band Transmission
6.3 Coherent Phase Shift Keying - BPSK
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6.1 Introduction
In Ch. 4 we studied digital baseband transmission where the generated data stream, represented in the form of discrete pulse-amplitude modulated signal (PAM) is transmitted directly over a low-pass channel.
In Ch.6 we will study digital pass-band transmission where the incoming digital signal is modulated onto a carrier (usually sinusoidal) with fixed frequency limits imposed by the band-pass channel available
The communication channel used in pass-band digital transmission may be microwave radio link, satellite channel etc.
Other aspects of study in digital pass-band transmission are line codes design and orthogonal FDM techniques for broadcasting.
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Definitions:
The modulation of digital signals is a process involving switching (keying) the amplitude, frequency or phase of a sinusoidal carrier in some way in accordance with the incoming digital data.
Three basic schemes exist:
amplitude shift keying (ASK)
frequency shift keying (FSK)
phase shift keying (PSK)
REMARKS:
In continuous wave modulation phase modulated and frequency modulated signals are difficult to distinguish between, this is not true for PSK and FSK.
PSK and FSK both have constant envelope while ASK does not.
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Figure 6.1 Illustrative waveforms for the three basic forms of signaling binary information. (a) Amplitude-shift keying. (b) Phase-shift keying. (c) Frequency-shift keying with continuous phase.
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6. Hierarchy of Digital Modulation Techniques
Depending on whether the receiver does phase-recovery or not the modulation techniques are divided into:
Coherent
Non-coherent
Phase recovery circuit - ensures synchronization of locally generated carrier wave (both frequency and phase), with the incoming data stream from the Tx.
Binary versus M-ary schemes
binary – use only two symbol levels;
M-ary schemes – pure M-ary scheme exists as M-ary ASK, M-ary PSK and M-ary FSK, using more then one level in the modulation process; Also hybrid M-ary schemes – quadrature-amplitude modulation (QAM); preferred over band-pass transmissions when the requirement is to preserve bandwidth at the expense of increased power
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Remarks:
Linearity
M-ary PSK and M-ary QAM are both linear modulation schemes; M-ary PSK – constant envelope; M-ary QAM – no
M-ary PSK – used over linear channels
M-ary QAM – used over non-linear channels
Coherence
ASK and FSK – used with non-coherent systems; no need of maintaining carrier phase synchronization
“noncoherent PSK” means no carrier phase information; instead pseudo PSK = differential PSK (DPSK);
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Probability of Error
Design goal – minimize the average probability of symbol error in the presence of AWGN.
Signal-space analysis is a tool for setting decision areas for signal detection over AWGN (i.e. based on maximum likelihood signal detection) (Ch.5!)
Based on these decisions probability of symbol error Pe is calculated
for simple binary coherent methods as coherent binary PSK and coherent binary FSK, there are exact formulas for Pe
for coherent M-ary PSK and coherent M-ary FSK approximate solutions are sought.
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Power Spectra
power spectra of resulting modulated signals is important for:
comparison of virtues and limitations of different schemes
study of occupancy of channel bandwidth
study of co-channel interference
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A modulated signal is described in terms of in-phase and
quadrature component as follows:
complex envelope
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The complex envelope is actually the baseband version of the modulated (bandpass) signal.
sI(t) and sQ(t) as components of are low-pass signals.
Let SB(f) denote the power spectral density of the complex envelope , known as baseband power spectral density.
The power spectral density Ss(f) of the original band-pass signal s(t) is a frequency shifted version of SB(f) except for a scaling factor:
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as far as the power spectrum is concerned it is sufficient to evaluate the baseband power spectral density SB(f) and since is a low-pass signal, the calculation of SB(f) should be simpler than the calculation of Ss(f).
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Bandwidth efficiency
Main goal of communication engineering – spectrally efficient schemes
maximize bandwidth efficiency = ratio of the data rate in bits per seconds to the effectively utilized channel bandwidth.
achieve bandwidth at minimum practical expenditure of average SNR
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The effectiveness of a channel with bandwidth B can be expressed as:
bandwidth
data rate
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Before (Ch.4) we discussed that the bandwidth efficiency is the product of two independent factors:
multilevel encoding – use of blocks of bits instead of single bits.
spectral shaping – bandwidth requirements on the channel are reduced by the use of suitable pulse-shaping filters
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Outline
6.1. Introduction
6.2. Pass-band Transmission
6.3 Coherent Phase Shift Keying
Binary Phase shift Keying (BPSK)
Quadriphase-Shift Keying (QPSK)
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17. 6.2 Pass-band transmission model
Functional blocks of the model
Transmitter side
message source, emitting a symbol every T seconds; a symbol belongs to an alphabet of M symbols, denoted by m1, m2, ….mM; the a priori probabilities P(m1), P(m2),…P(mM) specify the message source output; when symbols are equally likely we can express the probability pi as:
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signal transmission encoder , producing a vector si made up of N real elements, one such set for each of the M symbols of the source alphabet; dimension- wise N ≤ M;
si is fed to a modulator that constructs a distinct signal si(t) of duration T seconds as the representation of symbol mi generated by the message source; the signal si is an energy signal (what does this mean?); si is real valued
Channel:
linear channel wide enough to accommodate the transmission of the modulated signal with negligible or no distortion
the channel white noise is a sample function of AWGN with zero mean and N0/2 power spectral density
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19. Figure 6.2 Functional model of pass-band data transmission system.
Receiver side (blocks described in detail p )
detector
signal transmission decoder; reverses the operations performed in the transmitter;
Figure 6.2 Functional model of pass-band data transmission system.
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Outline
6.1. Introduction
6.2. Pass-band Transmission
6.3 Coherent Phase Shift Keying
Binary Phase shift Keying (BPSK)
Quadriphase-Shift Keying (QPSK)
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21. 6.3 Coherent Phase Shift Keying - Binary Phase Shift Keying (BPSK)
In a coherent binary PSK the pair of signals used to represent binary 0 and 1 are defined as:
duration of one bit
fc=nc/Tb
transmitted energy
per bit
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The equations (6.8) and (6.9) represent antipodal signals – sinusoidal signals that differ only in a relative phase shift of 180 degrees.
In BPSK there is only one basis function of unit energy expressed as:
So the transmitted signals can be expressed as:
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A coherent BPSK system can be characterized by having a signal space that is one dimensional (N= 1), with signal constellation consisting of two message points (M = 2)
The coordinates of the message points are:
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message point corresponding to s1
message point corresponding to s2
nc is an integer such that
Tsymbol = nc/Tbit
Figure 6.3 Signal-space diagram for coherent binary PSK system. The waveforms depicting the transmitted signals s1(t) and s2(t), displayed in the inserts, assume nc  2. Note that the frequency fc is chosen to ensure that each transmitted bit contains an integer number of cycles..
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25. Error Probability of Binary PSK
Decision rule: based on the maximum likelihood decision algorithm/rule which in this case means that we have to choose the message point closest to the received signal point
observation vector x lies in region Zi if
the Euclidean distance ||x-sk|| is minimum for k = i
For BPSK: N= 1, space is divided into two areas (fig.6.3)
the set of points closest to message point 1 at +E1/2
the set of points closest to message point 2 at – E1/2
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The decision rule is simply to decide that signal s1(t) (i.e. binary 1) was transmitted if the received signal point falls in region Z1, and decide that signal s2(t) (i.e. binary symbol 0) was transmitted if the received signal falls in region Z2.
Two kinds of errors are possible due to noise:
sent s1(t), received signal point falls in Z2
sent s2(t), received signal point falls in Z1
This can be expressed as: Zi: 0 < x1 < æ
and the observed element is expressed as a function of the received signal x(t) as:
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In Ch.5 it was deduced that memory-less AWGN channels, the observation elements Xi are Gaussian RV with mean sij and variance N0/2.
The conditional probability density function that xj (signal sj was received providing mi was sent) is given by:
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When we substitute for the case of BPSK
Then the conditional probability of the receiver in favor of 1 provided 0 was transmitted is:
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if we substitute and change the integration variable:
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Considering an error of the second kind:
signal space is symmetric about the origin
p01 is the same as p10
Average probability of symbol error or the bit error rate for coherent BPSK is:
So increasing the signal energy per bit makes the points and move farther apart which correspond to reducing the error probability.
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31. Generation and Detection of Coherent BPSK Signals
Transmitter side:
Need to represent the binary sequence 0 and 1 in polar form with constant amplitudes, respectively – and (polar non-return-to-zero – NRZ - encoding).
Carrier wave is with frequency fc=(nc/Tb)
Required BPSK modulated signal is at the output of the product modulator.
Receiver side
noisy PSK is fed to a correlator with locally generated reference signal
correlator output is compared to a threshold of 0 volts in the decision device
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Figure 6.4 Block diagrams for (a) binary PSK transmitter and (b) coherent binary PSK receiver.
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Power Spectra of BPSK
From the modulator – the complex envelope of the BPSK has only in-phase component
Depending on whether we have a symbol 1 or 0 during the signaling interval 0 ≤ t ≤ Tb the in-phase component is +g(t) or – g(t).
symbol
shaping function
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We assume that the input binary wave is random, with symbols 1 or 0 equally likely and that symbols transmitted during the different time slots are statistically independent.
So, (Ch.1) the power spectra of such a random binary wave is given by the energy spectral density of the symbol shaping function divided by the symbol duration.(See Ex.1.3 and 1.6)
g(t) is an energy signal – FT
Finally, the energy spectral density is equal to the squared magnitude of the signals FT.
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Outline
6.1. Introduction
6.2. Pass-band Transmission
6.3 Coherent Phase Shift Keying
Binary Phase shift Keying (BPSK)
Quadriphase-Shift Keying (QPSK)
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36. 6.3 Coherent Phase Shift Keying - QPSK
Reliable performance
Very low probability of error
Efficient utilization of channel bandwidth
Sending more then one bit in a symbol
Quadriphase-shift keying (QPSK) - example of quadrature-carrier multiplexing
Information is carried in the phase
Phase can take one of four equally spaced values – π/4, 3π/4, 5π/4, 7π/4
We assume gray encoding (10, 00, 01, 11)
Transmitted signal is defined as:
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37. Signal-Space Diagram of QPSK
From 6.23 we can redefine the transmitted signal using a trigonometric identity:
From this representation we can use Gram-Schmidt Orthogonal Procedure to create the signal-space diagram for this signal.
It allows us to find the orthogonal basis functions used for the signal-space representation.
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In our case there exist two orthogonal basis functions in the expansion of si(t). These are φ1(t) and φ2(t), defined by a pair of quadrature carriers:
Based on these representations we can make the following two important observations:
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There are 4 message points and the associated vectors are defined by:
Values are summarized in Table 6.1
Conclusion:
QPSK has a two-dimensional signal constellation (N = 2) and four message points (M = 4).
As binary PSK, QPSK has minimum average energy
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40. Figure 6.6 Signal-space diagram of coherent QPSK system.
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Example 6.1
Generate a QPSK signal for the given binary input.
Input binary sequence is:
Divided into odd- even- input bits sequences
Two waveforms are created: si1φ1(t) and si2 φ2(t) – individually viewed as binary PSK signals.
By adding them we get the QPSK signal
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42. Example 6.1 – cont’d
To define the decision rule for the detection of the transmitted data sequence the signal space is partitioned into four regions in accordance with:
observation vector x lies in region Zi if
the Euclidean distance ||x-sk|| is minimum for k = i
Result: Four regions – quadrants – are defined, whose vertices coincide with the origin.
Marked in fig. 6.6 (previous pages)
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Figure 6.7 (a) Input binary sequence. (b) Odd-numbered bits of input sequence and associated binary PSK wave. (c) Even-numbered bits of input sequence and associated binary PSK wave. (d) QPSK waveform defined as s(t)  si1f1(t)  si2f2(t).
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44. Error probability of QPSK
In a coherent system the received signal is defined as:
w(t) is the sample function of a white Gaussian noise process of zero mean and N0/2.
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45. The observation vector has two elements, x1 and x2, defined by:
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46. The observation vector has two elements, x1 and x2, defined by:
i=1 and 3 so
cos(π/4)
= 1/2
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i=2 and 4 so
sin(3π/4)
= 1/2
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The observable elements x1 and x2 are sample values of independent Gaussian RV with mean equal to +/-√E/2 and -/+√E/2 and variance equal to N0/2.
The decision rule is to find whether the received signal si is in the expected zone Zi or not.
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50. Calculation of the error probability:
QPSK is actually equivalent to two BPSK systems working in parallel and using carriers that are quadrature in phase.
According to 6.29 and 6.30 these two BPSK are characterized as follows:
The signal energy per bit is √E/2
The noise spectral density is N0/2.
Calculate the average probability of bit error for each channel as:
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In one of the previous classes we derived the formula for the bit error rate for coherent binary PSK as:
Using 6.20 we can find the average probability for bit error in each channel of the coherent QPSK as:
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The bit errors for the in-phase and quadrature channels of the coherent QPSK are statistically independent
The in-phase channel makes a decision on one of the two dibits constituting a symbol; the quadrature channel – for the other one.
Then the average probability of a correct decision is product of two statistically independent events p1 and p2.
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The average probability for a correct decision resulting from the combined action of the two channels can be expressed as (p1 * p2):
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Thus the average probability for a symbol error for coherent QPSK can be written as:
The term erfc2(√E/2N0)<< 1 so it can be ignored, then:
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Since there are two bits per symbol in the QPSK system, the energy per symbol is related to the energy per bit in the following way:
So, using the ratio Eb/N0 we can express the symbol error (6.37):
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Finally we can express the bit error rate (BER) for QPSK as:
Conclusions:
A coherent QPSK system achieves the same average probability of bit error as a coherent PSK system for the same bit error rate and the same Eb/N0 but uses half of the channel bandwidth. or
At the same channel bandwidth the QPSK systems transmits information at twice the bit rate and the same average probability of error.
Better usage of channel bandwidth!
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57. Generation and Detection of Coherent QPSK Signals
Algorithm (transmitter)
input binary data sequence transformed into polar form (non-return-to-zero encoder) – symbols 1 and 0 are represented by +√E/2 and -√E/2
divided into two streams by a demultiplexer (odd and even numbered bits) – a1(t) and a2(t)
in any signaling interval the amplitudes of a1(t) and a2(t) equal si1 and si2 depending on the particular bit that is sent
a1(t) and a2(t) modulate a pair of quadrature carriers (orthogonal basis functions φ1(t) = √2/Tcos(2πfct) and φ2(t)= √2/Tsin(2πfct) )
results in a pair of binary PSK which can be detected independently due to the orthogonallity of the basis functions.
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Algorithm (receiver)
pair of correlators with common input
locally generated pair of coherent reference signals φ1(t) and φ2(t).
correlator outputs – x1 and x2 produced in response to the input signal x(t)
threshold comparison for decision
in-phase – x1>0 decision for 1; x1<0 decision of 0
quadrature – x2>0 decision for 1; x2<0 decision of 0
combined in a multiplexer
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Figure 6.8 Block diagrams of (a) QPSK transmitter and (b) coherent QPSK receiver.
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60. Power Spectra of QPSK Signals
Assumptions;
binary wave is random;
1 and 0 symbols are equally likely;
symbols transmitted in adjacent intervals are statistically independent
Then:
depending on the dibit sent during the signaling interval Tb ≤ t ≤ Tb
the in-phase component equals +g(t) or – g(t)
similar situation exists for the quadrature component
Note: the g(t) denotes the symbol shaping function
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it follows that the in-phase and quadrature components have a common power spectral density E sinc2(Tf).
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The in-phase and quadrature components are statistically independent.
the baseband power spectral density of QPSK equals the sum of the individual power spectral densities of the in-phase and quadrature components
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