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Transcript chapter 6 pass-band data transmission

Chapter 6: Pass-band Data Transmission

CHAPTER 6 PASS-BAND DATA TRANSMISSION

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Chapter 6: Pass-band Data Transmission

Outline

•

6.1. Introduction

• 6.2. Pass-band Transmission • 6.3 Coherent Phase Shift Keying - BPSK Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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 the incoming digital signal is modulated onto a carrier (usually sinusoidal) with fixed frequency limits imposed by the band pass channel available • The etc.

broadcasting.

digital pass-band transmission communication channel

line codes design and orthogonal FDM techniques for where used in pass-band digital transmission may be microwave radio link, satellite channel • Other aspects of study in digital pass-band transmission are Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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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Chapter 6: Pass-band Data Transmission 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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Chapter 6: Pass-band Data Transmission

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 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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); Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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 P e is calculated – for simple binary coherent methods as coherent binary PSK and coherent binary FSK, there are exact formulas for P e – for coherent M-ary PSK and coherent M-ary FSK approximate solutions are sought.

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Chapter 6: Pass-band Data Transmission

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 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission A modulated signal is described in terms of in-phase and quadrature component as follows: 



f T c

) 



f t c

) ~  

f t c

)] (6.1) complex envelope ~ 



(6.2) 

f T c

)  cos(2 

f t c

) 

sin(2 

f t c

) (6.3) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission •

The complex envelope is actually the baseband version of the modulated (bandpass) signal

~ • s I (t) and s Q (t) as components of are low-pass signals.

Let

S B (f)

The power spectral density

S s (f)

of the original band-pass signal s(t) is a frequency shifted version of

S B (f)

except for a scaling factor:

 1 4 [

S B

(



f c

) 

S B

(



f c

)] (6.4) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

So,

• as far as the

power spectrum

is concerned it is sufficient to evaluate the

baseband power spectral density S

(f)

signal, the calculation of S B (f) should be simpler than the calculation of S s (f).

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Chapter 6: Pass-band Data Transmission

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 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission The effectiveness of a channel with bandwidth B can be expressed as:  

R

B

(6.5)

data rate bandwidth Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • 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 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

Outline

• • 6.1. Introduction

6.2. Pass-band Transmission

• 6.3 Coherent Phase Shift Keying – Binary Phase shift Keying (BPSK) – Quadriphase-Shift Keying (QPSK) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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 m 1 , m 2 , ….m

M; the

a priori

probabilities P(m 1 ), P(m 2 ),…P(m M ) specify the message source output; when symbols are equally likely we can express the probability p i as:

p i

  1

M i

)

for all i

(6.6) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission – signal transmission of

encoder

, producing a vector

s i

real elements, one such set for each of the of the source alphabet; dimension- wise

≤

;

made up symbols – s i is fed to a

modulator

that constructs a

distinct signal s i (t)

of duration T seconds as the representation of symbol m i generated by the message source; the signal s i

signal

(what does this mean?); s i is real valued is an

energy

• 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 N 0 /2 power spectral density Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • Receiver side (blocks described in detail p.326-327) – 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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Chapter 6: Pass-band Data Transmission

Outline

• • 6.1. Introduction • 6.2. Pass-band Transmission

6.3 Coherent Phase Shift Keying

–

Binary Phase shift Keying (BPSK)

– Quadriphase-Shift Keying (QPSK) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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: ( ) 

s t

1 ( )  2

E b T b

cos(2  2

E b T b

cos(2    )  

f t c

) (6.8) duration of one bit f c =n c /T b 2

E b T b

cos(2  ) (6.9) transmitted energy per bit Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • 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:  1  2

T b

cos(2 

f t c

), 0

T b

• So the transmitted signals can be expressed as: (6.10)

s t

1 ( ) 

E b

 1 0

T b

(6.11)

s t

2 ( )  

E b

 1 0

T b

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Chapter 6: Pass-band Data Transmission • 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:

11 

T b

0 

s t

1  1

t dt

 

E b

(6.13)

21  0 

T b s t

2  

E b

 1

t dt

(6.14) Digital Communication Systems 2012 R.Sokullu

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message point corresponding to s 1 Chapter 6: Pass-band Data Transmission message point corresponding to s 2 n c is an integer such that T symbol = n c /T bit Figure 6.3

Signal-space diagram for coherent binary PSK system. The waveforms depicting the transmitted signals s 1 (t) and s 2 (t), displayed in the inserts, assume n c  2.

Note that the frequency f number of cycles..

c is chosen to ensure that each transmitted bit contains an integer Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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 Z i if the Euclidean distance ||x-s k || 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 +E 1/2 – the set of points closest to message point 2 at – E 1/2 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • The decision rule is simply to decide that signal s 1 (t) (i.e. binary 1) was transmitted if the received signal point falls in region Z 1 , and decide that signal s 2 (t) (i.e. binary symbol 0) was transmitted if the received signal falls in region Z 2 .

• Two kinds of errors are possible due to noise: – sent s 1 (t), received signal point falls in Z 2 – sent s 2 (t), received signal point falls in Z 1 • This can be expressed as:

Z i : 0 < x 1 < æ

• and the observed element is expressed as a function of the received signal x(t) as:

x

1 

T b

0 

x t

 1

t dt

(6.15)

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Chapter 6: Pass-band Data Transmission

So,

• In Ch.5 it was deduced that memory-less AWGN channels, the observation elements

X i

are Gaussian RV with

mean s ij

and

variance N 0 /2.

• The conditional probability density function that

x j

(signal

s j

was received providing

m i

was sent) is given by:

f x j

(

x j

/

m i

)

 

1 exp[



1 (

1 

s ij

) ]

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Chapter 6: Pass-band Data Transmission • When we substitute for the case of BPSK

f x

1   1 

0 exp[  1

0 (

1 

21 1 

0 exp[  1

0 (

1 

E b

2 ) ] (6.16) • Then the conditional probability of the receiver in favor of 1 provided 0 was transmitted is:  10  0  

f x

1  1 

0 0   exp[  1

0 (

1 

E b

2 ) ]

1 (6.17) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • if we substitute and change the integration variable:

  10 

1 (

1  1 

E b

)] (6.18)

E b

/  

0 exp(  2 )  1 2

erfc

(

E

N

0 (6.19) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • Considering an error of the second kind: – signal space is symmetric about the origin – p 01 is the same as p 10 • Average probability of symbol error or the bit error rate for coherent BPSK is:

P



1 2

erfc

(

E N

) (6.20)

• 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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Chapter 6: Pass-band Data Transmission

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 f product modulator.

c =(n c /T b ) – Required BPSK modulated signal is at the output of the • 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 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission Figure 6.4

Block diagrams for (a) binary PSK transmitter and (b) coherent binary PSK receiver.

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Chapter 6: Pass-band Data Transmission

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 ≤ T b the in-phase component is +g(t) or – g(t).

   0, 2

E b

T b

T b otherwise

   (6.21) symbol shaping function Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • 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

S



2 E

sin ( (



T f



)

T f

)



2 E

sin

(

) (6.22)

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Chapter 6: Pass-band Data Transmission

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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Chapter 6: Pass-band Data Transmission

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:

  2

E T

cos[2 

f t c

 (2

 1)  4 ], 0 0,

elsewhere T

(6.23) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

Signal-Space Diagram of QPSK

• From 6.23 we can redefine the transmitted signal using a trigonometric identity:  2

E T i

  4 

f t c

)  2

E T i

  4 

f t c

) (6.24) • 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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Chapter 6: Pass-band Data Transmission • In our case there exist

two orthogonal basis functions

in the expansion of s i (t). These are φ 1 (t) and φ 2 (t), defined by a pair of quadrature carriers:  1  2 cos(2 

T f t c

), 0

(6.25)  2  2 sin(2 

T f t c

), 0

(6.26) • Based on these representations we can make the following two important observations: Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • There are 4 message points and the associated vectors are defined by:

s i

    

E E

cos[(2

sin[(2

 1)  1)   4 4 ] ]      ,

 1, 2, 3, 4 • Values are summarized in Table 6.1

(6.27) • Conclusion: – QPSK has a

two-dimensional signal constellation

(N = 2) and

four message points

(M = 4).

– As binary PSK, QPSK has minimum average energy Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

Figure 6.6

Signal-space diagram of coherent QPSK system.

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Chapter 6: Pass-band Data Transmission

Example 6.1

•

Generate a QPSK signal for the given binary input.

 Input binary sequence is: 01101000  Divided into odd- even- input bits sequences  Two waveforms are created:

φ 2 (t) s i1 φ 1 (t)

and

s i2

– individually viewed as binary PSK signals.

 By adding them we get the QPSK signal Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

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 Z i if the Euclidean distance ||x-s k || is minimum for k = i



Result:

Four regions – quadrants – are defined, whose vertices coincide with the origin.

 Marked in fig. 6.6 (previous pages) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

Figure 6.7

(

) Input binary sequence. (

) Odd-numbered bits of input sequence and associated binary PSK wave. (

) Even-numbered bits of input sequence and associated binary PSK wave. (

) QPSK waveform defined as

(

)

s i



1 (

)

 s i



2 (

 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

Error probability of QPSK

• In a coherent system the received signal is defined as: 

  

0 1, 2,3, 4 (6.28)

•

w(t)

is the sample function of a white Gaussian noise process of zero mean and

N 0 /2.

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Chapter 6: Pass-band Data Transmission The observation vector has two elements, x 1 defined by: and x 2 ,

1  

0 

x t

 1

E t dt

cos    2

 1   4   

1  



1 2 (6.29) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission The

observation vector

has two elements, x 1 defined by: and x 2 ,

1  

0 

x t

 1

E t dt

cos    2

 1   4   

1  



1 2 (6.29) i=1 and 3 so cos(π/4) = 1/2 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

x

2   

0 

x t

 2

E t dt

sin    2

i

 1   4   

w

2  

E



w

2 2 (6.30) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

x

2   

0 

x t

 2

E t dt

sin    2

i

 1   4   

w

2  

E



w

2 2 (6.30) i=2 and 4 so sin(3π/4) = 1/2 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission

So,

• The

observable elements

x

and

x

are sample values of independent Gaussian RV with

mean

equal to

+/-√E/2

and

-/+√E/2

and

variance

equal to

N

/2.

• The decision rule is to find

whether the received signal s

or not.

is in the expected zone Z

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Chapter 6: Pass-band Data Transmission

Calculation of the error probability:

• QPSK is actually equivalent to

two BPSK

working in parallel and using

quadrature in phase

carriers

systems that are • 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 N 0 /2.

• Calculate the average probability of bit error for each channel as: Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • In one of the previous classes we derived the formula for the

bit error rate for coherent binary PSK

as:

P



1 2

erfc

(

E

N

) (6.20)

• Using 6.20 we can find the average probability for bit error in

each channel

of the coherent QPSK as:

'  1 2

erfc

   1 2

erfc

 

/ 2  

N o E

N o

  (6.31) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • 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 p

and p

.

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Chapter 6: Pass-band Data Transmission • The average probability for a correct decision resulting from the combined action of the two channels can be expressed as (p 1 * p 2 ):

P c P

') 2 1 2

erfc

(

erfc

(

0 )] 2

0 )  1 4

erfc

2 (

0 ) (6.32) Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission • Thus the average probability for a

symbol error

for coherent QPSK can be written as:

P



erfc

(

E

N

0 )  1 4

erfc

2 (

E

N

0 ) (6.33) • The term erfc 2 (√E/2N 0 )<< 1 so it can be ignored, then:

P

erfc

(

E

2 N

) (6.34)

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Chapter 6: Pass-band Data Transmission • 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:



2

E b

(6.36)

• So, using the ratio E b /N 0

error (6.37)

: we can express the

symbol P e erfc

(

E N o b

) (6.37)

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Chapter 6: Pass-band Data Transmission • Finally we can express the

bit error rate (BER)

for QPSK as:

BER



1 2

erfc

(

E N o b

) (6.38)

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 E b /N 0

uses half of the channel bandwidth.

but

• 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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Chapter 6: Pass-band Data Transmission

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) –

a 1 (t)

and

a 2 (t)

– in any signaling interval the amplitudes of

a 1 (t)

and

a 2 (t)

equal s i1 and s i2 depending on the particular bit that is sent –

a 1 (t)

and

a 2 (t)

modulate a pair of quadrature carriers (orthogonal basis functions

φ 1 (t) = √2/Tcos(2πf c t)

and

φ 2 (t)= √2/Tsin(2πf c t)

) – results in a pair of binary PSK which can be detected independently due to the orthogonallity of the basis functions.

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Chapter 6: Pass-band Data Transmission • Algorithm (receiver) – pair of correlators with common input – locally generated pair of coherent reference signals

φ 1 (t)

and

φ 2 (t)

– correlator outputs – x 1 to the input signal

x(t)

and x 2 produced in response – threshold comparison for decision • in-phase – x 1 >0 decision for 1; x 1 <0 decision of 0 • quadrature – x 2 >0 decision for 1; x 2 <0 decision of 0 – combined in a multiplexer Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission Figure 6.8

Block diagrams of (a) QPSK transmitter and (b) coherent QPSK receiver.

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Chapter 6: Pass-band Data Transmission

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: 1. depending on the dibit sent during the signaling interval T b ≤ t ≤ T b 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 Digital Communication Systems 2012 R.Sokullu

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Chapter 6: Pass-band Data Transmission   0,

E T

, 0

T otherwise

(6.39)

So, it follows that the in-phase and quadrature components have a common power spectral density E sinc

(Tf).

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Chapter 6: Pass-band Data Transmission • 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

S B

 2 ( )  4

E b

sin

2 (2

T f b

) (6.40) Digital Communication Systems 2012 R.Sokullu

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