Quadriphase-Shift Keying (QPSK)

Comm. Systems Unit 1- Digital MODEM Sau-Hsuan Wu

Quadriphase-Shift Keying (QPSK) The transmitted signal

2cos 2 2 1 4 , 1, . . , 4

Or, equivalently, it can be rewritten as2

cos2 1

4 cos 22

sin2 1

4 sin 2

We define two orthonormal basis functions

2cos 2 0

2sin 2 0

Dept. of Electrical and Computer Eng., NCTU 26


Using Gray-coded input dibit10 → , 00 → , 01 → , 11 →

The 4 message pointsare


cos2 1

4

sin2 1

4

1,2,3,4


The QPSK signal waveform of s(t) si11(t) si22(t)



The received signal is given by , 0 , 1, 2, 3, 4

Notice that = si11(t) si22(t) The observation vector x = [x1, x2] has two dimensions

cos2 1

4 2

sin2 1

4 2

Both and ~ 0, /2 The noise is two dimensional as well The decision rule is that if x zi (quadrant i) si, i =1,…,4



To calculate the average BER, we note that a coherent QPSK system is equivalent to two coherent BPSK systems

For each channel, the signal energy per bit is Eb = E/2 Bit errors in the in-phase and quadrature channels of the

coherent QPSK are statistically independent, given by12 erfc 2

Accordingly, the average probability of correct decision is

1 1 erfc 214 erfc 2



Therefore, the average probability of symbol error is

1 erfc 214 erfc 2

≅ erfc 2 erfc , when 2 ≫ 1

For Gray coded symbols, the most probable number of bit errors is one as a symbol is most likely to be taken as adjacent symbols

∴ BER 212 erfc

A coherent QPSK achieves the same BER as a coherent BPSK QPSK transmits at twice the bit rate of BPSK for the same BW



2/ sin 2

Generation and detection of QPSK


Transmitter

Receiver


Problems of QPSK The carrier phase changes by 180 when both the in-phase

quadrature components changes sign The carrier phase changes by 90 whenever in-phase or

quadrature components changes sign This might result in changes in the carrier amplitude when

the QPSK signal is filtered during the course of transmission, prior to detection, thereby causing additional symbol errors


Possible transition paths of QPSK message points

00

01

10

11


00

01

10

11

Suppose that 5 symbols (dibits) of 00, 11, 00, 01, 11 are transmitted using QPSK signaling


00 11 00 01 11


Can we play some tricks to reduce phase changes? Control the patterns of phase transitions

00 10 11 01 00 00 01 11 11

Change the signal point on 1 before changing the one on 2


00 10

1101


What are the effects on the resultant signal?


(00) 10 01 00 1100 11 00 01 11


Offset QPSK Delay the bit stream responsible for generating the

quadrature components by half a symbol interval with respect to (w.r.t.) the bit stream for the in-phase components

2cos 2 0

2sin 2 2

32

The phase transitions is confined to 90 Phase transitions in OQPSK occur twice as frequently but

with half the intensity encountered in QPSK The BER of OQPSK is exactly the same to that of QPSK



The error probability in the in-phase or quadrature channel of a coherent OQPSK receiver is still ⁄ erfc /2

The amplitude fluctuations in OQPSK due to filtering are expected to be smaller than that in QPSK



Can we further reduce the amplitude fluctuations? The key is to further confine the intensity of phase transitions Alternatively use the two types of constellations of QPSK

Phase transitions are restricted to and


/4-shifted QPSK


-shifted QPSK The carrier phases for successive symbols are alternatively

picked from one of the two QPSK constellations Like QPSK, it can be differentially decoded, in which case

we call it -shifted DQPSKcos ∆ cos In Phasesin ∆ sin Quadrature

Correspondence between Gray-coded dibit and phase changes ∆ for -shifted DQPSK00 → , 01 → , 11 → , 10 →

Suppose we have four dibits 00 10 10 01, and the initial phase

, , 0,



Detection of -shifted DQPSK The receiver first computes the projections xI and xQ of x(t)

onto the basis functions 1(t) and 2(t), respectively Extract the phase angle k of the channel output Compute the phase difference ∆ =

If ∆ 180, then ∆ = ∆ +360 If ∆ 180, then ∆ = ∆ 360



M-ary PSK The phase of the carrier 2 1 / , 1, … ,

cos 2 1 , 0 , 1, … ,

The signal space diagram of 8PSK is shown to the right

The in-phase component iscos 1

The quadrature component issin 1

Both and ~ 0, /2 , are circularly Gaussian



The BER of MPSK Recap the BER of QPSK

erfc /2 erfc /2

Suppose that s1 is sent. The decision regionin signal space is illustrated to the right as

At high SNR erfc /2 ≫ erfc /2

≅ erfc /2 = taking + taking



Define Aik as the event of vector x being closer to vector skthan to si when si is sent

In AWGN channel, the probability of mistaking mk for mi, ki, is thus given by is often called the pairwise error probability, and is

denote by , Let P be the probability of symbol error when is

sent. From probability theory, we have the union boundP ∑ ≡ ∑ , , i=1,…,M

Since is identically distributed along any direction,let , we have

, exp/ erfc



As a result, the probability of symbol error

P12 erfc

2

Suppose that the signal constellation is circularly symmetric about the origin, then P is the same for all i=1,…,M

12 erfc

2,

Define the minimum distance of a signal constellation dminmin , ,

Since erfc(u) is monotonically decreasing w.r.t. u, we have 12 erfc

21

2 erfc2



ByChenroff Bound

erfc2

1exp 4

Therefore, we have1

2 erfc2

12

exp 4

The error probability decreases exponentially w.r.t. The probability of mistaking one symbol for either 1 or 2

most nearest symbols is much greater than any other errors For Gray coded M-ary modulation

Any pair of adjacent symbols differ in only one bit position Therefore, the most probable number of bit errors is one



The average symbol error probability is related to BER by

thbitisinerror

thbitisinerror BER log

Therefore,

Suppose m1 is sent and is large

For MPSK, d12 = d18=2 sin


≅ , ,

22 erfc 2

erfc sin


Lab 2 Generate a series of binary random numbers Modulate the binary numbers with QPSK, given that the

carrier frequency fc is 1 MHz, and the symbol energy E is 10dB and symbol frequency is 1KHz

Demodulate the transmitted signal using the block diagram on p.p. 34

Add the demodulated samples by AWGN noise with =1 Do symbol detection on the resultant samples Draw and compare the BERs with the theoretical values

Generate OQPSK and -shifted QPSK for a series of dibits

Repeat the above procedures for OQPSK and draw BER


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Quadriphase-Shift Keying (QPSK)

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