Band Pass Modulation Qpsk

Quadrature Phase Shift Keying In the BPSK description we used a single basis function of

φ1 (t ) =

2 t cos(2πf c t )rect   T T 

However, we could just as well have used the orthogonal basis function

φ 2 (t ) =

2 t sin( 2πf c t )rect   T T 

Essentially QPSK can be construed as two superimposed BPSK channels, one operating with φ1(t) and the other with φ2(t). Consequently we have an alphabet of 4 possible signals given by s i (t ) = si1φ1 (t ) + s i 2φ 2 (t ) for i=1,2,3 and 4 The table below gives a possible Grey coding of the QPSK signals i Input bit Phase of QPSK signal (deg) si1 1 10 45 c 2 00 135 -c 3 01 -135 -c 4 11 -45 c E where c = . The constellation points are as shown below: 2

s2

si2 c c -c -c

s1

c

s1(t) c

-c

s3

-c

Band Pass Modulation QPSK.doc

s4

1

The coherent QPSK receiver implementation is as shown below:

X

integrate

x1

bit1

φ 1 (t)

x(t)

X

integrate

bit2 x2

φ 2 (t)

The correlator outputs are given as T

x1 = ∫ x(t )φ1 (t )dt

T

and

0

x 2 = ∫ x(t )φ 2 (t )dt 0

To calculate the BER the QPSK scheme is construed as two BPSK channels superimposed. Since these channels are orthogonal to each other and do not mutually interfere, the BER is simply the same as that of the BPSK scheme as

 2 Eb Pe = Q  N0

   

where Eb is the bit energy equal to half the symbol energy E. Next consider the probability of symbol error. As QPSK consists of two BPSK channels a symbol error occurs if either BPSK channel has a bit error. Hence  2 Eb Pe = 2Q  N0

    = 2Q E   N   0   

This is accurate provided that the BER is small. However, we have double accounted for the probability that both BPSK channels are in error. Hence we have to subtract

Band Pass Modulation QPSK.doc

2

 2 Eb Q  N 0 

   

2

to account for the probability of such an event. Hence the exact Pe is given by  2 Eb Pe = 2Q  N 0 

  2 Eb  − Q   N 0  

   

2

Note that for typical BER values, this correction is insignificant. The initial estimate of twice the probability of bit error is known to larger than the actual probability of error. Hence it is an upper bound to the probability of error. As the probability of error is determined by the union of two independent probability events this type of bound is called a union bound. We could also have calculated the probability of symbol error directly by saying that: Pe is the probability of symbol error Pc is the probability that the symbol is correct Pe = 1 – Pc Where Pc is the probability that the symbol is correctly decoded. Pc implies that both bits are decoded correctly as Pc = (probability that channel 1 bit is correct) & (probability that channel 2 bit is correct) Since the noise on the two channels is independent Pc = (probability that channel 1 bit is correct) x (probability that channel 2 bit is correct) Pc = (1 – (probability of bit error))2 Hence   2 Eb Pe = 1 − 1 − Q  N  0  

2

  2 Eb   = 2Q   N 0  

 2 Eb   − Q  N  0  

   

2

PSD of QPSK

Band Pass Modulation QPSK.doc

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Since QPSK operates as two independent BPSK channels, the overall PSD of the QPSK transmission is the same as one of the BPSK channel transmission PSD. Hence the expression for Gx(f) is the same as with the BPSK case discussed earlier as:

Gx ( f ) =

(

1 sin c(( f − f c )T ) 2 + sin c(( f + f c )T ) 2 2

)

Offset QPSK

In the QPSK discussion, we used an idealized rectangular pulse shaping function. However, in practical links which are generally bandlimited, the pulse shape will look more like a sinc() function than a rect() function. This implies that the transmitter power will not be constant with time. Consider the QPSK case where the state transitions between state 1 to 3 or 2 to 4 such that the envelop of the signal passes through the origin (ie has zero modulus for an instant in time). Below is a sketch of the instantaneous power of QPSK with a band limited signal. transitions through origin

envelope of QPSK signal

0

T

2T

3T

t

If the transmitter power amplifier was perfectly linear then such a fluctuating evelope modulus would not be an issue. However, practical amplifiers are not that linear at their most efficient operating levels and hence intermodulation distortion will occur. The consequences of 3rd order intermodulation distortion is a spreading of the PSD which causes interference in the adjacent bands.

Band Pass Modulation QPSK.doc

4

PSD (dB)

3rd order intermodulation distortion

typical PSD specification

To minimize the effects of the transmitter nonlinearities, an attempt is made to make the modulation as constant modulus as possible. One possibility is to insist that the state only shift 90 degrees every bit period as illustrated below. However, this defeats the main purpose of QPSK which achieves 2 bits per symbol instead of just one.

s1(t)

A better alternative is to use Offset QPSK or OQPSK. In OQPSK we delay the BPSK modulation of the quadrature channel by T/2. Hence we have two basis functions 2 2 t  t 1 sin(2πf c t )rect  −  cos(2πf c t )rect   and φ 2 (t ) = T T T  T 2 Note that the OQPSK can still be considered as two independent BPSK channels as before. The only difference is that now the second BPSK channel is delayed by half a

φ1 (t ) =

Band Pass Modulation QPSK.doc

5

symbol period. This insures that the envelope modulus stays reasonably constant and never goes to zero as it did with QPSK. This is illustrated in the plot below:

I channel

Q channel

envelope of OQPSK

0

T

2T

t

The momentary drops in envelope power of 3 dB will result in some intermodulation distortion but this will be a lot less than before with QPSK. An application of OQPSK is found in IS95 reverse link transmitters which, being in the handset, have limited linearity (due to power consumption considerations). The reverse link is OQPSK modulated at the chip rate of 1.2288 Mcps. To ensure that the two basis functions remain orthogonal, it is necessary that T/2 represent an integer number of half cycles of the carrier frequency fc. Hence Tf c ∈ int Note that because both BPSK channels of the OQPSK modulation are independent and use the same pulse shaping function, the PSD of the transmitted signal is the same as with QPSK which as discussed is the same as BPSK. Example PSD of QPSK

Find the PSD of an offset QPSK signal produced by a random binary sequence in which symbols 1 and –1 are equally likely and the adjacent symbols are statistically independent. Band Pass Modulation QPSK.doc

6

Consider first normal QPSK where we can represent the modulated signal as two orthogonal BPSK channel signals operating on the independent I and Q channels. Recall that baseband BPSK with NRZ bipolar has a PSD of

S v ( f ) = T sin c 2 ( fT ) Hence as the I and Q channels are independent we have the total PSD of S v ( f ) = 2T sin c 2 ( fT ) Now this is modulated by

2 cos(2πf c t ) ) resulting in

S ( f ) = 12 S v ( f − f c ) + 12 S v ( f + f c ) S ( f ) = T2 sin c 2 (T ( f − f c )) + T2 sin c 2 (T ( f + f c )) Normalizing such that the symbol energy is 2Eb we have S ( f ) = Eb sin c 2 (T ( f − f c )) + Eb sin c 2 (T ( f + f c )) Next with OQPSK we can consider this as two NRZ BPSK channels as in QPSK except that the I channel is delayed with respect to the Q channel by T/2. As I and Q are still orthogonal, S(f) remains the same as for QPSK.

Example

Suppose we have a quadrature modulation scheme where the I and Q channels can be construed as being modulated independently by an arbitrary 1D modulation scheme. Assume that PeI – average probability of error in the I channel PeQ – average probability of error in the Q channel The probability of overall correct symbol detection is the probability that the I channel is correctly decoded and that the q channel is decoded correctly. As the I and Q modulation schemes are independent we have Pc = (1 − PeI )(1 − PeQ )

Band Pass Modulation QPSK.doc

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Pc = 1 − PeI − PeQ + PeI PeQ such that the probability of symbol error is Ps = 1 − Pc = PeI + PeQ − PeI PeQ Normally we can assume that PeI PeQ is very small such that Ps ≈ PeI + PeQ which is essentially the union bound.

Band Pass Modulation QPSK.doc

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