6.5.1 Amplitude Modulation (DSBSC)
Generally, in amplitude modulation, the amplitude of a high frequency carrier signal, cos(2πfct), is varied in accordance to the instantaneous amplitude of the modulat-ing message signal m(t). The resultmodulat-ing modulated carrier or AM signal can be represented as:
sAM(t) = Ac[1 + km(t)] cos(2πfct). (6.15) The modulation index k of an AM signal is defined as the ratio of the peak message signal amplitude to the peak carrier amplitude. For a sinusoidal modulating signal m(t) = AAm
c cos(2πfmt), the modulation index is given by k = Am
Ac
. (6.16)
This is a nonlinear technique and can be made linear by multiplying the carrier with the message signal.The resulting modulation scheme is known as DSBSC modula-tion. In DSBSC the amplitude of the transmitted signal, s(t), varies linearly with the modulating digital signal, m(t). Linear modulation techniques are bandwidth efficient and hence are very attractive for use in wireless communication systems where there is an increasing demand to accommodate more and more users within a limited spectrum. The transmitted signal DSBSC signal s(t) can be expressed as:
s(t) = Am(t)exp(j2πfct). (6.17)
If m(t) is scaled by a factor of a, then s(t), the output of the modulator, is also scaled by the same factor as seen from the above equation. Hence the principle of homogeneity is satisfied. Moreover,
s12(t) = A[m1(t) + m2(t)]cos(2πfct) (6.18)
= Am1(t)cos(2πfct) + Am2(t)cos(2πfct)
= s1(t) + s2(t)
where A is the carrier amplitude and fcis the carrier frequency. Hence the principle of superposition is also satisfied. Thus DSBSC is a linear modulation technique.
AM demodulation techniques may be broadly divided into two categories: co-herent and non-coco-herent demodulation. Coco-herent demodulation requires knowledge
Figure 6.1: BPSK signal constellation.
of the transmitted carrier frequency and phase at the receiver, whereas non-coherent detection requires no phase information.
6.5.2 BPSK
In binary phase shift keying (BPSK), the phase of a constant amplitude carrier signal is switched between two values according to the two possible signals m1 and m2 corresponding to binary 1 and 0, respectively. Normally, the two phases are separated by 180o. If the sinusoidal carrier has an amplitude A, and energy per bit Eo= 12A2cTb then the transmitted BP SK signal is
sBP SK(t) = m(t) s2Eb
Tb cos(2πfct + θc). (6.19) A typical BPSK signal constellation diagram is shown in Figure 6.1.
The probability of bit error for many modulation schemes in an AW GN channel is found using the Q-function of the distance between the signal points. In case of BP SK,
PeBP SK = Q(
s2Eb N0
). (6.20)
6.5.3 QPSK
The Quadrature Phase Shift Keying (QPSK) is a 4-ary PSK signal. The phase of the carrier in the QPSK takes 1 of 4 equally spaced shifts. Although QPSK can be viewed as a quaternary modulation, it is easier to see it as two independently modulated quadrature carriers. With this interpretation, the even (or odd) bits are
Figure 6.2: QPSK signal constellation.
Figure 6.3: QPSK transmitter.
used to modulate the in-phase component of the carrier, while the odd (or even) bits are used to modulate the quadrature-phase component of the carrier.
The QPSK transmitted signal is defined by:
si(t) = A cos(ωt + (i − 1)π/2), i = (1, 2, 3, 4) (6.21) and the constellation disgram is shown in Figure 6.2.
6.5.4 Offset-QPSK
As in QPSK, as shown in Figure 6.3, the NRZ data is split into two streams of odd and even bits. Each bit in these streams has a duration of twice the bit duration,
Figure 6.4: DQPSK constellation diagram.
Tb, of the original data stream. These odd (d1(t)) and even bit streams (d2(t)) are then used to modulate two sinusoidals in phase quadrature,and hence these data streams are also called the in-phase and and quadrature phase components. After modulation they are added up and transmitted. The constellation diagram of Offset-QPSK is the same as Offset-QPSK. Offset-Offset-QPSK differs from Offset-QPSK in that the d1(t) and d2(t) are aligned such that the timing of the pulse streams are offset with respect to each other by Tb seconds. From the constellation diagram it is observed that a signal point in any quadrant can take a value in the diagonally opposite quadrant only when two pulses change their polarities together leading to an abrupt 180 degree phase shift between adjacent symbol slots. This is prevented in O-QPSK and the allowed phase transitions are ± 90 degree.
Abrupt phase changes leading to sudden changes in the signal amplitude in the time domain corresponds to significant out of band high frequency components in the frequency domain. Thus to reduce these sidelobes spectral shaping is done at baseband. When high efficiency power amplifiers, whose non-linearity increases as the efficiency goes high, are used then due to distortion, harmonics are generated and this leads to what is known as spectral regrowth. Since sudden 180 degree phase changes cannot occur in OQPSK, this problem is reduced to a certain extent.
6.5.5 π/4 DQPSK
The data for π/4 DQPSK like QPSK can be thought to be carried in the phase of a single modulated carrier or on the amplitudes of a pair of quadrature carriers. The modulated signal during the time slot of kT < t < (k + 1)T given by:
s(t) = cos(2πfct + ψk+1) (6.22) Here, ψk+1 = ψk+ ∆ψk and ∆ψk can take values π/4 for 00, 3π/4 for 01, −3π/4 for 11 and −π/4 for 10. This corresponds to eight points in the signal constellation but at any instant of time only one of the four points are possible: the four points on axis or the four points off axis. The constellation diagram along with possible transitions are shown in Figure 6.4.