Basic Digital Passband Modulation

Reference : HA H. NGUYEN and ED SHWEDYK A First Course in Digital Communications , 2009

Binary amplitude-shift keying (BASK)

s(t) = m(t)c(t)where m(t) is the modulating signal (the baseband signal, an NRZ signal) and c(t) = V cos(2πfct) is the sinusoidal carrier.

Optimum Receiver

Threshold

BASK signaling scheme: (a) signal space plot; (b) optimum receiver implementation; (c) decision regions

Error probability

Power Spectral Density (PSD)

Approximately 95% of the total transmitted power lies in a band of 3/Tb (hertz), centered at fc.

PSD of BASK

Binary phase-shift keying (BPSK)A BPSK signal is generated by amplitude modulating the sinusoidal carrier with a NRZ-L signal of amplitude ±1. The transmitted signal is s(t) = m(t)c(t) (where m(t) is a NRZ-L signal) with a resultant phase that is either 0 or π radians.

Signal space plot of BPSK.

PSD for the BPSK

The PSD of BPSK is similar to that of BASK except that there are no impulse functions at ±fc, reflecting the fact that there is no power at the carrier. This is reasonable since BPSK is really a “double” sideband suppressed carrier modulation.

Binary frequency-shift keying (BFSK)

Simple BFSK modulators: (a) by gating two oscillators; (b) using a voltage controlled oscillator (VCO).

Two oscillators

s1(t) and s2(t) are orthogonal over the interval [0, Tb]

If the two phases are the same

If the two phases are different

VCO

where n and m are positive integers, and n m.

Signal space plot and decision regions of BFSK.

Performance comparison of BASK, BPSK, and BFSK

BPSK is 3 dB more efficient than BFSK, which has the same performance as BASK.

BFSK occupies a larger bandwidth than BPSK and BASK (recall that BPSK and BASK occupy the same bandwidth).

Each of the three modulation techniques has a spectrum that decays as 1/f ^2 for frequencies away from the carrier, reflecting the fact that for each modulation the transmitted signal has discontinuities.

Digital modulation techniques for spectral efficiency

QPSK signals and a mapping to the messages

Quadrature phase-shift keying (QPSK)

An example of a QPSK signal

The basic idea behind QPSK exploits the fact that cos(2πfct) and sin(2πfct) are orthogonal over the interval [0, Tb] when fc = k/Tb, k integer.

Therefore only two orthonormal functions are needed to represent the four signals

Signal space plot of QPSK modulation.

Optimum Receiver for QPSKThe optimum receiver is derived by expanding the received signal r(t) = si(t) + w(t) over the interval of Ts seconds into a series as follows:

The criterion will be to find a receiver that minimizes the symbol (message) error probability.

Rather than minimizing the error, consider instead the equivalent criterion, that of maximizing the probability of a correct decision.

Assign the observation vector r = (r1, r2, . . . , rm) in the m-dimensional signal space to the region for which the integrand Pi f(r|si(t)) is the largest.

P1 = P2 = P3 = P4

Minimum-distance receiver

P[error] = P[error|si(t)] = 1 − P[correct|si(t)].

Decision regions of the minimum-distance receiver of QPSK

Signal space diagram of QPSK: to compute P[correct|s1(t)] one finds the volume of f(ˆr1, ˆr2|s1(t)) over the shaded quadrant

same as that of BPSK

An alternative representation of QPSK

Inphase carrier V cos(2πfct) Quadrature carrier V sin(2πfct),

A different block diagram of a QPSK modulator

Signal space plots for inphase and quadrature bit streams

Receiver implementation for QPSK

Offset quadrature phase-shift keying (OQPSK)

To prevent the phase change of π in QPSK, offset (or staggered) quadrature phase-shift keying (OQPSK) is used so that signal amplification can be done more efficiently.

OQPSK differs from QPSK only in that in OQPSK the aI(t) and aQ(t) bit streams are offset by one

bit interval Tb.

Minimum shift keying (MSK)

The transmitted signals s(t) have sudden jumps at multiples of symbol duration for QPSK), or multiples of bit duration (for OQPSK).

The two carriers V cos(2πfct) and Vsin(2πfct) are weighted by sinusoids of frequency 1/(4Tb).

The signals are orthogonal over the interval of Tb seconds, or any integer multiple of Tb

The bit error probability of MSK is the same as that of BPSK, QPSK, and OQPSK

The expression in (7.64) shows that s(t) not only has a

constant envelope, but also a continuous phase. Furthermore, the transmitted signal is of either

frequency f2 = fc + 1/4Tb or frequency f1 = fc − 1/4Tb depending on the ratio aQ(t)/aI (t).

Thus the transmitted signal may be considered to be a frequency shift keying signal with continuous phase (CPFSK). Note also that the frequency separation is f2 − f1 = 1/2Tb, which is the minimum separation possible for the two sinusoidal carriers to be “coherently” orthogonal. This explains the name “minimum shift keying” of the modulation scheme.