Tele4653 l3

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TELE4653 Digital Modulation &Coding

Digital Modulation

Wei Zhang

w.zhang@unsw.edu.au

School of Electrical Engineering and Telecommunications

The University of New South Wales

Outline

Offset QPSK

TELE4653 - Digital Modulation & Coding - Lecture 2. March 15, 2010. – p.1/20

Modulation with Memory

Modulation is the mapping between the digital sequence

and the signal sequence to be transmitted over the channel.

Modulation with memory: the mapping depends on the

current and the past bits.

Example: differential encoding.

bk = ak ⊕ bk−1

from Digital Communications (5th Ed.) – John G. Proakis and Masoud Salehi

Why do we need Continuous-Phase FSK (CPFSK)?

A conventional FSK signal is generated by shifting the

carrier by m∆f , 1 ≤ m ≤ M . It can be accomplished by

having M separate oscillators tuned to the desired

frequencies.

The abrupt switching from one oscillator output to another

results in large spectral side lobes of the signal.

To address spectral side lobes, the frequency is changed

continuously. CPFSK.

The signal waveform of CPFSK is given by

s(t) =

Tcos [2πfct + φ(t; I) + φ0] (1)

where φ(t; I) represents the time-varying phase of the carrier, as

φ(t; I) = 4πTfd

−∞

d(τ)dτ (2)

with a PAM signald(t) =

Ing(t − nT ). (3)

In denotes the sequence of amplitudes and g(t) is the

rectangular pulse of amplitude of 1

2Tand duration of T .

Although d(t) contains discontinuities, φ(t; I) is continuous.

The phase φ(t; I) in the interval nT ≤ t ≤ (n + 1)T is

φ(t; I) = 2πfdTn−1∑

k=−∞

Ik + 4πfdTq(t − nT )In (4)

= θn + 2πhInq(t − nT ) (5)

where h = 2fdT is the modulation index, θn = πh∑

k=−∞Ik

represents the accumulation of all symbols, and

q(t) =

0 t < 0

2T0 ≤ t ≤ T

2t > T

For continuous-phase modulation (CPM) signals,

φ(t; I) = 2π

k=−∞

Ikhkq(t − kT ), nT ≤ t ≤ (n + 1)T (7)

where {Ik} is the sequence of M -ary symbols selected from

{±1,±3, · · · ,±(M − 1)}, {hk} is a sequence of modulation

indices, and q(t) is some normalized waveform shape as

q(t) =

g(τ)dτ (8)

Full-response CPM if g(t) = 0 for t > T , and Partial-responseCPM if g(t) 6= 0 for t > T .

Minimum-shift keying (MSK) is a special case of binary CPFSK

(and CPM) in which h = 1

2and g(t) is a rectangular pulse of

duration T . The phase of the carrier in the interval

nT ≤ t ≤ (n + 1)T is [obtained from Eq. (5)]

φ(t; I) = θn + πIn

t − nT

, nT ≤ t ≤ (n + 1)T (9)

and the MSK signal is

s(t) = A cos [2πfct + φ(t; I)] (10)

= A cos

t −1

2nπIn + θn

, (11)

for nT ≤ t ≤ (n + 1)T .

For binary CPFSK, i.e., In = {±1}, the signal may be written as

si(t) = A cos

2πfit + θn +1

2nπ(−1)i−1

, i = 1, 2 (12)

wheref1 = fc −

4T(13)

f2 = fc +1

4T(14)

Note ∆f = f2 − f1 = 1/2T , i.e., the minimum frequency

separation that is necessary to ensure the orthogonality of

signals s1(t) and s2(t). This explains why binary CPFSK with

2is called the MSK.

Offset QPSK

For conventional QPSK signals, the possible 180◦ phase

change can occur when both I and Q components change

simultaneously.

To prevent 180◦ phase changes that cause abrupt changes

in the signal, resulting in large spectral side lobes, offsetQPSK (OQPSK) is introduced, by misalignment of the I and

Q components. The OQPSK signal can be written as

s(t) = A

∞∑

n=−∞

I2ng(t − 2nT )

cos 2πfct

∞∑

n=−∞

I2n+1g(t − 2nT − T )

sin 2πfct

OQPSK vs. MSK

Conventional QPSK contains phase jumps of ±180◦ or

±90◦.

Offset QPSK contains phase jumps of ±90◦. It has constant

frequency, but there exist jumps in its waveform.

MSK may be represented as a form of OQPSK.

MSK has continuous phase, so there exist no jumps in the

waveform. But there are jumps in its instantaneous

frequency.

GMSK can smooth the frequency jumps of MSK by shaping

the lowpass signal before being applied to the MSK

modulator.

Tele4653 l3

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