Lect Mfsk Qam

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Frequency-Shift Keying (FSK) Modulation and Quadrature Amplitude Modulation (QAM) on Mac

24. Frequency-Shift Keying (FSK) Modulation and Quadrature

Amplitude Modulation (QAM)

Binary Frequency-Shift Keying (BFSK) [1-3]

A binary frequency-shift keying (BFSK) signal can be defined by

s(t ) = A f t t T

A f t elsewhere

cos ,

cos ,

2 0 0

2 1

π

π

≤ ≤

(24.1)

where A is a constant, f 0 and f 1 are the transmitted frequencies, and T is the bit

duration. The signal has a power P = A2 /2, so that A = 2P . Thus equation (24.1)

can be written as

s(t ) =2 2 0 0

2 2 1

P f t t T

P f t elsewhere

cos ,

cos ,

π

π

≤ ≤

=

PT T

f t t T

PT T

f t elsewhere

22 0 0

22 1

cos ,

cos ,

π

π

≤ ≤

= E T f t t T

E T

f t elsewhere

2

2 0 0

22 1

cos ,

cos ,

π

π

≤ ≤

(24.2)

where E = PT is the energy contained in a bit duration. For orthogonality, f 0 = m / T

and f 1 = n / T for integer n > integer m and f 1 - f 0 must be an integer multiple of

1/2T. We can take φ 1(t ) =2

T cos 2π f 0 t and φ 2 ( t ) =

2

T sin 2π f 1 t as the

orthonormal basis functions [3]. The applicable signal constellation diagram of the

orthogonal BFSK signal is shown in Figure 24.1.

Figure 24.1 Orthogonal BFSK signal constellation diagram.

Figure 24.2 shows the BFSK signal sequence generated by the binary sequence

0 1 0 1 0 0 1.

Figure 24.2 (a) Binary sequence, (b) BFSK signal, and (c) binary modulating and

BASK signals.

24.1




It can be seen that phase continuity is maintained at transitions. Further, the BFSK signal

is the sum of two BASK signals generated by two modulating signals m0(t ) and

m1(t ). Therefore, the Fourier transform of the BFSK signal s(t ) is

S( f ) = A2 −∞

∞∫ [m0(t ) e

j 2π f 0t ] e- j2π ft dt +

A2 −∞

∞∫ [m0(t ) e

- j 2π f 0t ] e- j2π ft dt +

A2 −∞

∞∫ [m1(t ) e

j 2π f 1t ] e- j2π ft dt +

A2 −∞

∞∫ [m1(t ) e

- j 2π f 1t ] e- j2π ft dt

=

A

2 M 0( f - f 0) +

A

2 M 0( f + f 0) + A2

M 1( f - f 1) + A

2 M 1( f + f 1) (24.3)

Figure 24.3 shows the amplitude spectrum of the BFSK signal when m0(t ) and m1(t )

are periodic pulse trains.

Figure 24.3 (a) Modulating signals, (b) Spectrum of (a), and (c) spectrum of BFSK

signal (positive frequencies only).

An alternative representation of the BFSK signal consists of letting f 0 = f c - ∆ f and f 1

= f c + ∆ f . Then

f 1 - f 0 = 2∆ f (24.4)

and

s(t ) = Acos2π ( f c + ∆ f )t (24.5)

where f c is the carrier frequency, ∆ f = β B is the frequency deviation, β is the

modulation index, and B = 1/ T is the bandwidth of the modulating signal. When

∆ f >> 1/ T , we have a wideband BFSK signal. The bandwidth is approximately equal to

2 ∆ f . When ∆ f << 1/ T , we have a narrowband BFSK signal. The bandwidth is

approximately equal to 2 B.

Figure 24.4 shows the modulator and coherent demodulator for BFSK signals [2].

Figure 24.4 (a) BFSK modulator and (b) coherent demodulator.

24.2




M-ary Frequency-Shift Keying ( M -FSK) [2-4]

An M-ary frequency-shift keying ( M -FSK) signal can be defined by

s(t ) = A f it t T elsewhere

cos( ' ),

,

2 0

0

π θ + ≤ ≤

(24.6)

for i = 0, 1, ..., M - 1. Here, A is a constant, f i is the transmitted frequency, θ ' is

the initial phase angle, and T is the symbol duration. It has a power P = A2 /2, so that

A = 2P . Thus equation (24.6) can be written as

s(t ) = 2P cos(2π f it + θ ' ), 0 < t < T

= PT T

2cos(2π f it + θ ' ), 0 < t < T

= E T

2cos(2π f it + θ ' ), 0 < t < T (24.7)

where E = PT is the energy of s(t ) contained in a symbol duration for i = 0, 1, ...,

M - 1. For convenience, the arbitrary phase angle θ ' is taken to be zero. If we choose

f 0 = k / T , f 1 = (k + 2)/ T , f 3 = (k + 4)/ T , ..., k > 0, we can take

φ 1 ( t ) =2

T cos 2π f 0 t , φ 2 ( t ) =

2

T sin 2π f 1 t , ... as the orthonormal basis

functions [3]. Figure 24.5 shows the signal constellation diagram of an orthogonal 3-FSK

signal.

Figure 24.5 Orthogonal 3-FSK signal constellation diagram.

Figure 24.6 shows the 4-FSK signal generated by the binary sequence 00 01 10 11.

Figure 24.6 4-FSK modulation: (a) binary signal and (b) 4-FSK signal.

Figure 24.7 shows the modulator and coherent demodulator for M -FSK signals [4]. Themapping table simply maps the detected index i onto a binary vector.

Figure 24.7 (a) M -FSK modulator and (b) coherent demodulator.

24.3




M-ary Quadrature Amplitude Modulation ( M -QAM)

An M-ary quadrature amplitude modulation ( M -QAM) signal can be defined by

s(t ) = Ai f ct i t T

elsewhere

cos( ),

,

2 0

0

π θ + ≤ ≤

(24.8)

= Ai i f ct Ai i f ct t T

elsewhere

cos cos sin sin ,

,

θ π θ π 2 2 0

0

− ≤ ≤

(24.9)

for i = 0, 1, ..., M - 1. Here, Ai is the amplitude, f c is the carrier frequency, θ i is

the phase angle, and T is the symbol duration. It has a power Pi = Ai2 /2, so that

Ai = 2Pi . Thus equation (24.9) can be written as

s(t ) = PiT cosθ i2

T cos2π f ct - PiT sinθ i

2

T sin2π f ct

= E i cosθ i2

T cos2π f ct - E i sinθ i

2

T sin2π f ct (24.10)

where E i = PiT is the energy of s(t ) contained in a symbol duration for i = 0, 1, ...,

M - 1. If we take φ 1

( t ) =2

T cos 2π f

ct and φ

2(t ) = -

2

T sin 2π f

ct as the

orthonormal basis functions, the applicable signal constellation diagrams of the 16-QAM

and 4-QAM signals are shown in Figure 24.8.

Figure 24.8 (a) 16-QAM and (b) 4-QAM signal constellation diagrams.

Figure 24.9 shows the modulator and a possible implementation of the coherent

demodulator for M -QAM signals.

Figure 24.9 (a) M -QAM modulator and (b) coherent demodulator.

References

[1] M. Schwartz, Information Transmission, Modulation, and Noise, 4/e, McGraw

Hill, 1990.

[2] P. Z. Peebles, Jr., Digital Communication Systems, Prentice Hall, 1987.

[3] H. Taub and D. L. Schilling, Principles of Communication Systems, 2/e, McGraw

Hill, 1986

24.4




[4] F. Xiong, Digital Modulation Techniques, Artech House, 2000.

24.5




0

E

s

1

E

s 0 =

2T

cos 2 π f 0t

= 2T

cos 2 π f 1 t

φ 1

t ( )

φ 2

t ( )

Figure 24.1 Orthogonal BFSK signal constellation diagram.

0 1 0 1 0 0 1Binary

sequence

0

A

-A

TimeBFSKsignal

s ( t )

T 0

(a)

Time1

(b)

(c)

0

A

-A

TimeBASKsignal

0

A

-A

TimeBASKsignal

m ( t )0

Time1

0

Time1

m ( t )1

Figure 24.2 (a) Binary sequence, (b) BFSK signal, and (c) binary modulating and

BASK signals.

24.6




(c)

(b)

f 0

B 1T

3T

-1T

Envelope1

T 2

2T

=-2T

-3T

(a)

0

...2B

01T 0 -3

T 0 -

1

T 2

f

1T

0 +f f f 3T

0 +f

S ( f )

f 1T

1 + f 3T

1 +

f f

2B

1f 1

T 1 -f 3

T 1 -

Envelope1

T 2

f c

2∆ f

0 1 0 1

0t

m ( t )

1

0M ( f )

0M ( f )

1or

t

T 0 2T 3T

1

m ( t )1

∆β =

B

f

Figure 24.3 (a) Modulating signals, (b) Spectrum of (a), and (c) spectrum of BFSK

signal (positive frequencies only).

(a) (b)

m ( t )

f t cosA 2π 0

s ( t )

f t cosA 2π 1

m ( t )

0

1

f t cos 2π 0s ( t )

~~

f t cos 2π 1

~~

1Am 0.5 f t cos 4π ( t )+ Am ( t )0.5

1

1

+ Am ( t )0.50Am 0.5 f t cos 4π ( t )0

0Am ( t )0.5

0

Am ( t )0.51

+

- 0

m ( t )0or

m ( t )1

ΣΣ

Figure 24.4 (a) BFSK modulator and (b) coherent demodulator.

24.7




0E

E

s

1

s 0 =

2T

cos 2 π f 0 t

= 2T

cos 2 π f 1 t

=2T

cos 2 π f 2t

E

s 2

φ 1 t ( )

φ 2

t ( )

φ 3

t ( )

Figure 24.5 Orthogonal 3-FSK signal constellation diagram.

0

A

-A

Time

0 0 0 1 1 0 1 1

Time

T

(b)

T

0

4-FSKsignal

(a)

Binarysequence

1

s ( t )

Figure 24.6 4-FSK modulation: (a) binary signal and (b) 4-FSK signal.

24.8




(a)

s (

t )

f t cos 2π 0s ( t )

f t cos 2π 1:

f t cos 2π -1M

:

l 0

Choosemax l

i

l 1

l -1M

(b)

0

T dt (.)

Serial

toparallelconverter

Frequencymodulator

D/Aconverter:

log2

M bits

Binary

sequence

Mappingtable

Binarysequence

0

T dt (.)

0

T dt (.)

Figure 24.7 (a) M -FSK modulator and (b) coherent demodulator.

24.9




s 1s 3

s 7 s 6

s 5 s 4

s 8 s 9 s

s s s s

0s

s 2s 0

1110

15 14 13 12

θ 4

E 4s 0 s 1

s 3

0s

2

E 1

θ 1

(a)

(b)

=2T

cos 2 π f c t φ

1t ( )

= - 2T

sin 2 π f c

t φ 2

t ( )

φ 2 t ( )

φ 1

t ( )

Figure 24.8 (a) 16-QAM and (b) 4-QAM signal constellation diagrams.

Binarysequence

(a)

Serialto

parallelconverter

:

log2

M bits

f t cos

A

2π c

f t sin 2π c

s ( t )+

cos θ i

-

i

A sinθ i i

f t cos 2π c s ( t )

f t sin 2π c

~~

~~

cos θ i

A0.5

sin θ i

A-0.5

cos θ i

f t cos 4π c -cos θ i

f t sin 4π c +sin θ i

A0.5

A0.5

A0.5

sin θ i

f t sin 4π c

+cos θ i

f t cos 4π c

-sin θ i

A0.5

A0.5

A0.5

= tan-1

θ i (.)^ Mapping

table

i

i

i

i

i

i

i

i

(b)

Assignamplitude

andphaseθ

i

A i

Binarysequence

Σ

Figure 24.9 (a) M -QAM modulator and (b) coherent demodulator.

24.10

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Lect Mfsk Qam