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Chapter 8: M-ary Signaling Techniques

A First Course in Digital Communications

Ha H. Nguyen and E. Shwedyk February 2009

A First Course in Digital Communications 1/46

Chapter 8: M-ary Signaling Techniques

Introduction

There are benefits to be gained when M-ary (M = 4) signaling methods are used rather than straightforward binary signaling. In general, M-ary communication is used when one needs to design a communication system that is bandwidth efficient. Unlike QPSK and its variations, the gain in bandwidth is accomplished at the expense of error performance. To use M-ary modulation, the bit stream is blocked into groups of λ bits ⇒ the number of bit patterns is M = 2λ. The symbol transmission rate is rs = 1/Ts = 1/(λTb) = rb/λ symbols/sec ⇒ there is a bandwidth saving of 1/λ compared to binary modulation. Shall consider M-ary ASK, PSK, QAM (quadrature amplitude modulation) and FSK.

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Chapter 8: M-ary Signaling Techniques

Optimum Receiver for M-ary Signaling

• ✁

i

m ) (t si ) (t w ) (t r

i

m ˆ

w(t) is zero-mean white Gaussian noise with power spectral density of N0

2 (watts/Hz).

Receiver needs to make the decision on the transmitted signal based on the received signal r(t) = si(t) + w(t). The determination of the optimum receiver (with minimum error) proceeds in a manner analogous to that for the binary case.

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Chapter 8: M-ary Signaling Techniques

Represent M signals by an orthonormal basis set, {φn(t)}N

n=1,

N ≤ M: si(t) = si1φ1(t) + si2φ2(t) + · · · + siNφN(t), sik = Ts si(t)φk(t)dt. Expand the received signal r(t) into the series r(t) = si(t) + w(t) = r1φ1(t) + r2φ2(t) + · · · + rNφN(t) + rN+1φN+1(t) + · · · For k > N, the coefficients rk can be discarded. Need to partition the N-dimensional space formed by

• r = (r1, r2, . . . , rN) into M regions so that the message error

probability is minimized.

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Chapter 8: M-ary Signaling Techniques

1 1 1

• r

) ( Choose m t s ℜ

2 2 2

• r

) ( Choose m t s ℜ

M M M

m t s

• r

) ( Choose ℜ

) , , , ( space n

• bservatio

l dimensiona

2 1 M

N r r r r

−

The optimum receiver is also the minimum-distance receiver: Choose mi if N

k=1(rk − sik)2 < N k=1(rk − sjk)2;

j = 1, 2, . . . , M; j = i.

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Chapter 8: M-ary Signaling Techniques

M-ary Coherent Amplitude-Shift Keying (M-ASK)

si(t) = Vi

• 2

Ts cos(2πfct), 0 ≤ t ≤ Ts = [(i − 1)∆]φ1(t), φ1(t) =

• 2

Ts cos(2πfct), 0 ≤ t ≤ Ts, i = 1, 2, . . . , M.

) (

1 t

φ

∆ ∆ 2 ∆ − ) 1 (k ∆ − ) 1 (M ∆ − ) 2 (M ) (

1 t

s ) (

2 t

s ) (

3 t

s ) (t sk ) (

1 t

sM − ) (t sM

s

kT t =

( )

−

• s

s

kT T k

t

) 1 (

d watts/Hz 2 strength WGN, N ) (t si

) (

1 t

φ ) (t r ) (t w

1

r

i

m ˆ

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Chapter 8: M-ary Signaling Techniques

Minimum-Distance Decision Rule for M-ASK

Choose    sk(t), if

• k − 3

2

• ∆ < r1 <
• k − 1

2

• ∆, k = 2, 3, . . . , M − 1

s1(t), if r1 < ∆

2

sM(t), if r1 >

• M − 3

2

• ∆

.

∆ ∆ − ) 1 (k

1

r

( )

) (

1

t s r f

k

) ( Choose t sk ) ( Choose t sM

1

Choose ( ) s t ⇐

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Chapter 8: M-ary Signaling Techniques

Error Performance of M-ASK

( )

) (

1

t s r f

k 1

r 2 ∆ 2 ∆

P[error] =

M

• i=1

P[si(t)]P[error|si(t)]. P[error|si(t)] = 2Q

• ∆/
• 2N0
• ,

i = 2, 3, . . . , M − 1. P[error|si(t)] = Q

• ∆/
• 2N0
• ,

i = 1, M. P[error] = 2(M − 1) M Q

• ∆/
• 2N0
• .

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Chapter 8: M-ary Signaling Techniques

Modified M-ASK Constellation

The maximum and average transmitted energies can be reduced, without any sacrifice in error probability, by changing the signal set to one which includes the negative version of each signal. si(t) = (2i − 1 − M)∆ 2

• Vi
• 2

Ts cos(2πfct), 0 ≤ t ≤ Ts, i = 1, 2, . . ., M.

2 ∆ 2 ∆ − ) (

1 t

φ 2 3∆ 2 3∆ −

∆ ) (

1 t

φ

∆ 2 ∆ − ∆ − 2

Es = M

i=1 Ei

M = ∆2 4M

M

• i=1

(2i − 1 − M)2 = (M 2 − 1)∆2 12 . Eb = Es log2 M = (M 2 − 1)∆2 12 log2 M ⇒ ∆ =

• (12 log2 M)Eb

M 2 − 1

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Chapter 8: M-ary Signaling Techniques

Probability of Symbol Error for M-ASK

P [error] = 2(M − 1) M Q

• 6Es

(M2 − 1)N0

• = 2(M − 1)

M Q

• 6 log2 M

M2 − 1 Eb N0

• .

P [bit error] = 1 λP [symbol error] = 2(M − 1) M log2 M Q

• 6 log2 M

M2 − 1 Eb N0

• (with Gray mapping)

5 10 15 20 25 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) P[symbol error] M=4 (W=1/2Tb) M=8 (W=1/3Tb) M=16 (W=1/4Tb) M=2 (W=1/Tb)

W is obtained by using the W Ts = 1 rule-of-thumb. Here 1/Tb is the bit rate (bits/s).

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Chapter 8: M-ary Signaling Techniques

Example of 2-ASK (BPSK) and 4-ASK Signals

Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb 10Tb −1 1 Baseband information signal Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb 9Tb 10Tb −2 2 BPSK Signalling 2Tb 4Tb 6Tb 8Tb 10Tb −2 2 4−ASK Signalling A First Course in Digital Communications 11/46

Chapter 8: M-ary Signaling Techniques

M-ary Phase-Shift Keying (M-PSK)

si(t) = V cos

• 2πfct − (i − 1)2π

M

• ,

0 ≤ t ≤ Ts, i = 1, 2, . . . , M; fc = k/Ts, k integer; Es = V 2Ts/2 joules si(t) = V cos (i − 1)2π M

• cos(2πfct)+V sin

(i − 1)2π M

• sin(2πfct).

φ1(t) = V cos(2πfct) √Es , φ2(t) = V sin(2πfct) √Es . si1 =

• Es cos

(i − 1)2π M

• , si2 =
• Es sin

(i − 1)2π M

• .

The signals lie on a circle of radius √Es, and are spaced every 2π/M radians around the circle.

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Chapter 8: M-ary Signaling Techniques

Signal Space Plot of 8-PSK

si(t) = V cos

• 2πfct − (i − 1)2π

M

• ,

0 ≤ t ≤ Ts, i = 1, 2, . . . , M; fc = k/Ts, k integer; Es = V 2Ts/2 joules

) (

1 t

φ 000 ) (

1

↔ t s ) (

2 t

φ 4 π 100 ) (

8

↔ t s 001 ) (

2

↔ t s 011 ) (

3

↔ t s ) ( 010

4 t

s ↔ ) ( 110

5 t

s ↔ ) ( 111

6 t

s ↔ 101 ) (

7

↔ t s

s

E

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Chapter 8: M-ary Signaling Techniques

Signal Space Plot of General M-PSK

si(t) = V cos

• 2πfct − (i − 1)2π

M

• ,

0 ≤ t ≤ Ts, i = 1, 2, . . . , M; fc = k/Ts, k integer; Es = V 2Ts/2 joules

) (

1 t

φ ) (

1 t

s

s

E ) (

2 t

φ M π 2 ) (

2 t

s ) (t sM M π 2 −

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Chapter 8: M-ary Signaling Techniques

Optimum Receiver for M-PSK

) (

2 t

φ ) (

1 t

φ

s

T t =

s

T t =

( ) ( )

smallest the choose and , , 2 , 1 for Compute

2 2 2 2 1 1

M i s r s r

i i

= − + −

( )

• s

T

t d

( )

• s

T

t d ) (t r

1

r

2

r

i

m ˆ

1

r ) (

1 t

s

s

E

2

r M π ) (

2 t

s ) ( Choose 1 Region

1 t

s ) ( Choose 2 Region

2 t

s

P [error] = P [error|s1(t)] = 1 −

r1,r2∈Region 1

f(r1, r2|s1(t))dr1dr2.

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Chapter 8: M-ary Signaling Techniques

Lower Bound of P[error] of M-PSK

1

r ) (

1 t

s

s

E

2

r M π ) (

2 t

s ) ( Choose 1 Region

1 t

s ) ( Choose 2 Region

2 t

s

1

r ) (

1 t

s

2

r M π ) (

2 t

s

1

ℜ

( )

sin

s

E M π

( )

,0

s

E

P[error|s1(t)] > P[r1, r2 fall in ℜ1|s1(t)], or P[error|s1(t)] > Q

• sin

π M 2Es/N0

• .

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Chapter 8: M-ary Signaling Techniques

Upper Bound of P[error] of M-PSK

1

r ) (

1 t

s

s

E

2

r M π ) (

2 t

s ) ( Choose 1 Region

1 t

s ) ( Choose 2 Region

2 t

s

1

r ) (

1 t

s

2

r M π ) (

2 t

s

1

ℜ

( )

sin

s

E M π

( )

,0

s

E

1

r ) (

1 t

s

2

r M π − ) (t sM

2

ℜ

( )

sin

s

E M π

( )

,0

s

E

P[error] < P[r1, r2 fall in ℜ1|s1(t)] + P[r1, r2 fall in ℜ2|s1(t)], or P[error] < 2Q

• sin

π M 2Es/N0

• ,

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Chapter 8: M-ary Signaling Techniques

Symbol Error Probability of M-PSK

5 10 15 20 25 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) P[symbol error] M=16 M=32 M=8 M=4 M=2 Exact Lower bound Upper bound

With a Gray mapping, the bit error probability is approximated as: P[bit error]M-PSK ≃

1 log2 M Q

• λ sin2 π

M

2Eb

N0

• .

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Chapter 8: M-ary Signaling Techniques

Comparison of BPSK and M-PSK

P[error]M-PSK ≃ Q

• λ sin2 π

M 2Eb N0

• ,

where Es = λEb. P[error]BPSK = Q(

• 2Eb/N0).

λ M M-ary BW/Binary BW λ sin2(π/M) M-ary Energy/Binary Energy 3 8 1/3 0.44 3.6 dB 4 16 1/4 0.15 8.2 dB 5 32 1/5 0.05 13.0 dB 6 64 1/6 0.0144 17.0 dB

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Chapter 8: M-ary Signaling Techniques

M-ary Quadrature Amplitude Modulation (M-QAM)

M-QAM constellations are two-dimensional and they involve inphase (I) and quadrature (Q) carriers: φI(t) =

• 2

Ts cos(2πfct), 0 ≤ t ≤ Ts, φQ(t) =

• 2

Ts sin(2πfct), 0 ≤ t ≤ Ts, The ith transmitted M-QAM signal is: si(t) = VI,i

• 2

Ts cos(2πfct) + VQ,i

• 2

Ts sin(2πfct), 0 ≤ t ≤ Ts i = 1, 2, . . . , M =

• Ei
• 2

Ts cos(2πfct − θi) VI,i and VQ,i are the information-bearing discrete amplitudes of the two quadrature carriers, Ei = V 2

I,i + V 2 Q,i and

θi = tan−1(VQ,i/VI,i).

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Chapter 8: M-ary Signaling Techniques

(1,3) Rectangle 4 = M (1,7) Triangle 8 = M Rectangle (4,4)

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Chapter 8: M-ary Signaling Techniques

Rectangle Triangle Hexagon (8,8) (4,12) (1,5,10) 16 = M

1

R

2

R

2

R

1

R

1

R

2

R

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Chapter 8: M-ary Signaling Techniques

A Simple Comparison of M-QAM Constellations

With the same minimum distance of all the constellations, a more efficient signal constellation is the one that has smaller average transmitted energy.

8 = M

Es for the rectangular, triangular, (1,7) and (4,4) constellations are found to be 1.50∆2, 1.125∆2, 1.162∆2 and 1.183∆2, respectively.

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Chapter 8: M-ary Signaling Techniques

Rectangular M-QAM

( )

I t

φ ( )

Q t

φ 64 = M 32 = M 16 = M 8 = M 4 = M

The signal components take value from the set of discrete values {(2i − 1 − M)∆/2}, i = 1, 2, . . . , M

2 .

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Chapter 8: M-ary Signaling Techniques

Modulation of Rectangular M-QAM

Each group of λ = log2 M bits can be divided into λI inphase bits and λQ quadrature bits, where λI + λQ = λ. Inphase bits and quadrature bits modulate the inphase and quadrature carriers independently.

• ❍

bits Inphase

,

Select

I i

V

,

Select

Q i

V ) 2 cos( 2 t f T

c s

π ) 2 sin( 2 t f T

c s

π bits Quadrature bits Infor. ) (t si ASK Inphase ASK Quadrature

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Chapter 8: M-ary Signaling Techniques

Demodulation of Rectangular M-QAM

Due to the orthogonality of the inphase and quadrature signals, inphase and quadrature bits can be independently detected at the receiver.

Inphase ASK decision Multiplexer Decision 2 ( ) sin(2 )

Q c s

t f t T φ π = 2 ( ) cos(2 )

I c s

t f t T φ π = ) ( ) ( ) ( t t s t

i

w r + =

s

t T =

s

t T = Quadrature ASK decision (b) Receiver

( )

d

s

T

t

• ∫

( )

d

s

T

t

• ∫

The most practical rectangular QAM constellation is one which λI = λQ = λ/2, i.e., M is a perfect square and the rectangle is a square.

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Chapter 8: M-ary Signaling Techniques

Symbol Error Probability of M-QAM

For square constellations: P[error] = 1 − P[correct] = 1 −

• 1 − P√

M[error]

2 , P√

M[error] = 2

• 1 −

1 √ M

• Q
• 3Es

(M − 1)N0

• ,

where Es/N0 is the average SNR per symbol. For general rectangular constellations: P[error] ≤ 1 −

• 1 − 2Q
• 3Es

(M − 1)N0 2 ≤ 4Q

• 3λEb

(M − 1)N0

• where Eb/N0 is the average SNR per bit.

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Chapter 8: M-ary Signaling Techniques

5 10 15 20 25 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) P[symbol error] M=64 M=16 M=2 M=4 M=256 Exact performance Upper bound

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Chapter 8: M-ary Signaling Techniques

Performance Comparison of M-PSK and M-QAM

For M-PSK, approximate P[error] ≈ Q

• 2Es

N0 sin π M

• .

For M-QAM, use the upper bound 4Q

• 3λEb

(M−1)N0

• .

Comparing the arguments of Q(·) for the two modulations gives: κM = 3/(M − 1) 2 sin2(π/M). M 10 log10 κM 8 1.65 dB 16 4.20 dB 32 7.02 dB 64 9.95 dB 256 15.92 dB 1024 21.93 dB

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Chapter 8: M-ary Signaling Techniques

Performance Comparison of M-ASK, M-PSK, M-QAM

5 10 15 20 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) P[symbol error] M−ASK M−PSK M−QAM M=4, 8, 16, 32 4−QAM or QPSK

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Chapter 8: M-ary Signaling Techniques

M-ary Coherent Frequency-Shift Keying (M-FSK)

si(t) =

• V cos(2πfit),

0 ≤ t ≤ Ts 0, elsewhere , i = 1, 2, . . ., M, where fi are chosen to have orthogonal signals over [0, Ts]. fi =    (k ± i)

• 1

2Ts

• ,

(coherently orthogonal) (k ± i)

• 1

Ts

• ,

(noncoherently orthogonal) , i = 0, 1, 2, . . .

) (

1 t

φ ) (

1 t

s ) (

2 t

s ) (

3 t

s ) (

2 t

φ ) (

3 t

φ

s

E

s

E

s

E

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Chapter 8: M-ary Signaling Techniques

Minimum-Distance Receiver of M-FSK

Choose mi if

M

• k=1

(rk − sik)2 <

M

• k=1

(rk − sjk)2 j = 1, 2, . . . , M; j = i, ⇒ Choose mi if ri > rj, j = 1, 2, . . . , M; j = i.

s

E t s t ) ( ) (

1 1

= φ

s

T t =

s

T t = Decision

( )

• s

T

t d

( )

• s

T

t d

s M M

E t s t ) ( ) ( = φ

) (t r

1

r

M

r

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Chapter 8: M-ary Signaling Techniques

Symbol Error Probability of M-FSK

P[error] = P[error|s1(t)] = 1 − P[correct|s1(t)]. P[correct|s1(t)] = P[(r2 < r1) and · · · and (rM < r1)|s1(t) sent]. = ∞

r1=−∞

P[(r2 < r1) and · · · and (rM < r1)|{r1 = r1, s1(t)}]f(r1|s1(t))dr. P[(r2 < r1) and · · · and(rM < r1)|{r1 = r1, s1(t)}] =

M

• j=2

P[(rj < r1)|{r1 = r1, s1(t)}] P[rj < r1|{r1 = r1, s1(t)}] = r1

−∞

1 √πN0 exp

• − λ2

N0

• dλ.

P[correct] = ∞

r1=−∞

r1

λ=−∞

1 √πN0 exp

• − λ2

N0

• dλ

M−1 × 1 √πN0 exp

• −(r1 − √Es)2

N0

• dr1.

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Chapter 8: M-ary Signaling Techniques

−2 2 4 6 8 10 12 14 16 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Eb/N0 (dB) P[symbol error] M=2 M=4 M=8 M=16 M=32 M=64

P [error] = 1− 1 √ 2π ∞

−∞

• 1

√ 2π y

−∞

e−x2/2dx M−1 exp  − 1 2

• y −
• 2 log2 MEb

N0 2 dy.

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Chapter 8: M-ary Signaling Techniques

Bit Error Probability of M-FSK

Due to the symmetry of M-FSK constellation, all mappings from sequences of λ bits to signal points yield the same bit error probability. For equally likely signals, all the conditional error events are equiprobable and occur with probability Pr[symbol error]/(M − 1) = Pr[symbol error]/(2λ − 1). There are λ

k

• ways in which k bits out of λ may be in error

⇒ The average number of bit errors per λ-bit symbol is

λ

• k=1

k λ k Pr[symbol error] 2λ − 1 = λ 2λ−1 2λ − 1 Pr[symbol error]. The probability of bit error is simply the above quantity divided by λ: Pr[bit error] = 2λ−1 2λ − 1 Pr[symbol error].

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Chapter 8: M-ary Signaling Techniques

Union Bound on the Symbol Error Probability of M-FSK

P[error] = P[(r1 < r2) or (r1 < r3) or, · · · , or (r1 < rM)|s1(t)]. Since the events are not mutually exclusive, the error probability is bounded by: P[error] < P[(r1 < r2)|s1(t)]+ P[(r1 < r3)|s1(t)] + · · · + P[(r1 < rM)|s1(t)]. But P[(r1 < rj)|s1(t)] = Q

• Es/N0
• , j = 3, 4, . . . , M.

Then P[error] < (M−1)Q

• Es/N0
• < MQ
• Es/N0
• < Me−Es/(2N0).

where the bound Q(x) < exp

• − x2

2

• has been used.

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Chapter 8: M-ary Signaling Techniques

An Upper Bound on Q(x)

Q(x) = ∞

x

1 √ 2π exp

• −λ2

2

• dλ < exp
• −x2

2

• 1

2 3 4 5 6 7 8 10

−15

10

−10

10

−5

10 x Q(x) and its simple upper bound Q(x) exp(−x2/2)

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Chapter 8: M-ary Signaling Techniques

Interpretations of P[error] < Me−Es/(2N0)

1

Let M = 2λ = eλ ln 2 and Es = λEb. Then P[error] < eλ ln 2e−λEb/(2N0) = e−λ(Eb/N0−2 ln 2)/2. As λ → ∞, or equivalently, as M → ∞, the probability of error approaches zero exponentially, provided that Eb N0 > 2 ln 2 = 1.39 = 1.42 dB.

2

Since Es = λEb = V 2Ts/2, then P[error] < eλ ln 2e−V 2Ts/(4N0) = e−Ts[−rb ln 2+V 2/(4N0)] If −rb ln 2 + V 2/(4N0) > 0, or rb <

V 2 4N0 ln 2 the probability or

error tends to zero as Ts or M becomes larger and larger.

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Chapter 8: M-ary Signaling Techniques

Comparison of M-ary Signaling Techniques

A compact and meaningful comparison is based on the bit rate-to bandwidth ratio, rb/W (bandwidth efficiency) versus the SNR per bit, Eb/N0 (power efficiency) required to achieve a given P[error]. M-ASK with single-sideband (SSB) transmission, W = 1/(2Ts) and rb W

• SSB-ASK = 2 log2 M

(bits/s/Hz). M-PSK (M > 2) must have double sidebands, W = 1/Ts and rb W

• PSK = log2 M,

(bits/s/Hz), (Rectangular) QAM has twice the rate of ASK, but must have double sidebands ⇒ QAM and SSB-ASK have the same bandwidth efficiency. For M-FSK with the minimum frequency separation of 1/(2Ts), W =

M 2Ts = M 2(λ/rb) = M 2 log2 M rb, and

rb W

• FSK = 2 log2 M

M .

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Chapter 8: M-ary Signaling Techniques

USSB Transmission of BPSK Signal

USSB( )

s t

c

cos(2 ) f t π

c

sin(2 ) f t π + + ( ) m t ( ) ( ) ( ) sgn( ) h t H f H f j f ← → = LTI Filter ˆ( ) m t

ˆ m(t) = m(t) ∗ h(t) = m(t) ∗

• − 1

πt

• = − 1

π ∞

−∞

m(t − λ) λ dλ

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Chapter 8: M-ary Signaling Techniques

Example of USSB-BPSK Transmitted Signal

Tb 2Tb 3Tb 4Tb 5Tb 6Tb −V V

t (a) BPSK signal

Tb 2Tb 3Tb 4Tb 5Tb 6Tb 7Tb 8Tb −V V

t (b) USSB−BPSK signal

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Chapter 8: M-ary Signaling Techniques

Power-Bandwidth Plane (At P[error] = 10−5)

−10 −5 −1.6 0 5 10 15 20 25 30 0.1 0.2 0.3 0.5 1 2 3 5 10 SNR per bit, Eb/N0 (dB) rb/W (bits/s/Hz) PSK QAM and ASK (SSB) FSK Bandwidth−limited region: rb/W>1 Power−limited region: rb/W<1 M=8 M=16 M=32 M=64 M=8 M=16 M=32 M=64 M=16 M=64 M=4 M=2

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Chapter 8: M-ary Signaling Techniques

Two Statements

Consider information transmission over an additive white Gaussian noise (AWGN) channel. The average transmitted signal power is Pav, the noise power spectral density is N0/2 and the bandwidth is

• W. Two statements are:

1

For each transmission rate rb, there is a corresponding limit

• n the probability of bit error one can achieve.

2

For some appropriate signalling rate rb, there is no limit on the probability of bit error one can achieve, i.e., one can achieve error-free transmission. Which statement sounds reasonable to you?

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Chapter 8: M-ary Signaling Techniques

Shannon’s Channel Capacity

C = W log2

• 1 + Pav

WN0

• ,

where W is bandwidth in Hz, Pav is the average power and N0/2 is the two-sided power spectral density of the noise. Shannon proved that it is theoretically possible to transmit information at any rate rb, where rb ≤ C, with an arbitrarily small error probability by using a sufficiently complicated modulation scheme. For rb > C, it is not possible to achieve an arbitrarily small error probability. Shannon’s work showed that the values of Pav, N0 and W set a limit on transmission rate, not on error probability! The normalized channel capacity C/W (bits/s/Hz) is: C W = log2

• 1 + Pav

WN0

• = log2
• 1 + C

W Eb N0

• .

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Chapter 8: M-ary Signaling Techniques

Shannon’s Capacity Curve Eb

N0 = 2C/W −1 C/W

−10 −5 −1.6 0 5 10 15 20 25 30 0.1 0.2 0.3 0.5 1 2 3 5 10 SNR per bit, Eb/N0 (dB) rb/W (bits/s/Hz) PSK QAM and ASK (SSB) FSK Bandwidth−limited region: rb/W>1 Power−limited region: rb/W<1 M=8 M=16 M=32 M=64 M=8 M=16 M=32 M=64 M=16 M=64 M=4 M=2 Channel capacity limit, C/W Shannon limit

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Spectrum Efficiency of DVB-S2 Standard

More information: www.dvb.org

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

• 4
• 2

2 4 6 8 10 12 14 16

SNR (dB) Spectrum Efficiency

Example 1: 50Mbit/s in 36MHz at 4dB with QPSK Example 2: 80 Mbit/s in 36MHz at 9.5dB with 8PSK

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