EE456 Digital Communications Professor Ha Nguyen September 2016 - - PowerPoint PPT Presentation
Chapter 8: M -ary Signaling Techniques EE456 Digital Communications Professor Ha Nguyen September 2016 EE456 Digital Communications 1 Chapter 8: M -ary Signaling Techniques Chapter 8: M -ary Signaling Techniques EE456 Digital
Chapter 8: M-ary Signaling Techniques
Professor Ha Nguyen September 2016
EE456 – Digital Communications 1
Chapter 8: M-ary Signaling Techniques
EE456 – Digital Communications 2
Chapter 8: M-ary Signaling Techniques
There are benefits to be gained when QPSK is used rather than the straightforward BPSK signaling. In general, an M-ary communication system can be designed for either bandwidth efficient or power efficient applications. Whether being bandwidth efficient or power efficient depends on how the M-ary signal set is constructed. Unlike QPSK (M = 4), when designed for bandwidth-efficient applications with M > 4, 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.
EE456 – Digital Communications 3
Chapter 8: M-ary Signaling Techniques
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
) ( Choose m t s ℜ
2 2 2
) ( Choose m t s ℜ
M M M
m t s
) ( Choose ℜ
✕) , , , ( space n
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
si(t) = Vi
Ts cos(2πfct), 0 ≤ t ≤ Ts = [(i − 1)∆]φ1(t), φ1(t) =
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
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
Choose sk(t), if
2
2
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 ⇐ P [error] =
M
P [si(t)]P [error|si(t)] P [error|si(t)] = 2Q
i = 2, 3, . . . , M − 1 P [error|si(t)] = Q
i = 1, M P [error] = 2(M − 1) M Q
For a given M, P [error] depends on the noise power (N0) and the minimum distance ∆. This means that moving the origin of the signal constellation does not affect the performance!
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Chapter 8: M-ary Signaling Techniques
The maximum and average transmitted energies can be reduced, without any sacrifice in error probability, by moving the origin to the center of the constellation, i.e., allowing both positive and negative values of the signal component. si(t) = (2i − 1 − M)∆ 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
(2i − 1 − M)2 = (M2 − 1)∆2 12 . Eb = Es log2 M = (M2 − 1)∆2 12 log2 M ⇒ ∆ =
M2 − 1
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Chapter 8: M-ary Signaling Techniques
P [error] = 2(M − 1) M Q
(M2 − 1)N0
M Q
M2 − 1 Eb N0
P [bit error] = 1 λP [symbol error] = 2(M − 1) M log2 M Q
M2 − 1 Eb N0
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
s
kT t = ) (t r
1
~ r ∆
s
kT t =
( )
−
s
kT T k
t
) 1 (
d ) (t r
1
r ∆ ) 2 cos( ) (
2 1
t f t
c Ts
π φ = ) (t h 1
s
T t ) 2 cos( t f V
c
π
1
~ r
1
r In practice, the thresholds (i.e., decision boundaries) for the Slicer are computed (estimated) directly from the output of the matched filter. The minimum distance ˜ ∆ between closest signal points is related to the “average” of |˜ r1| as: |˜ r1| ≈ 1 M/2
M/2
˜ ∆ 2 + ˜ w1
1 M/2
M/2
˜ ∆ 2 + ˜ w1
∆ 4 ⇒ ˜ ∆ = 4|˜ r1| M The minimum distance ˜ ∆ can also be computed from the “average” of ˜ r2
1 as:
r2
1 ≈
1 M/2
M/2
˜ ∆ 2 2 = (M2 − 1) ˜ ∆2 12 ⇒ ˜ ∆ =
1
M2 − 1 .
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Chapter 8: M-ary Signaling Techniques
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
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Chapter 8: M-ary Signaling Techniques
si(t) = V cos
M
= V cos (i − 1)2π M
(i − 1)2π M
si1 =
(i − 1)2π M
(i − 1)2π M
) (
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
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
) (
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 − P [correct|s1(t)] = 1 −
f(r1, r2|s1(t))dr1dr2
.
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Chapter 8: M-ary Signaling Techniques
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 outside Region 1 |s1(t)] > P [(r1, r2) fall in ℜ1|s1(t)], P [error|s1(t)] > Q
π M 2Es/N0
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Chapter 8: M-ary Signaling Techniques
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 outside Region 1 |s1(t)] = P [(r1, r2) ∈ ℜ1 OR (r1, r2) ∈ ℜ2|s1(t)] < P [(r1, r2) ∈ ℜ1|s1(t)] + P [(r1, r2) ∈ ℜ2|s1(t)], P [error] < 2Q
π M 2Es/N0
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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=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
M
2Eb
N0
EE456 – Digital Communications 17
Chapter 8: M-ary Signaling Techniques
M-QAM are two-dim constellations and they involve inphase (I) and quadrature (Q) carriers: φI(t) =
Ts cos(2πfct), 0 ≤ t ≤ Ts, φQ(t) =
Ts sin(2πfct), 0 ≤ t ≤ Ts, The ith transmitted M-QAM signal is: si(t) = VI,i
Ts cos(2πfct) + VQ,i
Ts sin(2πfct), 0 ≤ t ≤ Ts i = 1, 2, . . . , M =
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).
In general, QAM symbols have different energies. The average symbol energy is calculated as: Es =
M
EiP [si(t)] = M
i=1 Ei
M , for equally-likely signals
EE456 – Digital Communications 18
Chapter 8: M-ary Signaling Techniques
(1,3) Rectangle 4 = M (1,7) Triangle 8 = M Rectangle (4,4)
EE456 – Digital Communications 19
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 EE456 – Digital Communications 20
Chapter 8: M-ary Signaling Techniques
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.
EE456 – Digital Communications 21
Chapter 8: M-ary Signaling Techniques
( )
I t
φ ( )
Q t
φ 64 = M 32 = M 16 = M 8 = M 4 = M
Signal components belong to the set of discrete values {(2i − 1 − M)∆/2}, i = 1, 2, . . . , M
2 .
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.
EE456 – Digital Communications 22
Chapter 8: M-ary Signaling Techniques
s
t T =
s
t T = ( )
I t
φ ( )
Q t
φ 3
I
λ = 2
Q
λ = 5
Q I
λ λ λ = + = Inphase ASK 2 8
I
I
M
λ
= = Rectangular QAM 2 32 M
λ
= = Quadrature ASK 2 4
Q
Q
M
λ
= =
,
Select
I i
V bits λ bits
I
λ bits
Q
λ
2
( ) cos(2 )
s
I c T
t f t φ π =
2
( ) sin(2 )
s
Q c T
t f t φ π =
,
Select
Q i
V Matched filter Slicer Slicer ( ) t w ( )
I t
φ ( )
Q t
φ Matched filter ( )
i
s t
, ,
( ) ( ), 1,2, , ( )
Q i Q i I i I
V s V t i t M t φ φ + = =
, Q i Q
V + w
, I i I
V + w bits λ bits
I
λ bits
Q
λ
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Chapter 8: M-ary Signaling Techniques
1 2 3 4 5 6 7 8 −10 −5 5 10 t/Ts Inphase 8-ASK signal 1 2 3 4 5 6 7 8 −10 −5 5 10 t/Ts Quadrature 4-ASK signal 1 2 3 4 5 6 7 8 −10 −5 5 10 t/Ts 32-QAM signal 011 011 00 00 101 101 111 111 001 001 100 100 000 000 110 110 010 01 01 01 10 10 11 11 00 00 11 00 010 00 11 01 EE456 – Digital Communications 24
Chapter 8: M-ary Signaling Techniques
For square constellations: PM−QAM[error] = 1 − PM−QAM[correct] = 1 −
M−ASK[error]
2 , P√
M−ASK[error] = 2
1 √ M
(M − 1)N0
where Es/N0 is the average SNR per symbol. For general rectangular constellations: PM−QAM[error] ≤ 1 −
(M − 1)N0 2 ≤ 4Q
(M − 1)N0
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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
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
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)
2Ts
(coherently orthogonal) (k ± i)
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
EE456 – Digital Communications 28
Chapter 8: M-ary Signaling Techniques
Choose mi if
M
(rk − sik)2 <
M
(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
EE456 – Digital Communications 29
Chapter 8: M-ary Signaling Techniques
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
P [(rj < r1)|{r1 = r1, s1(t)}]. P [rj < r1|{r1 = r1, s1(t)}] = r1
−∞
1 √πN0 exp
N0
P [correct] = ∞
r1=−∞
r1
λ=−∞
1 √πN0 exp
N0
M−1 × 1 √πN0 exp
N0
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Chapter 8: M-ary Signaling Techniques
P [error] = 1 − 1 √ 2π ∞
−∞
√ 2π y
−∞
e−x2/2dx M−1 exp − 1 2 y −
N0
2
dy.
−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
With M-FSK, the required Eb/N0 to achieve a given error probability decreases as M
M-FSK is a power-efficient, not bandwidth-efficient modulation scheme!
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Chapter 8: M-ary Signaling Techniques
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
There are λ
k
number of bit errors per λ-bit symbol is
λ
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
f
1
f
M
f
2
f
3
f
1 M
f
−
f ∆
For FSK with M = 2λ frequencies, only one of M frequencies is activated over
that is activated over any symbol duration is determined by the mapping from λ information bits to the frequency value. With a rectangular pulse shaping, the spectrum of each activated frequency is proportional to sinc2(fTs), i.e., the first zero crossings are at frequencies 1/Ts away from the activated frequency. FSK is not a spectral-efficient modulation scheme!
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Chapter 8: M-ary Signaling Techniques
A compact and meaningful comparison of different M-ary techniques 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]. To reduce bandwidth, M-ASK can be transmitted with a single-sideband (SSB)
rb W
= 2 log2 M (bits/s/Hz). M-PSK and M-QAM (M > 2) must be transmitted with double sidebands, hence W ≈ 1/Ts and rb W
= log2 M, (bits/s/Hz), 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
= 2 log2 M M .
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Chapter 8: M-ary Signaling Techniques
−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
1 For each transmission rate rb, there is a corresponding limit on the
2 For some appropriate signalling rate rb, there is no limit on the
EE456 – Digital Communications 36
Chapter 8: M-ary Signaling Techniques
C = W log2
Pav W N0
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
Pav W N0
W Eb N0
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Chapter 8: M-ary Signaling Techniques
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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Chapter 8: M-ary Signaling Techniques
Table 1: IMT-Advanced Compliant Systems - Air Interface Radio Aspects Standard LTE-Advanced WirelessMAN (3GPP LTE Rel. 10, 11) (Mobile WiMAX Rel. 2) Frequency band (800–2620) MHz (450–3600) MHz Radio access mode TDD & FDD TDD & FDD Channel bandwidth Scalable (1.25–20) MHz Scalable (1.25–20) MHz Multiple access tech. DL: OFDMA, UL: SC-FDMA OFDMA Peak data rate DL: 1 Gbit/s, UL: 500 Mbit/s DL: 1 Gbit/s, UL: 100 Mbit/s Multiple antenna tech. 2/4/8 antennas DL: 1/2/4 , UL: 1/2 Frame Length 10 ms 20 ms (super-frame) Channel Coding 1/3-rate turbo code RS over GF(256) (outer) 1/3-rate tail biting conv. code Block conv. or Parity check (inner) (for BCH) BTC, CTC & LDPC (optional) Interleaving Internal sub-block interleavers Block interleaver (in rate matching) Data modulation QPSK, 16QAM, 64QAM QPSK, 16QAM, 64QAM
EE456 – Digital Communications 39
Chapter 8: M-ary Signaling Techniques
The probability of error (bit of symbol) has been obtained for various pass-band modulation schemes in terms of SNR per bit, Eb/N0. Such probability analysis was obtained under the simplest channel model, namely AWGN. This channel model ignores any attenuation by the transmission medium/environment and
In reality, there is always channel attenuation, even the attenuation is varying over time (think about wireless channels). Usually the received signal power is usually much smaller than the transmitted signal power. When considering channel attenuation, it is important to understand that it is the received SNR per bit, Eb/N0, measured at the receiver side that determines the error probability. Consider bandwidth-efficient applications such as cellular phone or cable TV systems, in which the channel quality changes dynamically. There are two design options:
1
In order to maintain the same transmission rate (i.e., stay with the same constellation size M) at the same quality of service (i.e., same error probability), the transmitted power should be adjusted according to the channel condition: higher transmit power when the channel quality is poor and vice versa.
2
If the transmit power is held fixed, in order to maintain the same quality of service (i.e., same error probability), the transmitter needs to adapt the modulation scheme according to the channel condition: lower constellation size (smaller M), which also means slower transmission rate, when the channel quality is poor and vice versa. Many practical systems operate with constant transmit power (easier for circuit design) and exercise adaptive modulation. To enable adaptive modulation, the information about the channel quality, usually measured at the receiver, needs to be sent back to the transmitter on a reverse channel. For example, the latest DOCSIS 3.1 standard for cable TV specifies 14 modulation choices: BPSK, QPSK, 8-QAM, 16-QAM, 32-QAM, 64-QAM, 128-QAM, 256-QAM, 512-QAM, 1024-QAM, 2048-QAM, 4096-QAM, 8192-QAM, and 16384-QAM.
EE456 – Digital Communications 40
82 Revised per PHY3.1-N-14.1202-3 on 12/11/14 by PO.
1 1
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BPSK 1 1 QPSK 2 1/√2 8-QAM 3 1/√10 16-QAM 4 1/√10 32-QAM 5 1/√20 64-QAM 6 1/√42 128-QAM 7 1/√82 256-QAM 8 1/√170 512-QAM 9 1/√330 1024-QAM 10 1/√682 2048-QAM 11 1/√1322 4096-QAM 12 1/√2730 8192-QAM 13 1/√5290 16384-QAM 14 1/√10922