698
CHAPTER 10 E ERROR-CONTROL CODING
10.14 Consider the (31, 15) Reed-Solomon code.
(a) How many bits are there in a symbol of the code?
(b) What is the block length in bits?
(c) What is the minimum distance of the code?
(d) How many symbols in error can

668 CHAPTER 10 n ERROR-CONTROL CODING
factor exp(—dfmerEbl2No), where r is the code rate and time is the free distance of the
convolutional code. Therefore, as a ﬁgure of merit for measuring the improvement in error
performance made by the use of coding w

674 CHAPTER 10 E ERROR-CONTROL CODING
TABLE 10.9 Asymptotic coding gain of Ungerboeck 8—PSK codes,
with respect to uncoded 4-PSK
Number of states /4 8 16 32 64 128 256 512
Coding gain (dB) 3 3.6 4.1 4.6 4.8 5 5.4 57
respect to uncoded 4—PSK. N

680
CHAPTER 10 E ERROR-CONTROL CODING
soft estimate of the message bits x. This estimate is re—interleaved to produce the total log.
likelihood ratio lz(x). The extrinsic information lz(x) fed back to the ﬁrst decoding Stage
is therefore
72(x) = lzlx)- 71

10.8 Turbo Codes 677
10°
Shannan limit
10-1
Turbu code
10'2
2’.
2
§ 10-3
5
E
Eb/No, dB
FIGURE 10.27 Noise performances of 1/2 rate, turbo code and uncoded transmission for
AWGN channel; the ﬁgure also includes Shannon’s theoretical limit on channel ca

662
CHAPTER 10 w ERROR-CONTROL CODING
does not continue to grow as the number of incoming message bits increases; rather, it
remains constant at 2"”, where K is the constraint length of the code.
Consider, for example, the trellis diagram of Figure 10.15

686
CHAPTER 10 E ERROR-CONTROL CODING
where m is the k—by—l message vector, and h is the (n—k)-by-1 parity vector; see Equatifm
(10.9). Correspondingly, the parity-check matrix A is partitioned as
A1
AT =
(10.39)
where A1 is a square matrix of dimensions

10.10 Low-Density Parity-Check Codes 689
have a matrix overlap (i.e., inner product of any two columns in matrix A) not to exceed 1;
such a constraint, over and above the regularity constraints, is expected to improve the per-
formance of LDPC codes. Unfo

8. 1 0 Adaptive Antenna Arrays for Wireless Communications 5 57
Spatial processing is provided by the antenna array, and the temporal processing is pro-
vided by a bank of finite-duration impulse response (FIR) ﬁlters. For obvious reasons, this
structure

560 CHAPTER 8 m MULTIUSER RADIO COMMUNICATIONS
munication systems is noise; these systems have sufﬁcient channel bandwidth to permit
the use of pulse—code modulation (PCM) as the standard method for converting speech
into a 64 kb/s stream, which provides

650
CHAPTER 10 E ERROR-CONTROL CODING
ALMNH
Flip-flop Module-2
adder
Message bits
FIGURE 10.10 Encoder for the (7, 4) cyclic code generated by g(X) = l + X + X3.
Using the coefﬁcients of these three polynomials as t

10.5 Convolutiml Codes 659
:1
Level j = O 1
L+2
FIGURE 10. l 5 Trellis for the convolutional encoder of Figure 10.1311.
return to the state a. Clearly, not all the states can be reached in these two portions of the
trellis. However, in the central por

ERROR-CONTROL
: CODING
This chapter is the natural sequel to the preceding chapter on Shannon’s information
theory. In particular, in this chapter we present error-control coding techniques that
provide different ways of implementing Shannon’s channel-cod

9.13 Rate Distortion Theory 61 l
the function SN( f)/ )H( f ) l2 is identical to the way in which water distributes itself in a
vessel.
Consider now the idealized case of a band—limited signal in additive white Gaussian
noise of power spectral density N (

l 0. l 0 Law-Density Parity-Check Codes 6 83
1 iteralion
BEH
1o—5
10‘6
1 1.5 2 2.5
EblNo, as
FIGURE 10.30 Results of the computer experiment on turbo decoding, for increasing number of
iterations.
The only channel impairment assumed in the experiment

638 CHAPTER 10 E ERROR-CONTROL CODING
. ® a.’
I’
(a) ([7)
FIGURE 10.6 (a) Hamming distance al(c,-7 c1) 2 2: + 1. (b) Hamming distance d(ci, c«) < 2.1:.
The received vector is denoted by r.
code vector closest to the received vector r. If, on the other h

10.3 Linear Block Codes 635
5* EXAMPLE 10.1 Repetition Codes
Repetition codes represent the simplest type of linear block codes. In particular, a single mes-
sage bit is encoded into a block of 71 identical bits, producing an (n, 1) block code. Such a
cod

632 CHAPTER 10 ERROR-CONTROL CODING
and division by 0 is not permitted. Modulo—Z addition is the EXCLUSIVE-OR operation
in logic, and modulo-2 multiplication is the AND operation.
a 10.3 Linem- Block Codes
A code is said to be linear if any two code words

644
CHAPTER 10 a ERROR-CONTROL CODING
According to Equation (10.1), we want the code polynomial to be in the form
600 = 17(X) + X’H‘WKX) (10.38)
Hence, the use of Equations (10.35) and (10.3 8) yields
dixlgixl = bixl + X”_km(X)
Equivalently, in light of m

10.2 Discrete-Memryless Channel: 629
The three types of ARQ described here offer trade-offs of their own between the
need for a half-duplex or full-duplex link and the requirement for efﬁcient use of com-
munication resources. In any event, they all rely

10.4 Cyclic Codes 647
Let q(X) denote the quotient and s(X) denote the remainder, which are the results of
dividing r(X) by the generator polynomial g(X). We may therefore express r(X) as follows:
r(X) = q(X)g(X) + s(X) (10.47)
The remainder s(X) is a pol

656
CHAPTER 10 H ERROR-CONTROL CODING
an output multiplexer. An example of such an encoder is shown in Figure 10.1%)
where k = 2, n = 3, and the two shift registers have K = 2 each. The code rate is 2/3. In
this second example, the encoder processes the i

10.7 Trellis-Coded Modulation 67 l
Input : 8-PSK
I signal mapper Must significant bit
: O O 1 1 1 1
I 1 1 O O 1 1
l 0 1 o 1 o 1
I Modulo-2 | r
I adder101234567
Wot i:
ll 1 Signal number U W
Rate—U2 convolutional encoder
(a)
E

78.8
8.9
Problems 563
In this problem, we revisit Example 8.1 based on the receiver conﬁguration of Figure
8.10. Suppose that a lossy waveguide is inserted between the receiving antenna and the
low-noise ampliﬁer. The waveguide loss is 1 dB, and its physi

692
CHAPTER 10 ERROR-CONTROL CODING
a single one. As shown in Figure 10.32, similar irregular interleavers are Used in both
convolutional encoding paths to generate the parity—check bits 21 and 22 in response to the
message bits x. Irregular turbo codes a

ESE 471 Assignment #3
Baseband Signaling: Physical Signal Mapping
Assigned Date: March 29, 2016
Due Date: April 11, 2016 in class.
Continuing on with various components inside the transmitter, we are now at the last box in the
transmitter diagram, which i

1. (25 pts). Given a receiver (i.e., the first receiver) operating at a radio frequency (i.e.,;,) of2.8 Ghz has a local
oscillator frequency of2.86 Ghz. A second receiver operates at a radio frequency equalling to
the image frequency of the first receiver