Channel Coding II Exercises – SoSe 2012 – - Universität Bremen

Channel Coding II
Exercises
– SoSe 2012 –
Dirk Wübben, Carsten Bockelmann, Matthias Woltering
NW1, Raum N 2400, Tel.: 0421/218-62392
E-mail: {wuebben,bockelmann,woltering}@ant.uni-bremen.de
Universität Bremen, FB1
Institut für Telekommunikation und Hochfrequenztechnik
Arbeitsbereich Nachrichtentechnik
Prof. Dr.-Ing. A. Dekorsy
Postfach 33 04 40
D–28334 Bremen
WWW-Server: http://www.ant.uni-bremen.de
Version of June 20, 2012

June 20, 2012
I
Exercises
1 Concatenated codes 1
1.3 Serial concatenation of codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Exercise 1.1: Decoding sequence of linear product codes . . . . . . . . . . . . . . . . 1
1.7 Decoding of concatenated codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Exercise 1.2: L-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Exercise 1.3: Comparison of the exact solution
and approximation of LLR combining . . . . . . . . . . . . . . . . . . . . . . . 1
Exercise 1.4: Soft-output-decoding of SPC-codes . . . . . . . . . . . . . . . . . . . . 2
Exercise 1.5: BCJR Decoding of a convolutional code . . . . . . . . . . . . . . . . . . 2
Exercise 1.6: Decoding of a product code . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Trelliscoded modulation (TCM) 3
Exercise 2.1: Field of signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Exercise 2.2: TCM-calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Exercise 2.3: Transmission scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 CONCATENATED CODES June 20, 2012 1
1 Concatenated codes
1.3 Serial concatenation of codes
Exercise 1.1
Decoding sequence of linear product codes
Considering concatenated codes as per chapter 1, the succession of the encoding and therewith of the
decoding of linear constituent codes is reversible. This has the effect, that all columns and all rows
represent valid code words of the respective code.
Show this for linear block codes with the help of matrix calculus and check your results by means of the
example given in the lecture slides (concatenation of (3,2,2)- and (4,3,2)-SPC). Note that the block
interleaver has to be taken into your considerations.
1.7 Decoding of concatenated codes
Exercise 1.2
L-algebra
a) The definition of log-likelihood-ratio (LLR) is assumed to be known (see lecture slides). Prove
( n∑
)
L ⊕u i = ln
i=1
n∏
(e L(ui) ∏
+1)+ n (e L(ui) −1)
i=1
i=1
n∏
(e L(ui) ∏
+1)− n (e L(ui) −1)
i=1
i=1
(1)
by induction.
b) Calculate the expected valueE{ˆx} for the estimation of a BPSK symbolxgiven the corresponding
LLRL(ˆx).
Plot the value E{ˆx} for different signal-to-noise-ratios by assuming the range −2 : 0.1 : 2 for the
receive symbols y and the range −2 : 2 : 10 dB for E b /N 0 . Hint: For equiprobable input symbols
the LLRL(ˆx) is determined by the value L(y|x) = 4α Es
N 0
y (we assume α = 1 in this exercise).
Exercise 1.3
Comparison of the exact solution
and approximation of LLR combining
Create a MATLAB-function [exact, approx] = llr(L), which determines the LLRs of the combination
of several statistically independent values exactly (2) and approximately (3).
( n
)
∏
L(u 1 ⊕...⊕u n ) = 2artanh tanh(L(x i )/2)
≈
n∏
i=1
i=1
(2)
sgn(L(x i ))·min
i
(|L(x i )|) (3)
Compare the exact solution and the approximate one with by assuming the symbols y 1 = −2 : 0.1 : 2
and y 2 = 0.2 : 0.2 : 1. Determine the LLRs and plot the result against y 1 for E b /N 0 = 2dB.

1 CONCATENATED CODES June 20, 2012 2
Exercise 1.4
Soft-output-decoding of SPC-codes
a) The information word u = (1, 0, 1) is encoded with a (4,3,2)-SPC-Code, BPSK-modulated and
subsequently transmitted over an AWGN-channel with a signal-to-noise ratio of E s /N 0 = 2 dB.
At the receiver the sequence y = (−0.8, 1.1, 0.3, 0.4) is observed. Determine the LLRs and
decode the receiving vector with the routine llr.m from exercise 1.3. What is the result
b) Now decode with the approximation solution. Compare the results with a).
c) Determine the probabilities for a correct decoding decision.
Exercise 1.5
BCJR Decoding of a convolutional code
Assume, that y = [−0.6727,−0.8254,0.8133,−0.2742,0.4117,1.1832,−1.1364,−0.8861] has been
received after transmission over an AWGN channel with a noise variance of σ 2 n = 1. On the transmitter
side the binary information word u = [1100] has been encoded with a[5,7] 8 convolutional code, BPSK
modulated (x = 1−2·c) and finally transmitted. Perform BJCR decoding under the assumption that the
last state is known to be(0,0). For ease of calculation use the Max-Log-MAP in the logarithmic domain.
(Hint: Draw a full trellis first, then follow: Chapter 1, CC II)
Exercise 1.6
Decoding of a product code
Given is a product code consisting of two(3,2,2) single parity check codes. The following2×2-matrix
u describes the four information bits ( ) 0 0
u =
0 1
and the BPSK-modulated code bits are shown in the matrix x:
⎛ ⎞
+1 +1 +1
x = ⎝ +1 −1 −1 ⎠
+1 −1
After transmission over an AWGN-channel with E b /N 0 = 2dB, we get the following receive matrix:
⎛
y = ⎝
−1.5 +1.5 +1.2
+1.1 +1.0 −1.5
+0.5 −2.5
Decode this product code step by step using MATLAB.
⎞
⎠

2 TRELLISCODED MODULATION (TCM) June 20, 2012 3
2 Trelliscoded modulation (TCM)
Exercise 2.1
Field of signals
Given is the 8-ASK/PSK constellation in Fig. 1, that consists of two 4-PSK-constellations staggered by
45 ◦ with the radii r and 2r.
x 5
2r
x
x0
1
r
45°
x 4
x 6
x x
2
3
x 7
Fig. 1: Field of signals of the 8-ASK-PSK constellation
a) Determine the minimal Euclidean distance ∆ 0 for E S = 1.
b) Determine the asymptotic coding gain compared with an 8-PSK and a 2-PSK.
Exercise 2.2
TCM-calculation
Given is the represented convolutional encoder with the code rate R = K/(K + 1) = 1/2 and the
memory m = 2, that shall be used for TCM together with aq = 2 M+1 = 8-ary ASK-constellation.
u i (2)
c i (2)
+
c i (1)
x(i)
8-ASK
u i (1)
D
D
c i (0)
Fig. 2: Field of code words ford min = 8
a) Determine the corresponding partitioning of the field of signals for 8-ASK.
b) Determine the corresponding Trellis segment.
c) Determine the asymptotic coding gain compared with a 4-ASK.
Exercise 2.3
Transmission scheme

2 TRELLISCODED MODULATION (TCM) June 20, 2012 4
Write a simulation program for a 8-PSK transmission. Construct the transmitter (source, modulator), the
channel (AWGN) and the receiver (demodulator, decision).
Subsequently determine the bit error rate and the symbol error rate in dependence onE b /N 0 andE s /N 0 .