Problem Set #10 in ECE 456: Block Interleaving and Frequency-Hopping Communication Systems | Assignments Electrical and Electronics Engineering

University Problem Set #10 ECE 456

of Illinois Page 1 of 2 Fall 2002

Assigned: Thursday November 5, 2002

Due: Thursday November 14, 2002

Reading: Blahut, Chapter 9

Material discussed in Tuesday's lecture can be found in some of the texts

on reserve in Grainger Library

GMD decoding: Blahut (first edition, Chapter 15)

Concatenated codes: Blahut, pp. 296-297, McEliece (first edition, pp. 183-186)

Block interleaving via delay-scaled encoders: Blahut, pp. 113-114,|

Peterson & Weldon, pp. 357-359)

Block Interleaving via arrays: None of the books on reserve do it the dumb way.

For an “official” standard, see Consultative Committee for Space Data Systems,

“Recommendations for Space Data Systems Standard: Telemetry Channel

Coding,” Blue Book, Issue 2, CCSDS 101.0-B2, January 1987.

Problems:

1. We saw two different strategies for block interleaving in class. Block interleaving to depth

L that is implemented via L×n arrays sends data down the channel in different order than

block interleaving to depth L that is implemented via delay-scaling the encoder to depth L.

(a) If the code being interleaved can correct any t errors in an n-symbol codeword, what are the

longest bursts that can be guaranteed to be corrected by the supercodes obtained by the two

types of interleaving? How many short bursts of length L or less can be corrected? What

if each short burst is guaranteed to be confined to some set of L successive symbols that are

symbols from the same position/coordinate in the L interleaved codewords

(b) Assumng that the code being interleaved is a binary burst error correcting code that can

correct all (single) bursts of lengths b or smaller, and that the code satisfies the Rieger

bound with equality. What are the (single) burst error correcting capabilities of the

supercodes obtained by the two types of interleaving? Do the supercodes satisfy the

Rieger bound?

2.(a) Blahut Problem 10.5. Note that this might be a trick problem… Also, use L for the

interleaving depth instead of j (so that your answers will match up with the parts below).

There are nLm bits (numbered 0 through nLm–1) in a super-codeword, and the Lm bits

numbered (i–1)Lm through iLm–1 are the L m-bit i-th code symbols of the Reed-Solomon

codewords that have been interleaved. You probably made one of the following two

assumptions about the Lm bits numbered (i–1)Lm through iLm–1 :

(i) The bits numbered (i–1)Lm + jm through (i–1)Lm + (j+1)m–1 are the i-th symbol

in the j-th codeword, 0 ≤ j ≤ L–1, that is, the Lm-bit block can be subdivided into L

bytes of m bits each… This is called symbol-level interleaving in the sense that

successive m-bit bytes in the Lm-bit block are the symbols of the Reed-Solomon

codewords

(ii) The i-th symbol in the j-th codeword consists of the bits numbered

(i–1)Lm + j, (i–1)Lm + L + j, … (i–1)Lm + (m–1)L + j,

that is, the Lm-bit block can be subdivided into m bytes of L bits each, and each

byte consists of one bit from the i-th symbols of the L codewords. This is called

bit-level interleaving in the sense that the Reed-Solomon codeword symbols are

themselves interleaved at the bit level.

(b) Draw sketches of each type of interleaving to prove that you understood all the above

verbiage (even if there are errors in the wording…)

(d) Now consider that the burst is long enough that a decoding error occurs. Which of the

two forms of interleaving would you prefer, and why?

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University Problem Set #10 ECE 456

of Illinois Page 1 of 2 Fall 2002

Assigned: Thursday November 5, 2002 Due: Thursday November 14, 2002 Reading: Blahut, Chapter 9 Material discussed in Tuesday's lecture can be found in some of the texts on reserve in Grainger Library GMD decoding: Blahut (first edition, Chapter 15) Concatenated codes: Blahut, pp. 296-297, McEliece (first edition, pp. 183-186) Block interleaving via delay-scaled encoders: Blahut, pp. 113-114,| Peterson & Weldon, pp. 357-359) Block Interleaving via arrays: None of the books on reserve do it the dumb way. For an “official” standard, see Consultative Committee for Space Data Systems, “Recommendations for Space Data Systems Standard: Telemetry Channel Coding,” Blue Book, Issue 2, CCSDS 101.0-B2, January 1987.

Problems: 1. We saw two different strategies for block interleaving in class. Block interleaving to depth L that is implemented via L×n arrays sends data down the channel in different order than block interleaving to depth L that is implemented via delay-scaling the encoder to depth L. (a) If the code being interleaved can correct any t errors in an n-symbol codeword, what are the longest bursts that can be guaranteed to be corrected by the supercodes obtained by the two types of interleaving? How many short bursts of length L or less can be corrected? What if each short burst is guaranteed to be confined to some set of L successive symbols that are symbols from the same position/coordinate in the L interleaved codewords (b) Assumng that the code being interleaved is a binary burst error correcting code that can correct all (single) bursts of lengths b or smaller, and that the code satisfies the Rieger bound with equality. What are the (single) burst error correcting capabilities of the supercodes obtained by the two types of interleaving? Do the supercodes satisfy the Rieger bound?

2.(a) Blahut Problem 10.5. Note that this might be a trick problem… Also, use L for the interleaving depth instead of j (so that your answers will match up with the parts below).

There are nLm bits (numbered 0 through nLm–1) in a super-codeword, and the Lm bits numbered (i–1)Lm through iLm–1 are the L m-bit i-th code symbols of the Reed-Solomon codewords that have been interleaved. You probably made one of the following two assumptions about the Lm bits numbered (i–1)Lm through iLm–1 : (i) The bits numbered (i–1)Lm + jm through (i–1)Lm + (j+1)m–1 are the i-th symbol in the j-th codeword, 0 ≤ j ≤ L–1, that is, the Lm-bit block can be subdivided into L bytes of m bits each… This is called symbol-level interleaving in the sense that successive m-bit bytes in the Lm-bit block are the symbols of the Reed-Solomon codewords (ii) The i-th symbol in the j-th codeword consists of the bits numbered (i–1)Lm + j, (i–1)Lm + L + j, … (i–1)Lm + (m–1)L + j, that is, the Lm-bit block can be subdivided into m bytes of L bits each, and each byte consists of one bit from the i-th symbols of the L codewords. This is called bit-level interleaving in the sense that the Reed-Solomon codeword symbols are themselves interleaved at the bit level. (b) Draw sketches of each type of interleaving to prove that you understood all the above verbiage (even if there are errors in the wording…) (c) Which form of interleaving has larger (single) burst error correcting capability? (d) Now consider that the burst is long enough that a decoding error occurs. Which of the two forms of interleaving would you prefer, and why?

University Problem Set #10 ECE 456

of Illinois Page 2 of 2 Fall 2002

3. In a multi-user frequency-hopping communication system, each transmitter changes its carrier frequency at regular intervals of time. The transmitter is said to hop to the next frequency and hence the name… The hops are controlled by frequency-hopping (FH) patterns that specify which frequency is to be used next, and each transmitter is assigned its own FH pattern. FH patterns are usually periodic. It is also assumed that signals that are using different carrier frequencies f 0 , f 1 , … do not interfere with each other. (a) Consider the degenerate case in which the i-th transmitter is assigned the FH pattern (fi, fi, fi, fi, fi, fi, … ) of period 1. What kind of communication system is this? (b) Consider a slightly less degenerate case in which the i-th transmitter’s FH pattern is (fi, fi+1, fi+2, … , fq–2, f 0 , f 1 , … , fi–1 ) of period q–1. Assuming that all the FH patterns in the transmitted signals are perfectly synchronized at the receiver, show that the system will perform the same as the one in part (a). (c) Show that your answer to part (b) is wrong for the case of frequency-selective channels in which the various carrier frequencies might be experiencing different levels of fading. Which of (a) and (b) is the better system in this case? Which is the fairer system? (d) If the FH patterns of part (b) are not perfectly synchronized at the receiver, then two or more transmitted signals may collide at the receiver in the sense that they are all using the same carrier frequency. In such cases, all the data transmitted by the colliding transmitters is lost (Holy erasure channel, Batman!). In fact, all the colliding transmitters will continue to collide forever which is no fun.

Now suppose that q is a prime power (you knew that was coming, didn’t you?), that there are q frequencies that we associate with the q elements of GF(q), and that the FH patterns are codewords from a low-rate (q–1,k) cyclic Reed-Solomon code over GF(q). We also assume that each FH pattern is in a different cyclic equivalence class (i) Why did we assume that each FH pattern is in a different cyclic equivalence class? What would happen if this restriction is not applied? (ii) Show that no FH pattern uses all q frequencies. For the special case of a (q–1,2) RS code with check polynomial h(x) = (x–1)(x–α) where ord(α) = q–1, show that there are q FH patterns of period 6, and that each pattern leaves a different frequency unused. As an example, exhibit 7 FH patterns of length 6 over GF(7). Hint: 3 is a primitive element of GF(7). (ii) What is the maximum number of collisions between two FH patterns? Give the answer for the general general and also for the (q–1, 2) codes of part (d)(ii). (e) Interleaved Reed-Solomon codes are used for error correction in FH communication systems in which all L i-th symbols of the L interleaved codewords are transmitted at one frequency, i.e. during the time that the transmitter is hopped to one frequency. Thus, collisions erase all the symbols transmitted during that hop. Discuss what advantages, if any, this has in erasures-and-errors decoding of Reed-Solomon codes (Hint: Section 7.5) (f) Under conditions of total asynchronicity, a collision between two FH patterns usually occurs in the tail end of some hop and the beginning of some other hop (not necessarily the next hop), with the total time duration of the collision being one hop duration (that is, L symbols). Explain why this collision causes a single erasure in all but one of the interleaved Reed-Solomon codewords, and two erasures in the remaining codeword. Would this also hold if the transmission during a hop included header/trailer or synchronization information?

Problem Set #10 in ECE 456: Block Interleaving and Frequency-Hopping Communication Systems, Assignments of Electrical and Electronics Engineering

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