Communication Theory and Engineering

Transcription

1 Communication Theory and Engineering Master's Degree in Electronic Engineering Sapienza University of Rome A.A

2 Practice work 8 Soft vs. Hard decoding

3 Hard vs. Soft decoding Hard decoding: the detector selects the codeword closest to the received sequence, in terms of Hamming distance Soft decoding: the detector selects the codeword closest to the received sequence, in terms of Euclidean distance From the channel samples Estimated symbols Receiver Detector Line decoder Channel B front-end (slicer) k Q k A k decoder Ĉ k Estimated decoded bits ˆBk Hard decoding From the channel Receiver front-end samples Soft decoder Q k ˆBk Estimated decoded bits Soft decoding

4 Example: -bit parity check with soft decoding Antipodal transmission, +a and a, over discrete-time AWGN channel In the -bit parity check coding, that is a block code (n, k=n-), we have:! d H,min = 2 d E,min = 2a 2 Q a 2 Pr σ blockerror ( 2 k )Q a 2 c σ c As shown in lecture, one can estimate the probability of bit errors by setting the arguments of Q(.) equal, that is by ignoring the constant multipliers: a c 2 σ c = a u SNR u = 2SNR c The code gain is 3 db, but...

5 beware of bandwidth! The coded system has a rate of n/(n-) Example: -bit parity check with soft decoding σ c 2 = n 2 n and since a c 2 σ c = a u one has: a c 2 a u 2 = 2 n n If n=3 the power required for the coded system is.25 db smaller than for the uncoded system over the same white noise channel The theoretical coding gain is 3 db, but.75 db is lost due to the presence of larger noise on the coded channel

6 MATLAB Example: -bit parity check with soft decoding What does it happen by considering constant multipliers? One can identifies upper and lower bounds (UB and LB) for Pr bit error! Pr bit error,soft decoding (n ) Q a 2(n )/n Pr σ bit error,soft decoding (2 n )Q u a 2(n )/n Coded vs. uncoded systems: Compute the coding gain with fixed Pr bit error For -bit parity check (n=3, k=2), and Pr bit error = 0-5 Uncoded Coded with no multipliers Coded UB Coded LB Q ( a ) =0 5 ( a ) uncoded =? Q ( a 4 /3 ) =0 5 ( a ) approx,cod =? 0.5Q ( a 4 /3 ) =0 5 ( a ) σ UB,cod =? u 3Q ( a 4 /3 ) =0 5 ( a ) σ LB,cod =? u a ( ) ncoded G db = 20log u 0 ( a ) σ xxx,coded u Hint: check for MATLAB functions to compute the inverse Q(.) function!

7 MATLAB Example: -bit parity check vs. Hamming code Antipodal transmission, +a and a, over discrete-time AWGN channel Analyze soft decoding performance varying Pr bit error = [0-3, 0-5, 0-7, 0-9, 0 -, 0-3, 0-5 ] for -bit parity check and report on a single graph UB, LB and approx. of coding gain as Pr bit error changes. Analyze soft decoding performance varying Pr bit error = [0-3, 0-5, 0-7, 0-9, 0 -, 0-3, 0-5 ] for Hamming code (7,4) and report on a single graph UB, LB and approx. of coding gain as Pr bit error changes. Remind: for Hamming code (7,4) G =! d H,min = 3 d E,min = 2a 3

8 Hard decoding For a hard decoder, the decoding takes place after the slicer In the presence of additive noise, the equivalent binary channel is a BSC channel with Pr bit error equal to p. A maximum likelihood (ML) decision maker selects the codeword ĉ closest to c* in terms of Hamming distance. In this way, t errors can be corrected: t = ( d H,min ) / 2 For example: For parity check code with bit, t = 0. This code is not useful if you are using an Hard decoding For the Hamming code (7, 4), t =. This code corrects all single-bit errors

9 Correction capacity of Hard decoding t = ( d )/2 H,min One can be certain of correcting up to t bit errors, but some error patterns with more bit errors than t may be correctable also, unless the code is a so-called perfect code: All bit patterns of length n are within Hamming distance t of a codeword No bit pattern of length n is Hamming distance t or less from more than one codeword

10 MATLAB Example: Is Hamming code (7,4) a perfect code? Analyze via MATLAB the properties of the Hamming code (7,4) and demonstrate that it is a perfect code. All bit patterns of length n=7 are within Hamming distance t= of a codeword No bit pattern of length n=7 is Hamming distance t= or less from more than one codeword Hint: check for MATLAB functions to compute the Hamming distance!

11 Hard Decoding Performance The probability of m bit errors in a block of n bits is a binomial distribution P(m,n) =! n m pm ( p) n m = n! m!(n m)! pm ( p) n m Perfect codes A symbol decoding error if more than t bits are incorrect n Pr blockerror = P(m,n)= P(m,n) m=t+ t m=0 Not-perfect codes Some error patterns with more than t bits errors will be corrected by n the ML decoder Pr blockerror P(m,n) m=t+ Quasiperfect codes Although some error patterns with t+ bit errors are corrected, none with t+ 2 or more are corrected. For these we can get a lower bound: Pr blockerror n m=t+2 P(m,n)

12 MATLAB Example: Hamming code (7,4) with hard decoding Being Hamming code (7, 4) a perfect code, it follows that if c is transmitted and there are 2 bits error then c* is at distance from a code ĉ c and therefore there is a decoding error. Pr blockerror = t m=0 P(m,n) Pr blockerror = ( p) 7 7p( p) 6 As shown in lecture, for p=0-2 the probability of error on the block is about 2*0-3. It seems that the code reduces the probability of error by a factor of 5. However, the coded system requires a bandwidth 7/4 times larger than the uncoded system: σ c = 7 4 So the actual gain obtained is smaller than 5

13 MATLAB Example: Hamming code (7,4) with hard decoding Antipodal transmission +a and a, and discrete-time AWGN channel, coded with Hamming code (7, 4), and decoded in hard mode. a. Find UB and LB of the probability of error on the bit in terms of probability of error on the coded block. b. Show that to achieve a Pr bit error of 0-5 after decoding, the coded system has a coding gain of less than one db.

14 MATLAB Example: Hamming code (7,4) with hard decoding Antipodale transmission +a and a, and discrete-time AWGN channel, coded with Hamming code (7.4), and decoded in hard mode. c. Repeat the analysis for Pr bit error equal to 0-7, and discuss the results with respect to those obtained in case of soft decoding (Practice work 8).