Solution 01101 10110 G so 2 5 1 4 2 1 322 1 1 x c x c x x c x c x c 5 2 3 4 1 2

# Solution 01101 10110 g so 2 5 1 4 2 1 322 1 1 x c x c

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Solution 01101 10110 G so 2 5 1 4 2 1 3 2 2 1 1 x c x c x x c x c x c 5 2 3 4 1 2 3 2 1 1 c c s c c s c c c s Applying all possible codewords with single bit error, we get All syndrome vectors are unique, so all single bit errors can be corrected. (the codeword 00000 at the bottom is for no error condition) 20.19.01.2006 Final Exam Calculate the error correcting capability of the block code whose codewords are given. 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 1 0 1 0 1 1 0 0 0 1 1 1 1 0 1 1 0 0 1 1 1 0 0 0 0 1 1 1 1 1 1 1 000 001 010 100 101 110 00000 00001 00010 00100 01000 10000 _ syndrome bit error
151227621 DIGITAL COMMUNICATIONS 15 Solution Minimum free distance is found to be 3 in the table (the Hamming distance of the closest codewords, which happens to be equal to the minimum weight here). Using 2 1 f d t the error correcting capability of this code is calculated as 1 2 1 3 t . That is, the code can correct 1 bit errors. 21. 19.01.2006 Final Exam Outputs of a convolutional encoder with a constraint length of 3 are calculated using 1 2 i i i x x O and 2 1 1 2 i i i x x O . Draw state transition graph (trellis section) and mark outputs on it. Solution 00 01 10 11 00 01 10 11 01 10 00 11 01 11 00 10 Current State Next State
151227621 DIGITAL COMMUNICATIONS 16 22.19.01.2006 Final Exam Find Adaptive ∆M output binary stream for the following waveform. Use only two levels; ∆1=±1 and ∆2=±2. A binary 0 at the output represents a ‘decrement’ and a binary 1 represents an ‘increment’. Solution (it is assumed that encoding starts with 1 bit delay). Output : 1111100101000101 23. 24.03.2007 1 st Midterm Find the total energy in the given signal. Solution T A t A dt A E T T T T 2 2 2 2 2 / 2 / 2
151227621 DIGITAL COMMUNICATIONS 17 24. 24.03.2007 1 st Midterm A sawtooth periodic signal is given. Calculate the zero frequency value (f=0 Hz). Solution 5 . 0 2 16 1 8 1 ) 2 1 8 1 ( 8 1 0 4 4 2 4 4 0 t t dt t b f (obvious from the figure) 25. 24.03.2007 1 st Midterm Vertical lines are located at sample (measurement) points. Labels on vertical scale are output values of the quantization step. Lighter lines between these values are the midpoints. a) Find output sequence. b) Find ΔM values using the values found in a. Solution
151227621 DIGITAL COMMUNICATIONS 18 26. 30.04.2007 2 nd Midterm Determine if the following code is systematic? 11011 11 10100 10 01111 01 00000 00 Solution If GX C where X is the information word, G is the generator matrix composed of identity and parity matrices as P I G and C is the corresponding codeword, then the code is called systematic. That is, if the information bits are embedded in the codewords, the code is said to be systematic. We shall see that first 2 bits of the codewords are identical to the corresponding information bits, so the code is systematic.

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• Fall '18
• Mr. Bhullar
• Hamming Code, Error detection and correction, Parity bit

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