EECS 270 Fall 2014, Lecture 21 Page 1 of 7 Error Correction continued and start on FPGA architecture Today we ll finish up error correction and start in on FPGA architecture. But first we ll have an announcement and a bit of review from last time. Announcement: And review: GA4 problem 2 had an error a needed device was left off the list. Yes, it could have been built out of gates (and some of you who already did it may have done so) but that wasn t the intent of the question. 1. Draw the state transition diagram which corresponds to the following logic equations. Assume the initial state is 001 and that our states are one-hot encoded. o NS2=S2*A+S1*!A o NS1=S2*!A+S0 o NS0=S1*A o X=S1+S2 Question: how could you tell if this weren t one-hot encoded? 2. What is the Hamming Distance of the following set: A={11111, 00011, 10100}? What does that say about the potential error correction and detection if we knew the sender only sent data from set A?
EECS 270 Fall 2014, Lecture 21 Page 2 of 7 Error correction reviewed Parity gave as a method of detecting a one-bit flip. This is useful because we can, at the least, not use bad data, and we could perhaps even ask for the data to be resent. But what if we want to fix the data without needing to ask for a resend? It turns out we can do that if the set of messages we are using has a Hamming distance of 3 or more. Why is that? Let s review the specific scheme with 4 data bits and 3 parity bits A, B, C, D are data bits while X, Y, Z as parity bits Let P(x 1..x n ) be the function that generates even ones parity over the inputs x 1 to x n. Let X=P(A,B,C) Let Y=P(A,B,D) Let Z=P(A,C,D) From here, we re going to assume only one bit will go bad in a given message? What happens if more than one bit gets flipped? Our message will be received incorrectly. But let s ignore that case for now. Which parity bits will incorrect if the following bit were flipped in transmission? A B C D X Y Z So what s going on? How many bits of error correction would we need if we want to send X data bits with 1 bit of correction? The equation: P=# of parity bits, D= # of data bits.
EECS 270 Fall 2014, Lecture 21 Page 3 of 7 Other thoughts 1. How can we use that set to detect 2-bit errors? 2. How do we build the encoder (takes in 4 data bits, generates 7 bit message)? 3. How do we build the decoder (takes in 7 bit message, generates corrected 4-bit data)? -- This is question 1 of GA4. Moving forward on error correction The most obvious question is: how do we make codes for more than 4 data bits? And the answer is fairly simple. If the goal is to make it so that we have 8 data bits, the equation says we need 5 parity bits. But how to generate them? The basic theme is that each data bit must be: Covered by at least 2 parity bits Be covered by a set of parity bits that is unique that is no other data bit can be covered by exactly the same parity bits. An easy (and fairly standard) way to do this is to use the integers in binary. Start counting at 1. Have each number that is a power of 2 (1, 2, 4, 8, etc.) be a parity bit. The other bits are data bits. Notice the powers of 2 have exactly one 1 in them. Of course, all numbers have a unique set of 1 s (not shared with any other number. And finally, when we receive the data, we check which parity bits are different than we d expect. We just OR their numbers together and see which bit was flipped. So if we find 0001 and 0100 were different than expected, we d say that 0101 was the bit that was flipped! 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Notice that if we just do 1 bit of data, we have standard triple redundancy. What is the set of legal codes? Notice if we want just 4 bits of data, we have the Hamming(7,4) scheme discussed above. 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0_ 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 And a few last words The schemes we ve used here have been generating sets. What I mean is that the sender is only sending data that meets certain requirements the parity bits are generated in a fixed way based on the data bits. And so there are so sets of bits that the sender will send, and some it won t send. In the case of even one s parity over 2 data bits, the set of things that might be sent are {000, 110, 101, 011}. That set has a Hamming Distance of 2. The Hamming(7,4) scheme and the Hamming (15,11) scheme both generate sets that have a Hamming Distance of 3. Why can we only do detection of a 1 bit error with a set that has a Hamming Distance of 2? Why can we detect 2 bits of error with a Hamming Distance of 3? Why can we correct 1 bit of error? What Hamming distance would we need if we wanted to correct 2 bits of data?
EECS 270 Fall 2014, Lecture 21 Page 4 of 7 FPGA internals: Lookup tables. Let s start looking at the internals of a Field Programmable Gate Array (FPGA). Basic issue: we want programmable hardware but we can t just move transistors around they are fixed in location. So how do we make it so we can create a real hardware device (not immolated in software)? Answer: We use memory. Memory is one thing that we can easily change on the fly. Consider the following truth table of an XNOR gate where F=x XNOR y. We can implement it as a memory. We can even implement 2 logic functions if the memory is two-bits wide (i.e. word size is 2). Create a memory that implements the following (from the text) k p s w 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1
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