Signal Processing for Communication Systems
Key focus : Convolutional codes: Error-correcting codes effective in noisy channels and burst error correction. Know about sliding window & shift registers and how they are used in a convolutional encoder. Linear block codes Linear block codes, like Hamming codes, are mainly employed in hard decision decoding, utilizing the inherent algebraic structure of the code … Read more
Key focus: Euclidean & Hamming distances are used to measure similarity or dissimilarity between two sequences. Used in Soft & Hard decision decoding. Distance is a measure that indicates either similarity or dissimilarity between two words. Given a pair of words a=(a0,a1, … ,an-1) and b=(b0,b1,…,bn-1) , there are variety of ways one can characterize … Read more
Markov chains are useful in computing the probability of events that are observable. However, in many real world applications, the events that we are interested in are usually hidden, that is we don’t observe them directly. These hidden events need to be inferred. For example, given a sentence in a natural language we only observe the … Read more
Key focus: Markov chains are a probabilistic models that describe a sequence of observations whose occurrence are statistically dependent only on the previous ones. ● Time-series data like speech, stock price movements.● Words in a sentence.● Base pairs on the rung of a DNA ladder. States and transitions Assume that we want to model the … Read more
The Shannon power efficiency limit is the limit of a band-limited system irrespective of modulation or coding scheme. It informs us the minimum required energy per bit required at the transmitter for reliable communication. It is also called unconstrained Shannon power efficiency Limit. If we select a particular modulation scheme or an encoding scheme, we … Read more
Reed-Solomon (RS) codes are powerhouse error-correcting tools, but they have a specific Achilles’ heel: if a burst inside one codeword exceeds the correction radius \(t = \lfloor (n-k)/2 \rfloor\), a bounded-distance decoder fails on that codeword. Block Interleaving is the mathematical bridge that allows RS codes to survive massive, contiguous bursts by spreading the damage … Read more
Convolutional Encoding Unlike block codes (like Reed-Solomon), Convolutional Codes do not have a fixed block size. Instead, they process a continuous stream of bits. The output at any given time depends not only on the current input bit but also on the previous $K-1$ bits, where $K$ is the Constraint Length.+1. Convolutional codes are a … Read more
Keywords: maximum likelihood estimation, statistical method, probability distribution, MLE, models, practical applications, finance, economics, natural sciences. Introduction Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by finding the set of values that maximize the likelihood function of the observed data. In other words, MLE is … Read more
Detection path: ML estimation → ML decoding → Hard vs soft decision → Hamming codes In one sentence: Maximum-likelihood decoding picks the codeword that maximizes the channel likelihood of the received word — on a BSC that reduces to nearest Hamming neighbor when the crossover probability is below one half. Introduction Maximum likelihood decoding … Read more
Hard decision keeps 0/1; soft decision keeps reliability (LLRs). Why soft decoding gains ~2 dB and how it ties to ML decoding.