Graphs and Their Representations What is a Graph? Graph : A non-linear data structure made up of vertices (nodes) and edges (connections). Vertices : Points in the graph (also called nodes). Edges : Connections between two vertices. Non-linear : Unlike arrays or linked lists, graphs allow different paths to connect vertices. Applications of Graphs: Social Networks : People as vertices, relationships as edges. Maps/Navigation : Locations as vertices, roads as edges. Internet : Web pages as vertices, hyperlinks as edges. Biology : Neural networks, disease spread modeling. Graph Representations: Graph Representation : How a graph is stored in memory. Different representations impact space and speed of operations. Adjacency Matrix : A 2D array representing edges between vertices. Types of Graph Representations: 1. Adjacency Matrix: 2D a...

In Data Structures and Algorithms (DSA), Trees Trees are a fundamental data structure that help in efficiently organizing and managing hierarchical data. They are used in many applications like databases, file systems, search engines, and network routing, among others. How Trees Help in DSA: Efficient Searching : Binary search trees (BST) allow fast searching, insertion, and deletion of data in O(log n) time (on average), making it much faster than linear search on an array. Hierarchical Data Representation : Trees help in representing hierarchical data, like file systems or organizational charts, where each node represents a unit, and the edges represent the relationship between them. Balanced Trees : Variants like AVL trees or Red-Black trees maintain balance, ensuring efficient performance in searching and updates. Priority Queue : Heaps (a type of binary tree) are used in pri...

Successive Interference Cancellation (SIC) Ah, I see your point now more clearly. You're asking a crucial question: How can a device subtract the strong signal from the received signal if it doesn't know what the strong signal is? This is the essence of Successive Interference Cancellation (SIC) , and I’ll explain in detail how this actually works in practice: The Key Idea Behind SIC In SIC , the process of subtracting the strong signal to improve the weak signal relies on the fact that the receiver already knows the strong signal once it has decoded it . The confusion usually comes from the fact that it sounds like the receiver is magically subtracting something it doesn't know. Here's how it works: Step-by-Step Breakdown of SIC 1. Superposed Transmission Both users transmit their signal...

Basic Idea Behind Multiuser Alamouti STBC In a MIMO (Multiple Input, Multiple Output) system, we use multiple antennas at both the transmitter and the receiver to improve performance (better data rate, reliability, etc.). The Alamouti Space-Time Block Code (STBC) is a method used to send data in such a way that it becomes more robust to noise and fading. Single-User Alamouti Example: Let’s first recall the basic Alamouti code for one user with two antennas: At Time 1: Antenna 1 sends \( s_1 \) (the first data symbol). Antenna 2 sends \( s_2 \) (the second data symbol). At Time 2: Antenna 1 sends \( -s_2^* \) (the complex conjugate of \( s_2 \)). Antenna 2 sends \( s_1^* \) (the complex conjugate of \( s_1 \)). This is the Alamouti STBC for one user. Multiuser Ala...

  MATLAB Code for Multi-User STBC (using Alamouti's Scheme)  clc; clear; % Parameters N = 1e4; % Symbols per user U = 2; % Number of users SNR_dB = 0:5:30; alpha = 0.8; % Modification factor power = [0.7 0.3]; % Power allocation (sum <= 1) % Generate QPSK symbols for each user data = cell(U,1); s1 = cell(U,1); s2 = cell(U,1); for u = 1:U data{u} = randi([0 3], N, 2); s = pskmod(data{u}, 4, pi/4); s1{u} = s(:,1); s2{u} = s(:,2); end % Channels (independent Rayleigh per user) h1 = cell(U,1); h2 = cell(U,1); for u = 1:U h1{u} = (randn(N,1)+1j*randn(N,1))/sqrt(2); h2{u} = (randn(N,1)+1j*randn(N,1))/sqrt(2); end SER = zeros(length(SNR_dB),U); % SNR loop for k = 1:length(SNR_dB) SNR = 10^(SNR_dB(k)/10); noise_var = 1/SNR; n1 = sqrt(noise_var/2)*(randn(N,1)+1j*randn(N,1)); n2 = sqrt(noise_var/2)*(randn(N,1)+1j*randn(N,1)); % Superposed transmission (all users) x1 = zeros(N,1...

LIFO vs FIFO The basic difference is simply the order in which items come out . LIFO (Last In, First Out) What goes in last comes out first. Used by a stack Operations: push , pop Example Push: A → B → C Pop: C (comes out first) Real-life analogy Stack of plates 🍽️ You put a plate on top → you take the top plate first. FIFO (First In, First Out) What goes in first comes out first. Used by a queue Operations: enqueue , dequeue Example Enqueue: A → B → C Dequeue: A (comes out first) Real-life analogy Line at a ticket counter 🎟️ The person who arrives first is served first. Side-by-side comparison Fe...

K-Nearest Neighbors (KNN) Algorithm: Simple Math and Example Let's break down the mathematical concept behind the KNN algorithm and go through a simple example. 1. Mathematical Concept KNN works by finding the K closest points in the feature space (based on distance metrics such as Euclidean distance) to classify or predict a new data point. Step 1: Choose a number K (the number of nearest neighbors). Step 2: Calculate the distance between the new data point and all other points in the training dataset. Step 3: Sort the distances and pick the K smallest distances. Step 4: For classification, assign the class label based on the majority of the K neighbors. Step 5: For regression, calculate the average value of the K neighbors and assign it as the prediction. 2. Distance Metric: Euclidean Distance For classifi...

1. SVM Objective The core goal of the Support Vector Machine (SVM) is to find the decision boundary (hyperplane) that maximizes the margin between the two classes. The equation of the decision boundary is typically written as: w · x + b = 0 Where: w = [w 1 , w 2 ] is the weight vector (which is perpendicular to the hyperplane). x = [x 1 , x 2 ] is the feature vector (the input data). b is the bias term (the offset from the origin). This equation defines a hyperplane in a multidimensional space. 2. Finding the Margin In SVM, the objective is to maximize the margin , which is the distance between the decision boundary (the hyperplane) and the support vectors . The margin is mathematically defined as: Margin = 2 / |w| Where |w| is the norm (magnitude) of the weight vector. The margin boundaries (parallel to the...