Mutual Information and Channel Capacity Explanation 1. Channel Capacity and Mutual Information Channel capacity C is the maximum rate at which information can be reliably transmitted over a channel. From information theory: C ≥ I(X; Y) where: I(X; Y) is the mutual information between channel input X and output Y . Q is the input distribution. W is the channel transition probability. Mutual Information expressed via Entropy (Equation 9) Mutual information between input and output can be expressed as: C = I(X; Y) = h(Y) - h(Y | X) where: h(Y) is the entropy of the output. h(Y|X) is the conditional entropy of the output given the input. 2. Conditional Entropy for the Channel (Equation 10) The conditional entropy of the output given the input (assuming Gaussian noise with ...

Understanding the Relationship Between PDF and CDF At its core, the CDF is just the accumulated area under the PDF . Definitions Let X be a continuous random variable. PDF: f X (x) CDF: F X (x) = P(X ≤ x) Mathematical Relationship The CDF is the integral of the PDF: F X (x) = ∫ -∞ x f X (t) dt Step-by-Step Method Identify the support of the PDF Determine where f X (x) is nonzero. Integrate the PDF This often gives: F X (x) = 0, x < a ∫ a x f X (t) dt, a ≤ x ≤ b 1, x > b Check endpoints F X (-∞) = 0 F X (∞) = 1 Example Giv...

Using Neural Networks for Tabular and Image Data: A Practical Guide Neural networks are versatile models that can learn patterns from a wide variety of data types. In this article, we explore how to feed tabular data and image data into neural networks for classification tasks, with practical examples in PyTorch. 1. Tabular Data: Features and Targets Tabular data is structured in rows and columns , where: Features: Input variables used by the model to learn patterns (e.g., age, salary, scores). Target: The output variable the model predicts (e.g., personality type, class label). Example: Personality Classification Suppose we have a synthetic dataset with 29 features representing various attributes of individuals and a target column personality_type . my_df = pd.read_csv('personality_synthet...

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Yes! You’ve got the concept mostly correct , but we can tighten the explanation and make it a bit more precise and clear. Here’s a clean way to describe it: How FastAPI Works 1. FastAPI is the application It’s where you define routes , API endpoints , and business logic . Think of it as the “brain” of your web service. 2. Uvicorn is the server Uvicorn is an ASGI server that listens for HTTP requests from clients (like browsers or mobile apps). It’s responsible for receiving requests and sending responses , but it does not know your business logic. 3. The request flow A client calls an API endpoint (e.g., GET /users or POST /predict ). Uvicorn receives the request and passes it to the FastAPI app. FastAPI processes the request: ...

  Bubble Sort is a sorting algorithm that arranges the elements of an array in ascending order (from lowest to highest). Conceptually, the algorithm works by repeatedly comparing adjacent elements in the array. If the current element is greater than the next one, the two elements are swapped. The process starts by comparing the first and second elements, then the second and third, and continues through the array. Each pass places the largest unsorted element in its correct position. This process is repeated until the entire array is sorted. To implement the Bubble Sort algorithm in a programming language, the following components are required: An array containing the values to be sorted. An inner loop that iterates through the array, comparing adjacent elements and swapping them if the first element is greater than the next. With each pass, this loop runs one fewer iteration because the largest elements are already in their correct positions. An outer loop that determines ...

Finding the PDF from a Given Graph When a probability density function (PDF) is given as a graph, the goal is to write its mathematical expression from the picture. Step 1: Identify the Support The support is the interval where the graph is above the x-axis. From the graph: -1 ≤ x ≤ 1 Outside this interval, the PDF is zero. Step 2: Identify Key Points Read the important coordinates from the graph: (-1, 0) (0, 1) (1, 0) The graph is piecewise linear, forming a triangle. Step 3: Find the Equation of Each Line Left side: from -1 to 0 Slope: m = (1 - 0) / (0 - (-1)) = 1 Equation of the line: f X (x) = x + 1    for -1 ≤ x ≤ 0 Right side: from 0 to 1 Slope: m = (0 - 1) / (1 - 0) = -1 ...

Why MSE Is Often Used in Wireless Communication Understanding MSE and MMSE In wireless communication systems, signals transmitted over the air are distorted by noise, multipath effects, and interference. To measure how well a receiver can recover the original transmitted signal, engineers frequently use Mean Squared Error (MSE) , which calculates the average of the squares of errors between estimated and true values. The Minimum Mean Square Error (MMSE) approach finds the estimator that minimizes this squared error.  Channel Estimation and Equalization In systems like MIMO and OFDM, receivers must estimate channel characteristics to undo the distortion caused by the propagation environment. Using pilot symbols or known training sequences, the receiver compares the observed signal with expected values and computes estimates of the channel. Minimizing the MS...