DC Component (Zero-Frequency) – Complete Hand Calculation Guide The DC component (zero-frequency component) represents the average value or total area of a signal, depending on the transform used. 1. DC Component of a Periodic Signal The DC component of a periodic signal is its average value over one full period . Definition For a periodic signal x(t) with period T : DC component = (1/T) ∫₀ᵀ x(t) dt Mean value Average value Zero-frequency component (in Fourier analysis) Continuous-Time X_DC = (1/T) ∫_{t₀}^{t₀ + T} x(t) dt You can integrate over any full period. Discrete-Time X_DC = (1/N) Σ x[n], n = 0 to N-1 2. Examples (Periodic Signals) Example 1: Sine Wave x(t) = A sin(ωt) X_DC = 0 A pure sine (or cosine) has zero DC component. Example 2: Offset Sine Wave x(t) = A sin(ωt) + 3 X_DC = 3 The DC component e...

Python Code to Build Tree and Get Paths Build Binary Tree and Get Root-to-Leaf Paths class TreeNode: def __init__(self, val): self.val = val self.left = None self.right = None def build_tree(arr): if not arr: return None nodes = [TreeNode(v) for v in arr] for i in range(len(arr)): left = 2*i + 1 right = 2*i + 2 if left < len(arr): nodes[i].left = nodes[left] if right < len(arr): nodes[i].right = nodes[right] return nodes[0] def root_to_leaf_paths(root): paths = [] def dfs(node, path): if not node: return path.append(node.val) if not node.left and not node.right: paths.append(list(path)) else: dfs(node.left, path) dfs(node.right, path) path.p...

OFDM Transmission in Practical LTE and 5G Systems Understanding the difference between baseband processing and real RF transmission in modern communication systems. 1. What We Study vs What Is Actually Transmitted In textbooks, the OFDM signal is written as: \[ x(t) = \sum_{k} X[k] e^{j 2\pi f_k t} \] This represents a complex baseband signal . However, antennas cannot transmit complex numbers. They transmit real RF signals . 2. Practical Transmitter Chain (LTE / 5G) In systems like LTE (4G) and 5G NR , the transmission process works as follows: Step 1: Digital Baseband Processing Bits QAM Mapping OFDM IFFT Add Cyclic Prefix The result is: \[ x_{BB}[n] = I[n] + j...

Alternate Mark Inversion (AMI) Alternate Mark Inversion (AMI) is a type of line coding technique used in digital communication systems to transmit binary data over physical media such as copper wires or optical links. It improves synchronization and reduces certain transmission problems compared to simple encoding schemes. What Is Line Coding? Line coding is the process of converting binary data (0s and 1s) into electrical signals that can be transmitted over a communication channel. Different encoding techniques improve signal quality, timing recovery, and error detection. How Alternate Mark Inversion Works In AMI encoding : A binary 0 is represented by no voltage (0 volts). A binary 1 is represented by a non-zero voltage , but the polarity alternates between positive and negative. ...

  MATLAB Code for Raised Cosine Filter Pulse Shaping clc; clear; close all ; %% ===================================================== %% PARAMETERS %% ===================================================== N = 64; % Number of OFDM subcarriers cpLen = 16; % Cyclic prefix length modOrder = 4; % QPSK oversample = 8; % Oversampling factor span = 10; % RRC filter span in symbols rolloff = 0.25; % RRC roll-off factor %% ===================================================== %% Generate Baseband OFDM Symbols %% ===================================================== data = randi([0 modOrder-1], N, 1); % Random bits txSymbols = pskmod(data, modOrder, pi/4); % QPSK modulation % IFFT to get OFDM symbol tx_ofdm = ifft(txSymbols, N); % Add cyclic prefix tx_cp = [tx_ofdm(end-cpLen+1:end); tx_ofdm]; %% ===================================================== %% Oversample the Baseband Signal %% ===============================================...

Why Complex Exponentials Are Eigenfunctions of Every LTI System Eigenfunction of an LTI System For any LTI system , complex exponentials are eigenfunctions. If the input is: $$ x(t) = e^{j\omega_0 t} $$ then the output is: $$ y(t) = H(j\omega_0)\, e^{j\omega_0 t} $$ where \(H(j\omega_0)\) is the system’s frequency response. Why is it called an eigenfunction? Because it satisfies the eigenvalue equation: $$ T\{x(t)\} = \lambda x(t) $$ Eigenfunction → \( e^{j\omega_0 t} \) Eigenvalue → \( H(j\omega_0) \) So the eigenvalue is not \( e^{j\omega_0 t} \). The eigenvalue is the scalar \( H(j\omega_0) \). What about...

RLC Network Question and Solution Question A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form: Σ (k = 1 to 3) aₖ cos(kω₀ t), where aₖ ≠ 0, ω₀ ≠ 0. The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network? Σ (k = 1 to 3) bₖ cos(kω₀ t + φₖ), where bₖ ≠ aₖ for all k ...

  In OFDM (Orthogonal Frequency Division Multiplexing) , we transmit multiple orthogonal subcarriers simultaneously. Since the subcarriers are orthogonal , they do not interfere with each other, which is one of the main advantages of OFDM. Practically, OFDM converts a wideband signal into multiple narrowband orthogonal subcarriers. For typical wireless communication, if the signal bandwidth (or symbol duration) exceeds the coherence bandwidth of the channel, the signal experiences frequency-selective fading . Fading distorts the signal, making it difficult to recover the original information. By using OFDM, we transmit the same wideband signal across multiple orthogonal narrowband subcarriers, reducing the effect of fading. For example, if we want to transmit a signal of bandwidth 1024 kHz , we can divide it into N = 8 subcarriers . Each subcarrier is then spaced by: Δf = Total Bandwidth N = 1024 8 kHz...