Constellation Point Distance and Power Requirement in M-ary PSK In digital communication systems, especially M-ary PSK (Phase Shift Keying) , the arrangement of constellation points plays a crucial role in determining error performance and power efficiency. 1. Constellation Point Distance In an M-ary PSK system, the constellation points are equally spaced on a circle of radius √E s , where: E s = Energy per symbol M = Number of constellation points The minimum distance between adjacent constellation points is: d min = 2 √E s sin(π / M) Key Observations: As M increases, sin(π/M) decreases. The minimum distance between points decreases. Smaller distance → constellation points are closer → higher probability of error. 2. Probability of Error and Symbol Energy The proba...

Can a Sinusoid Be Wide-Sense Stationary (WSS)? A noisy sinusoidal is not itself a Wide-sense stationary (WSS) signal. Sinusoidal signal + Random phase = WSS In the code below it demonstrates that how sinusoidal signal can be turned into wide-sense stationary signal where it captures the property like constant mean and same auto-correlation value over time. MATLAB Code % Generate Noisy Sinusoid (Single Script) clc; clear; close all; %% Parameters N = 1000; % Number of samples freq = 5; % Frequency in Hz sampleRate = 100; % Sampling rate in Hz meanValue = 2; % Desired mean variance = 0.5; % Desired variance %% Amplitude from variance amp = sqrt(2 * variance); %% Random phase theta = 2 * pi * rand(); %% Time vector n = 0:N-1; t = n / sampleRate; %% Generate sinusoidal signal signal = meanValue + amp * cos(2 * pi * freq * t + theta); %% Plot signal figure; plo...

Quantum Key Distribution (QKD): An Intuitive Explanation This explanation rebuilds the ideas behind Quantum Key Distribution (QKD) from the ground up, focusing on intuition first and math second. Big Picture Quantum Key Distribution is a way for two parties (Alice and Bob) to: Create a shared random secret key Detect if anyone (Eve) tried to eavesdrop The core idea: measuring a quantum system changes it . Everything else in QKD follows from this rule. 1. What Are Quantum “States”? A photon can be polarized in different directions: Horizontal (↔) Vertical (↕) Diagonal (↗) Other diagonal (↘) We label these using symbols: |0⟩ = horizontal |1⟩ = vertical These form one basis (a way of asking questions). ...

Lossless Transmission Line Calculator Assumptions: R = 0, G = 0 (Lossless line) Input Parameters Inductance per unit length (L) [H/m] Capacitance per unit length (C) [F/m] Frequency (f) [Hz] Load Impedance Z L [Ω] Incident Voltage V⁺ [V] Line Length λ/4 (odd multiple) λ/2 (any multiple) Calculate Results Results will appear here.

Instructions for Fast Fourier Transform (FFT) Simulator Step 1: Click on "Generate Input Signal" to generate the signal. Step 2: Enter the input frequency value in Hz in the parameters section. Step 3: Select the base signal from the dropdown menu. Step 4: Choose an operation such as addition, multiplication, or convolution from the dropdown menu. Step 5: Click the "Simulate" button to run the simulation. Step 6: Reset the simulator by clicking the "Reset Simulator" button. Generate Input Signal Parameters Input Signal Frequency (Hz) Pulse Width (s) Magnitude Plot Phase Plot Base Signal Select an input Sine Wave Cosine...

Dynamic Uniform Mid-Rise Quantizer Signal (comma separated): Vmin: Vmax: Levels (L): Quantize

Uniform Mid-Rise Quantizer (8 Levels, Range [-8, +8]) Your function implements a uniform mid-rise quantizer over: \[ V_{\min} = -8, \quad V_{\max} = 8 \] with: \[ L = 8 \text{ total quantization levels} \] Step Size (Resolution) \[ \Delta = \frac{V_{\max} - V_{\min}}{L} \] Substitute values: \[ \Delta = \frac{16}{8} = 2 \] Each quantization interval has width 2 . Decision Boundaries (Partitions) \[ P_i = V_{\min} + i\Delta \] For \( L = 8 \): \[ [-8, -6, -4, -2, 0, 2, 4, 6, 8] \] Representation Levels (Midpoints) \[ C_i = V_{\min} + (i + 0.5)\Delta \] The 8 quantized output levels: \[ [-7, -5, -3, -1, 1, 3, 5, 7] \] 4 negative levels 4 positive levels No level exactly at 0 → mid-rise quantizer Quantization Rule Step 1 — Compute interval index \[ k = \left\lfloor \frac{x - V_{\min}}{\Delta} \right\rfloor \] Step 2 — Compute quantized value \[ Q(x) = V_{\min} + (k + 0.5)\Delta \] Step 3 — Clip to valid range \[ Q(x)...

Derivation of Frequency Separation for Orthogonality Step 1: Define BFSK Signals s₁(t) = √(2E b /T) cos(2πf₁t) s₂(t) = √(2E b /T) cos(2πf₂t) Defined over: 0 ≤ t ≤ T For orthogonality: ∫₀ᵀ s₁(t)s₂(t) dt = 0 Step 2: Remove Constants ∫₀ᵀ cos(2πf₁t) cos(2πf₂t) dt = 0 Step 3: Use Trigonometric Identity cos A cos B = ½ [ cos(A − B) + cos(A + B) ] Applying identity: ½ ∫₀ᵀ [ cos(2π(f₁ − f₂)t) + cos(2π(f₁ + f₂)t) ] dt Step 4: Focus on Frequency Difference The second term integrates to zero for high carrier frequencies. ∫₀ᵀ cos(2πΔf t) dt ...