LTE 2-User MIMO (Uplink) – Step-by-Step Example This example shows how two mobile users transmit simultaneously and how the base station separates their signals using linear algebra. 1. Transmission x = [ x1 x2 ] x1: Signal from User 1 x2: Signal from User 2 2. Channel Matrix H = [ 1 0.5 0.3 1 ] Signals from both users mix across both receive antennas. 3. Received Signal y = Hx y1 = x1 + 0.5x2 y2 = 0.3x1 + x2 The base station receives only mixed signals — no clean separation exists. 4. Detection Using Zero Forcing (ZF) x̂ = H⁻¹ y Inverse of H det(H) = 1 - (0.5 × 0.3) = 0.85 H⁻¹ = (1 / 0.85) × [ 1 -0.5 -0.3 1 ] H⁻¹ = [ 1.176 -0.588 -0.353 1.176 ] 5. Recover Signals x̂1 = 1.176y1 - 0.588y2 x̂2 = -0.353y1 + 1.176y2 6. Substitution For x̂1: x̂1 = 1.176(x1 + 0.5x2) - 0.588(0.3x1 + x2) = x1 For x̂2: x̂2 = -0.353(x1 + 0.5x2) + 1.176(0.3x1 + x2) = x2 Final Result...

Elliptically Polarized Wave Power Calculator Enter Wave Parameters Electric field amplitude Ex (V/m): Electric field amplitude Ey (V/m): Calculate Average Power Average Power per unit area ⟨S⟩ = -- W/m² Workflow & Math Behind For an elliptically polarized wave traveling in the +z direction , the electric field is: 𝐄(z, t) = x̂ Ex cos(kz - ωt) + ŷ Ey cos(kz - ωt + δ) Step-by-Step Workflow: Step 1: Identify the amplitudes along x and y axes: Ex and Ey . Step 2: Compute the magnetic field using: H = (1/η) (ẑ × E) Step 3: Compute the instantaneous Poynting vector: S = E × H Step 4: Time-average over one cycle to get average power per unit area. Math Formula: The time-averaged power is: ⟨S⟩ = (Ex² + Ey²) / (2 η) Where: ...

Channel Estimation Lab Fundamental Channel Estimation Experiment with how a wireless channel (h) distorts a known pilot signal (x) and how we recover it. Noise Level (Variance): 0.10 Channel Magnitude (|h|): 1.0 Channel Phase (degrees): 45 ° Actual Channel (h): Estimated Channel (ĥ): Estimation Error: Formula: y = hx + n | ĥ = mean(y/x) 1. Received Signal (y) Notice the rotation (phase shift) and scaling (magnitude) compared to the original pilot at (1,0). 2. Equalized Signal (y / ĥ) After dividing by our estimate ĥ, the points should cluster back around (1,0). How the Simulator Works ...

Radiation Resistance Simulator Enter Antenna Parameters Antenna length (l in meters): Wavelength (λ in meters): Calculate Radiation Resistance Radiation Resistance (Rr) = -- Ω

Radiation Resistance (in terms of λ) Definition Radiation resistance represents how effectively an antenna radiates power. It can be expressed in terms of wavelength (λ). Formula (Short Dipole) For a short dipole antenna (l ≪ λ): Rr = 80π² (l / λ)² Simplified Formula Rr ≈ 790 (l / λ)² Since 80π² ≈ 790 Half-Wave Dipole For a half-wave dipole antenna: Rr ≈ 73 Ω Example If l = λ / 10: Rr = 790 (1/10)² Rr = 790 × 0.01 Rr = 7.9 Ω Summary Depends on (l / λ)² Larger antenna → higher radiation resistance Important for antenna design Short dipole: Rr = 80π² (l / λ)² Approx: Rr ≈ 790 (l / λ)² Half-wave dipole: ≈ 73 Ω

Pandas Quick Reference 1. Import Pandas import pandas as pd 2. Reading Data Load CSV files into a DataFrame: df = pd.read_csv("data.csv") # read file df = pd.read_csv("data.csv", index_col=0) # use first column as index 3. Inspecting Data Check dimensions: df.shape # (rows, columns) View top/bottom rows: df.head() # first 5 rows df.tail() # last 5 rows Column names: df.columns Quick summary: df.info() # types, non-null counts df.describe() # summary statistics for numeric columns 4. Data Types (dtypes) Check the type of a column or all columns: df["price"].dtype # single column df.dtypes # all columns Convert types: df["points"].astype("float64") 5. Index df.index.dtype df.index 6. Missing Data (NaN) Detect missing values: pd.isnull(df["coun...

  At the core of machine learning lies a key challenge: building models that perform well not just on training data, but also on new, unseen data. Two common problems that affect model performance are underfitting and overfitting . Overfitting: Overfitting occurs when a model learns the training data too well, including noise and random fluctuations. In decision trees, this happens when the tree becomes very deep, creating many splits and leaves. Each leaf ends up containing very few data points, so predictions become highly specific to the training data. While this leads to very accurate results on training data, the model performs poorly on new or validation data because it fails to generalize. Underfitting: Underfitting happens when a model is too simple to capture important patterns in the data. For example, a very shallow decision tree with only a few splits groups many different data points together. As a result, predictions are inaccurate even on the training...

AC Circuit Interactive DSP Tool AC Circuit Interactive DSP Tool Adjust Parameters Resistance (R): 20 Ω Inductance (L): 0.04 H Capacitance (C): 10 μF Frequency (f): 50 Hz Results Download Report Phasor Diagram Yellow = static phasor (initial phasor), Blue = rotating phasor Waveform Rotating Phasor Controls R: 20 Ω L: 40 mH C: 10 μF f: 50 Hz Download Report Rotating Phasor Live Signal Waveform Bode Plot 📐 Math Behind AC Circuits In an AC series RLC circuit, we have: Resistance: R (Ω) Inductor: L (H) → Reactance \( X_L = 2 \pi f L \) Capacitor: C (F) → Reactance \( X_C = \frac{1}{2 \pi f C} \) Angular Frequency: \( \omega = 2 \pi f \) The total impedance of the series ci...