Question 37 Let α, β be two non-zero real numbers and v 1 , v 2 be two non-zero real vectors of size 3 × 1. Suppose that v 1 and v 2 satisfy v 1 T v 2 = 0 , v 1 T v 1 = 1 , and v 2 T v 2 = 1 . Let A be the 3 × 3 matrix given by: A = αv 1 v 1 T + βv 2 v 2 T The eigenvalues of A are ________. (A) 0, α, β (B) 0, α + β, α - β (C) 0, (α + β)/2, √(αβ) (D) 0, 0, √(α² + β²) Solution To solve for the eigenvalues of matrix A , we test the given vectors as potential eigenvectors: Testing v 1 : Multiplying A by v 1 : Av 1 = (αv 1 v 1 T + βv 2 v 2 T )v 1 = αv 1 (v 1 T v 1 ) + βv 2 (v 2 T v 1 ) Using the given conditions ( v 1 T v 1 = 1 and v 2 T v 1 = 0), w...

Modulation vs. Multiplexing: Key Differences in Telecommunications Modulation vs. Multiplexing: Understanding the Foundation of Modern Communication In the world of telecommunications, Modulation and Multiplexing are often used interchangeably by beginners, but they represent two entirely different layers of signal processing. While one focuses on the integrity and speed of a single signal , the other focuses on the efficiency of the medium . Feature Modulation Schemes Multiplexing Techniques Core Definition Mapping information bits onto a physical carrier wave. Combining multiple data streams into one shared medium. Primary Objective Optimizing data rate and no...

Unified Passband PA Lab Power Amplifier Nonlinearity Lab Unified Passband Simulation: Visualizing Carrier Clipping & Spectral Regrowth Modulation (Passband) Pure Sine (Ideal) BPSK (Constant Env) QPSK (Constant Env) 16QAM (Variable Env) Input Amplitude (A) PA Saturation (V-limit) Re-Simulate Time Domain: Carrier Wave Clipping AM-AM Curve: Linear vs. Real PA Spectrum: Harmonic Regrowth (dB) How the Simulation Works The mathematical proof for \( P = \frac{A^2}{2R} \) assumes a perfect, infinite world. However, in RF hardware, we are limited by the physical rails...

Why Signal Power is A²/2R: Mathematical Proof & Intuitive Guide Why is the Power of a Carrier Signal Equal to \( \frac{A^2}{2R} \)? If you are studying electrical engineering, telecommunications, or RF (Radio Frequency) design, you have undoubtedly encountered the formula for the average power of a sinusoidal signal: \( P = \frac{A^2}{2R} \) . At first glance, it looks similar to the standard DC power formula (\( P = V^2/R \)), but with a mysterious factor of 2 in the denominator. Where does that 2 come from? Is it just a constant we have to memorize? In this guide, we’ll break down the mathematical derivation and the intuition behind it. The Short Answer: The factor of 2 comes from the average value of the square of a sine (or cosine) wave over one complete cycle. A sine wave doesn't stay at its peak; it oscillates, delivering exactly half the power of a constant DC signal wi...

  Full Carrier DSB-SC (AM) Signal For a full carrier AM (DSB-FC) signal: Carrier Power: \( P_c = \frac{A_c^2}{2R} \) Total Sideband Power: \( P_{SB} = \frac{m^2}{2} P_c \) Therefore, the ratio of total sideband power to carrier power is: \( \frac{P_{SB}}{P_c} = \frac{m^2}{2} \) If the modulation index doubles: \( m' = 2m \) The new ratio becomes: \( \frac{(2m)^2}{2} = \frac{4m^2}{2} = 4\left(\frac{m^2}{2}\right) \) Hence, the ratio of total sideband power to carrier power increases by a factor of 4 . Answer: 4

  Solution Let S = X 1 + X 2 . Since X 3 is independent of S, P(X 1 + X 2 < X 3 ) = E[P(X 3 > S | S)]. Because X 3 ~ U(0, 1), P(X 3 > S) = 1 − S,  for 0 ≤ S ≤ 1 0,  for S > 1 The sum S = X 1 + X 2 has the triangular density f S (s) = s,  for 0 ≤ s ≤ 1 f S (s) = 2 − s,  for 1 ≤ s ≤ 2 Hence, P(X 1 + X 2 < X 3 ) = ∫ 0 1 (1 − s)s ds = ∫ 0 1 (s − s²) ds = [s²/2 − s³/3] 0 1 = 1/2 − 1/3 = 1/6 . Final Answer P(X 1 + X 2 < X 3 ) = 1/6

IF Choice Trade-off Simulator IF Trade-off Simulator Adjust the Intermediate Frequency (IF) to see how it affects Selectivity, Image Rejection, and Tracking (based on MCQ 61). Signal Frequency ($f_s$): 1000 kHz Intermediate Frequency ($f_{IF}$): 455 kHz Image Frequency ($f_i = f_s + 2f_{IF}$) 1910 kHz Image Separation 910 kHz Q-Factor (Stability) 2.2 Analysis based on your MCQ: Online Signal Processing Simulations Main Page >

PAPR Simulator: OFDM vs. DFT-s-OFDM OFDM vs. DFT-s-OFDM Simulator Analyze Peak-to-Average Power Ratio (PAPR) for 5G/LTE Uplink Design Modulation Order (M-QAM) 4-QAM (QPSK) 16-QAM 64-QAM 256-QAM Subcarriers (N) 64 carriers Symbol Smoothing Over-sample (4x) Run Simulation Why the difference? In OFDM , symbols are independent on each carrier. Summing them causes random spikes. In DFT-s-OFDM , the DFT "spreads" each data symbol across all carriers, acting like a single-carrier signal. ...