Time-Bandwidth Product, GMSK, and Pulse Shaping: A Comprehensive Guide Understanding Time-Bandwidth Product (TBP): From Raised Cosine to GMSK Exploring the trade-off between signal duration, spectral width, and system performance. 1. What is the Time-Bandwidth Product (TBP)? The Time-Bandwidth Product (TBP) is a fundamental metric in signal processing that defines the relationship between a signal's duration ($\Delta t$) and its spectral width ($\Delta f$). It is the signal-processing equivalent of the Heisenberg Uncertainty Principle . $$TBP = B \times T$$ Where $B$ is the bandwidth and $T$ is the symbol duration (or pulse width). No signal can be simultaneously "t...

IIR Filter & Feedback Lab IIR Filter & Feedback Lab Exploring Recursive Systems: Where Output depends on Output 1 Input Signal $x[n]$ Signal Type Unit Impulse δ[n] Sine Wave Unit Step u[n] Frequency (for Sine) Notice: With an Impulse input, the output of an IIR filter can ring forever (Infinite response). 2 Feedback Coefficients Numerator $b$ (Feed-forward) Denominator $a$ (Feedback - $a_0$ is always 1) System Stable Try setting $a$ to 1.0, -1.1 to see an unstable system! ...

Ultimate FIR Filter Interactive Lab FIR Filter & LTI System Lab A visual workflow for understanding Discrete-Time Convolution 1 Define Input Signal $x[n]$ Signal Type Clean Sine Wave Sine Wave + White Noise Unit Impulse δ[n] Square Wave Frequency (Hz) This is your raw data. In an LTI system, we want to modify this signal's characteristics. 2 Define Filter $h[k]$ Filter Presets (Coefficients) Moving Average (Low Pass) Simple Differentiator (High Pass) Gaussian-like (Smooth LP) ...

  The Yule-Walker equations are fundamentally derived using the properties of a Wide-Sense Stationary (WSS) random process. Without the WSS assumption, the classical Yule-Walker equations cannot be obtained in their standard form. What is a Wide-Sense Stationary (WSS) Process? A random process \(X(t)\) is said to be Wide-Sense Stationary (WSS) if it satisfies three important conditions. 1. Constant Mean $$ E[X(t)] = \mu $$ where the mean \(\mu\) does not depend on time. 2. Constant Variance $$ Var(X(t))=\sigma^2 $$ The variance remains unchanged over time. 3. Autocovariance Depends Only on Lag Instead of depending on two different time instants, $$ C_X(t_1,t_2), $$ the covariance depends only on their difference, $$ C_X(\tau) = C_X(t_1-t_2). $$ Similarly, the autocorrelation function becomes $$ R_X(\tau) = E[X(t)X(t+\tau)]. $$ This property is the key assumption used in deriving the Yule-Walker equations. What are the Yule-Walker Equa...

Advanced LTI Research Simulator LTI System Analysis & Stochastic Signal Processing Simulation with Exact Spectral Estimation & Statistical Verification 1. Signal Source (x[n]) Define the input stochastic process or deterministic signal. Gaussian White Noise (WGN) Sine Wave (Normalized f=0.05) Unit Impulse δ[n] Analysis Window (N) 512 Samples 1024 Samples 2. System Impulse Response (h[n]) Enter coefficients separated by commas (FIR Filter). 0.1, 0.2, 0.4, 0.2, 0.1 3. Statistical Estimator Toggle ACF bias and variance behavior. Unbiased (1/(N-|k|)) Biased (1/N) ...

Effect of an LTI Filter on Signal Mean and Variance | Complete Guide Why Do Mean and Variance Change After Passing Through an LTI Filter? One of the most common questions in Digital Signal Processing (DSP) is: "Why does the mean or variance of a random signal change after passing through a Linear Time-Invariant (LTI) filter?" Understanding this concept is essential for topics such as random processes, communication systems, estimation theory, adaptive filtering, and noise analysis. This article explains the mathematics, intuition, and physical interpretation behind the change in signal statistics after LTI filtering. What is an LTI Filter? A Linear Time-Invariant (LTI) filter is completely characterized by its impulse response h[n] . If the input signal is x[n] then the output is obtained using convolution: y[n] = h[n] * x[n] Every output sample is a weighted combination of several input samples. Therefore, the st...

  + 3 V 1 Ω 10 Ω I 2 Ω 2 Ω 1 Ω + 3 V Answer: There is no closed return path. So, I = 0 GATE EC Previous Year Papers with Solutions → Electronics and Communication Study Material →

Question Consider a system of linear equations Ax = b, where A = [ 1   -√2   3 -1   √2   -3 ] b = [ 1 3 ] This system of equations admits Option Description (A) a unique solution for x (B) infinitely many solutions for x (C) no solutions for x (D) exactly two solutions for x Solution Step 1: Write the System The given matrix equation is A = [ 1   -√2   3 -1   √2   -3 ] b = [ 1 3 ] Therefore, the system of equations is x₁ − √2 x₂ + 3x₃ = 1 −x₁ + √2 x₂ − 3x₃ = 3 Step 2: Compare the Two Rows of A Observe that (-1, √2, -3) = -(1, -√2, 3). Thus, the second row of the coefficient matrix is exactly the negative of the first row. Therefore, rank(A) = 1. Step 3: Check Whether the System is Consistent If the second row is the negative of the first row, then the second entry of the vector b should also be the negative of the ...