To derive the Yule–Walker equations, we multiply the AR model by $x[n-m]$ and take expectations: (m represents the lag (or time delay)) $$ E[x[n]x[n-m]] = \sum_{k=1}^{p} \phi_k E[x[n-k]x[n-m]] + E[e[n]x[n-m]] $$ Since $e[n]$ is white noise and uncorrelated with past samples: $$ E[e[n]x[n-m]] = 0 \quad \text{for } m \ge 1 $$ Using the definition of autocorrelation, we obtain: $$ R[m] = \sum_{k=1}^{p} \phi_k R[m-k], \quad m = 1,2,\dots,p $$ These equations can be written in matrix form as: $$ \underbrace{ \begin{bmatrix} R[0] & R[1] & \cdots & R[p-1] \\ R[1] & R[0] & \cdots & R[p-2] \\ \vdots & \vdots & \ddots & \vdots \\ R[p-1] & R[p-2] & \cdots & R[0] \end{bmatrix} }_{\text{Known autocorrelation matrix}} \; \underbrace{ \begin{bmatrix} \phi_1\\ \phi_2\\ \vdots\\ \phi_p \end{bmatrix} }_{\text{Unknown AR coefficients}} = \underbrace{ \begin{bmatrix} R[1]\\ R[2]\\ \vdots\\ R[p] \end{bmatrix} }_{\text{Kno...

Why does the matrix contain only \(R(n)\) intead of X(n)? The unknowns in the Yule–Walker equations are the AR coefficients \(\phi_1, \phi_2, \ldots, \phi_p\). To estimate these coefficients, we need equations that contain known quantities . The known quantities are the autocorrelation values \(R[0], R[1], \ldots, R[p]\), which are computed from the signal. Therefore, the Yule–Walker equations can be written as: $$ \underbrace{ \begin{bmatrix} R[0] & R[1] & \cdots & R[p-1] \\ R[1] & R[0] & \cdots & R[p-2] \\ \vdots & \vdots & \ddots & \vdots \\ R[p-1] & R[p-2] & \cdots & R[0] \end{bmatrix} }_{\text{Known autocorrelation matrix}} \; \underbrace{ \begin{bmatrix} \phi_1\\ \phi_2\\ \vdots\\ \phi_p \end{bmatrix} }_{\text{Unknown AR coefficients}} = \underbrace{ \begin{bmatrix} R[1]\\ R[2]\\ \vdots\\ R[p] \end{bmatrix} }_{\text{Known autocorrelation vector}} $$ The only unknowns are the coefficients \(\phi_1, \phi_2, \ldots...

WSS Simulator | Wide Sense Stationary & LTI System Analysis Wide Sense Stationary (WSS) Simulator Explore the statistical properties of random processes through LTI systems with real-time autocorrelation analysis. ⚙️ Signal Parameters Samples (100 - 5k) Input Distribution WSS (Gaussian) Noisy Sinusoid Uniform Noise Laplace Noise Binary Noise Pink Noise ...

The technique used here is generally called Similarity-Based Recommendation or Nearest-Neighbor Recommendation . The matrix itself is called a Similarity Matrix . The fact that it is stored as a Compressed Sparse Matrix is an implementation detail used to save memory and improve performance. 1. Similarity Matrix (Core ML Concept) A similarity matrix stores how similar every pair of movies is. Avatar Titanic Alien Avatar 1.00 0.42 0.87 Titanic 0.42 1.00 0.21 Alien 0.87 0.21 1.00 Each value represents the similarity between two movies. 1.0 = Identical movie (itself) 0.9 = Very similar 0.5 = Moderately similar 0.1 = Barely similar 0.0 = Not similar Similarity matrices are commonly used in: Content-Based Filtering Item-Based Collaborative Filtering k-Nearest Neighbors (k-NN) Recommendation Systems 2. Compressed Sparse Matrix (Storage Technique) A Sparse Ma...

QPSK Modulation Lab Digital Signal Processing Simulator System Parameters Input Bitstream (Dibits) Carrier Frequency ($f_c$) 5 Hz Symbol Rate ($R_s$) 1 Baud Signal Amplitude ($A$) 1.0 Quadrature Phase Shift Keying (QPSK) transmits 2 bits per symbol by shifting the phase of a carrier into one of four states: 45°, 135°, 225°, or 315°. $$ I_k, Q_k = \text{Map}\{b_{2k}, b_{2k+1}\} \in \left\{ \pm \frac{1}{\sqrt{2}} \right\} $$ The factor \( \tfrac{1}{\sqrt{2}}...