/en/tags/dsp/Recent content in DSP on Mi&Bee BlogHugo -- gohugo.ioen蓝宝石的傻话Thu, 14 May 2026 10:00:00 +0800/en/posts/physical-world/acoustics-dsp-basics/Mon, 04 May 2026 10:00:00 +0800/en/posts/physical-world/acoustics-dsp-basics/<h2 id="destructive-interference">Destructive Interference</h2> <p>Active noise cancellation (ANC) builds on a simple physical principle: destructive interference. Two sound waves of the same frequency and opposite phase cancel each other out.</p> <p>The original noise signal:</p> <p>$$ p_n(t) = A \cos(2\pi ft + \phi) $$</p> <p>The anti-noise generated by the ANC system:</p> <p>$$ p_c(t) = -A \cos(2\pi ft + \phi) = A \cos(2\pi ft + \phi + \pi) $$</p> <p>Superposition:</p> <p>$$ p_{\text{total}} = p_n(t) + p_c(t) = 0 $$</p>/en/posts/physical-world/adaptive-filtering-lms-nlms/Thu, 07 May 2026 10:00:00 +0800/en/posts/physical-world/adaptive-filtering-lms-nlms/<h2 id="the-lms-algorithm">The LMS Algorithm</h2> <p>The core problem in adaptive filtering is: given a reference signal <code>x(n)</code> and a desired signal <code>d(n)</code>, find a filter coefficient vector <code>w</code> such that the output <code>y(n)</code> approximates <code>d(n)</code>.</p> <p>The Least Mean Square (LMS) algorithm is the most classic solution to this problem, proposed by Widrow and Hoff in 1960. The core idea is to take one step down the steepest gradient of the error surface at each iteration — stochastic gradient descent.</p>/en/posts/physical-world/variable-step-frequency-domain/Thu, 14 May 2026 10:00:00 +0800/en/posts/physical-world/variable-step-frequency-domain/<h2 id="the-convergence-trade-off-of-fixed-step-size">The Convergence Trade-off of Fixed Step Size</h2> <p>Standard LMS and NLMS algorithms use a fixed step size parameter $\mu$. The choice of step size directly impacts algorithm performance, but a fundamental contradiction exists:</p> <ul> <li><strong>Large step size</strong>: Fast convergence and quick adaptation to environmental changes, but large steady-state error and low filtering precision</li> <li><strong>Small step size</strong>: Small steady-state error and high filtering precision, but slow convergence and sluggish response to abrupt changes</li> </ul> <p>This contradiction is particularly pronounced in applications such as echo cancellation and active noise control — the system needs fast convergence during startup, yet wishes to maintain low error in steady state. A fixed step size cannot satisfy both phases simultaneously.</p>