Differential Equations Decoded 44 Convolution the Multiplication Trick That Solves Linear Systems

Free to reuse. Free to remix. No attribution required. Make your own at https://www.patreon.com/cw/MadSciHub QUICK SUMMARY Convolution combines an impulse response h(t) with any input f(t) via (h * f)(t) = integral from 0 to t of h(t - tau) f(tau) dtau, and the Convolution Theorem says this integral becomes a plain product H(s) F(s) in the Laplace domain. One formula runs every linear circuit, every audio effect, every post-synaptic current in your brain, and the C in every CNN. KEY CONCEPTS 1. Convolution Integral - (h * f)(t) = integral from 0 to t of h(t - tau) f(tau) dtau. 2. Impulse Response h(t) - The system fingerprint, one function that fully characterizes any linear time invariant system. 3. Convolution Theorem - L of h star f equals H(s) times F(s), turning a time-domain integral into s-domain multiplication. 4. Transfer Function H(s) - Laplace transform of the impulse response. DEFINITIONS - Convolution: An operation on two functions defined by the integral from 0 to t of h(t - tau) f(tau) dtau. - Impulse Response: Output of a linear time invariant system when driven by a Dirac delta. - Transfer Function: The Laplace transform of the impulse response, denoted H(s). - Causal System: Output at time t depends only on inputs at times tau less than or equal to t. - LTI System: Output to a sum of inputs equals the sum of outputs; impulse response does not change over time. - Dummy Variable: Integration variable like tau that ranges over past times and vanishes when the integral is evaluated. HOW IT WORKS 1. Find the impulse response h(t) by solving the system response to a Dirac delta. 2. Identify the input f(t). 3. Write the convolution integral from 0 to t of h(t - tau) f(tau) dtau. 4. To dodge the integral, take Laplace transforms and form H(s) times F(s). 5. Multiply the two transforms in s-domain. 6. Inverse Laplace the product to get y(t). 7. Or evaluate the convolution integral directly when time-domain functions are friendly. 8. For inverse Laplace of products, pick convolution OR partial fractions based on which is faster. KEY ARGUMENTS 1. A clap in a stairwell produces a reverb tail - the impulse response. Any sound in that stairwell is the input convolved with the reverb tail. 2. Every input is secretly a continuous pile of weighted delta kicks, so the response is a pile of weighted impulse responses added up. 3. The convolution integral is the formal version of that pile of weighted, time-shifted impulse responses. 4. Causality forces the bounds to be 0 to t, not 0 to infinity - future kicks cannot be ringing now. 5. The Convolution Theorem emerges from factoring the Laplace kernel after substituting u equals t minus tau. 6. Convolution is NOT ordinary multiplication - writing L of h times f as H(s) times F(s) is the number one exam error on this chapter. 7. With the theorem, any linear ODE with arbitrary forcing has a one-line solution: y(t) equals the convolution of h with f. 8. You now have two tools for H(s) times F(s) - convolution or partial fractions - pick the faster one per problem. 9. Post-synaptic currents, convolutional neural networks, and reaction diffusion pattern formation all run the same integral. KEY TAKEAWAYS - Convolution is addition of weighted, time-shifted impulse responses - just bookkeeping for past kicks. - The star in h * f is an operator, not multiplication - respect it or lose 60 percent of the problem. - Laplace turns the integral into a product: time convolution equals s-domain multiplication. - Integration limits are always 0 to t for causal signals - never 0 to infinity. - One impulse response h(t) fully describes a linear time invariant system. - Choose convolution when both factors are simple table entries; choose partial fractions when denominators are hairy. MEMORY HOOKS - A stairwell clap plus a song equals every signal processing problem on Earth in one metaphor. - Time convolution versus s multiplication equals Roman versus Arabic numerals - representation does the work. - A hundred trillion synapses in your skull are computing a hundred trillion convolutions right now. SOURCE https://tutorial.math.lamar.edu/classes/de/ConvolutionIntegrals.aspx #DifferentialEquations #Convolution #LaplaceTransform #ConvolutionTheorem #ImpulseResponse #TransferFunction #SignalProcessing #StudyWithMe #ExamPrep #CollegeMath #EngineeringMath #Decoded #madscilecture #decoded #differentialequations #math #science