andy
64276f270d
Affected files: .obsidian/graph.json .obsidian/workspace-mobile.json .obsidian/workspace.json STEM/AI/Literature.md STEM/AI/Neural Networks/MLP.md STEM/AI/Properties.md STEM/Quantum/Orbitals.md STEM/Quantum/Schrödinger.md STEM/Quantum/Wave Function.md STEM/Signal Proc/Convolution.md STEM/Signal Proc/Image/Image Processing.md STEM/img/hydrogen-electron-density.png STEM/img/hydrogen-wave-function.png STEM/img/orbitals-radius.png STEM/img/radial-equations.png STEM/img/radius-electron-density-wf.png STEM/img/wave-function-nodes.png STEM/img/wave-function-polar-segment.png STEM/img/wave-function-polar.png
30 lines
1.0 KiB
Markdown
30 lines
1.0 KiB
Markdown
Integral operator
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- Satisfies mathematical properties of integral operator
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- Product of two after one has been reversed and shifted
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$$x(t)=x_1(t)\circledast x_2(t)=\int_{-\infty}^\infty x_1(t-\tau)\cdot x_2(\tau)d\tau$$
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# Properties
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1. $x_1(t)\circledast x_2(t)=x_2(t)\circledast x_1(t)$
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- Commutativity
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2. $(x_1(t)\circledast x_2(t))\circledast x_3(t)=x_1(t)\circledast (x_2(t)\circledast x_3(t))$
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- Associativity
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3. $x_1(t)\circledast [x_2(t)+x_3(t)]=x_1(t)\circledast x_2(t)+ x_1(t)\circledast x_3(t)$
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- Distributivity
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4. $Ax_1(t)\circledast Bx_2(t)=AB[x_1(t)\circledast x_2(t)]$
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- Associativity with Scalar
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5. Symmetrical graph about origin
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# Applications
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1. Communications systems
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- Shift signal in frequency domain (Frequency modulation)
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2. System analysis
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- Find system output given input and transfer function
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# Polynomial Multiplication
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- Convolving coefficients of two poly gives coefficients of product
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# Discrete
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$$G[i,j]=H[u,v]\circledast F[i,j]$$
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$$G[i,j]=\sum^k_{u=-k}\sum^k_{v=-k} H[u,v]F[i-u,j-v]$$ |