Andy Pack
f29c435494
Affected files: .obsidian/graph.json .obsidian/workspace.json Gaming/Steam controllers.md Gaming/Ubisoft.md STEM/Signal Proc/Convolution.md STEM/Signal Proc/Fourier Transform.md STEM/Signal Proc/Pole-Zero.md STEM/Signal Proc/System Classes.md STEM/Signal Proc/Transfer Function.md STEM/Speech/Linguistics/Consonants.md STEM/Speech/Linguistics/Linguistics.md STEM/Speech/Linguistics/Terms.md STEM/Speech/Linguistics/Vowels.md STEM/Speech/Literature.md STEM/Speech/NLP/Jargon.md STEM/Speech/NLP/NLP.md STEM/Speech/NLP/Recognition.md STEM/Speech/Perception/Perception.md STEM/Speech/Speech Processing/Applications.md STEM/Speech/Speech Processing/Source-Filter.md STEM/Speech/Speech Processing/Vocal Tract.md Work/Applications/Anthropic/Cover letter.md Work/Applications/Anthropic/In line with values.md Work/Applications/Anthropic/Why Work.md Work/Companies.md Work/Freelancing.md Work/Products.md Work/Tech.md
44 lines
1.4 KiB
Markdown
44 lines
1.4 KiB
Markdown
---
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tags:
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- signals
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- maths
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---
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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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6. $y(t)=x_1(t-a)\circledast x_2(t-b)$
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- $x(t)=x_1(t)\circledast x_2(t)$
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- $y(t)=x(t-a-b)$
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7. $x(t)=x_1(t)\circledast x_2(t)$
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- $x_1$ between $a_1$ and $b_1$
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- $x_2$ between $a_2$ and $b_2$
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- Starting point of $x(t)=a_1+a_2$
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- Ending point of $x(t)=b_1+b_2$
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8. $\overline{x \circledast y}=\bar x \circledast \bar y$
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9. $(x \circledast y)'=x'\circledast y=x\circledast y'$
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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](Transfer%20Function.md)
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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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$$G[i,j]=\sum^k_{u=-k}\sum^k_{v=-k} H[u,v]F[i-u,j-v]$$ |