The even impulse pair is the Fourier transform of 
,
![AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x)=1/2delta(x+1/2)+1/2delta(x-1/2).](/images/equations/ImpulsePair/NumberedEquation1.svg) |
(1)
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It satisfies
![AdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x)*f(x)=1/2f(x+1/2)+1/2f(x-1/2),](/images/equations/ImpulsePair/NumberedEquation2.svg) |
(2)
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where 
denotes convolution, and
![int_(-infty)^inftyAdjustmentBox[I, BoxMargins -> {{0.13913, -0.13913}, {-0.5, 0.5}}]I(x)dx=1.](/images/equations/ImpulsePair/NumberedEquation3.svg) |
(3)
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The odd impulse pair is the Fourier transform of 
,
![AdjustmentBox[I, BoxMargins -> {{0.101266, -0.101266}, {0.375, -0.375}}, BoxBaselineShift -> -0.375]AdjustmentBox[I, BoxMargins -> {{0, 0}, {-0.25, 0.25}}, BoxBaselineShift -> 0.25](x)=1/2delta(x+1/2)-1/2delta(x-1/2).](/images/equations/ImpulsePair/NumberedEquation4.svg) |
(4)
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Impulse Pair
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Cite this as:
Weisstein, Eric W. "Impulse Pair." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ImpulsePair.html