Other Transform Pairs

Monday, 25 March 2019

4:54 PM

ℱ﷐﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯﷯

Consider ﷐ℱ﷮−1﷯﷐𝛿﷐ω−﷐ω﷮0﷯﷯﷯

﷐ℱ﷮−1﷯﷐𝛿﷐ω−﷐ω﷮0﷯﷯﷯=﷐1﷮2𝜋﷯﷐−∞﷮∞﷮𝛿﷐ω−﷐ω﷮0﷯﷯﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔
=﷐1﷮2𝜋﷯﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯

ℱ﷐﷐𝑒﷮𝑗﷐𝜔﷮0﷯𝑡﷯﷯=2𝜋𝛿(𝜔−﷐ω﷮0﷯)

Recall
﷐cos﷮﷐ω﷮0﷯𝑡﷯=﷐﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯+﷐𝑒﷮−𝑗﷐ω﷮0﷯𝑡﷯﷮2﷯
﷐sin﷮﷐ω﷮0﷯𝑡﷯=﷐﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯−﷐𝑒﷮−𝑗﷐ω﷮0﷯𝑡﷯﷮2𝑗﷯

So, using the linearity property
ℱ﷐﷐cos﷮﷐ω﷮0﷯𝑡﷯﷯=﷐1﷮2﷯﷐ℱ﷐﷐𝑒﷮𝑗﷐𝜔﷮0﷯𝑡﷯﷯+ℱ﷐﷐𝑒﷮−𝑗﷐𝜔﷮0﷯𝑡﷯﷯﷯
=﷐1﷮2﷯﷐2𝜋𝛿﷐ω−﷐ω﷮0﷯﷯+2𝜋𝛿﷐ω+﷐ω﷮0﷯﷯﷯

Similarly, ℱ﷐﷐sin﷮﷐ω﷮0﷯𝑡﷯﷯=−𝑗𝜋﷐𝛿 ﷐𝜔−﷐ω﷮0﷯﷯−𝛿﷐𝜔+﷐𝜔﷮0﷯﷯﷯

Ink Drawings
Ink Drawings
Ink Drawings
Ink Drawings

Created with Microsoft OneNote 2016.