University:
California State University, Northridge
Course:
PHYS 100A | General Physics I
Academic year:

2020
Views:

462

Pages:

1
Author:

isaphumanb6

Table of Fourier Transforms 1 ∞ 𝑓(𝑡) = ∫ 𝐹(𝑗𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔 2𝜋 −∞ ∞ 𝐹(𝑗𝜔) = ∫ 𝑓(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡 −∞ 1. Transformation f(t) <-> F(jω) 𝑎1 𝑓1 (𝑡) + 𝑎2 𝑓2 (𝑡) <-> 𝑎1 𝐹1 (𝑗𝜔) + 𝑎2 𝐹2 (𝑗𝜔) 2. Linearity 3. Symmetry 𝐹(𝑗𝑡) <-> 2𝜋𝑓(−𝜔) 4. Scaling 𝑓(𝑎𝑡) <-> (|𝑎|) 𝐹 ( ) 1 𝑗𝜔 𝑎 𝑓(𝑡 − 𝑎) <-> 𝑒 −𝑗𝜔𝑎 𝐹(𝑗𝜔) 5. Delay 6. Modulation 𝑒 𝑗𝑎𝑡 𝑓(𝑡) <-> 𝐹(𝑗𝜔 − 𝑗𝑎) 7. Convolution 𝑓1 (𝑡) ∗ 𝑓2 (𝑡) <-> 𝐹1 (𝑗𝜔)𝐹2 (𝑗𝜔) 8. Multiplication 𝑓1 (𝑡)𝑓2 (𝑡) <-> (2𝜋) 𝐹1 (𝑗𝜔) ∗ 𝐹2 (𝑗𝜔) 1 𝑓 (𝑛) (𝑡) <-> (𝑗𝜔)𝑛 𝐹(𝑗𝜔) 9. Time Differentiation 1, |𝑡| < 10. 𝑓(𝑡) = { 𝑇 2 𝜔𝑇 ) 2 } <-> 𝑇 𝑠𝑖𝑛𝑐 ( 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝐴 (1 − 11. 𝑓(𝑡) = { |𝑡| 𝑇 ) , |𝑡| < 𝑇 } <-> 𝐴𝑇 𝑠𝑖𝑛𝑐 2 ( 𝜔𝑇 ) 2 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 12. 𝑘𝛿(𝑡) <-> 𝑘 13. 𝑘 <-> 2𝜋𝑘𝛿(𝑡𝜔) 14. 𝑐𝑜𝑠(𝜔0 𝑡) <-> 𝜋[𝛿(𝜔 − 𝜔0 ) + 𝛿(𝜔 + 𝜔0 )] 15. 𝑠𝑖𝑛(𝜔0 𝑡) <-> 𝑗𝜋[𝛿(𝜔 + 𝜔0 ) − 𝛿(𝜔 − 𝜔0 )] ∞ 16. ∑∞ 𝑘=−∞ 𝛿(𝑡 − 𝑘𝑇) <-> 𝜔0 ∑𝑘=−∞ 𝛿(𝜔 − 𝑘𝜔0 ) ; 𝜔0 = 2𝜋 𝑇 𝑗𝑘𝜔0 𝑡 17. ∑∞ <-> 2𝜋 ∑∞ 𝑘=−∞ 𝐹𝑘 𝑒 𝑘=−∞ 𝐹𝑘 𝛿(𝜔 − 𝑘𝜔0 ) 2 Power 𝑃𝑎𝑣 = |𝐹0 |2 + 2 ∑∞ 𝑘=1|𝐹𝑘 | 1 ∞ ∞ Energy = 2𝜋 ∫−∞|𝐹(𝑗𝜔)|2 𝑑𝜔 = ∫−∞ 𝑓 2 (𝑡)𝑑𝑡 1 𝑇 Exponential Fourier Series: 𝐹𝑛 = 𝑇 ∫0 𝑓(𝑡) 𝑒 −𝑗𝑛𝜔0 𝑡 𝑑𝑡