1. Lecture 35 Wave spectrum Fourier series Fourier analysis
Fourier transformation

2. Fourier series
Any periodic function can be decomposed into the sum of a set of simple oscillation functions. 𝑓 𝑡 = 𝑎 0 + 𝑛=1 ∞ [ 𝑎 𝑛 cos 𝑛𝜔𝑡 + 𝑏 𝑛 sin 𝑛𝜔𝑡 ] Where 𝜔=2𝜋/𝑇, 𝑎′𝑠 𝑎𝑛𝑑 𝑏 ′ 𝑠 are numerical constants which tell us how much of each component oscillation is present. Eg. cos 𝜔𝑡+𝜙 = cos 𝜙 cos 𝜔𝑡 + sin 𝜙 sin 𝜔𝑡

3. Square wave
𝑓 𝑡 = 4 sin 𝜃 𝜋 + 4 sin 3𝜃 3𝜋 + 4 sin 5𝜃 5𝜋 + 4 sin 7𝜃 7𝜋 +…

5. Sawtooth wave
𝑓 𝑡 =− 2 sin 𝜃 𝜋 + 2 sin 2𝜃 2𝜋 − 2 sin 3𝜃 3𝜋 + 2 sin 4𝜃 4𝜋 +…

6. 𝑓 𝑡 = 𝑎 0 + 𝑛=1 ∞ [ 𝑎 𝑛 cos 𝑛𝜔𝑡 + 𝑏 𝑛 sin 𝑛𝜔𝑡 ] For given 𝑓(𝑡), how can we find out what amount of each harmonic is required?

7. Complete orthonormal set
Sines and cosines form an orthonormal set, for 𝑚≠𝑛 0 𝑇 sin 𝑚𝜔𝑡 sin (𝑛𝜔𝑡) 𝑑𝑡 = 1 𝜔 0 2𝜋 sin 𝑚𝜔𝑡 sin 𝑛𝜔𝑡 𝑑(𝜔𝑡) =0 0 𝑇 sin 𝑚𝜔𝑡 cos (𝑛𝜔𝑡) 𝑑𝑡 =0 0 𝑇 cos 𝑚𝜔𝑡 cos (𝑛𝜔𝑡) 𝑑𝑡 =0 And 0 𝑇 sin 2 𝑚𝜔𝑡 𝑑𝑡 = 1 𝜔 0 2𝜋 sin 2 𝑚𝜔𝑡 𝑑(𝜔𝑡) = 𝑇 2 0 𝑇 cos 2 𝑚𝜔𝑡 𝑑𝑡 = 𝑇 2

8. 𝑓 𝑡 = 𝑎 0 + 𝑛=1 ∞ [ 𝑎 𝑛 cos 𝑛𝜔𝑡 + 𝑏 𝑛 sin 𝑛𝜔𝑡 ] 𝑎 0 = 1 𝑇 0 𝑇 𝑓 𝑡 𝑑𝑡 𝑎 𝑛 = 2 𝑇 0 𝑇 𝑓 𝑡 cos (𝑛𝜔𝑡) 𝑑𝑡 𝑏 𝑛 = 2 𝑇 0 𝑇 𝑓 𝑡 sin (𝑛𝜔𝑡) 𝑑𝑡

9. Exponential expression of Fourier series
𝑓 𝑡 = 𝑎 0 + 𝑛=1 ∞ [ 𝑎 𝑛 cos 𝑛𝜔𝑡 + 𝑏 𝑛 sin 𝑛𝜔𝑡 ] 𝑎 𝑛 ±𝑖 𝑏 𝑛 cos 𝑛𝜔𝑡 ±𝑖 sin 𝑛𝜔𝑡 = 𝑎 𝑛 cos 𝑛𝜔𝑡 + 𝑏 𝑛 sin 𝑛𝜔𝑡 ±𝑖 𝑎 𝑛 sin⁡(𝑛𝜔𝑡)∓𝑖 𝑏 𝑛 cos⁡(𝑛𝜔𝑡) 𝑓 𝑡 = 𝑛=−∞ ∞ 𝑐 𝑛 𝑒 𝑖𝑛𝜔𝑡 𝑐 𝑛 = 1 2 𝑎 𝑛 −𝑖 𝑏 𝑛 , 𝑓𝑜𝑟 𝑛>0 𝑎 0 , 𝑓𝑜𝑟 𝑛=0 1 2 𝑎 𝑛 +𝑖 𝑏 𝑛 , 𝑓𝑜𝑟 𝑛<0

10. Complete orthonormal set
0 𝑇 𝑒 𝑖𝑛𝜔𝑡 𝑒 −𝑖𝑚𝜔𝑡 𝑑𝑡 = 0 𝑇 𝑒 𝑖 𝑛−𝑚 𝜔𝑡 𝑑𝑡 = 0, 𝑓𝑜𝑟 𝑛≠𝑚 𝑇, 𝑓𝑜𝑟 𝑛=𝑚
𝑓 𝑡 = 𝑛=−∞ ∞ 𝑐 𝑛 𝑒 𝑖𝑛𝜔𝑡
𝑐 𝑛 = 1 T 0 𝑇 𝑓 𝑡 𝑒 −𝑖𝑛𝜔𝑡 𝑑𝑡

11. Energy theorem
The energy in a wave is proportional to the square of its amplitude.
0 𝑇 𝑓 2 𝑡 dt= 0 𝑇 𝑎 0 + 𝑛=1 ∞ 𝑎 𝑛 cos 𝑛𝜔𝑡 + 𝑏 𝑛 sin 𝑛𝜔𝑡 𝑑𝑡
=𝑇 𝑎 𝑇 2 𝑛=1 ∞ ( 𝑎 𝑏 0 2 )
The total energy in a wave is the sum of the energies in all of the Fourier components.

12. Fourier transform
𝑓 𝑡 = 𝑛=−∞ ∞ 𝑐 𝑛 𝑒 𝑖𝑛𝜔𝑡 , 𝑐 𝑛 = 1 T 0 𝑇 𝑓 𝑡 𝑒 −𝑖𝑛𝜔𝑡 𝑑𝑡 If 𝑇→∞, 𝜔→0 𝑓 𝑡 = −∞ ∞ 𝑐 𝜂 𝑒 2𝜋𝑖𝜂𝑡 𝑑𝜂 𝑐 𝜂 = −∞ ∞ 𝑓 𝑡 𝑒 −2𝜋𝑖𝜂𝑡 𝑑𝑡 c(𝜂) is called the Fourier transform of 𝑓(𝑡). The Fourier transform expresses a function of time (or signal) in terms of the amplitude (and phase) of each of the frequencies that make it up.

15. Example of noise reduction using FT
Input image
Spectrum
Band-pass
filter
Output image

16. Gaussian wave packet Gaussian distribution
𝑓 𝑥 = 1 𝜎 2𝜋 𝑒 − 𝑥−𝜇 𝜎 2
A Gaussian wave packet
𝜓 𝑥,𝑡 = −∞ ∞ 1 𝜎 𝑘 2𝜋 𝑒 − 𝑘− 𝑘 𝜎 𝑘 2 𝑒 𝑖 𝑘𝑥−𝜔𝑡 𝑑𝑘

17. Fourier transform of Gaussian is a Gaussian
For simplicity, consider the Fourier transformation of a Gaussian function
𝜓 𝑥 =𝐴 𝑒 − 𝑥 2 2 (Δ𝑥) 2
Centered at 𝑥=0, with a width ∆𝑥
𝜓 𝑥 = −∞ ∞ 𝜓(𝑘) 𝑒 𝑖𝑘𝑥 𝑑𝑘
𝜓 𝑘 = −∞ ∞ 𝜓 𝑥 𝑒 𝑖𝑘𝑥 𝑑𝑥 ∝ 𝑒 − 𝑘 2 Δ𝑥 =𝑒 − 𝑘 2 2 (Δ𝑘) 2
Gaussian ←→ Gaussian

18. Uncertainty principle
In quantum mechanics, the momentum is
𝑝=ℏ𝑘
For Gaussian wave packet
Δ𝑘=1/Δ𝑥
Therefore
Δ𝑝=ℏ/Δ𝑥 𝑜𝑟 Δ𝑥Δ𝑝=ℏ

19. What is the Square Kilometre Array (SKA)
Next Generation radio telescope – compared to best current instruments it is ...
~100 times sensitivity
~ 106 times faster imaging the sky
More than 5 square km of collecting area on sizes 3000km
E-MERLIN
eVLA 27 27m dishes
Longest baseline 30km
GMRT 30 45m dishes
Longest baseline 35 km