Fourier Analyses Time series Sampling interval Total period Question: How perturbations with different frequencies contribute to the turbulent kinetic energy?

Decomposition illustration

a. What is a Fourier Series? Fourier Transform a. What is a Fourier Series? Decompose a single using a series of sine and cos waves Time-Amplitude domain Frequency-Amplitude domain 3Hz 10Hz 50Hz

How to find frequencies of a signal?

(Euler’s formula)

Discrete Fourier Transform Observations: N Sampling interval: Period First harmonic frequency: nth harmonic frequency: All frequency: time at kth observation:

Time-Amplitude domain Frequency-Amplitude domain If time series F(k) is known, then, the coefficient c(n) can be found as: Forward Transform:

Example: Index (k): 0 1 2 3 4 5 6 7 Time (UTC): 1200 1215 1230 1245 1300 1315 1330 1345 Q(g/kg): 8 9 9 6 10 3 5 6 n 0 1 2 3 4 5 6 7 c(n) 7.0 0.28-1.03i 0.5 -0.78-0.03i 1.0 -0.78+0.03i 0.5 0.28+1.03i For frequencies greater than 4, the Fourier transform is just the complex Conjugate of the frequencies less than 4.

c(0) =7.0 c(1)=0.28-1.03i c(2)=0.5 c(3)=-0.78-0.03i c(4)=1.0 c(5)=-0.78+0.03i c(6)=0.5 c(7)=0.28+1.03i

Aliasing We have ten observations (10 samples) in a second and two different sinusoids that could have produced the samples. Red sinusoid has 9 cycles spanning 10 samples, so the frequency Blue sinusoid has 1 cycle spanning 10 samples, so the frequency Which one is right? Two-point rule Two data points are required per period to determine a wave. 2 observations: 1 wave 4 observations: 2 waves If sampling rate is , the highest wave frequency can be resolved is , which is called Nyquist frequency

Folding Folding occurs at Nyquist frequency. What problem does folding cause? What will cause aliasing or folding? The sensor can respond to frequencies higher than the rate that the sensor is sampled. The true signal has frequencies higher than the sampling rate.

Leakage

Fast Fourier Transform (FFT) FFT is nothing more than a discrete Fourier transform that has been restructured to take advantage of the binary computation processes of digital computer. As a result, everything is the same but faster! Relationship between decimal and binary numbers 0 1 2 3 4 5 6 7 8 9 10 0 1 10 11 100 101 110 111 1000 1001 1010 The decimal numbers n and k can be represented by If N=8, then, j=0, 1, 2, 3 7: binary 1 1 1; 5: binary 101; 3: binary 11

Energy Spectrum Note that n starts from 1, because the mean (n=0) does not contribute any information about the variation of the signal. For frequencies higher than Nyquist frequency, values are identically equal to those at the lower frequencies. They are folded back and added to the lower frequencies. Discrete spectral intensity (or energy)

Example Index (k): 0 1 2 3 4 5 6 7 Time (UTC): 1200 1215 1230 1245 1300 1315 1330 1345 Q(g/kg): 8 9 9 6 10 3 5 6