MATH 412: Homework # 5 Tim Ahn July 22, 2016

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1 MATH 412: Homework # 5 Tim Ahn July 22, 2016 Supplementary Exercises Supp #23 Find the Fourier matrices F for = 2 and = 4. set.seed(2) = 2 omega = 2*pi/ F = exp(omega*outer(0:(-1),0:(-1))*(0+1i)) F ## [,1] [,2] ## [1,] 1+0i 1+0i ## [2,] 1+0i -1+0i = 4 omega = 2*pi/ F = exp(omega*outer(0:(-1),0:(-1))*(0+1i)) F ## [,1] [,2] [,3] [,4] ## [1,] 1+0i 1+0i 1+0i 1+0i ## [2,] 1+0i 0+1i -1+0i 0-1i ## [3,] 1+0i -1+0i 1-0i -1+0i ## [4,] 1+0i 0-1i -1+0i 0+1i Supp #24 Given the signal (x 0,..., x 3 ) = (1, 2, 0, 1) for length = 4. a) Write down the even and odd numbered components x, x. Even : x = (1, 0) Odd : x = (2, 1) b) Compute the discrete Fourier transforms c = F 2 x, c = F 2 x using matrix multiplications. 1

2 = 2 x1 = c(1,0) x2 = c(2,1) omega = 2*pi/ F = exp(omega*outer(0:(-1),0:(-1))*(0+1i)) Fstar = Conj(t(F)) c1 = F%*%x1 c2 = F%*%x2 c1 ## [,1] ## [1,] 1+0i ## [2,] 1+0i c2 ## [,1] ## [1,] 3+0i ## [2,] 1+0i c) Find the twiddle factors z j 4 = ( z 4 ) j and use them to compute c. c = F 2 x c = F 2 x ( c ) j = z j 4 (c ) j z j 4 = e j 4 2πi d) Compute the discrete Fourier transform of x from c and c. x 0 = c low1 + c low2 x 1 = c high1 + c high2 x 2 = c low1 c low2 x 3 = c low1 c high2 Supp #26 Make a random signal of length = 1024, compute its DFT and then compute its inverse DFT y with R. The imaginary part of y may be extracted with the command Im(y) and the real part may be extracted with the command Re(y). Verify that the imaginary part is very small in magnitude. Does it look like it is a completely random sequence of independent values? Use a plot. < x <- rnorm() # Random signal of length = 1024 c <- fft(x)/ # DFT 2

3 y <- fft(c*, inverse = T)/ z <- Im(y) plot(z, ylab = "Im(y)") # Inverse DFT # Imaginary part of y Im(y) 1.5e e e Index hist(z, breaks=30, xlab = "Im(y)", main = "Histogram of Im(y)") 3

4 Histogram of Im(y) Frequency e e e e 15 Im(y) The values shown in the two plots are extremely small, and it does appear to have a similar normal distribution as that of the random normal sample that the original vector was selected from. Supp #27 Download orbital data (eccentricity, precession, obliquity only) for the time window from 50 Myr before present to 48 Myr before present from the website and find the power spectra using R. Describe the main frequency components. Are there major differences between those data and the ones from -Myr to +1Myr? library(data.table) orbit <- read.table("c:/users/ahn/desktop/georgetown/2016 Summer/MATH 412/Homework/orbit.txt", quote="\" orbit2 <- read.table("c:/users/ahn/desktop/georgetown/2016 Summer/MATH 412/Homework/orbit2.txt", quote=" colnames(orbit) <- c("years", "eccentricity", "precession", "obliquity") colnames(orbit2) <- c("years", "eccentricity", "precession", "obliquity") # Eccentricity plot(eccentricity ~ years, orbit, type = 'l', xlab = "Years (Kyr)", main = "Eccentricity (-50Myr to -48M 4

5 eccentricity Eccentricity ( 50Myr to 48Myr) Years (Kyr) plot(eccentricity ~ years, orbit2, type = 'l', xlab = "Years (Kyr)", main = "Eccentricity (-Myr to +Myr) Eccentricity ( Myr to +Myr) eccentricity Years (Kyr) mylength = length(orbit$eccentricity) myvara = orbit$eccentricity myvar2 = orbit2$eccentricity myfreq1 = 2:50 plot(mylength/(myfreq1-1),abs(fft(myvara))[myfreq1], type = 'b', lwd = 2, xlab = "period (Kyr)", main = "Eccentricity from -50Myr to -48Mry") 5

6 abs(fft(myvara))[myfreq1] Eccentricity from 50Myr to 48Mry period (Kyr) plot(mylength/(myfreq1-1),abs(fft(myvar2))[myfreq1], type = 'b', lwd = 2, xlab = "period (Kyr)", main = "Eccentricity from -Myr to +Mry") abs(fft(myvar2))[myfreq1] Eccentricity from Myr to +Mry period (Kyr) # Precession myvarb = orbit$precession myvar2 = orbit2$precession myfreq2 = 10:200 plot(precession ~ years, orbit, type = 'l', xlab = "Years (Kyr)", main = "Precession (-50Myr to -48Myr)" 6

7 precession Precession ( 50Myr to 48Myr) Years (Kyr) plot(precession ~ years, orbit2, type = 'l', xlab = "Years (Kyr)", main = "Precession (-Myr to +Myr)") Precession ( Myr to +Myr) precession Years (Kyr) plot(mylength/(myfreq2-1),abs(fft(myvarb))[myfreq2], type = 'b', lwd = 2, xlab = "period (Kyr)", main = "Precession from -50Myr to -48Mry") 7

8 abs(fft(myvarb))[myfreq2] Precession from 50Myr to 48Mry period (Kyr) plot(mylength/(myfreq2-1),abs(fft(myvar2))[myfreq2], type = 'b', lwd = 2, xlab = "period (Kyr)", main = "Precession from -Myr to +Mry") abs(fft(myvar2))[myfreq2] Precession from Myr to +Mry period (Kyr) # Obliquity myvarc = orbit$obliquity myvar2 = orbit2$obliquity myfreq3 = 2:100 plot(obliquity ~ years, orbit, type = 'l', xlab = "Years (Kyr)", main = "Obliquity (-50Myr to -48Myr)") 8

9 Obliquity ( 50Myr to 48Myr) obliquity Years (Kyr) plot(obliquity ~ years, orbit2, type = 'l', xlab = "Years (Kyr)", main = "Obliquity (-Myr to +Myr)") Obliquity ( Myr to +Myr) obliquity Years (Kyr) plot(mylength/(myfreq3-1),abs(fft(myvarc))[myfreq3], type = 'b', lwd = 2, xlab = "period (Kyr)", main = "Obliquity from -50Myr to -48Mry") 9

10 abs(fft(myvarc))[myfreq3] Obliquity from 50Myr to 48Mry period (Kyr) plot(mylength/(myfreq3-1),abs(fft(myvar2))[myfreq3], type = 'b', lwd = 2, xlab = "period (Kyr)", main = "Obliquity from -Myr to +Mry") abs(fft(myvar2))[myfreq3] Obliquity from Myr to +Mry period (Kyr) # Power Spectra plot((myfreq1-1)/mylength,abs(fft(myvara))[myfreq1]^2, type = 'b', lwd = 2, xlab = "frequency (1/Kyr)", main = "Power Spectrum of the Earth's Eccentricity (-50Myr to -48Myr)") 10

11 abs(fft(myvara))[myfreq1]^ Power Spectrum of the Earth's Eccentricity ( 50Myr to 48Myr) frequency (1/Kyr) plot((myfreq2-1)/mylength,abs(fft(myvarb))[myfreq2]^2, type = 'b', lwd = 2, xlab = "frequency (1/Kyr)", main = "Power Spectrum of the Earth's Precession (-50Myr to -48Myr)") abs(fft(myvarb))[myfreq2]^ Power Spectrum of the Earth's Precession ( 50Myr to 48Myr) frequency (1/Kyr) plot((myfreq3-1)/mylength,abs(fft(myvarc))[myfreq3]^2, type = 'b', lwd = 2, xlab = "frequency (1/Kyr)", main = "Power Spectrum of the Earth's Obliquity (-50Myr to -48Myr)") 11

12 abs(fft(myvarc))[myfreq3]^ Power Spectrum of the Earth's Obliquity ( 50Myr to 48Myr) frequency (1/Kyr) There appears to be components for eccentricity at 100k years and also at around 400k years. For precession and obliquity, there are components at nearly 20k and 40k years, respectively. All these trends are reflected in both time periods and show no significant difference. With regard to frequency, peaks appear about every 100k for eccentricity, which indicates a frequency of about 20 during the time period for both periods. For precession, the peaks occur at intervals of slightly more than 20k years indicating a frequency of slightly more than 100. Obliquity peaks occur about at 40k years with a frequency of Chapter 11 Problem #3(i)(ii) Let x = (x 0,..., x ) T be a column vector, and let c = F x be its DFT. Also assume that = rm for some integers r and M with 1 < r <. Find the DFT of the following vectors in terms of c: (i) u = (u 0, u 1,..., u ) T = (x, x 2,..., x 0 ) T (time reversal) If we set u j = x j 1, (j = 0, 1, 2,...), then u = (x, x 2,..., x 1, x 0 ). l = k 1 k = l 1 12

13 (Fu) j = = = = u=0 l=0 l=0 = z j z jk u k = z ( l 1) j z j l=0 k=0 x l z lj zj x l z jk x l 1 z lj zj x l (since z j = 1) l=0 = z j c l z lj x l (ii) v = (v 0, v 1,..., v ) T = (x k, x k+1,..., x, x 0,..., x k 1 ) T for any k {1,..., 1} (shift) c = F x = z jk v = = =...? k=0 k=0 k=0 c z nk z jk x x znk x z k(n j) 13