Geophysical Data Analysis

Problem set # 1

Due Thursday, Sept 14

 

1. Prove the orthogonality relations for sine and cosine functions used in deriving the real Fourier series.

 

2. Analytically compute the coefficients of the Fourier series of the step function:

f(x) = 1        0 < x < 1/2

f(x) = -1    -1/2 < x < 0

Why does the series contain only sine terms? Now plot the first 5 terms of this series and their sum using your favorite plotting program. Also plot the sum of the first 10 and 20 terms.

 

3. Show that the following are Fourier transform pairs:

a) f(t-a) and e-iwa F(w)

b) F(w-a) and eiat f(t)

c) df/dt and iwF(w)

 

4. Do problems 2-3, 2-6, and 2-9 and 3-1 in Lynn

 

5. Convolve two simple functions "by hand" . Assume the signals are boxcar functions of unit amplitude, one five seconds duration, the other three seconds duration, and the the signals are sampled once per second. Plot your result.