Page 1 of 3

NAME (PRINT): ____________________________________________________

Homework #3: Spectral leakage and finite recording

Show your steps in answering the problems. No steps, no credit.

Due date: 5:30pm, 10/8/2020

In seismic exploration, one of the source types is a vibroseis. Imagine that you were driving the vibroseis truck and you set its vibrating frequency to be at 1Hz but you didn’t tell this to Bob and Alice. The “true” ground shaking (e.g., displacement) can be described by a simple formula

. You wanted to test whether Bob and Alice could figure out the vibroseis frequency using Fourier series spectral analysis. You told them to take a geophone and record the shaking signal and do the Fourier analysis and report their spectra back to you. Bob recorded the signal from t=0s and stopped at t=1.5s. Alice recorded the same signal and but her recording time was from t=0s to t=2s.

(1) [2pt] Plot Bob’s recorded signal from t=0s to t=1.5s.

(2) [2pt] Bob would perform Fourier series analysis to his signal. According to Fourier, Bob had to assume that his signal be periodic with a period T=1.5s. Plot Bob’s periodic signal a few more cycles, from t=-3s to t=3s. Of course, in Fourier’s mind, the periodic signal goes to both positive and negative infinity in time. We call Bob’s periodic signal .

s t( ) = sin 2π t( )

s t( )

sBob t( )

Geol-4381 Geophysical Signals & Analysis

Page 2 of 3

(3) [2pt] Since Bob’s periodic signal has a period of 1.5s, what are the discrete

angular frequencies in his Fourier series?

(4) [4pt] Help Bob compute , , , and ? Plot the spectrum

versus for n=1,2,3,4.

(5) [2pt] Plot Alice’s recorded signal from t=0s to t=2s.

(6) [2pt] Alice would perform Fourier series analysis to her signal. According to Fourier, Alice had to assume that her signal be periodic with a period T=2.0s. Plot

ω n

sBob t( ) = a0 + an cos ω nt( ) + bn sin ω nt( )⎡⎣ ⎤⎦ n=1

+∞

∑

a0 a1 a2 a3, a4 b1,b2,b3,b4

an 2 +bn

2 ω n

Geol-4381 Geophysical Signals & Analysis

Page 3 of 3

Alice’s periodic signal, for a few more cycles, from t=-4s to t=4s. We call Alice’s periodic signal .

(7) [2pt] Since Alice’s periodic signal has a period of 2s, what are the discrete angular frequencies for her Fourier series?

(8) [4pt] Help Alice calculate an (n=0,1,2 to 8) and bn (n=1,2 to 8)? Plot the spectrum

versus for n=1 to 8.

(9) [5pt] What is the frequency content in the true signal? You took Bob and Alice’s spectra and compare them. Does the Fourier series depend on the recording length T? What causes the difference?

END

sAlice t( )

ω n

sAlice t( ) = a0 + an cos ω nt( ) + bn sin ω nt( )⎡⎣ ⎤⎦ n=1

+∞

∑

an 2 +bn

2 ω n