Home / Problem Set 2 / Highest Harmonic

Highest Harmonic

The questions below are due on Monday March 01, 2021; 10:00:00 PM.
 
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Note that this exercise is intended to be solved by hand, without the use of computation.

Consider Fourier series representations

f_i(t)=c_0+\sum_{k=1}^\infty c_k\cos(k\omega_it)+d_k\sin(k\omega_it)
for each of the following periodic functions of time, where \omega_i represents the fundamental frequency of the signal f_i(t).

\begin{aligned} f_1(t)&=\cos(3t)+\cos(5t)+\cos(9t)\\ f_2(t)&=1+\cos^2(9t)\\ f_3(t)&=\cos(17t)+\cos(18t)\\ f_4(t)&=\cos(16t)+\cos(18t) \end{aligned}
1 How many unique frequencies are contained in each of these signals? What are those frequencies (in radians per second)?

How many unique frequencies are needed to construct f_{1\dots4}\,? Do not count the k=0 term (constant offsets).
f_1
f_2
f_3
f_4

What are the radian frequencies (in rad/sec) present in f_1, assuming t is in seconds?
Enter your answer as a python list, e.g. [1, 2, 3].
What are the radian frequencies (in rad/sec) present in f_2, assuming t is in seconds?
Enter your answer as a python list, e.g. [1, 2, 3].
What are the radian frequencies (in rad/sec) present in f_3, assuming t is in seconds?
Enter your answer as a python list, e.g. [1, 2, 3].
What are the radian frequencies (in rad/sec) present in f_4, assuming t is in seconds?
Enter your answer as a python list, e.g. [1, 2, 3].

2 What is the fundamental period of each of these signals?

Enter an expression for the fundamental period of each signal. You can use \pi by typing pi.
f_1
f_2
f_3
f_4

3 For which values of k are c_k or d_k non-zero in each of these signals, when we find its Fourier Series over one fundamental period? (Ignore the c_0 term.)

What are the k values of the frequency components of f_1?
Enter your answer as a python list, e.g. [1, 2, 3].
What are the k values of the frequency components of f_2?
Enter your answer as a python list, e.g. [1, 2, 3].
What are the k values of the frequency components of f_3?
Enter your answer as a python list, e.g. [1, 2, 3].
What are the k values of the frequency components of f_4?
Enter your answer as a python list, e.g. [1, 2, 3].

4 What's the highest frequency present in each signal? What's the highest harmonic number?

Enter an expression for the highest frequency in each signal (in rad/sec).
f_1
f_2
f_3
f_4

Enter the highest harmonic (value of k) in each signal.
f_1
f_2
f_3
f_4