# Highest Harmonic

If you are a current student, please Log In for full access to the web site.

Note that this link will take you to an external site (

`https://shimmer.csail.mit.edu`) to authenticate, and then you will be redirected back to this page.

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.

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.

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.

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.

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.

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.

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.

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.

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 |