# Trigonometric Series for Square Wave

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Note that this exercise is intended to be solved by hand, without the use of computation.

## 1) Square Wave

Consider the following signal, which is periodic with a fundamental period of T=1 second:

Let c_k and d_k be the Fourier series coefficients (trigonometric form) associated with this signal, such that

x(t)= c_0 + \sum_{k=1}^\infty \big(c_k\cos (k\omega_ot) + d_k\sin (k\omega_ot)\big)

What is the DC term, c_0?

c_0 =~

For k \geq 1, what is the value of c_k? Your answer may include the variable k, and you can use

`pi`

to refer to \pi.
c_k =~

For k \geq 1, what is the value of d_k? Your answer may include the variable k, and you can use

`pi`

to refer to \pi.
d_k =~

Which of the following are true?

c_k = 0 for all k | |

d_k = 0 if k is even | |

|d_k| decreases like k^2 | |

there are an infinite number of non-zero d_k |