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MMLU-Pro / 10539 / Consider a periodic signal x(t) with period (T) equals to ten. Over one…

Problem

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question

Consider a periodic signal x(t)x(t) with period (T)(T) equals to ten. Over one period (i.e., −5≤t<5)-5 \leq t<5), it is defined as $$ x(t)=\left\{\begin{array}{cc} 2 & -5 \leq t<0 \\ -2 & 0 \leq t<5 \end{array}\right. $$ In Fourier series, the signal x(t)x(t) is written in the form of x(t)=∑k=−∞∞ckej2πktT x(t)=\sum_{k=-\infty}^{\infty} c_k e^{\frac{j 2 \pi k t}{T}} where the Fourier series coefficient ckc_k is obtained as, ck=1T∫−T2T2x(t)e−j2πktTdt c_k=\frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} x(t) e^{-\frac{j 2 \pi k t}{T}} d t Determine the value of c0c_0 (i.e., k=0)\left.k=0\right)
Plain-text mathematical notation (without MathML)
Consider a periodic signal x(t) with period (T) equals to ten. Over one period (i.e., −5≤t<5), it is defined as $$ x(t)=\left\{\begin{array}{cc} 2 & -5 \leq t<0 \\ -2 & 0 \leq t<5 \end{array}\right. $$ In Fourier series, the signal x(t) is written in the form of x(t)=∑_(k=−∞)^(∞)c_(k)e^((j2πkt)/(T)) where the Fourier series coefficient c_(k) is obtained as, c_(k)=(1)/(T)∫_(−(T)/(2))^((T)/(2))x(t)e^(−(j2πkt)/(T))dt Determine the value of c₀ (i.e., k=0)
Original LaTeX notation
Consider a periodic signal $x(t)$ with period $(T)$ equals to ten. Over one period (i.e., $-5 \leq t<5)$, it is defined as $$ x(t)=\left\{\begin{array}{cc} 2 & -5 \leq t<0 \\ -2 & 0 \leq t<5 \end{array}\right. $$ In Fourier series, the signal $x(t)$ is written in the form of $$ x(t)=\sum_{k=-\infty}^{\infty} c_k e^{\frac{j 2 \pi k t}{T}} $$ where the Fourier series coefficient $c_k$ is obtained as, $$ c_k=\frac{1}{T} \int_{-\frac{T}{2}}^{\frac{T}{2}} x(t) e^{-\frac{j 2 \pi k t}{T}} d t $$ Determine the value of $c_0$ (i.e., $\left.k=0\right)$

category

computer science

options

  1. 1
  2. 3
  3. 2
  4. -3
  5. 0
  6. -1
  7. 0.5
  8. 1.5
  9. -0.5
  10. -2

src

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Source and history

upstream_id / Original problem ID
10539
task_id
6ab92c62-4291-5911-94e3-4090ca011a9c
task_key
default--test--10539
task_revision_id
1
config / split
default / test

Official source

initial import