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MMLU-Pro / 10612 / A train pulls out of the station at constant velocity. The received signal…

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question

A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as 1/i21/i^2. The total received signal at time ii is Yi=1iXi+ZiY_i = \frac{1}{i}X_i + Z_i where Z1,Z2,…Z_1, Z_2, \ldots are i.i.d. drawn from N(0,1)N(0,1). The transmitter constraint for block length nn is 1n∑i=1nxi2(w)≤2\frac{1}{n}\sum_{i=1}^n x_i^2(w) \leq 2 for w∈{1,2,…,2nR}w \in \{1,2,\ldots, 2^{nR}\}. Use Fano's inequality to find the capacity for this channel.
Plain-text mathematical notation (without MathML)
A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as 1/i². The total received signal at time i is Y_(i)=(1)/(i)X_(i)+Z_(i) where Z₁,Z₂,… are i.i.d. drawn from N(0,1). The transmitter constraint for block length n is (1)/(n)∑_(i=1)^(n)x_(i)²(w)≤2 for w∈{1,2,…,2^(nR)}. Use Fano's inequality to find the capacity for this channel.
Original LaTeX notation
A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as $1/i^2$. The total received signal at time $i$ is $Y_i = \frac{1}{i}X_i + Z_i$ where $Z_1, Z_2, \ldots$ are i.i.d. drawn from $N(0,1)$. The transmitter constraint for block length $n$ is $\frac{1}{n}\sum_{i=1}^n x_i^2(w) \leq 2  $ for $w \in \{1,2,\ldots, 2^{nR}\}$. Use Fano's inequality to find the capacity for this channel.

category

computer science

options

  1. 1.5
  2. 2.5
  3. 1.0
  4. 0.0
  5. 0.5
  6. 0.75
  7. 3.0
  8. 1.25
  9. 0.25
  10. 2.0

src

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Source and history

upstream_id / Original problem ID
10612
task_id
3e0558f7-bf31-56cc-a413-cfa365987825
task_key
default--test--10612
task_revision_id
1
config / split
default / test

Official source

initial import