Virtual Labs

Consider a binary erasure channel with parameter

\epsilon=0.3

(denoted by

BEC(0.3)

). Suppose that a

n

-length vector,

\boldsymbol{x}\in\mathbb{F}_2^n

, is transmitted through this channel. Which of the following is correct?

a: An erasure at the first position increases the probability of getting an erasure in the second position. Explanation

Explanation

b: The event that

i

of the

n

bits are erased has higher probability than the event that

j

bits are erased, if

j>i

Explanation

c: Bits can be flipped or erased in

\boldsymbol{x}

. Explanation

Explanation

d: Exactly

0.3n

bits will be erased in

\boldsymbol{x}

. Explanation

Explanation

e: None Explanation

Explanation

Consider that we are transmitting some

n

-length bit vector

\boldsymbol{x}

through a binary symmetric channel with bit flip probability

p=0.9

. What is the probability that exactly the first half of the bits in

\boldsymbol{x}

are flipped?

0

Explanation

\binom{n}{n/2}0.9^{n/2}\times 0.1^{n/2}

Explanation

\binom{n}{n/2}0.09^n

Explanation

0.09^{n/2}

Explanation

Assume that the input to

BSC(0.3)

(0,1,0,1)

. Consider three possible output sequences (1)

(1,1,0,0)

, (2)

(0,1,1,1)

and (3)

(0,1,1,0)

. Which of the following is true?

a: (1) and (2) have the same probability of occurrence, but (3) has a higher probability than both Explanation

Explanation

b: (1) and (2) have the same probability of occurrence, but (3) has a lower probability than both Explanation

Explanation

c: (1) and (3) have the same probability of occurrence, but (2) has a higher probability than both Explanation

Explanation

d: All three are equally probable Explanation

Explanation

Consider that we are transmitting the sequence

\boldsymbol{x}=(+1,-1,-1)

through an AWGN channel with noise characteristics

\mathcal{N}(0,0.04)

(that is, zero-mean and variance =

0.04

). Identify which of the following output sequence is the most probable one:

\boldsymbol{y_1}=(1.3,-1.1,-1.3)

(b)

\boldsymbol{y_2}=(1.4,-1.2,-1.1)

\boldsymbol{y_1}

. Explanation

Explanation

\boldsymbol{y_2}

. Explanation

Explanation

\boldsymbol{y_1}

and

\boldsymbol{y_2}

are equally probable. Explanation

Explanation

d: We cannot answer this question. Explanation

Explanation

Binary-input Memoryless Channels