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Problem 2 - Information Theory Consider the following channel X, DI . a) Calculate entropy of...
3. The Nyquist formula informs us about the capacity in a noiseless channel and the Shannon's...
3. The Nyquist formula informs us about the capacity in a noiseless channel and the Shannon's formula informs us of the capacity of a noisy channel. In practice, we mostly encounter noisy channels so why do we still need the Nyquist formula? [2 marks] Nyquist formula: Bit rate 2 x bandwidth x log2 L Shannon formula: C-Bx log: (1+ SNR) a. Answer:
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] =...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] = m and Pr(X-1-1-m, and the error probability during the transmission from X and Y is p. 0 1-p Figure 1: A typical binary symmetric channel, where the input is X and the output is Y. a) Given that p-1/3 and m-3/4, find H(X), H (Y), H (YİX), and 1(X:Y). (8 marks) b) Still given p = 1 /3....
Information bits {0,1} are sent over binary symmetric communication channel with conditional probabilities P(YX) as shown...
Information bits {0,1} are sent over binary symmetric communication channel with conditional probabilities P(YX) as shown below. The priory probabilities of 0 and 1 are P(X=0)=0.3, P(X=1)=0.7. The error probability {=0.2. transmitter X 0 1-€ receiver Y 0 ៩ w 1-€ a) If 1 is transmitted, what are the probabilities of receiving 0 and 1? P(Y=0|X=1) and P(Y=1X=1) b) If 0 is received, what are the probabilities that 0 and 1 information bit is transmitted? P(X=0 Y=0) and P(X=1 Y=0)
1. Suppose log10(x) = 25. Calculate log2 (x) by calculating x and then calculating log2 (x)....
1. Suppose log10(x) = 25. Calculate log2 (x) by calculating x and then calculating log2 (x). Be sure to show the instructions you gave to R, as well as the value of x and the final result. 2. Find log2 (x), as in (1) without calculating x, by multiplying by something. 3. Suppose log2 (x) = 25. Calculate log10(x) for y = 25 by calculating x and then calculating log10(x). Be sure to show the instructions you gave to R,...
In information theory Shannon entropy is defined by H(x) = -Sum(P(x)*log(P(x)) where P is probability mass...
In information theory Shannon entropy is defined by H(x) = -Sum(P(x)*log(P(x)) where P is probability mass function of random variable x, and log to base 2. Given loaded die with P(6)=0.5 and P(1)=P(2)=P(3)=P(4)=P(5), compute entropy of observed rolls 654266. Note: To answer the question you do not need to know any more information about Shannon entropy.
tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y. tion? (2) Calculate E(X...
tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y.
tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y.
The following periodic signal is input to an ideal low pass filter of bandwidth 25 KHz. 1. x(t) 2...
The following periodic signal is input to an ideal low pass filter of bandwidth 25 KHz. 1. x(t) 2 a) Determine the average power of the signal x(t). b) If T 0.1 ms, give the output of the filter as a function of time, y(t) e) Determine the average power of the signal y(t) d) Determine the bandwidth of the signal y(), considered as a baseband signal. e) Now assume that the signal x() (with T-0.1 ms) is instead input...
1. Express the limit as a derivative and evaluate. 17 lim 16+h-2 lim 2. Calculate y. tan x 1 + co...
1. Express the limit as a derivative and evaluate. 17 lim 16+h-2 lim 2. Calculate y. tan x 1 + cos x y sin(cos x) y= sec(1 +x2) x cos y + sin 2y xy Use an Implicit Differentiation] 3. Find y" if x, y,6-1. [Use Implicit Differentiation] 4. Find an equation of the tangent to the curve at the given point. 121 12+ 1 [Use Implicit Differentiation] 4. Find the points on the ellipse x2 + tangent line has...
Question 1) Consider the following Cascade of two Binary symmetric channels (CBSC) with probabilities as indicated in the figure below 1. Find P(Y=1 / X=1 ), P(Y=0 / X=1) 2. Find P(Y=1 / X=0 ), P(Y=0...
Question 1)
Consider the following Cascade of two Binary symmetric
channels (CBSC) with probabilities as indicated in the
figure below
1. Find P(Y=1 / X=1 ), P(Y=0 / X=1)
2. Find P(Y=1 / X=0 ), P(Y=0 / X=0)
3. Find The Channel Matrix for
each BSC separately
4. Find The overall
Channel Matrix of the cascade channels
5. Assume that P1 = P2 =
Pe , Prove that the Channel Matrix is
M2
6. Use the assumptions and results in...
Problem 1 A sinusodial signal x(t)- sin2t (t in seconds) is input to a system with frequency resp...
Problem 1 A sinusodial signal x(t)- sin2t (t in seconds) is input to a system with frequency response: H(G What signal y(t) is observed at the output? Problem 2 The inverse Fourier transform of a system frequency response is given by h(t)t. The signal x(t) 3 cos(4t 0.5) is input to the system (t in seconds). (a) What is the expression of the signal y(t) at the system output? (b) What is the power attenuation in dB caused by the...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] = m and Pr(X-1-1-m, and the error probability during the transmission from X and Y is p. 0 1-p Figure 1: A typical binary symmetric channel, where the input is X and the output is Y. a) Given that p-1/3 and m-3/4, find H(X), H (Y), H (YİX), and 1(X:Y). (8 marks) b) Still given p = 1 /3....
Information bits {0,1} are sent over binary symmetric communication channel with conditional probabilities P(YX) as shown below. The priory probabilities of 0 and 1 are P(X=0)=0.3, P(X=1)=0.7. The error probability {=0.2. transmitter X 0 1-€ receiver Y 0 ៩ w 1-€ a) If 1 is transmitted, what are the probabilities of receiving 0 and 1? P(Y=0|X=1) and P(Y=1X=1) b) If 0 is received, what are the probabilities that 0 and 1 information bit is transmitted? P(X=0 Y=0) and P(X=1 Y=0)
tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y.
tion? (2) Calculate E(X), E(X2), and Var(X). (3) Calculate F(a) P(X s a) for a (0, 1]. (4) Let Y =-log X. Calculate F(y)-P(Y v) for u 20. Calculate the density of Y.
The following periodic signal is input to an ideal low pass filter of bandwidth 25 KHz. 1. x(t) 2 a) Determine the average power of the signal x(t). b) If T 0.1 ms, give the output of the filter as a function of time, y(t) e) Determine the average power of the signal y(t) d) Determine the bandwidth of the signal y(), considered as a baseband signal. e) Now assume that the signal x() (with T-0.1 ms) is instead input...
1. Express the limit as a derivative and evaluate. 17 lim 16+h-2 lim 2. Calculate y. tan x 1 + cos x y sin(cos x) y= sec(1 +x2) x cos y + sin 2y xy Use an Implicit Differentiation] 3. Find y" if x, y,6-1. [Use Implicit Differentiation] 4. Find an equation of the tangent to the curve at the given point. 121 12+ 1 [Use Implicit Differentiation] 4. Find the points on the ellipse x2 + tangent line has...
Question 1)
Consider the following Cascade of two Binary symmetric
channels (CBSC) with probabilities as indicated in the
figure below
1. Find P(Y=1 / X=1 ), P(Y=0 / X=1)
2. Find P(Y=1 / X=0 ), P(Y=0 / X=0)
3. Find The Channel Matrix for
each BSC separately
4. Find The overall
Channel Matrix of the cascade channels
5. Assume that P1 = P2 =
Pe , Prove that the Channel Matrix is
M2
6. Use the assumptions and results in...
Problem 1 A sinusodial signal x(t)- sin2t (t in seconds) is input to a system with frequency response: H(G What signal y(t) is observed at the output? Problem 2 The inverse Fourier transform of a system frequency response is given by h(t)t. The signal x(t) 3 cos(4t 0.5) is input to the system (t in seconds). (a) What is the expression of the signal y(t) at the system output? (b) What is the power attenuation in dB caused by the...
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