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Foundation of Digital Communications Examples of typical exam-like exercises Comments The exam exercises will cover the following three main areas of the course Probability theory and random variables Deterministic signals, Fourier transforms, linear systems, power and energy Random processes The following is a collection of typical exercises, divided on the three main topics Exercises on Probability theory and random variables Exercise 1 A random variable f ( x) has the following probability density function: 5 5 x e 2 1. Draw a qualitative plot of the probability density function f (x ) 2. Evaluate the mean of the random variable E[ ] 3. Evaluate the probability P ( 5) 4. Evaluate the probability P [5,5] Exercise 2 A random variable f ( x) has the following probability density function: 5 5 x e 2 The random variable is sent through a nonlinear device, and the output is the random variable relation of the device is the following: . The input-output if 0 then 1 if 0 then 1 1. Evaluate the probability P ( 1) and P ( 1) 2. 3. Evaluate the cumulative distribution function of the random variable Evaluate the probability density function of the random variable Exercise 3 is a gaussian random variable with mean equal to 2 and variance equal to 1. 1. Write the analytical expression for the probability P ( 6) 2. Estimate the numerical value of the probability P ( 6) Note: it is possible to use the following numerical approximation for the erfc() function erfc( x) e x 2 x 1 The random variable is sent to a nonlinear device, generating at its output the random variable 2 . 1. Evaluate the mean E[ ] 2. Evaluate the variance 2 Exercise 4 Evaluate the exact error probability of the cascade of two binary symmetric channels (BSC). The first BSC has 4 has a transition probability p 10 , while the second has a transition probability be assumed to be statistically independent. p 106 . The two BSC can Exercises on deterministic signal Exercise 1 Consider the following deterministic signal x(t ) x(t ) 2A A T 0 1. 2. 3. 4. 2 T t Evaluate its Fourier transform Evaluate its energy and its average power Classify the signal (finite energy or finite power) Evaluate the corresponding spectral density (energy or power, considering that one of the two is meaningless) Exercise 2 Consider the same deterministic signal x(t ) introduced in the previous exercise. This signal is sent through a linear and time invariant filter with transfer function: A for t [0, T ] h(t ) 0 outside 1. Evaluate and draw the output y (t ) of the filter 2. Evaluate the spectral density (energy or power, considering that one of the two is meaningless) of y (t ) Suggestion: try to use the properties of convolution and linear systems Exercise 3 Evaluate the convolution of the following two signals: x(t ) e 4t h(t ) A for t 0,4 and zero outside Exercise 4 Let’s consider the signal y(t ) x(t ) cos(2f 0t ) , where the Fourier transform of x(t ) is equal to 1 in the interval [-B,+B] and zero outside. The frequency f 0 is much greater than B. z (t ) A y 2 (t ) B . The signal z(t) is then sent to an ideal low pass filter with bandwidth equal to 3 B, generating the signal g (t ) . The signal y(t) is sent to a nonlinear device which generates the output 2 1. Estimate the spectrum of g (t ) . 2. Write the time-domain expression of the signal g (t ) as a function of x(t ) Exercises on random processes Exercise 1 Consider the following random process: y (t ) a0 a1 x(t ) cos 2 2f 0t where: a0 , a1 , f 0 are numeric constants; x(t ) is a WSS and ergodic random process with zero mean and autocorrelation Rx ( ) ; is a random variable uniformly distributed in the interval [0,2 ] and statistically independent from x(t ) . 1. Evaluate the mean of y (t ) 2. 3. 4. Evaluate the autocorrelation of y (t ) Determine if the process is ergodic for its mean Evaluate the power spectral density of y (t ) Exercise 2 Consider a digital transmission signal, expressed as: y (t ) i r (t iT i i B ) where: i are statistically independent random variables that can assume the two values A with equal probability r (t ) is a deterministic function, shown below r (t ) A t TB TB 1. Evaluate the mean of y (t ) 2. Evaluate the power spectral density of y (t ) is a random variable uniformly distributed in the interval [0,2 ] and statistically independent from i . Exercise 3 We have a LTI filter characterized by the following transfer function. H( f ) 2 1 f f1 f1 3 f1 of the filter is equal to 10 GHz. The input of the system is a white gaussian noise with N0 12 power spectral density equal to , where N 0 10 [W/Hz]. 2 The cut-off frequency Evaluate: 1. The average power of 2. nout (t ) in Watt; Evaluate the analytical and numerical value of the probability Pnout (t ) A , where A 2 0.4 Watt. Note: it is possible to use the following numerical approximation for the erfc() function erfc( x) e x 2 x 4