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- Question : 1P - 2.1 Prove the following properties of Hilbert transforms: a. If x(t) = x(?t), then x
- Question : 2P - 2.2 Let x(t) and y(t) denote two bandpass signals, and let xl(t) and yl(t) denote their lowpass equivalents with respect to some frequency f0. We know that in general xl(t) and yl(t) are complex signals. 1. Show that ? ?? x(t)y(t) dt = 1 2 Re" ?? ? xl(t)yl?(t) dt# 2. From this conclude that Ex = 1 2Exl , i.e., the energy in a bandpass signal is one-half the energy in its lowpass equivalent.
- Question : 3P - 2.3 Suppose that s(t) is either a real- or complex-valued signal that is represented as a linear combination of orthonormal functions { fn(t)}, i.e., s
- Question : 4P - 2.4 Suppose that a set of M signal waveforms {slm(t)} is complex-valued. Derive the equations for the Gram-Schmidt procedure that will result in a set of N ? M orthonormal signal waveforms.
- Question : 5P - 2.5 Carry out the Gram-Schmidt orthogonalization of the signals in Figure 2.2
- Question : 6P - 2.6 Assuming that the set of signals {?nl(t), n = 1, . . . , N} is an orthonormal basis for rep resentation of {sml(t), m = 1, . . . , M}, show that the set of functions given by Equa tion 2.2
- Question : 7P - 2.7 Show that ? (t) = ?? (t) where ? (t) denotes the Hilbert transform and ? and ? are given by Equation 2.2
- Question : 8P - 2.8 Determine the correlation coefficients ?km among the four signal waveforms {si(t)} shown in Figure 2.2
- Question : 9P - 2.9 Prove that sl(t) is generally a complex-valued signal, and give the condition under which it is real. Assume that s(t) is a real-valued bandpass signal
- Question : 10P - 2.10 Consider the three waveforms fn(t) shown in Figure P2.10.a. Show that these waveforms are orthonormal. b. Express the waveform x(t) as a linear combination of fn(t), n = 1, 2, 3, if x(t) = ??? ?1 0 ? t < 1 1 1 ? t < 3 ?1 3 ? t < 4 and determine the weighting coefficients.
- Question : 11P - 2.11 Consider the four waveforms shown in Figure P2.11. a. Determine the dimensionality of the waveforms and a set of basis functions. b. Use the basis functions to represent the four waveforms by vectors s1, s2, s3, and s4. c. Determine the minimum distance between any pair of vectors.
- Question : 12P - Determine a set of orthonormal functions for the four signals shown in Figure P2.12
- Question : 13P - A random experiment consists of drawing a ball from an urn that contains 4 red balls numbered 1, 2, 3, 4 and three black balls numbered 1, 2, 3. The following events are defined. 1. E1 = The number on the ball is even. 2. E2 = The color of the ball is red, and its number is greater than 1. 3. E3 = The number on the ball is less than 3. 4. E4 = E1 ? E3 5. E5 = E1 ? (E2 ? E3)Chapter Two: Deterministic and Random Signal Analysis Answer the following questions. 1. What is P(E2)? 2. What is P(E3|E2)? 3. What is P(E2|E4E3)? 4. Are E3 and E5 independent?
- Question : 14P - In a certain city three car brands A, B, C have 20%, 30% and 50% of the market share, respectively. The probability that a car needs major repair during its first year of purchase for the three brands is 5%, 10%, and 15%, respectively. 1. What is the probability that a car in this city needs major repair during its first year of purchase? 2. If a car in this city needs major repair during its first year of purchase, what is the probability that it is made by manufacturer A?
- Question : 15P - The random variables Xi, i = 1, 2, . . . , n, have joint PDF p(x1, x2, . . . , xn). Prove that p(x1, x2, x3, . . . , xn) = p(xn|xn?1, . . . , x1)p(xn?1|xn?2, . . . , x1)
- Question : 16P - A communication channel with binary input and ternary output alphabets is shown in Figure P2.16. The probability of the input being 0 is 0.4. The transition probabilities are shown on the figure1. If the channel output is A, what is the best decision on channel input that minimizes the error probability? Repeat for the cases where channel output is B and C. 2. If a 0 is transmitted and an optimal decision scheme (the one derived in part 1) is used at the receiver, what is the probability of error? 3. What is the overall error probability for this channel if the optimal decision scheme is used at the receiver.
- Question : 17P - 2.17 The PDF of a random variable X is p(x). A random variable Y is defined as Y = aX + b where a < 0. Determine the PDF of Y in terms of the PDF of X.
- Question : 18P - 2.18 Suppose that X is a Gaussian random variable with zero mean and unit variance. Let Y = a X 3 + b, a > 0 Determine and plot the PDF of Y
- Question : 19P - 2.19 The noise voltage in an electric circuit can be modeled as a Gaussian random variable with mean equal to zero and variance equal to 10?8. 1. What is the probability that the value of the noise exceeds 10?4? What is the probability that it exceeds 4
- Question : 20P - 2.20 X is a N (0, ? 2) random variable. This random variable is passed through a system whose input-output relation is given by y = g(x). Find the PDF or the PMF of the output random variable Y in each of the following cases. 1. Square-law device, g(x) = ax 2. 2. Limiter, g(x) = ??? ?b x ? ?b b x ? b x |x| < b 3. Hard limiter, g(x) = ??? a x > 0 0 x = 0 b x < 0 4. Quantizer, g(x) = xn for an ? x < an+1, 1 ? n ? N, where xn lies in the interval [an, an+1] and the sequence {a1, a2, . . . , aN+1} satisfies the conditions a1 = ??, aN+1 = ? and for i > j we have ai > a j.
- Question : 21P - hows that for an N (m, ? 2) random variable we have E [(X ? m)n] = 1 0 for
- Question : 22P - . Let Xr and Xi be statistically independent zero-mean Gaussian random variables with identical variance. Show that a (rotational) transformation of the form Yr + jYi = (Xr + j Xi)e j? results in another pair (Yr, Yi) of Gaussian random variables that have the same joint PDF as the pair (Xr, Xi). b. Note that " Y Yri # = A" X Xri # where A is a 2
- Question : 23P - Show that if X is a Gaussian vector, the random vector Y = AX, where the invertible matrix A represents a linear transformation, is also a Gaussian vector whose mean andChapter Two: Deterministic and Random Signal Analysis covariance matrix are given by mY = AmX CY = AC X At
- Question : 24P - The random variable Y is defined as Y = n i=1 Xi where the Xi, i = 1, 2, . . . , n, are statistically independent random variables with Xi = 1 with probability 0 with probability 1p? p a. Determine the characteristic function of Y . b. From the characteristic function, determine the moments E(Y ) and E(Y 2).
- Question : 25P - 2.25 This problem provides some useful bounds on Q(x). 1. By integrating e? u2+2v2 on the region u > x and v > x in R2, where x > 0, then changing to polar coordinates and upper bounding the integration region by the region r > ?2x in the first quadrant, show that Q(x) ? 1 2 e? x22 for all x ? 0. 2. Apply integration by parts to ? x e? y2 2 dy y2 and show that x ?2?(1 + x 2) e? x2 2 < Q(x) < 1 ?2? x e? x2 2 for all x > 0. 3. Based on the result of part 2 show that, for large x, Q(x) ? 1 x?2? e? x2 2
- Question : 26P - Let X1, X2, X3, . . . denote iid random variables each uniformly distributed on [0, A], where A > 0. Let Yn = min{X1, X2, . . . , Xn}. 1. What is the PDF of Yn? 2. Show that if both A and n go to infinity such that nA = ?, where ? > 0 is a constant, the density function of Yn tends to an exponential density function. Specify this density function.
- Question : 27P - The four random variables X1, X2, X3, X4 are zero-mean jointly Gaussian random variables with covariance Ci j = E(Xi X j) and characteristic function X(?1, ?2, ?3, ?4). Show that E(X1 X2 X3 X4) = C12C34 + C13C24 + C14C23
- Question : 28P - 2.28 Let X(t) = E et X denote the moment generating function of random variable X. 1. Using the Chernov bound, show that ln P [X ? ?] ? ? max t?0 (?t ? ln X(t)) 2. Define I (?) = max t?0 (?t ? ln X(t)) as the large-deviation rate function of the random variable X, and let X1, X2, . . . , Xn be iid. Define Sn = (X1 + X2 +
- Question : 29P - 2.29 From the characteristic functions for the central chi-square and noncentral chi-square random variables given in Table 2.3
- Question : 30P - The PDF of a Cauchy distributed random variable X is p(x) = a/? x 2 + a2 , ?? < x < ? a. Determine the mean and variance of X. b. Determine the characteristic function of X.
- Question : 31P - Let R0 denote a Rayleigh random variable with PDF f R0(r0) = 0 otherwise ?r02 e? 2r?022 r0 ? 0Chapter Two: Deterministic and Random Signal Analysis and R1 be Ricean with PDF f R1(r1) = 0 otherwise ?r12 I0 ??r21 e? r122+?? 2 2 r1 ? 0 Furthermore, assume that R0 and R1 are independent. Show that P(R0 > R1) = 1 2 e ? ?2 4? 2
- Question : 32P - Suppose that we have a complex-valued Gaussian random variable Z = X + jY , where (X, Y ) are statistically independent variables with zero mean and variance E X 2 = E Y 2 = ? 2. Let R = Z + m, where m = mr + jmi and define R as R = A + j B. Clearly, A = X + mr and B = Y + mi. Determine the following probability density functions: 1. pA,B(a, b) 2. pU, (u, ?), where U = ?A2 + B2 and = tan?1 B/A 3. pU (u) Note: In part 2 it is convenient to define ? = tan?1(mi/mr) so that mr = mr2 + mi2 cos ?, mi = mr2 + mi2 sin ? Furthermore, you must use Equation 2.3
- Question : 33P - The random variable Y is defined as Y = 1 n n i=1 Xi where Xi, i = 1, 2, . . . , n, are statistically independent and identically distributed random variables each of which has the Cauchy PDF given in Problem 2.30. a. Determine the characteristic function of Y. b. Determine the PDF of Y. c. Consider the PDF of Y in the limit as n ? ?. Does the central limit theorem hold? Explain your answer
- Question : 34P - Show that if Z is circular, then it is zero-mean and proper, i.e., E [Z] = 0 and E ZZt = 0.
- Question : 35P - 2.35 Show that if Z is a zero-mean proper Gaussian complex vector, then Z is circular.
- Question : 36P - 2.36 Show that if Z is a proper complex vector, then any transform of the form W = AZ + b is also a proper complex vector.
- Question : 37P - 2.37 Assume that random processes X(t) and Y (t) are individually and jointly stationary. a. Determine the autocorrelation function of Z(t) = X(t) + Y (t). b. Determine the autocorrelation function of Z(t) when X(t) and Y (t) are uncorrelated. c. Determine the autocorrelation function of Z(t) when X(t) and Y (t) are uncorrelated and have zero means.
- Question : 38P - The autocorrelation function of a stochastic process X(t) is RX(?) = 1 2 N0?(?) Such a process is called white noise. Suppose x(t) is the input to an ideal bandpass filter having the frequency response characteristic shown in Figure P2.38. Determine the total noise power at the output of the filter.
- Question : 39P - A lowpass Gaussian stochastic process X(t) has a power spectral density S( f ) = 0 otherwise N0 | f | < B Determine the power spectral density and the autocorrelation function of Y (t) = X 2(t).
- Question : 40P - The covariance matrix of three random variables X1, X2, and X3 is ?? C11 0 C13 0 C22 0 C31 0 C33 ?? The linear transformation Y = AX is made where A = ? ? 1 0 0 0 2 0 1 0 1? ? Determine the covariance matrix of Y
- Question : 41P - Let X(t) be a stationary real normal process with zero mean. Let a new process Y (t) be defined by Y (t) = X 2(t) Determine the autocorrelation function of Y (t) in terms of the autocorrelation function of X(t). Hint: Use the result on Gaussian variables derived in Problem 2.27.
- Question : 42P - For the Nakagami PDF, given by Equation 2.3
- Question : 43P - The input X(t) in the circuit shown in Figure P2.43 is a stochastic process with E[X(t)] = 0 and RX(?) = ? 2?(?); i.e., X(t) is a white noise process. a. Determine the spectral density SY ( f ). b. Determine RY (?) and E[Y 2(t)].Chapter Two: Deterministic and Random Signal Analysis
- Question : 44P - 2.44 Demonstrate the validity of Equation 2.8
- Question : 45P - 2.45 Use the Chernoff bound to show that Q(x) ? e?x2/2.
- Question : 46P - 2.46 Determine the mean, the autocorrelation sequence, and the power density spectrum of the output of a system with unit sample response h(n) = ??????? 1 n = 0 ?2 n = 1 1 n = 2 0 otherwise when the input x(n) is a white noise process with variance ?x2.
- Question : 47P - 2.47 The autocorrelation sequence of a discrete-time stochastic process is R(k) = 1 2 |k|. Determine its power density spectrum.
- Question : 48P - 2.48 A discrete-time stochastic process X(n) ? X(nT ) is obtained by periodic sampling of a continuous-time zero-mean stationary process X(t), where T is the sampling interval; i.e., fs = 1/T is the sampling rate. a. Determine the relationship between the autocorrelation function of X(t) and the auto correlation sequence of X(n). b. Express the power density spectrum of X(n) in terms of the power density spectrum of the process X(t). c. Determine the conditions under which the power density spectrum of X(n) is equal to the power density spectrum of X(t).
- Question : 49P - The random process V (t) is defined as V (t) = X cos 2? fct ? Y sin 2? fct where X and Y are random variables. Show that V (t) is wide-sense stationary if and only if E(X) = E(Y ) = 0, E(X 2) = E(Y 2), and E(XY ) = 0.
- Question : 50P - Consider a band-limited zero-mean stationary stochastic process X(t) with power density spectrum SX( f ) = 1 0 otherwise | f | ? W X(t) is sampled at a rate fs = 1/T to yield a discrete-time process X(n) ? X(nT ). a. Determine the expression for the autocorrelation sequence of X(n). b. Determine the minimum value of T that results in a white (spectrally flat) sequence.c. Repeat (b) if the power density spectrum of X(t) is SX( f ) = 1 0 otherwise ? | f |/W | f | ? W
- Question : 51P - Show that the functions fk(t) = sinc"2W t ? 2kW # , k = 0,
- Question : 52P - The noise equivalent bandwidth of a system is defined as B eq = 1 G ? 0 |H( f )|2 d f where G = max |H( f )|2. Using this definition, determine the noise equivalent bandwidth of the ideal bandpass filter shown in Figure P2.38 and the low-pass system shown in Figure P2.43.
- Question : 53P - Suppose that N(t) is a zero-mean stationary narrowband process. The autocorrelation function of the equivalent lowpass process Z(t) = X(t) + jY (t) is defined as RZ (?) = E Z?(t)Z(t + ?) a. Show that E [Z(t)Z(t + ?)] = 0 b. Suppose Rz(?) = N0?(?), and let V = T 0 Z(t) dt Determine E V 2 and E |V|2
- Question : 54P - Determine the autocorrelation function of the stochastic process X(t) = A sin(2? fct + ) where fc is a constant and is a uniformly distributed phase, i.e., p(?) = 1 2? , 0 ? ? ? 2?Chapter Two: Deterministic and Random Signal Analysis
- Question : 55P - Let Z(t) = X(t) + jY (t) be a complex random process, where X(t) and Y (t) are realvalued, independent, zero-mean, and jointly stationary Gaussian random processes. We assume that X(t) and Y (t) are both band-limited processes with a bandwidth of W and a flat spectral density within their bandwidth, i.e., SX( f ) = SY ( f ) = 0 otherwise N0 | f | ? W 1. Find E[Z(t)] and RZ (t + ?, t), and show that Z(t) is WSS. 2. Find the power spectral density of Z(t). 3. Assume ?1(t), ?2(t), . . . , ?n(t) are orthonormal, i.e., ? ?? ? j(t)?k?(t) dt = 1 0 otherwise j = k and all ? j(t)
- Question : 56P - Let X(t) denote a (real, zero-mean, WSS) bandpass process with autocorrelation function RX(?) and power spectral density SX( f ), where SX(0) = 0, and let X
- Question : 57P - A noise process has a power spectral density given by Sn( f ) = 10 0 ?8 1 ? 10 | f 8| || ff || < > 10 108 8 This noise is passed through an ideal bandpass filter with a bandwidth of 2 MHz centered at 50 MHz. 1. Find the power content of the output process. 2. Write the output process in terms of the in-phase and quadrature components, and find the power in each component. Assume f0 = 50 MHz. 3. Find the power spectral density of the in-phase and quadrature components. 4. Now assume that the filter is not an ideal filter and is described by |H( f )|2 = 0 otherwise 10 | f 6| ? 49 49 MHz < | f | < 51 MHz Repeat parts 1, 2, and 3 with this assumption.

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