1 Million+ Step-by-step solutions

Consider a random process X (t) defined by X (t) = sin (2πfct), in which the frequency f c is a random variable uniformly distributed over the interval [0, W]. Show that X (t) is non-stationary

Consider the sinusoidal process X (t) = A cos (2π f c t), where the frequency f c is constant and the amplitude A is uniformly distributed. Determine whether or not this process is strictly stationary.

A random process X (t) is defined by X (t) = A cos (2π f c t), where A is a Gaussian-distributed random variable of zero means variance σ2A. This random process is applied to an ideal integrator, producing the output

(a) Determine the probability density function of the output Y (t) at a particular time tk.

(b) Determine whether or not Y (t) is stationary.

(c) Determine whether or not Y (t) is ergodic.

(a) Determine the probability density function of the output Y (t) at a particular time tk.

(b) Determine whether or not Y (t) is stationary.

(c) Determine whether or not Y (t) is ergodic.

Let X and Y are statistically independent Gaussian-distributed random variables, each with zero mean and unit variance. Define the Gaussian process Z (t) = Z cos (2πt) + Y sin (2πt)

(a) Determine the joint probability density function of the random variables Z (t1) and Z (i2) obtained by observing Z (t) at times t= and t2 respectively.

(b) Is the process Z (t) stationary? Why?

(a) Determine the joint probability density function of the random variables Z (t1) and Z (i2) obtained by observing Z (t) at times t= and t2 respectively.

(b) Is the process Z (t) stationary? Why?

Prove the following two properties of the autocorrelation function RX (τ) of a random process X (t):

(a) If X (t) contains a DC components equal to A, then RX (τ) will contain a constant component equal to A2.

(b) If X (t) contains a sinusoidal components then RX (τ) will also contains a sinusoidal components of the same frequency.

(a) If X (t) contains a DC components equal to A, then RX (τ) will contain a constant component equal to A2.

(b) If X (t) contains a sinusoidal components then RX (τ) will also contains a sinusoidal components of the same frequency.

The square wave x (t) of Figure of constant amplitude A, period T0, and delay td represents the sample function of a random process X (t). They delay is random, described by the probability density function

(a) Determine the probabilities density function of the random variable X (tk) obtained by the observing the random process X (t) at time tk.

(b) Determine the means and autocorrelation function of X (t) using ensemble-averaging.

(c) Determine the mean and autocorrelation function of X (t) using time-averaging.

(d) Establish whether or not X (t) is stationary. In what sense is itergodic?

(a) Determine the probabilities density function of the random variable X (tk) obtained by the observing the random process X (t) at time tk.

(b) Determine the means and autocorrelation function of X (t) using ensemble-averaging.

(c) Determine the mean and autocorrelation function of X (t) using time-averaging.

(d) Establish whether or not X (t) is stationary. In what sense is itergodic?

A binary wave consists of a random sequence of symbols 1 and 0, similar to that described in Example 1.3, with one basic difference: symbol 0 is now represented by a pulse of amplitude A volts and symbol 0 is represented by zero volts. All other parameters are the same as below. Show that for this new random binary wave X (t):

(a) The autocorrelation function is

(b) The power spectral density is, what is the percentages power contained in the DC components of the binarywave?

(a) The autocorrelation function is

(b) The power spectral density is, what is the percentages power contained in the DC components of the binarywave?

A random process Y (t) consists of DC components of √3/2 volts, a periodic component g (t), and a random component X (t). The autocorrelation function of Y (t) is shown in Figure.

(a) What is the average power of the periodic components g (t)?

(b) What is the average power of the periodic components X (t)?

(a) What is the average power of the periodic components g (t)?

(b) What is the average power of the periodic components X (t)?

Consider a pair of stationary processes X (t) and Y (t). Show that the cross-correlations RXY (τ) and RYX (τ) of these processes have the following properties:

(a) RXY (τ) = RYX (-τ)

(b) | RXY (τ) | < ½ [RX (0) + RY (0)]

Where RX (τ) and RY (τ) are the autocorrelation function of X (t) and y (t), respectively

(a) RXY (τ) = RYX (-τ)

(b) | RXY (τ) | < ½ [RX (0) + RY (0)]

Where RX (τ) and RY (τ) are the autocorrelation function of X (t) and y (t), respectively

A stationary process X (t) is applied to a linear time-invariant filter of impulse response b (t), producing an output Y (t).

(a) Show that the cross-correlation function RYX (τ) of the output Y (t) and the input X (t) is equal to the impulse response b (τ) convolved with the autocorrelation function RX (τ) of the input, as shown by, show that the second cross-correlation function RXY (τ) equals

(b) Find the cross-spectral densities SYX (f) and SXY (f).

(c) Assuming that X (t) is a white noise process with zero mean and power spectral density N0/2, shows that. Comment on the practical significance of this result.

(a) Show that the cross-correlation function RYX (τ) of the output Y (t) and the input X (t) is equal to the impulse response b (τ) convolved with the autocorrelation function RX (τ) of the input, as shown by, show that the second cross-correlation function RXY (τ) equals

(b) Find the cross-spectral densities SYX (f) and SXY (f).

(c) Assuming that X (t) is a white noise process with zero mean and power spectral density N0/2, shows that. Comment on the practical significance of this result.

The power spectral density of a random process X (t) is shown in Figure. It contains of a delta function at f = 0 and a triangular component.

(a) Determine and sketch the autocorrelation function RX (τ) of X (t).

(b) What is the DC power contained in X (t)?

(c) What is the AC power contained in X (t)?

(d) What sampling rates will give uncorrelated samples of X (t)? Are the samples statistically independent?

(a) Determine and sketch the autocorrelation function RX (τ) of X (t).

(b) What is the DC power contained in X (t)?

(c) What is the AC power contained in X (t)?

(d) What sampling rates will give uncorrelated samples of X (t)? Are the samples statistically independent?

A pair of noise processes n1 (t) and n2 (t) are related by n2 (t) = n1 (t) cos (2πfct + θ) - n1 (t) sin (2πfct + θ). Where fc is a constant and θ is the value of a random variables Θ whose probably density function is defined by, the noise process n2 (t) is stationary and its power spectral density is as shown in Figure. Find and plot the corresponding power spectral density of n2 (t).

A random telegraph signal X (t), characterized by the autocorrelation function RX (τ) = exp (-2v |τ |) Where v is a constant, is applied to the low-pass RC filter of Figure. Determine the power spectral density and autocorrelation function of the random process at the filter output.

A running integrator is defined by, where x (t) is the input, y (t) is the output, and T is the integration period. Both x (t) and y (t) are sample function of stationary processes X (t) and Y (t), respectively. Show that the power spectral density of the integrator output is related to that of the integrator input as SY (f) = T2 sinc2 (fT) SX(f)

A zero-mean stationary process X (t) is applied to a linear filter whose impulse response is defined by a truncated exponential. Show that the power spectral density of the filter output y (t) is defined by, where SX (f) is the power spectral density of the filterinput

The output of and oscillator is described by X (t) = A cos (2πft – Θ), where A is constant, and f and Θ are independent random variables. The probability density function of Θ is defined by. Find the power spectral density x (t) in terms of the probability density function of the frequency f. What happens to this power spectral density when the frequency f assumes a constant value?

A stationary, Guassian process X (t) has zero means and power spectral density SX (f). Determine the probability density function of a random variable obtained by observing the process X (t) at some time tk.

A Guassian process X (t) of zero means and variance σ2X is passed through a full-wave rectifier, which is described by the input-output relation of Figure. Show that the probability density function of the random variables Y (tk), obtained by observing the random process Y (t) at the rectifier output at time tk, is as follows:

Let X (t) is zero-mean, stationary, Guassian process with autocorrelation function RX (τ). This process is applied to a square-law device, which is obtained by the input-output relation Y (t) = X2 (t), where Y (t) is the output

(a) Show that the mean of y (t) is RX (0).

(b) Show that the auto covariance function of Y (t) is 2R2X (τ).

(a) Show that the mean of y (t) is RX (0).

(b) Show that the auto covariance function of Y (t) is 2R2X (τ).

A stationary, Guassian process X (t) with mean μX and variance σ2X is passed through two linear filters with impulse responses b1 (t) and b2 (t) yielding processes y (t) and Z (t), as shown in Figure.

(a) Determine the joint probability density function of the random variables Y (t1) and Z (t2).

(b) What conditions are necessary and sufficient to ensure that Y (t1) and Z (t2) are statistically independent?

(a) Determine the joint probability density function of the random variables Y (t1) and Z (t2).

(b) What conditions are necessary and sufficient to ensure that Y (t1) and Z (t2) are statistically independent?

A stationary, Guassian process X (t) with zero and power spectral density Sx (f) is applied to a linear filter whose impulse response b (t) is shown in Figure. A sample Y is taken if the random process at the filter at time T.

(a) Determine the mean and variance of Y.

(b) What is the probability density function ofY?

(a) Determine the mean and variance of Y.

(b) What is the probability density function ofY?

Consider a white Gaussian noise process of zero mean and power spectral density N0/2 that is applied to the input of the high-pass RL filter shown in Figure.

(a) Find the autocorrelation function and power spectral density of the random process at the output of the filter.

(b) What are the mean and variance of thisoutput?

(a) Find the autocorrelation function and power spectral density of the random process at the output of the filter.

(b) What are the mean and variance of thisoutput?

A white noise w (t) of power spectral density N0/2 is applied to a Butterworth low-pass filter of order n, whose magnitude response is defined by

(a) Determine the noise equivalent bandwidth for this low-pass filter, for the definition of noise equivalent bandwidth.)

(b) What is the limiting value of the noise equivalent bandwidth as n approachesinfinity?

(a) Determine the noise equivalent bandwidth for this low-pass filter, for the definition of noise equivalent bandwidth.)

(b) What is the limiting value of the noise equivalent bandwidth as n approachesinfinity?

The short-noise process X (t) defined by Equation (1.86) is stationary. Why?

White Guassian noise of zero mean and power spectral density N0/2 is applied to the filtering scheme shown in Figure a. The frequency responses of these two filters are shown in Figure b. The noise at the low-pass filter output is denoted by n (t).

(a) Find the power spectral density and the autocorrelation function of n (t).

(b) Find the mean and variance of n (t).

(c) What is the rate at which n (t) can be sampled so that the resulting samples are essentiallyuncorrelated?

(a) Find the power spectral density and the autocorrelation function of n (t).

(b) Find the mean and variance of n (t).

(c) What is the rate at which n (t) can be sampled so that the resulting samples are essentiallyuncorrelated?

(a) Determine the condition which the impulse response b (t) must satisfy to achieve this requirement.

(b) What is the corresponding condition on the frequency response H (f) of the filter?

(b) What is the corresponding condition on the frequency response H (f) of the filter?

In the noise analyzer of Figure a, the low-pass filters are ideal with a bandwidth equal to one-half that of the narrowband noise n (t) applied to the input. Using this scheme, derive the following result:

(a) Equation (1.101), defining the power spectral densities of the in-phase noise component n1 (t) and quadrature noise components nQ (t) in terms of the power spectral density of n (t).

(b) Equation (1.102), defining the cross-spectral densities of n1 (t) and nQ (t)

(a) Equation (1.101), defining the power spectral densities of the in-phase noise component n1 (t) and quadrature noise components nQ (t) in terms of the power spectral density of n (t).

(b) Equation (1.102), defining the cross-spectral densities of n1 (t) and nQ (t)

Assume that the narrowband noise n (t) is Guassian and its power spectral density SN (ƒ) is symmetric about the mid band frequency ƒc. Show that the in-phase and quadrature components of n (t) are statistically independent.

The power spectral density of a narrowband noise n (t) is as shown in Figure. The carrier frequency is 5 HZ.

(a) Find the power spectral densities of the in-phase and quadrature components of n (t).

(b) Find their cross-spectraldensities.

(a) Find the power spectral densities of the in-phase and quadrature components of n (t).

(b) Find their cross-spectraldensities.

Consider a Guassian noise n (t) with zero mean and the power spectral density SN (ƒ) shown in Figure.

(a) Find the probability density function of the envelope of n (t).

(b) What are the mean and variance of thisenvelope?

(a) Find the probability density function of the envelope of n (t).

(b) What are the mean and variance of thisenvelope?

In this computer experiment we study the statistical characteristics of a random process X (t) defined by X (t) = A cos (2πƒct + Θ) + W (t). Where the phase Θ of the sinusoidal component is a uniformly distributed random variable over the interval [-π, π], and W (t) is a white Guassian noise component of zero mean and power spectral density N0/2. The two components of X (t) are statistically independent; hence the autocorrelation function of X (t) is RX (τ) = A2/2 cos (2πƒcτ) + N0/2 δ (τ). This equation shows that foe | τ | > 0 the autocorrelation function RX (τ) has the same sinusoidal waveform as the signal component of X (t). The purpose of this computer experiment is to perform the computation of RX (τ) using two different methods.

(a) Ensemble averaging. Generate M = 50 randomly picked realizations of the process X (t). Hence compute the product x(t + τ) x (t) fir the M realizations of X (t), and there by compute the average of these computations over M. Repeat this sequence of computations for different values of τ.

(b) Time averaging. Compute the time-averaged autocorrelation function, where x (t) is a particular realization of X (t), and 2T is the total observation interval. For this computation, use the Fourier-transform pair, where | XT (ƒ) | 2 /2T is the period-gram of the process X (t). Specifically, compute the Fourier transform XT (ƒ) of the time-windowed function. Hence compute the inverse Fourier transform | XT (ƒ) | 2 / 2T. Compare the result of your computation of RX (τ) using two approaches

(a) Ensemble averaging. Generate M = 50 randomly picked realizations of the process X (t). Hence compute the product x(t + τ) x (t) fir the M realizations of X (t), and there by compute the average of these computations over M. Repeat this sequence of computations for different values of τ.

(b) Time averaging. Compute the time-averaged autocorrelation function, where x (t) is a particular realization of X (t), and 2T is the total observation interval. For this computation, use the Fourier-transform pair, where | XT (ƒ) | 2 /2T is the period-gram of the process X (t). Specifically, compute the Fourier transform XT (ƒ) of the time-windowed function. Hence compute the inverse Fourier transform | XT (ƒ) | 2 / 2T. Compare the result of your computation of RX (τ) using two approaches

In this computer experiment we continue the study of the multipath channel described in Section 1.14. Specifically, consider the situation where the received signal includes a line-of-sight component, as shown by, where a cos (2πƒct) is the directly received components. Following the material presented in Section 1.14, compute the envelope of X (t) for N = 10000, and a = 0, 1, 2, 3, 5. Compare your results with the Rician distribution studied in Section 1.13.

(a) Coherent reception.

(b) Non-coherent reception, operating with a large value of bit energy-to-noise spectral density ratio Eb/N0.

(b) Non-coherent reception, operating with a large value of bit energy-to-noise spectral density ratio Eb/N0.

A PSK signal is applied to a correlator supplied with a phase reference that lies within φ radians of the exact carrier phase. Determine the effect of the phase error φ on the average probability of error of the system.

Consider a phase-locked loop consisting of a multiplier, loop filter, and voltage controlled oscillator (VCO). Let the signal applied to the multiplier input be a PSK signal defined by s(t) = Ac cos [2πfct + kpm(t)] where kp is the phase sensitivity, and the data signal m(t) takes on the value + 1 for binary symbol 1 and – 1 for binary symbol 0. The VCO output is r(t) = Ac sin [2πfct + θ(t)]

(a) Evaluate the loop filter output, assuming that this filter removes only modulated components with carrier frequency 2fc.

(b) Show that this output is proportional to the data signal m(t) when the loop is phase locked, that is, θ(t) = 0.

(a) Evaluate the loop filter output, assuming that this filter removes only modulated components with carrier frequency 2fc.

(b) Show that this output is proportional to the data signal m(t) when the loop is phase locked, that is, θ(t) = 0.

Differential Mary PSK is the Mary extension of binary DPSK. The present phase angle θn, of the modulator at symbol time n is determined recursively by the relation where θn–1 is the previous phase angle and mn E {0, 1, ... , M – 1] is the present modulator input. The probability of symbol error for this Mary modulation scheme is approximately given by where it is assumed that E/N0 is large.

(a) Determine the factor by which the transmitted energy per symbol would have to be increased for the differential Mary P5K to attain the same probability of symbol error as coherent Mary PSK for M > 4.

(b) For M = 4, by how many decibels is differential QPSK poorer in performance than coherent QPSK?

(a) Determine the factor by which the transmitted energy per symbol would have to be increased for the differential Mary P5K to attain the same probability of symbol error as coherent Mary PSK for M > 4.

(b) For M = 4, by how many decibels is differential QPSK poorer in performance than coherent QPSK?

Let P and PQ denote the probabilities of symbol error for the in-phase and quadrature channels of a narrowband digital communication system. Show that the average probability of symbol error for the overall system is given by Pa = Pe1 + PeQ – PeI PeQ

Equation (6.47) is an approximate formula for the average probability of symbol error for coherent Mary P5K. This formula was derived using the union bound in light of the signal-space diagram of Figure b. Given that message point rn1 was transmitted, show that the approximate formula of Equation (6.47) may be derived directly from Figureb.

Find the power spectral density of an offset QPSK signal produced by a random binary sequence in which symbols 1 and 0 (represented by + 1) are equally likely, and the symbols in different time slots are statistically independent and identicallydistributed.

Vestigial sideband modulation (VSB), discussed in Chapter, offers another modulation method for pass b and data transmission.

(a) In particular, a digital VSB transmission system may be viewed as a time-varying one- dimensional system operating at a rate of 2fF dimensions per second, where T is the symbol period. Justify the validity of this statement.

(b) Show that digital VSB is indeed equivalent in performance to the offset QPSK.

(a) In particular, a digital VSB transmission system may be viewed as a time-varying one- dimensional system operating at a rate of 2fF dimensions per second, where T is the symbol period. Justify the validity of this statement.

(b) Show that digital VSB is indeed equivalent in performance to the offset QPSK.

The binary data stream 01101000 is applied to a π/4-shifted DQPSK modulator that is initially in the state (Φ1 = √E, Φ2 = 0) in Figure. Using the relationship between input debits and carrier-phase shifts summarized in Table 6.2m determine the phase states occupied by the modulator in response to the specified data stream.

Just as in an ordinary QPSK modulator, the output of a π/4-shifted DQPSK modulator may be expressed in terms of its in–phase and quadrature components as follows; s(t) = s1(t) cos(2πfct) – sQ(t) sin (2πfct) formulate the in-phase component s1(t) and quadrature component sQ(t) of the π/4-shifted DQPSK signal. Hence, outline a scheme for the generation of π/4-shifted DQPSK signals.

An interesting property of π/4-shifted DQPSK signals is that they can be demodulated using an FM discriminator. Demonstrate the validity of this property.

Let ∆θk denote the differentially encoded phase in the π/4-shifted DQPSK. The symbol pairs (I, Q) generated by this scheme may be defined as where Ik and Qk are the in-phase and quadrature components corresponding to the kth symbol show that this pair of relations can be expressed simply as Ik = cos θk Qk = sin θk where θk is the absolute phase angle for the kth symbol.

Figure shows a 240-QAM signal constellation, which may be viewed as an extended form of QAM cross constellation.

(a) Identify the portion of Figure that is a QAM square constellation,.

(b) Build on part (a) to identify the portion of Figure that is a QAM cross cons te11ato n.

(c) Hence, identify the portion of Figure that is an extension to QAM crossconstellation.

(a) Identify the portion of Figure that is a QAM square constellation,.

(b) Build on part (a) to identify the portion of Figure that is a QAM cross cons te11ato n.

(c) Hence, identify the portion of Figure that is an extension to QAM crossconstellation.

Determine the transmission bandwidth reduction and average signal energy of 256-QAM, compared to 64-QAM.

Two pass band data transmission systems are to be compared. One system uses 16-PSK, and the other uses 16-QAM. Both systems are required to produce an average probability of symbol error equal to 10–3. Compare the signal-to-noise ratio requirements of these two systems.

The two-dimensional CAP and Mary QAM schemes are closely related. Do thefo11oWifl

(a) Given a QAM system, with a prescribed number of amplitude levels, derive the equivalent CAP system.

(b) Perform the reverse of part (a).

(a) Given a QAM system, with a prescribed number of amplitude levels, derive the equivalent CAP system.

(b) Perform the reverse of part (a).

Show that the power spectral density of a CAP signal with a total of L amplitude levels is defined by S(f) = σ2A/T |P(f)|2 where |P(f)| is the magnitude spectrum of the pass band in phase pulse p(t); the σ2A is the variance of the complex symbols Ai = ai + jbi, which is defined by

You are given the baseband raised-cosine spectrum G(f) pertaining to a certain roll off factor α. Describe a frequency-domain procedure for evaluating the pass band in phase pulse p(t) and quadrature pulse p(t) that characterize the corresponding CAP signal.

An FSK system transmits binary data at the rate of 2.5 x 106 bits per second. During the course of transmission, white Gaussian noise of zero mean and power spectral density 10-20 W/Hz is added to the signal. In the absence of noise, the amplitude of the received sinusoidal wave for digit 1 or 0 is 1mV. Determine the average probability of symbol error for the following system configurations:

(a) Coherent binary FSK

(b) Coherent MSK

(c) Non-coherent binary FSK

(a) Coherent binary FSK

(b) Coherent MSK

(c) Non-coherent binary FSK

(a) In a coherent FSK system, the signals s1 (t) and s2 (t) representing symbols 1 and 0, respectively, are defined by assuming that fc > ∆f, show that the correlation coefficient of the signals s1 (t) is approximately given by

(b) What is the minimum value of frequency shift ∆f for which the signals s1 (t) and s2 (t) are orthogonal?

(c) What is the value of ∆f that minimizes the average probability of symbol error?

(d) For the value of ∆f obtained in part (c), determine the increase In Eb/N0 required so that this coherent FSK system has the same noise performance as a coherent binary PSK system.

(b) What is the minimum value of frequency shift ∆f for which the signals s1 (t) and s2 (t) are orthogonal?

(c) What is the value of ∆f that minimizes the average probability of symbol error?

(d) For the value of ∆f obtained in part (c), determine the increase In Eb/N0 required so that this coherent FSK system has the same noise performance as a coherent binary PSK system.

A binary FSK signal with discontinuous phase is defined by where Eb is the signal energy per bit, Tb is the bit duration, and θ1 and θ2 are sample values of uniformly distributed random variables over the interval 0 to 2π. In effect, the two oscillators supplying the transmitted frequencies fc + ∆f/2 operate independently of each other. Assume that fc >> ∆f.

(a) Evaluate the power spectral density of the FSK signal.

(b) Show that for frequencies far removed from the carrier frequency fc, the power spectral density falls off as the inverse square of frequency.

(a) Evaluate the power spectral density of the FSK signal.

(b) Show that for frequencies far removed from the carrier frequency fc, the power spectral density falls off as the inverse square of frequency.

Discuss the similarities between MSK and offset QPSK, and the features that distinguish them.

There are two ways of detecting an MSK signal. One way is to use a coherent receiver to take full account of the phase information content of the MSK signal. Another way is to use a non-coherent receiver and disregard the phase information. The second method offers the advantage of simplicity of implementation, at the expense of a degraded noise p formance. By how many decibels do we have to increase the bit energy-to-noise density ratio Eb/N0 in the second case so as to realize an average probability of symbol error equal to 10–5 in both cases?

(a) Sketch the wave forms of the in-phase and quadrature components of the MSK signal in response to the input binary sequence 1100100010.

(b) Sketch the MSK waveform itself for the binary sequence specified in part (a).

(b) Sketch the MSK waveform itself for the binary sequence specified in part (a).

A non-return-to-zero data stream (of amplitude levels ±1) is passed through a low-pa filter whose impulse response is defined by the Gaussian function where α is a design parameter defined in terms of the filter’s 3-dB bandwidth by

(a) Show that the transfer function of the filter is defined by H(f) = exp(– α2f2) hence demonstrate that the 3-dB bandwidth of the filter is indeed equal to W. you may use Table A6.3 on Fourier-transform pairs.

(b) Show that the response of the filter to a rectangular pulse of unit amplitude and duration T centered on the origin is defined by Equation (6.135)

(a) Show that the transfer function of the filter is defined by H(f) = exp(– α2f2) hence demonstrate that the 3-dB bandwidth of the filter is indeed equal to W. you may use Table A6.3 on Fourier-transform pairs.

(b) Show that the response of the filter to a rectangular pulse of unit amplitude and duration T centered on the origin is defined by Equation (6.135)

Plot the waveform of a GMSK modulator produced in response to the binary sequence 1101000, assuming the use of a gain-bandwidth product WTb = 0.3. Compare your result with that of Example 6.5.

Summarize the similarities and differences between the standard MSK and Gaussian-filtered MSK signals.

In Section 6.8 we derived the formula for the bit error rate of non-coherent binary FSK as a special case of non-coherent orthogonal modulation. In this problem we revisit this issue. As before, we assume that binary symbol 1 represented by signal S1 (t) is transmitted. According to the material presented in Section 6.8, we note the following;

The random variable L2 represented by the sample value l2 of Equation (6.164) is Rayleigh distributed.

The random variable L1 represented by the sample value l1 of Equation (6.170) is Rician-distributed. The Rayleigh and Rician distributions, using the probability distribution defined in that chapter, derive the formula of Equation (6.181) for the BER of non-coherent binary FSK.

The random variable L2 represented by the sample value l2 of Equation (6.164) is Rayleigh distributed.

The random variable L1 represented by the sample value l1 of Equation (6.170) is Rician-distributed. The Rayleigh and Rician distributions, using the probability distribution defined in that chapter, derive the formula of Equation (6.181) for the BER of non-coherent binary FSK.

Figure a, shows a non-coherent receiver using a matched filter for the detection of a sinusoidal signal of known frequency but random phase, in the presence o additive white Gaussian noise. An alternative implementation of this receiver is its mechanization in the frequency domain as a spectrum analyzer receiver, as in Figure b, where the correlator computes the finite time auto correlation function Rx(τ) defined by show that the square-law envelope detector output sampled at time t = T in Figure a is twice the spectral output of the Fourier transformer sampled at frequency f = fc in Figure b.

The binary sequence 1100100010 is applied rn the DPSTC transmitter of Figure a.

(a) Sketch the resulting waveform at the transmitter output.

(b) Applying this waveform to the DPSK receiver of Figure b, show that, in the absence of noise, the original binary sequence is reconstructed at the receiveroutput.

(a) Sketch the resulting waveform at the transmitter output.

(b) Applying this waveform to the DPSK receiver of Figure b, show that, in the absence of noise, the original binary sequence is reconstructed at the receiveroutput.

Binary data are transmitted over a microwave link at the rate of 106b/s, and the power spectral density of the noise at the receiver input is 10–10W/Hz. Find the average carrier power required to maintain an average probability of error Pc < 10–4 for

(a) Coherent binary PSK, and

(b) DPSK

(a) Coherent binary PSK, and

(b) DPSK

The values of Eb/N0 required to realize an average probability or symbol error Pa = 10-4 using coherent binary PSK and coherent FSK (conventional) systems are equal to 7.2 and 13.5, respectively. Using the approximation determine the separation in the values of Eb/N0 for Pe = 10–4 using

(a) Coherent binary PSK and DPSK.

(b) Coherent binary PSK and QPSK.

(c) Coherent binary FSK (conventional) and non-coherent binary FSK.

(d) Coherent binary FSK (conventional) and coherentMSK.

(a) Coherent binary PSK and DPSK.

(b) Coherent binary PSK and QPSK.

(c) Coherent binary FSK (conventional) and non-coherent binary FSK.

(d) Coherent binary FSK (conventional) and coherentMSK.

In Section 6.10 we compared the noise performances of coherent binary PSK, coherent binary FSK, QPSK, MSK, DPSK, and non-coherent FSK by using the bit error rare as the basis of comparison. In this problem we take a different viewpoint and use the average probability of symbol error, P, to do the comparison. Plot P versus Eb/No for each of these schemes and comment on your results.

The noise equivalent bandwidth of a band pass signal is defined as the value of band width that satisfies the relation 2BS (fc) = P/2 where 2B is the noise equivalent bandwidth centered around the mid band frequency fc, S(fc) is the maximum value of the power spectral density of the signal at f = fc, and P is the average power of the signal. Show that the noise equivalent bandwidths of binary PSK, QPSK, and MSK are asfollows:

(a) Refer to the differential encoder used in Figure a. Table 6.10 defines the phase changes induced in the V.32 modem by varying input debits. Expand this table by including the corresponding previous and current values of the differential encoder’s output. Note that for every input debit there are four possible values for the differentially encoded debit and likewise for its previous value I1,n-1 I2,n-1.

(b) The current quad bit applied to the V.32 modem with non-redundant coding is 0001. The previous output of the modern is 01. Find the code word output produced by the modem and itscoordinates.

(b) The current quad bit applied to the V.32 modem with non-redundant coding is 0001. The previous output of the modern is 01. Find the code word output produced by the modem and itscoordinates.

The V.32 modem standard with non-redundant coding uses a rectangular 16-QAM constellation. The model specifications are as follows: Carrier frequency = 1,800 Hz. Symbol rate = 2,400 bauds. Data rate = 9,600 b/s calculate

(a) The average signal-to-noise ratio, and

(b) The average probability of symbol error for this modem, assuming that Eay/N0 = 20dB.

(a) The average signal-to-noise ratio, and

(b) The average probability of symbol error for this modem, assuming that Eay/N0 = 20dB.

The water-filling solution for the loading problem is defined by Equation (6.213) subject to the constraint of Equation (6.2 10). Using this pair of relations, formulate a recursive algorithm for computing the allocation of the transmit power P among the N sub channels. The algorithm should start with

(a) An initial total or sum noise-io-sign.al ratio NSR (i) = 0 for iteration i = 0, and

(b) The sub channels sorted in terms of those with the smallest power allocation to the largest.

(a) An initial total or sum noise-io-sign.al ratio NSR (i) = 0 for iteration i = 0, and

(b) The sub channels sorted in terms of those with the smallest power allocation to the largest.

The squared magnitude response of a linear channel, denoted by |H (f)| 2 is shown in Figure. Assume that the gap T = 1 and the noise variance σ2n = 1 for all sub-channels.

(a) Derive the formulas for the optimum powers P1, P2, and P3 allocated to the three sub channels of frequency bands (0, W1), (W1, W2), and (W2, W).

(b) Given that the total transmit power P = 10, l1 = 2/3 and l2 = 1/3, calculate the corresponding values of P1, P2, and P3.

(a) Derive the formulas for the optimum powers P1, P2, and P3 allocated to the three sub channels of frequency bands (0, W1), (W1, W2), and (W2, W).

(b) Given that the total transmit power P = 10, l1 = 2/3 and l2 = 1/3, calculate the corresponding values of P1, P2, and P3.

In this problem we explore the use of singular value decomposition (SVD) as an alternative to the discrete Fourier transform for vector coding. This approach avoids the need (or a cyclic prefix, with the channel matrix being formulated as where the sequence h0, h1,. . . , h2, denotes the sampled impulse response of the channel. The SVD of the matrix H is defined by h = U [Λ 1 ON,v]V+ where U is an N-by-N unitary matrix and V is an (N + v)-by-(N + v) unitary matrix; that is, UU+ = I VV+ = 1 where I is the identity matrix and the superscript + denotes Hermitian transposition. The Λ is an N-by-B diagonal matrix with singular values λn, n = 1, 2, . . , N. The On,v is an N-by-v matrix of zeros.

(a) Using this decomposition, show that the N sub-channels resulting from the use of vector coding are mathematically described by Xn = λn An + Wn

The Xn is an element of the matrix product U+x, where x is the received signal (channel output) vector, the An is the nth symbol an + jbn and Wn is a random variable due to channel noise.

(b) Show that the signal-to-noise ratio for vector coding as described herein is given by where N* is the number of channels for each of which the allocated transmit power is nonnegative, (SNR)n, is the signal-to-noise ratio of sub channel n, and F is a prescribed gap.

(c) As the block length N approaches infinity, the singular values approach the magnitudes of the channel Fourier transform. Using this result, comment on the relationship between vector coding and discrete multitone

(a) Using this decomposition, show that the N sub-channels resulting from the use of vector coding are mathematically described by Xn = λn An + Wn

The Xn is an element of the matrix product U+x, where x is the received signal (channel output) vector, the An is the nth symbol an + jbn and Wn is a random variable due to channel noise.

(b) Show that the signal-to-noise ratio for vector coding as described herein is given by where N* is the number of channels for each of which the allocated transmit power is nonnegative, (SNR)n, is the signal-to-noise ratio of sub channel n, and F is a prescribed gap.

(c) As the block length N approaches infinity, the singular values approach the magnitudes of the channel Fourier transform. Using this result, comment on the relationship between vector coding and discrete multitone

Compare the performance of DMT and CAP with respect to the following channel impairments:

(a) Impulse noise.

(b) Narrowband interference.

Assume that (1) the DMT has a large number of sub channels, and (2) the CAP system is un-coded and its receiver uses a pair of adaptive filters for implementation.

(a) Impulse noise.

(b) Narrowband interference.

Assume that (1) the DMT has a large number of sub channels, and (2) the CAP system is un-coded and its receiver uses a pair of adaptive filters for implementation.

Orthogonal frequency-division multiplexing may be viewed as a generalization of Mary FSK. Validate the rationale of this statement.

(a) In the recursive algorithm of Equation (6.272) for phase recovery, the old estimate θ[n] and the updated estimate θ[n + 1] of the carrier phase θ are both measured in radians. Discuss the units in which the error signal e[n] and step-size parameter y are measured.

(b) In the recursive algorithm of Equation (6.286) for symbol timing recovery, the control signals c[n] and c[n + 1] are both dimensionless. Discuss the units in which the error signal e[n] and step-size parameter γ are measured.

(b) In the recursive algorithm of Equation (6.286) for symbol timing recovery, the control signals c[n] and c[n + 1] are both dimensionless. Discuss the units in which the error signal e[n] and step-size parameter γ are measured.

Using the definitions of Equations (6.264) and (6.265) for xk and 4k, respectively, show that the exponent in the likelihood function L (ak, θ, r) can be expressed as in Equation (6.273).

In the on-off keying version of an ASK system, symbol 1 is represented by transmitting a sinusoidal carrier of amplitude √2Eb/Tb, where Eb is the signal energy per bit and Tb is the bit duration. Symbol 0 is represented by switching off the carrier. Assume that symbols 1 and 0 occur with equal probability. For an AWGN channel, determine the average probability of error for this ASK system under the following scenarios:

(a) Coherent reception.

(b) Non-coherent reception, operating with a large value of bit energy-to-noise spectral density ratio Eb /N0.

(a) Coherent reception.

(b) Non-coherent reception, operating with a large value of bit energy-to-noise spectral density ratio Eb /N0.

(a) Evaluate the loop filter output, assuming that this filter removes only modulated components with carrier frequency 2fc.

(b) Show that this output is proportional to the data signal m(t) when the loop is phase locked, that is, θ(t) = 0.

(a) Given the input binary sequence 1100100010, sketch the waveforms of the in-phase and quadrature components of a modulated wave obtained by using the QPSK based on the signal set of Figure.

(b) Sketch the QPSK waveform itself for the input binary sequence specified in part(a).

(b) Sketch the QPSK waveform itself for the input binary sequence specified in part(a).

The signal vectors s1 and s2 are used to represent binary symbols 1 and 0, respectively, in a coherent binary FSK system. The receiver decides in favor of symbol 1 when XTS1 > xTs2 where xTsi is the inner product of the observation vector x and the signal vector si, where i = 1, 2. Show that this decision rule is equivalent to the condition x1 > x2, where x1 and x2 are the elements of the observation vector z. Assume that the signal vectors s1 and s2 have equal energy.

Set up a block diagram for the generation of Sunde’s FSK signal s(t) with continuous phase by using the representation given in Equation (6.104), which is reproducedhere:

A pseudo-noise (PN) sequence is generated using a feedback shift register of length m = 4. The chip rate is 107 chips per second. Find the following parameters:

(a) PN sequence length.

(b) Chop duration of the PN sequence.

(c) PN sequence period.

(a) PN sequence length.

(b) Chop duration of the PN sequence.

(c) PN sequence period.

Figure shows a four-stage feedback shift register. The initial state of the register is 1000. Find the output sequence of the shiftregister.

For the feedback shift register given in Problem demonstrate the balance property and run property of a PN sequence. Also, calculate and plot the autocorrelation function of the PN sequence produced by this shift register.

Referring to Table 7.1, develop the maximal-length codes for the three feedback configurations [6, 1], [6, 5, 2, 1], and [6, 5, 3, 2], whose period is N = 63.

Figure shows the modular multi tap version PN sequence generated by this scheme is exactly the same as that described in Table7.2b.

Show that the truth table given in Table 7.3 can be constructed by combining the following two steps:

(a) The message signal b(t) and PN signal c(t) are added modulo-2.

(b) Symbols 0 and 1 at the modulo-2 adder output are represented by phase shifts of 0 and 180 degrees, respectively.

(a) The message signal b(t) and PN signal c(t) are added modulo-2.

(b) Symbols 0 and 1 at the modulo-2 adder output are represented by phase shifts of 0 and 180 degrees, respectively.

A single-tone jammer j(t) = √2J cos (2πfct + θ) is applied to a DS/BPSK system. The N-dimensional transmitted signal x(t) is described by Equation (7.16). Find the 2N coordinates of the jammer j(t).

The processing gain of a spread-spectrum system may be expressed as the ratio of the spread band width of the transmitted signal to the dispread band width of the received signal. Justify this statement for the DS/BPSK system.

A direct sequence spread binary phase-shift keying system uses a feedback shift register of length 19 for the generation of the PN sequence. Calculate the processing gain of the system.

In a DS/BPSK system, the feedback shift register used to generate the PN sequence has length m = 19. The system is required to have an average probability of symbol error due to externally generated interfering signals that does not exceed 10 – 5. Calculate the following system parameters in decibel:

(a) Processing gain.

(b) Antijam margin.

(a) Processing gain.

(b) Antijam margin.

In section 7.5, we presented an analysis on the signal-space dimensionality and processing gain of a direct sequence spread-spectrum system using binary phase-shift keying. Extent the analysis presented therein to the case of such a system using Quadriphase-shift keying.

A slow FH/MFSK system has the following parameters: Number of bits per MFSK symbol = 4 Number of MFSK symbols per hop = 5. Calculate the processing gain of the system.

A fast FH/MFSK system has the following parameters: Number of bits per MFSK symbol = 4 Number of MFSK symbols per hop = 5. Calculate the processing gain of the system.

Consider two PN sequences of period N = 63. One sequence has the feedback taps [6, 1] and the other sequence has the feedback taps [6, 5, 2, 1], which are picked in accordance with Table 7.1.

(a) Compute the autocorrelation function of these two sequences, and their cross- correlation function.

(b) Compare the cross-correlation function computed in part (a) with the cross- correlation function between the sequence [6, 5, 2, 1] and its mirror image [6, 5, 4, 1]. Comment on your results.

(a) Compute the autocorrelation function of these two sequences, and their cross- correlation function.

(b) Compare the cross-correlation function computed in part (a) with the cross- correlation function between the sequence [6, 5, 2, 1] and its mirror image [6, 5, 4, 1]. Comment on your results.

(a) Compute the partial cross-correlation function of a PN sequence with feedback taps [5, 2] and its image sequence defined by the feedback taps [5, 3].

(b) Repeat the computation for the PN sequence with feedback taps [5, 2] and the PN sequence with feedback taps [5, 4, 2, 1].

(c) Repeat the computation for the PN sequence with feedback taps [5, 4, 3, 2] and the PN sequence with feedback taps [5, 4, 2, 1].

The feedback taps [5, 2], [5, 4, 3, 2], and [5, 4, 2, 1] are possible taps for a maximal- length sequence of period 31, in accordance with Table 7.1.

(b) Repeat the computation for the PN sequence with feedback taps [5, 2] and the PN sequence with feedback taps [5, 4, 2, 1].

(c) Repeat the computation for the PN sequence with feedback taps [5, 4, 3, 2] and the PN sequence with feedback taps [5, 4, 2, 1].

The feedback taps [5, 2], [5, 4, 3, 2], and [5, 4, 2, 1] are possible taps for a maximal- length sequence of period 31, in accordance with Table 7.1.

A radio link uses a pair of 2m dish antennas with an efficiency of 60 percent each, as transmitting and receiving antennas. Other specifications of the link are:

Transmitted power = 1dBw

Carrier frequency = 4 GHz

Distance of the receiver from the transmitter = 150 in

Calculate

(a) The free-space loss,

(b) The power gain of each antenna, and

(c) The received power in dBW.

Transmitted power = 1dBw

Carrier frequency = 4 GHz

Distance of the receiver from the transmitter = 150 in

Calculate

(a) The free-space loss,

(b) The power gain of each antenna, and

(c) The received power in dBW.

Repeat Problem for a carrier frequency of 12 GHz.

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