- 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 (2? 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
- A random process X (t) is defined by X (t) = A cos (2? f c t), where A is a Gaussian-distributed random variable of zero means variance ?2A. This random process is applied to an ideal integrator,
- 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
- 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
- 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
- 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
- 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
- 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)
- 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 (?) of the output Y (t)
- 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
- A pair of noise processes n1 (t) and n2 (t) are related by n2 (t) = n1 (t) cos (2?fct + ?) - n1 (t) sin (2?fct + ?). Where fc is a constant and ? is the value of a random variables ? whose probably
- A random telegraph signal X (t), characterized by the autocorrelation function RX (?) = exp (-2v |? |) Where v is a constant, is applied to the low-pass RC filter of Figure. Determine the power
- 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),
- 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
- The output of and oscillator is described by X (t) = A cos (2?ft ??? ?), where A is constant, and f and ? are independent random variables. The probability density function of ? is defined by. Find
- 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
- A Guassian process X (t) of zero means and variance ?2X is passed through a full-wave rectifier, which is described by the input-output relation of Figure. Show that the probability density function
- 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
- A stationary, Guassian process X (t) with mean ?X and variance ?2X is passed through two linear filters with impulse responses b1 (t) and b2 (t) yielding processes y (t) and Z (t), as shown in
- 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
- 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
- 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
- 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
- (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?
- 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
- 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
- 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
- Consider a Guassian noise n (t) with zero mean and the power spectral density SN (??) shown in Figure. (a) Find the probability density function of the envelope of n (t). (b) What are the mean and
- In this computer experiment we study the statistical characteristics of a random process X (t) defined by X (t) = A cos (2??ct + ?) + W (t). Where the phase ? of the sinusoidal component is a
- 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
- (a) Coherent reception. (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
- 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
- The signal component of a coherent PSK system is defined by s(t) Ack sin(2πfct) + Ac √1 – k2 cos (2πfct) where 0
- 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
- 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
- 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
- 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
- The binary data stream 01101000 is applied to a ?/4-shifted DQPSK modulator that is initially in the state (?1 = ?E, ?2 = 0) in Figure. Using the relationship between input debits and carrier-phase
- 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) –
- 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
- 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
- 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
- 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
- Show that the power spectral density of a CAP signal with a total of L amplitude levels is defined by S(f) = ?2A/T |P(f)|2 where |P(f)| is the magnitude spectrum of the pass band in phase pulse p(t);
- 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
- 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
- (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
- A binary FSK signal with discontinuous phase is defined by where Eb is the signal energy per bit, Tb is the bit duration, and ?1 and ?2 are sample values of uniformly distributed random variables
- 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
- (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
- 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 ? is a design parameter defined in terms
- 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
- 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
- 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
- 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
- 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.
- 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
- 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
- (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
- 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
- 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
- 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 ?2n = 1 for all sub-channels. (a) Derive the formulas for
- 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
- 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
- Orthogonal frequency-division multiplexing may be viewed as a generalization of Mary FSK. Validate the rationale of this statement.
- Figure shows the block diagram of a continuous-rime Mth power loop for phase recovery in an Mary PSK receiver.(a) Show that the output of the Mth power-law device contains a tone of frequency Mfm
- (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
- 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.
- 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
- 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
- (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)
- The binary data stream 01101000 is applied to a ?/4-shifted DQPSK modulator that is initially in the state (?1 = ?E, ?2?= 0) in Figure. Using the relationship between input debits and carrier-phase
- 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
- 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 reproduced here:
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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)
- (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
- 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 = 1dBwCarrier
- Repeat Problem for a carrier frequency of 12 GHz.