- 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
- 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
- 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
- Let X and Y are statistically independent Gaussian-distributed random variables, each with zero mean and unit variance. Define the Gaussian process Z
- 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
- 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
- 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
- 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
- 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
- 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
- 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)
- 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
- A random telegraph signal X (t), characterized by the autocorrelation function RX (?) = exp (-2v |? |) Where v is a constant, is applied to the
- 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
- 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
- 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
- A stationary, Guassian process X (t) has zero means and power spectral density SX (f). Determine the probability density function of a random
- 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
- 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
- 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)
- 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
- 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
- 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)
- 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
- (a) Determine the condition which the impulse response b (t) must satisfy to achieve this requirement.(b) What is the corresponding condition on the
- 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
- 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 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
- 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
- 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
- In this computer experiment we continue the study of the multipath channel described in Section 1.14. Specifically, consider the situation where the
- (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
- Consider a phase-locked loop consisting of a multiplier, loop filter, and voltage controlled oscillator (VCO). Let the signal applied to the
- 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
- 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
- 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
- Vestigial sideband modulation (VSB), discussed in Chapter, offers another modulation method for pass b and data transmission.(a) In particular, 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
- 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
- An interesting property of π/4-shifted DQPSK signals is that they can be demodulated using an FM discriminator. Demonstrate the validity of this
- 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
- 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
- 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
- The two-dimensional CAP and Mary QAM schemes are closely related. Do thefo11oWifl(a) Given a QAM system, with a prescribed number of amplitude
- 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
- You are given the baseband raised-cosine spectrum G(f) pertaining to a certain roll off factor α. Describe a frequency-domain procedure for
- 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
- (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
- 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
- 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
- (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
- 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
- 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.
- 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
- 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
- The binary sequence 1100100010 is applied rn the DPSTC transmitter of Figure a. (a) Sketch the resulting waveform at the transmitter output. (b)
- 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
- 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)
- 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
- 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
- (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.
- The V.32 modem standard with non-redundant coding uses a rectangular 16-QAM constellation. The model specifications are as follows: Carrier frequency
- 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
- 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
- In this problem we explore the use of singular value decomposition (SVD) as an alternative to the discrete Fourier transform for vector coding. This
- Compare the performance of DMT and CAP with respect to the following channel impairments:(a) Impulse noise.(b) Narrowband interference.Assume that
- 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
- (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
- 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)
- 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
- 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
- Consider a phase-locked loop consisting of a multiplier, loop filter, and voltage controlled oscillator (VCO). Let the signal applied to the
- (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 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
- 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
- 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),
- 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
- 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
- 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
- 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)
- 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
- 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
- A direct sequence spread binary phase-shift keying system uses a feedback shift register of length 19 for the generation of the PN sequence.
- 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
- In section 7.5, we presented an analysis on the signal-space dimensionality and processing gain of a direct sequence spread-spectrum system using
- 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
- 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
- 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],
- (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,
- 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
- Repeat Problem for a carrier frequency of 12 GHz.