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telecommunication engineering

Questions and Answers of Telecommunication Engineering

An analog signal is sampled, quantized, and encoded into a binary PCM wave. The number of representation levels used is 128. A synchronizing pulse is added at the end of each code word representing a
A binary PAM wave is to be transmitted over a baseband channel with an absolute maximum bandwidth of 75 kHz. The bit duration is 10μs. Find a raised-cosine spectrum that satisfies these requirements.
The duobinary, ternary, and bipolar signaling techniques have one common feature: They all employ three amplitude levels. In what way does the duobinary technique differ from the other two?
The binary data stream 001101001 is applied to the input of a duobinary system.(a) Construct the duobinary coder output and corresponding receiver output, without a precoder.(b) Suppose that owing to
Repeat Problem 4.21, assuming the use of a precoder in the transmitter.
The scheme shown in Figure may be viewed as a differential encoder (consisting of the modulo-2 adder and the 1-unit delay element) connected in cascade with a spec form of correlative coder
Consider a random binary wave x (t) in which the 1s and 0s occur with equal probability, the symbols in adjacent time slots are statistically independent, and symbol 1 is represented by A volts and
The binary data stream 011100101 is applied to the input of a modified duobinary system.(a) Construct the modified duobinary coder output and corresponding receiver output without a precoder.(b)
Repeat Problem 4.25 assuming the use of a precoder in the transmitter.
Consider a baseband M-ary system using M discrete amplitude levels. The receiver model is as shown in Figure, the operation of which is governed by the following assumptions; (a) The signal component
Suppose that in a baseband M-ary PAM system with M equally likely amplitude levels, as described in Problem 4.27, the average probability of symbol error Pe is less than 10-6 so as to make the
The amplitude distribution of cross-talk in a digital subscriber line may be modeled as Gaussian. Justify the validity of such a model.
(a) Derive the formula for the power spectral density of a transmitted signal using the 2B1Q line code.(b) Plot the power spectrum of the following line codesManchester codeModified duobinarv
In this problem we use the LMS algorithm to formulate an adaptive echo canceller for use in a digital subscriber line. The basic principle of adaptive echo cancellation is to synthesize a replica of
Figure shows the cascade connection of a linear channel and a synchronous tapped delay-line equalizer. The impulse response of the channel is denoted by c (t), and that of the equalizer is denoted by
Consider Equation (4.108), which defines the impulse response of a minimum mean- square error receiver.(a) Justify the validity of Equation (4.109) that is the Fourier-transformed version of Equation
Some radio systems suffer from multipath distortion, which is caused by the existence of more than one propagation path between the transmitter and the receiver. Consider a channel the output of
Let the sequence [x (n T)) denote die input applied to a tapped-delay-line equalizer. Show char inter-symbol interference is eliminated completely by the equalizer provided that its frequency
The step-size parameter plays a critical role in the operation of the LMS algorithm. In this context, discuss the following two issues:(a) Stability. If exceeds a certain critical value, the
Let the vectors w (1) [n] and w (2) [n] denote the tap-weights of the feed-forward and feed. back sections of the decision-feedback equalizer in Figure. Formulate the LMS algorithm for
In Section 4.11 we studied the eye diagram of a quaternary (M 4) PAM baseband transmission system under both noisy and band-limited conditions. In that experiment, the channel was assumed linear. In
In this experiment we study the root raised-cosine pulse due to Chennakeshu and Saulniex (1993). This pulse, denoted by p (t), has the following properties:The pulse p (t) is symmetric in time, that
In Section 3.7 we described line codes for pulse-code modulation. Referring to the material presented therein, formulate the signal constellations for the following line codes:(a) Unipolar
An 8-level PAM signal is defined by si (t) = Ai rect (t/T – ½) where A, = ±1, ±3, ±5, ±7. Formulate the signal constellation of {si (t)} 8i=1
Figure displays the waveforms of four signals s1 (t), s2 (t), s3 (t) and s4 (t). (a) Using the Gram-Schmidt orthogonalization procedure, find an orthonormal basis for this set of signals. (b)
(a) Using the Gram-Schmidt orthogonalizarion procedure, find a set of orrhonormal basis functions to represent the three signals s1 (t), s2 (t), and s3 (t) shown in Figure. (b) Express each of these
An orthogonal set of signals is characterized by the property that the inner product any pair of signals in the set is zero. Figure shows a pair of signals s1 (t) and s2 (t) that satisfy this
A set of 2M biorthogonal signals is obtained from a set of M orthogonal signals by augmenting it with the negative of each signal in the set.(a) The extension of orthogonal to biorthogonal signals
(a) A pair of signals si (t) and s k (t) have a common duration T. Show that the inner product of this pair of signals is given by below, where si and sk are the vector representations of si (t) and
Consider a pair of complex-valued signals s1 (t) and s2 (t) that are respectively represented by s1 (t) = a11 ?1 (t) + a12 ?2?(t), ? ? 2 (t) = a21 ?1 (t) + a22 ?2?(t), ? ? 1 (t) and ?2 (t) are both
Consider a random process X (t) expanded in the form N, where W? (t) is a remainder noise term. The {?i?(t)} Ni = 1 form an orthonormal set over interval 0 ? t ? T, and the Xj are defined by. Let W?
Consider the optimum detection of the sinusoidal signal s (t) sin (8πt/T), 0 ≤ t ≤ T, in additive white Gaussian noise.(a) Determine the correlator output assuming a noiseless input.(b)
Figure shows a pair of signals s1 (t) and s2 (t) that are orthogonal to each other over the observation interval 0 ? t ? 3T. The received signal is defined by below, where w (t) is white Gaussian
In the Manchester code, binary symbol 1is represented by the doublet pulse s (t) shown in Figure P5.13, and binary symbol 0 is represented by the negative of this pulse. Derive the formula for the
In the Bayes test, applied to a binary hypothesis testing problem where we have to choose one of two possible hypotheses H0 and H1, we minimize the risk R defined by R = C00 p0 P (say H0 | H0 is
Continuing with the four line codes considered in Problem 5.1, identity the line codes that have minimum average energy and those that do not. Compare your answers with the observations made on these
Consider the two constellations shown in Figure. Determine the orthonormal matrix Q that transforms the constellation shown in Figure a into the one shown in Figure b.
(a) The two signal constellations shown in Figure exhibit the same average ability of symbol error. Justify the validity of this statement. (b) Which of these two constellations has minimum average
Simplex (transorthogonal) signals are equally likely highly-correlated signals with the most negative correlation that can be achieved with a set of M orthogonal signals. That is, the correlation
In this problem we explore the approximations to the probability of an error, Pe, for the pair of antipodal signals shown in Figure in the presence of additive white Gaussian noise of power spectral
Consider the special case of a signal constellation that has a symmetric geometry with respect to the origin. Assume that the M message points of the constellation, pertaining to symbols m1, m2 . mm,
Let x1[n] be a sequence obtained by expanding the sequence x[n] = (1/4)n u[n] by a factor of 4; i.e., Find and sketch a six-point sequence q[n] whose six-point DFT Q[k] satisfies the two
A bandlimited continuous-time signal has a bandlimited power spectrum that is zero for |Ω| ≥ 2π(104) rad/s. The signal is sampled at a rate of 20,000 samples/s over a time interval of 10 s. The
For each of the following systems, determine whether the system is (1) Stable, (2) Causal, (3) Linear, (4) Time invariant,? (5) Memoryless:
(a) The impulse response h[n] of a linear time-invariant system is known to be zero, except in the interval N0 ≤ n ≤ N1. The input x[n] is known to be zero, except in the interval N2 ≤ n
By direct evaluation of the convolution sum, determine the step response of a linear time- invariant system whose impulse response ish[n] = α–nu[–n], 0 < α
Consider the linear constant-coefficient difference equation y[n] – ¾ y[n – 1] + 1/8 y[n-2] = 2x[n-1]. Determine y[n] for n ≥ 0 when x[n] = δ[n] and y[n] = 0, n < 0.
A causal linear time-invariant system is described by the difference equationy[n] – 5y[n – 1] + 6y[n – 2] = 2x[n – 1].(a) Determine the homogeneous response of the system, i.e., the possible
(a) Find the frequency response H(ej?) of the linear time-invariant system whose input and output satisfy the difference equation y[n] ? ? y[n ? 1] = x[n] + 2x[n ? 1] + x[n ? 2]. (b) Write a
Determine whether each of the following signals is periodic. If the signal is periodic, state its period.
An LTI system has impulse response h[n] = 5 (–1/2)nu[n]. Use the Fourier transform to find the output of this system when the input is x[n] = (1/3)n u[n].
Consider the difference equation (a) What are the impulse response, frequency response, and step response for the causal LTI system satisfying this difference equation.(b) What is the general
Determine the output of a linear time-invariant system if the impulse response h[n] and the input x[n] are as follows:(a) x[n] = u[n] and h[n] = anu[–n – 1], with α > 1.(b) x[n] = u[n – 4]
Consider an LTI system with frequency response Determine the output y[n] for all n if the input x[n] for all n is x[n] = sin(?n/4).
Consider a system with input x[n] and output y[n] that satisfy the difference equation y[n] = ny[n – 1] + x[n].The system is causal and satisfies initial-rest conditions; i.e., if x[n] = 0 for
Indicate which of the following discrete-time signals are eigenfunctions of stable, linear time-invariant discrete-time systems:(a) ej2πn/3(b) 3n(c) 2nu[– n –1](d) Cos(ω0n)(e) (1/4)n(f) (1/4)n
A single input-output relationship is given for each of the following three systems:(a) System A: x[n] = (1/3)n, y[n] = 2(1/3)n.(b) System B: x[n] = (1/2)n,
Consider the system illustrated in figure. The output of an LTI system with an impulse response h[n] = (1/4)nu[n + 10] is multiplied by a unit step function u[n] to yield the output of the over all
Consider the following difference equation:y[n] – ¼ y[n – 1] – 1/8 y[n – 2] = 3x[n].(a) Determine the general form of the homogeneous solution to this difference equation.(b) Both a causal
(a) Determine the Fourier transform of the sequence? (b) Consider the sequence Sketch w[n] and express W(ej?), the Fourier transform of w[n], in terms of R(ej?), the Fourier transform of r[n]. (c)
For each of the following impulse responses of LTI systems, indicate whether or not the system is causal:(a) h[n] = (1/2)nu[n](b) h[n] = (1/2)nu[n – 1](c) h[n] = (1/2)[n](d) h[n] = u[n + 2] – u[n
For each of the following impulse responses of LTI systems, indicate whether or not the system is stable:(a) h[n] = 4nu[n](b) h[n] = u[n] – u[n – 10](c) h[n] = 3nu[–n –1](d) h[n] = sin(π
Consider the difference equation representing a causal LTI systemy[n] + (1/α) y[n – 1] = x[n – 1].(a) Find the impulse response of the system, h[n], as a function of the constant a.(b) For what
Consider an arbitrary linear system with input x[n] and output y[n]. Show that if x[n] = 0 for all n, then y[n] must also be zero for all n.
For each of the pairs of sequences in figure use discrete convolution to find the response to the input x[n] of the linear time-invariant system with impulse response h[n]
Using the definition of linearity (Eqs. (2.26a)?(2.26b)), show that the ideal delay system (Example 2.3) and the moving-average system (Example 2.4) are both linear systems.
The impulse response of a linear time-invariant system is shown in figure. Determine and carefully sketch the response of this system to the input x[n] = u[n ? 4].
A linear time-invariant system has impulse response h[n] = u[n]. Determine the response of this system to the input x[n] shown in figure and described as
Which of the following discrete-time signals could be eigenfunctions of any stable LTI system?(a) 5nu[n](b) ej2ωn(c) ejωn + e j2ωn(d) 5n(e) 5n. ej2ωn
Three systems A, B, and C have the inputs and outputs indicated in Figure. Determine whether each system could be LTI. If your answer is yes, specify whether there could be more than one LTI system
Determine which of the following signals is periodic. If a signal is periodic, determine its period.(a) x[n] = ej(2πn/5)(b) x[n] = sin(π n/19)(c) x[n] = nejπn(d) x[n] = ejn
A discrete-time signal x[n] is shown in Figure. Sketch and label carefully each of the following signals: (a) x[n ? 2] (b) x[4 ? n] (c) x[2n] (d) x[n]u[2 ? n] (e) x[n ? 1]?[n - 3]
For each of the following systems, determine whether the system is (1) stable, (2) causal, (3) linear, and (4) time invariant.
Consider the difference equationy[n] + 1/15 y[n – 1] – 2/5 y[n – 2] = x[n].(a) Determine the general form of the homogeneous solution to this equation.(b) Both a causal and an anti causal LTI
Consider an LTI system with frequency response Determine the output y [n] for all n if the input for all n is x[n] = cos (?n/2).
Consider an LTI system with |H(ej?)| = 1, and let arg[H(ej?)] be as shown in Figure. If the input is x[n] = cos (3?/2 n + ?/4), determine the output y[n].
The input-output pair shown in Figure is given for a stable LTI system.? (a) Determine the response to the input x1[n] in Figure. (b) Determine the impulse response of the system
The system T in Figure is known to be time invariant, when the inputs to the system are x1[n], x2 [n], and x3[n], the responses of the system are y1[n], y2[n], and y3[n], as shown. (a) Determine
The system L in Figure is known to be linear. Shown are three output signals y1[n], y2[n], and y3[n], in response to the input signals x1[n], x2[n], and x3[n], respectively. (a) Determine whether the
Consider a discrete-time linear time-invariant system with impulse response h[n]. If the input x[n] is a periodic sequence with period N (i.e., if x[n] = x[n + N]), show that the output y[n] is also
Consider a system with input x[n] and output y[n]. The input-output relation for the system is defined by the following two properties:1. y[n] – ay[n – 1] = x[n],2. y[0] = 1.(a) Determine whether
Consider the linear time-invariant system with impulse response Determine the steady-state response, i.e., the response for large n, to the excitation x[n] = cos(? n)u[n].
A linear time-invariant system has frequency response The input to the system is a periodic unit-impulse train with period N = 16; i.e., Find the output of the system.
Consider the system in Figure. (a) Find the impulse response h[n] of the overall system. (b) Find the frequency response of the overall system. (c) Specify a difference equation that relates the
For X(ej?) = 1/(1 ? ?e? j?), with ? 1
Let X(ejω) denote the Fourier transform of the signal x[n] shown in Figure. Perform the following calculations without explicitly evaluating X(ejω):(a) Evaluate X(ejω)|ω=0.(b) Evaluate
For the system in Figure P2.45-1, determine the output y[n] when the input x[n] is ?[n] and H(ej?) is an ideal lowpass filter as indicated, i.e.,
A sequence has the discrete-time Fourier transform
A linear time-invariant system is described by the input-output relation y[n] = x[n] + 2x[n – 1] + x[n – 2](a) Determine h[n], the impulse response of the system. (b) Is this a stable
Let the real discrete-time signal x[n] with Fourier transform X(ej?) be the input to a system with the output defined by (a) Sketch the discrete-time signal s[n] = 1 + cos(? n) and its
Consider a discrete-time LTI system with frequency response H(ej?) and corresponding impulse response h[n]. (a) We are first given the following three clues about the system: (i) The system is
Consider the three sequencesv[n] = u[n] – u[n – 6],w[n] = δ[n] +2δ[n – 2] + δ[n – 4],q[n] = v[n] * w[n].
A linear time-invariant system has impulse response h[n] = αnu[n].(a) Determine y1[n], the response of the system to the input x1[n] = ej(π/2)n.(b) Use the result of Part(a) to help to determine
The frequency response of an LTI system is H(ejω) = e–jω/4, –π < ω ≤ π.Determine the output of the system, y[n], when the input is x[n] = cos(5πn/2). Express your answer in
Consider the cascade of LTI discrete-time systems shown in figure.? The first system is described by the equation? and the second system is described by the equation? y[n] = w[n] ? w[n ? 1].? The
Consider an LTI system with frequency response H(e jω) = e–j [(ω/2) + (π/4)], – π < ω ≤ π.Determine y[n], the output of this system, if the input isx[n] = cos (15πn/4
For the system shown in Figure, System 1 is a memory less nonlinear system. System 2 determines the value of A according to the relation. Specifically, consider the class of inputs of the form x[n]
Consider a system S with input x[n] and output y[n] related according to the block diagram in Figure. The input x[n] is multiplied by e?j?0n, and the product is passed through a stable LTI system
An ideal lowpass filter with zero delay has impulse response h1p[n] and frequency response (a) A new filter is defined by the equation h1[n] = (?1)nh1p?[n] = ej?n h1p[n]. Determine an equation for
The LTI system is referred to as a 900 ?phase shifter and is used to generate what is referred to as an analytic signal w[n] as shown in Figure. Specifically, the analytic signal w[n] is a

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