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telecommunication engineering

Questions and Answers of Telecommunication Engineering

A causal all-pass system Hap(z) has input x[n] and output y[n]. (a) If x[n] is a real minimum-phase sequence (which also implies that x[n] = 0 for n (b) Show that Eq. (p5.67-1) holds even if x[n] is
Determine whether the following statement is true or false. If it is true, concisely state your reasoning. If it is false, give a counterexample.Statement: If the system function H(z) has poles
A signal x[n] is processed through an LTI system H(z) and then down sampled by a factor of 2 to yield y[n] as indicated in Figure. Also, as shown in the same figure, x[n] is first down sampled and
Consider a discrete-time LTI system with a real-valued impulse response h[n]. We want to find h [n], or equivalently, the system function H(z) from the autocorrelation chh[?] of the impulse response.
Let h[n] and H(z) denote the impulse response and system function of a stable all-pass LTI system. Let hi[n] denote the impulse response of the (stable) LTI inverse system. Assume that h[n] is real.
Consider a real-valued sequence x[n] for which X(ejω) = 0 for π/4 ≤ |ω| ≤ π. One sequence value of x[n] may have been corrupted, and we would like to recover it approximately or exactly. With
Show that if h[n] is an N-point FIR filter such that h[n] = h[N – 1 – n] and H(z0) = 0, then H(1/z0) = 0. This shows that even symmetric linear-phase FIR filters have zeros that are reciprocal
Determine the system function of the two networks in figure, and show that they have the samepoles.
The signal flow graph of Figure represents a linear difference equation with constant coefficients. Determine the difference equation that relates the output y[n] to the input x[n].
Consider the system in Figure (d).(a) Find the system function relating the z-transforms of the input and output.(b) Write the difference equation that is satisfied by the input sequence x[n] and the
A linear time-invariant system is realized by the flow graph shown in Figure. (a) Write the difference equation relating x[n] and y[n] for this flow graph. (b) What is the system function of the
Determine the impulse response of each of the systems inFigure.
Let x[n] and y[n] be N-point sequences (N > 3) related by the following difference equation: y[n] – ¼ y[n – 2] = x[n – 2] – ¼ x[n]. Draw a direct form II signal flow graph for
The signal flow graph in Figure represents an LTI system. Determine a difference equation that gives a relationship between the input x[n] and the output y[n] of this system. As usual, all branches
Figure shows the signal flow graph for a causal discrete-time LTI system. Branches without gains explicitly indicated have a gain of unity. (a) Determine h[1], the impulse response at n = 1. (b)
Consider the signal flow graph shown in Figure. (a) Using the node variables indicated, write the set of difference equations represented by this network. (b) Draw the flow graph of an equivalent
Consider a causal LTI system S with impulse response h[n] and system function (a) Draw a direct form II flow graph for the system S. (b) Draw the transposed form of the flow graph in Part(a).
For the linear time-invariant system described by the flow graph in Figure, determine the difference equation relating the input x[n] to the output y[n]
Draw the signal flow graph for the direct form I implementation of the LTI system with system function
Draw the signal flow graph for the direct form II implementation of the LTI system with system function
Draw the signal flow graph for the transposed direct form II implementation of the LTI system with system function
Consider the signal flow graph shown in figure. (a) Draw the signal flow graph that results from applying the transposition theorem to this signal flow graph. (b) Confirm that the transposed signal
Consider the causal LTI system with system function H(z) = 1 – 1/3 z–1 + 1/6 z–2 + z–3(a) Draw the signal flow graph for the direct form implementation of this system.(b) Draw the signal
For some choices of the parameter ?, the signal flow graph in Figure can be replaced by a second-order direct form II signal flow graph implementing the same system function. Give one such choice for
Consider the causal LTI system with the system function Draw a signal flow graph that implements this system as a parallel combination of first-order transposed direct form IIsections.
Draw a signal flow graph implementing the system function as a cascade of second-order transposed direct form II sections with realcoefficients. ? ?
For many applications, it is useful to have a system that will generate a sinusoidal sequence. One possible way to do this is with a system whose impulse response is h[n] = ejω0n u[n]. The real and
For the system function draw the flow graphs of all possible realizations for this system as cascades of first-ordersystem.
Consider a causal linear time-invariant system whose system function is (a) Draw the signal flow graphs for implementations of the system in each of the following forms: (i) Direct form I (ii)
Several flow graphs are shown in Figure. Determine the transpose of each flow graph, and verify that in each case the original and transposed flow graphs have the same systemfunction.
Consider the system in Figure. (a) Find the system function relating the z-transforms of the input and output. (b) Write the difference equation that is satisfied by the input sequence x[n] and the
A linear time-invariant system with system function is to be implemented using a flow graph of the form shown in Figure. (a) Fill in all the coefficients in the diagram of Figure. Is your solution
The flow graph shown in Figure is noncomputable; i.e., it is not possible to compute the output using the difference equations represented by the flow graph because it contains a closed loop having
The impulse response of a linear time-invariant system is lal." alt = "The impulse response of a linear time-invariant system is (a)" alt = "The impulse response of a linear time-invariant system
Consider an FIR system whose impulse response isThis system is an example of a class of filters known as frequency-sampling filters, problem 6.37 discusses these filters in detail. In this problem,
Consider the discrete-time system depicted in Figure, (a) Write the set of difference equations represented by the flow graph of Figure. (b) Determine the system function H1(z) = y1(z)/X(z) of the
The three networks in Figure are all equivalent implementations of the same two-input, two-output linear time-invariant system. (a) Write the difference equations for network A.? (b) Find values of
All branches of the signal flow graphs in this problem have unity gain unless specifically indicated otherwise. (a) The signal flow graph of system A, shown in Figure, represents a causal LTI system.
Consider an all-pass system whose system function is? (a) Draw the direct from I signal flow graph for the system. How many delays and multipliers do you need? (Do not count multiplying by ? 1.) (b)
The flow graph shown in Figure is an implementation of a causal, LTI system. (a) Draw the transpose of the signal flow graph. (b) For either the original system of its transpose, determine the
Consider a linear time-invariant system with two inputs, as depicted in Figure. Let h1[n] and h2[n] be the impulse responses from nodes 1 and 2, respectively, to the output, node 3. Show that if
The networks in Figure all have the same system function. Assume that the systems in the figure are implemented using (B + 1)-bit fixed-point arithmetic in all the computations. Assume also that all
A causal LTI system has a system function (a) Is this system stable? (b) If the coefficients are rounded to the nearest tenth, would the resulting system best able?
Consider a causal continuous-time system with impulse response hc(t) and system function? (a) Use impulse invariance to determine H1(z) for a discrete-time system such that h1[n] = hc(n T). (b) Use
A discrete-time lowpass filter is to be designed by applying the impulse invariance method to a continuous- time Butterworth filter having magnitude-squared functionThe specifications for the
We wish to use impulse invariance or the bilinear transformation to design a discrete-time filter that meets specifications of the following form: For historical reasons, most of the design
The system function of a discrete-time system is (a) Assume that this discrete-time filter was designed by the impulse invariance method with Td = 2; i.e., h[n] = 2hc(2n), where hc(t) is real. Find
We wish to use the Kaiser Window method to design a discrete-time filter with generalized linear phase that meets specifications of the following form:
We wish to use the Kaiser Window method to design a real-valued FIR filter with generalized linear phase that meets the following specifications: This specification is to be met by applying the
We are interested in implementing a continuous-time LTI lowpass filter H(j?) using the system shown in Fig. when the discrete-time system has frequency response Hd(ej?). The sampling time T = 10?4
Suppose we design a discrete-time filter using the impulse invariance technique with an ideal continuous-time lowpass filter as a prototype. The prototype filter has a cutoff frequency of Ωc =
We wish to design a discrete-time lowpass filter using the bilinear transformation on a continuous-time ideal lowpass filter. Assume that the continuous-time prototype filter has cutoff frequency
Suppose that we have an ideal discrete-time lowpass filter with cutoff frequency ωc = π/4. In addition, we are told that this filter resulted from applying impulse invariance to a continuous-time
An ideal discrete-time highpass filter with cutoff frequency ωc = π/2 was designed using the bilinear transformation with T = 1 ms. what was the cutoff frequency Ωc for the prototype
An ideal discrete-time lowpass filter with cutoff frequency ωc = 2π/5 was designed using impulse invariance from an ideal continuous-time lowpass filter with cutoff frequency Ωc = 2π (4000)
The bilinear transformation is used to design an ideal discrete-time lowpass filter with cutoff frequency ωc = 3π/5 from an ideal continuous-time lowpass filter with cutoff frequency Ωc = 2π
Wish to design an FIR lowpass filter satisfying the specifications 0.95 < H(ejω) < 1.05, 0 ≤ |ω| ≤ 0.25π,−0.1 < H(ejω)
We wish to design an FIR lowpass filter satisfying the specifications 0.98 < H(ejω) < 1.02, 0 ≤ |ω| ≤ 0.63π,−0.15 < H(ejω)
Suppose that we wish to design a bandpass filter satisfying the following specification: −0.02 < |H(ejω)| < 0.02, 0 ≤ |ω| ≤ 0.2π,
Suppose that we wish to design a highpass filter satisfying the following specification: −0.04 < |H(ejω)| < 0.04, 0 ≤ |ω| ≤ 0.2π,0.995
We wish to design a discrete-time ideal bandpass filter that has a passband π/4 ≤ ω ≤ π/2 by applying impulse invariance to an ideal continuous-time bandpass filter with passband 2π(300) ≤
Specify whether the following statement is true or false. Justify your answer, if the bilinear transformation is used to transform a continuous-time all-pass system to discrete-time system, the
Suppose that we are given a continuous-time lowpass filter with frequency response Hc(jω) such that 1− δ1 ≤ | Hc(jΩ) | ≤ 1 + δ1, |Ω| ≤ Ωp,|Hc(j Ω)|
A continuous-time filter with impulse response hc(t) and frequency-response magnitude is to be used as the prototype for the design of a discrete-time filter. The resulting discrete-time system is to
Assume that Hc(s) has an r th-order pole at s = s0, so that Hc(s) can be expressed as where Gc(s) has only first-order poles. Assume Hc(s) is causal. (a) Give a formula for determining the constants
Figure shows the frequency response Ae(ej?) of a discrete-time FIR system for which the impulse response is (a) Show that Ae(ej?) cannot correspond to an FIR filter generated by the parks-McClellan
Consider the system in Figure. 1. Assume that Xc(j?) = 0 for |?| ? ? / T and that denotes an ideal lowpass reconstruction filter. 2. The D/A converter has a built-in zero-order-hold circuit, so that
In this problem, we consider the effect of mapping continuous-time filters to discrete-time filters by replacing derivatives in the differential equation for a continuous-time filter by central
Let h[n] be the optimal type I equiripple lowpass filter shown in Figure, designed with weighting function W(ej?) and desired frequency response Hd(ej?). For simplicity, assume that the filter is
Suppose that you have used the parks-McClellan algorithm to design a causal FIR linear-phase system. The system function of this system is denoted H(z). The length of the impulse response is 25
Suppose xc(t) is a periodic continuous-time signal with period 1 ms and for which the Fourier series isThe Fourier series coefficients αk are zero for |k| > 9.xc(t) is sampled with a sample spacing
Suppose x[n] is a periodic sequence with period N. Then x[n] is also periodic with 3N. Let X[k] denote the DFS coefficients of x[n] considered as a periodic sequence with period N, and let
Figure shows three periodic sequences x1[n] through x3[n]. These sequences can be expressed in a Fourier series as (a) For which sequences can the time origin be chosen such that all the X[k] are
Consider the sequence x[n] given by x[n] = ?n u[n]. A periodic sequence x[n] is constructed from x[n] in the following way: (a) Determine the Fourier transform X(ej?) of x[n]. (b) Determine the
Compute the DFT of each of the following finite-length sequences considered to be of length N (where N is even):
Consider the complex sequence (a) Find the Fourier transform X(ej?) of x[n]. (b) Find the N-point DFT X[k] of the finite-length sequence x[n]. (c) Find the DFT of x[n] for case ?0 = 2?k0/N, where k0
Consider the finite-length sequence x[n] in Figure. Let X(z) be the z-transform of x[n]. If we sample X(z) at z = ej(2?/4)k, k = 0, 1, 2, 3, we obtain? X1[k] = X(z) |z = ej(2?/4)k,? ? ? ? ? ? k = 0,
Let X(ejω) denote the Fourier transform of the sequence x[n] = (1/2)n u[n]. Let y[n] denote a finite-duration sequence of length 10; i.e., y[n] = 0, n < 0, and y[n] = n ≥ 10. The 10-point DFT
Consider a 20-point finite-duration sequence x[n] such that x[n] = 0 outside 0 ≤ n ≤ 19, and let X(ejω) represent the Fourier transform of x[n].(a) If it is desired to evaluate X(ejω) at ω =
The two eight-point sequences x1[n] and x2[n] shown in Figure have DFTs X1[k] and X2[k], respectively. Determine the relationship between X1[k] and X2[k].
Figure shows two finite-length sequences x1[n] and x2[n]. Sketch their six-point circularconvolution.
Suppose we have two four-point sequences x[n] and h[n] as follows: x[n] = cos (πn/2). n = 0, 1, 2, 3. h[n] = 2n,
Consider the finite-length sequence x[n] in Figure. The five-point DFT of x[n] is denoted by x[k]. Plot the sequence y[n] whose DFT is
Two finite-length signals, x1[n] and x2[n], are sketched in Figure. Assume that x1[n] and x2[n] are zero outside of the region shown in the figure. Let x3[n] be the eight-point circular convolution
Figure shows two sequence x1[n] and x2[n]. The value of x2[n] at time n = 3 is not known, but is shown as a variable ?. Figure shows y[n], the four-point circular convolution of x1[n] and x2[n].
Figure shows a six-point discrete-time sequence x[n]. Assume that x[n] = 0 outside the interval shown. The value of x[4] is not known and is represented as b, that the sample shown for b in the
Figure shows two finite-length sequences x1[n] and x2[n]. What is the smallest N such that the N-point circular convolution of x1[n] and x2[n] are equal to the linear convolution of these sequences,
Figure shows a sequence x[n] for which the value of x[3] is an unknown constant c. The sample with amplitude c is not necessarily drawn to scale. Let X1[k] = X[k]ej2?3k/5. Where X[k] is the
Two finite-length sequences x[n] and x1[n] are shown in Figure. The DFTs of these sequences, X[k] and X1[k], respectively, are related by equation? X1[k] = X[k]e ?j (2?km/6), where m is an unknown
Two finite-length sequences x[n] and x1[n] are shown in Figure. The N-point DFTs of these sequences, X[k] and X1[k], respectively, are relate by the equation X1[k] = X[k]ej2?k2/N, where N is an
(a) Figure shows two periodic sequences, x1[n] and x2[n], with period N = 7. Find a sequence y1[n] whose DFS is equal to the product of the DFS of x1[n] and the DFS of x2[n], i.e.,? Y1[k] = X1[k]
x[n] denotes a finite-length sequence of length N. show thatx[((– n))N] = x[((N – n))N].
Consider a finite-duration sequence x[n] of length p such that x[n] = 0 for n < 0 and n ≥ P. We want to compute samples of the Fourier transform at the N equally spaced frequencies ωk =
Consider a real finite-length sequence x[n] with Fourier transform X(ejω) and DFT X[k]. If Jm{X[k]} = 0, k= 0,1,…., N – 1. Can we
Consider the finite-length sequence x[n] in Figure. The four-point DFT of x[n] is denoted X[k]. Plot the sequence y[n] whose DFT is? Y[k] = W43k X[k].
Consider the real finite-length sequence x[n] shown in Figure. (a) Sketch the finite-length sequence y[n] whose six-point DFT is? Y[k] = W64k X[k], where X[k] is the six-point DFT of x[n]. (b)
Figure shows two sequences, (a) Determine and sketch the linear convolution x1[n] * x2[n]. (b) Determine and sketch the 100-point circular convolution x1[n] (100) x2[n]. (c) Determine and sketch the
Figure shows a finite-length sequence x[n]. Sketch the sequences? x1[n] = x[((n ? 2))4].? ? ? ? ? ? ? ?? 0 ? n ? 3, ?and? x2[n] = x[((? n))4]. ? ? ? ? ? ? ? ? ? 0 ? n ? 3
Figure shows two finite-length sequences. Sketch their N-point circular convolution for N = 6 and for N = 10.
Consider a finite-length sequence x[n] of length N: i.e., x[n] = 0 outside 0 ≤ n ≤ N – 1.X(ejω) denotes the Fourier transform of x[n]. X[k] denotes

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