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telecommunication engineering

Questions and Answers of Telecommunication Engineering

The DFT of a finite-duration sequence corresponds to samples of its z-transform on the unit circle. For example, the DFT of a 10-point sequence x[n] corresponds to samples of X(z) at the 10 equally
Consider a finite-length sequence x[n] of length N as indicated in Figure. (The solid line is used to suggest the envelope of the sequence values between 0 and N ? 1.) Two finite-length sequence
The even part of a real sequence x[n] is defined by Suppose that x[n] is a real finite-length sequence defined such that x[n] = 0 for n (a) Is Re{x[k]} the DFT of xe[n]? (b) What is the inverse DFT
Determine a sequence x[n] that satisfies all of the following three conditions: Condition 1: The Fourier transform of x[n] has the form X(ej?) = 1 + A1 cos ? + A2 cos 2?, Where A1 and A2 are some
Consider the finite-length sequencex[n] = 2δ[n] + δ[n – 1] + [n – 3]. We perform the following operation on this sequence:(i) We compute the five-point DFT X[k].(ii) We compute a
x[n] is a real-valued finite-length sequence of length 10 and is nonzero in the interval from 0 to 9, i.e., x[n] = 0, n < 0, n ≥ 10,x[n] ≠ 0,
Two finite-length sequences x1[n] and x2[n], which are zero outside the interval 0 ? n ? 99, are circularly convolved to form a new sequence y[n]; i.e., ? If, in fact, x1[n] is nonzero only for 10 ?
Consider two finite-length sequences x[n] and h[n] for which x[n] = 0 outside the interval 0 ≤ n ≤ 49 and h[n] = 0 outside the interval 0 ≤ n ≤ 9.(a) What is the maximum possible number of
Consider two finite-duration sequences x[n] and y[n]. x[n] is zero for n 19, as indicated in Figure Let w[n] denote the linear convolution of x[n] and y[n], Let g[n] denote the 40-point circular
Two finite-duration sequences h1[n] and h2[n] of length 8 are sketched in Figure. The two sequences are related by a circular shift, i.e., h1[n] = h2[((n ? m)) 8]. (a) Specify whether the magnitudes
We want to implement the linear convolution of a 10,000-point sequence with an FIR impulse response that is 100 points long. The convolution is to be implemented by using DFT s and inverse DFT s of
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
Let x2[n] be a real-valued five-point sequence whose seven-point DFT is denoted by X2[k]. If Real{X2[k]} is the seven-point DFT of g[n], show that g[0] = x2[0], and determine the relationship between
Shown in Figure are three finite-length sequences of length 5. Xi(ej?) denotes the DTFT of xi[n], and Xi[k] denotes the five-point DFT of xi[n]. For each of the following properties, indicate which
Letx[n] = 0, n < 0, n > 7, be a real eight-point sequence, and let X[k] be its eight-point DFT.(a) Evaluate in terms of x[n].(b) Let u[n] = 0, n < 0, n > 7,be an
(a) Suppose x[n] = 0, n < 0, n > (N – 1), is an N-point sequence having at least one nonzero sample. Is it possible for such a sequence to have a
Suppose x1[n] is an infinite-length, stable (i.e., absolutely summable) sequence with z-transform given byx1(z) = 1/ 1−1/3z−1.Suppose x2[n] is a finite-length sequence of length N, and the
Read each part of this problem carefully to note the differences among parts. (a) Consider the signal which can be represented by the IDFT equation as where X8[k] is the eight-point DFT of x[n]. Plot
In deriving the DFS analysis equation (8.11), we used the identity of Eq. (8.7). To verify this identity, we will consider the two conditions k ? r = m N and k ? r ? m N separately. (a) For k ? r = m
In section 8.2, we stated the property that if? x1[n] = x[n ? m], ?then? X1[k] = WkmN X[k]. Where X[k] and X1[k] are the DFS coefficients of x[n] and x1[n], respectively. In this problem, we consider
(a) Table 8.1 lists a number of symmetry properties of the discrete Fourier series for periodic sequences, several of which we repeat here. Prove that each of these properties is true. In carrying
We stated in Section 8.4 that a direct relationship between X(ejω) and X[k] can be derived, where X[k] is the DFS coefficients of a periodic sequence and X(ejω) is the “Fourier transform of one
Let X[k] denote the N-point DFT of the N-point sequence x[n].(a) Show that ifx[n] = − x[N – 1 – n],then X[0] = 0. Consider separately the cases of N even and N odd.(b) Show that if N is even
In Section 2.8, the conjugate-symmetric and conjugate-antisymmetric components of a sequence x[n] were defined, respectively, asxe[n] = ½ (x[n] + x*[– n]).X0[n] = ½ (x[n] – x*[– n]).In
Show from Eqs. (8.65) and (8.66) that with x[n] as an N-point sequence and X[k] as its N-point DFT, This equation is commonly referred to as Parseval?s relation for the DFT.
x[n] is a real-valued, nonnegative, finite-length sequence of length N; i.e., x[n] is real and nonnegative for 0 ≤ n ≤ N – 1 and is zero otherwise. The N-point DFT of x[n] is X[k], and the
X[n] and y[n] are two real-valued, positive, finite-length sequences of length 256; i.e.,? x[n] > 0, 0 ? n ? 225, y[n] > 0, 0 ? n ? 255, x[n] = y[n] = 0, otherwise r[n] denotes the linear
Y[n] is the output of a stable LTI system with system function H(z) = 1/(z – bz –1), where b is a known constant. We would like to recover the input signal x[n] by operating on y[n]. The
A modified discrete Fourier transform (MDFT) was proposed (Vernet, 1971) that computes samples of the z-transform on the unit circle offset from those computed by the DFT. In particular, with XM[k]
In some applications in coding theory, it is necessary to compute a 63-point circular convolution of two 63-point sequences x[n] and h[n]. Suppose that the only computational devices a available are
We want to filter a very long string of data with an FIR filter whose impulse response is 50 samples long. We wish to implement this filter with a DFT using the overlap-save technique. The procedure
A problem that often arises in practice is one in which a distorted signal y[n] is the output that results when a desired signal x[n] has been filtered by an LTI system. We wish to recover the
In this problem, you will examine the use of the DFT to implement the filtering necessary for the discrete-time interpolation, or up sampling, of a signal. Assume that the discrete-time signal x[n]
Derive Eq. using Eqs, (8.153) using Eqs, (8.164) and(8.165).
Consider the following procedure? (a) Form the sequence v[n] = x2[2n] where x2[n] is given by Eq. (8.166). This yields? v[n] = x[2n]? ? ? ? ? ? ? ? ? n = 0, 1 ?. N/2 ? 1 v[N ? 1 ? n] = x[2n + 1],? ?
Derive Eq. (8.156) using Eqs (8.174) and(8.175).
(a) Use Parseval’s theorem for the DFT to derive a relationship between ∑ | Xc1[k] |2 and ∑ |x[n]|2.(b) Use Parseval’s theorem for the DFT to derive a relationship between ∑ |Xc2[k] ∑|2
Suppose that a computer program is available for computing the DFT i.e., the input to the program is the sequence x[n] and the output is the DFT X[k]. Show how the input and/or output sequences may
Figure shows the graph representation of a decimation-in-time FFT algorithm for N = 8. The heavy line shows a path from sample x[n] to DFT sample X [2]. (a) What is the ?fain? along the path that is
Figure shows the flow graph for an 8-point decimation-in?time FFT algorithm, Let x[n] be the sequence whose DFT is X[k]. In the flow graph, A[?], B[?], C[?], and D[?] represent separate arrays that
In implementing an FFT algorithm, it is sometimes useful to generate the powers of WN with a recursive difference equation, or oscillator. In this problem we consider a radix-2 decimation-in-time
Computing the DFT generally requires complex multiplications. Consider the product X + JY = (A + JB) (C + JD) = (AC – BD) + J (BC + AD). In this form, a complex multiplication requires four real
Consider the butterfly in figure. This butterfly was extracted from a signal flow graph implementing an FFT algorithm. Choose the most accurate statement from the following list: 1. The butterfly was
Consider the system shown in figure. If the input to the system, x[n], is a 32-point sequence in the interval 0 ? n ? 31, the output y[n] at n = 32 is equal to X(e?j?) evaluated at a specific
A finite-length signal x[n] is nonzero in the interval 0 ? n ? 19. This signal is the input to the system shown in figure, where the output of the system, y[n], for the interval n = 19 ?.. 28 can be
The butterfly flow graph in figure can be used to compute the DFT of a sequence of length N = 2v ?in-place,? i.e., using a single array of complex-valued registers. Assume this array of registers
Consider the system shown in Figure, with, it is desired that the output of the system, y[n + 11] = X(e j?n), where ?n = (2?/19) + n(2?/10) for n = 0,?,4, Give the correct value for the sequence r[n]
Assume that you wish to sort a sequence x[n] of length N = 16 into bit-reversed order for input to an FFT algorithm. Give the new sample order for the bit-reversed sequence.
The butterfly in Figure was taken from a decimation-in-time FFT with N = 16. Assume that the four stages of the signal flow graph are indexed by m = 1,?., 4. What are the possible values of r for
Suppose you have two programs for computing the DFT of a sequence x[n] that has N = 2v nonzero samples. Program A computes the DFT by directly implementing the definition of the DFT sum from Eq. and
The butterfly in figure was taken from a decimation-in-time FFT with N = 16. Assume that the four stages of the signal flow graph are indexed by m = 1,?, 4. Which of the four stages have butterflies
Suppose you are told that an N = 32 FFT algorithm has a “twiddle” factor W232 for one of the butterflies in its fifth (last) stage. Is the FFT a decimation-in-time or decimation-in-frequency
Suppose you have a signal x[n] with 1021 nonzero samples whose discrete-time Fourier transform you wish to estimate by computing the DFT. You find that it takes your computer 100 seconds to compute
Consider the signal flow graph in figure. Suppose that the input to the system x[n] is an 8-point sequence. Choose the values of a and b such that y[8] = X(ej6?/8).
Suppose that you time-reverse and delay a real-valued 32-point sequence x [n] to obtain x1[n] = x[32 ? n]. If x1[n] is used as the input for the system in figure, find an expression for y[32] in
In Section 9.2, we used the fact that W-kNN = 1 to derive a recurrence algorithm for computing a specific DFT value X [k] for a finite-length sequence x[n], = 0, 1, ,?, N ? 1. (a) Using the fact that
Construct a flow graph for a 16-point radix-2 decimation-in-time FFT algorithm. Label all multipliers in terms of powers of W16, and also label any branch transmittances that are equal to – 1.
This problem deals with the efficient computation of samples of the z-transform of a finite-length sequence. Using the chirp transform algorithm, develop a procedure for computing valued of X (z) at
The N-point DFT of the N-point sequence x[n] = e–j (π/N) n2, for N even, is X[k] = √Ne– jπ/4 ej(π/N) k2.Determine the 2N-point DFT of the 2N-point sequence y[n] = e– j(π/N) n2, assuming
We are given a finite-length sequence x[n] of length 627 (i.e., x [n] = 0 for n 626), and we have available an FFT program that will compute the DFT of a sequence of any length N = 2v. For the given
A finite-length signal of length L = 500 (x[n] = 0 for n < 0 and n > L – 1) is obtained by sampling a continuous-time signal with sampling rate 10,000 samples per second. We wish to compute
Suppose that a finite-length sequence x[n] has the N-point DFT X [k], and suppose that the sequence satisfies the symmetry condition x[n] = − x[((n + N/2)) N], 0 ≤ n ≤ N – 1, where N is even
Consider an N-point sequence x[n] with DFT X[k], k = 0, 1, ?, N ? 1. The following algorithm computes the even-indexed DFT values X[k], k0, 2, ?. , N ? 2, for N even, using only a single N/2 ?point
Let x [n] and h [n] be two real finite-length sequences such that x[n] = 0 for n outside the interval 0 ? n ? L ? 1, h[n] = 0 for n outside the interval 0 ? n ? p ? 1. We wish to compute the sequence
X[n] is a 1024-point sequence that is nonzero only for 0 ? n ? 1023. Let X[k] be the 1024-point DFT of x[n]. Given X[k], we want to compute x[n] in the ranges 0 ? n ? 3 and 1020 ? 1023 using the
In many applications (such as evaluating frequency responses and interpolation), it is of interest to compute the DFT of a short sequence that is ?zero-padded.? In such cases, a specialized ?pruned?
In computing the DFT, it is necessary to multiply a complex number by another complex number whose magnitude is unity, i.e., (X + jY) e j?. Clearly, such a complex multiplication changes only the
In the Goertzel algorithm for computation of the discrete Fourier transform, X[k] is computed as X[k] = Yk[n], where yk[n] is the output of the network shown in Figure. Consider the implementation of
Consider direct computation of the DFT using fixed-point arithmetic with rounding. Assume that the register length is B bits plus the sign (i.e., a total of B + 1 bits) and that the round-off noise
In implementing a decimation-in-time FFT algorithm, the basic butterfly computation is Xm[p] = Xm-1[p] + /WrN Xm-1[q], Xm[q] = Xm-1[p] – WrN Xm-1[q]. In using fixed-point arithmetic to implement
In deriving formulas tor the noise-to-signal ratio for the fixed-point radix-2 decimation-in-time FFT algorithm, we assumed that each output node was connected to (N ? 1) butterfly computations, each
In Section 9.7 we considered a noise analysis of the decimation-in-time FFT algorithm of figure. Carry out a similar analysis for the decimation-in-frequency algorithm of figure, obtaining equations
The input and output of a linear time-invariant system satisfy a difference equation of the form, Assume that an FFT program is available for computing the DFT of any finite-length sequence of length
Suppose that we wish to multiply two very large numbers (possibly thousands of bits long) on a 16-bit computer. In this problem, we will investigate a technique for doing these using FFTs. (a) Let
The discrete Hartley transform (DHT) of a sequence x[n] of length N is defined as, where HN[a] = CN[a] + SN[a], with CN[a] = cos (2?a/N), SN[a] = sin(2?a/N). Problem explores the properties of the
In this problem, we will write the FFT as a sequence of matrix operations. Consider the 8-point decimation-in-time FFT algorithm shown in figure. Let a and f denote the input and output vectors,
In real continuous-time signal xc(t) is bandlimited to frequencies below 5 kHz; i.e., Xc(jΩ) = 0 for |Ω| ≥ 2π(5000). The signal xc(t) is sampled with a sampling rate of 10,000 samples per
A continuous-time signal xc(t) is bandlimited to 5 kHz; i.e., Xc(jΩ) = 0 for |Ω| ≥ 2π (5000). xc(t) is sampled with period T, producing the sequence x[n] = xc(nT). To examine the spectral
A speech signal is sampled with a sampling rate of 16,000 samples/s (16 kHz). A window of 20-ms duration is used in time-dependent Fourier analyses of the signal, as described in section 10.3, with
A real-valued continuous-time segment of a signal xc(t) is sampled at a rate of 20,000 samples/sec, yielding a 1000-point finite-length discrete-time sequence x[n] that is nonzero in the interval 0
A continuous-time signal xc(t) = cos(Ω0t) is sampled with period T to produce the sequence x[n] = xc(nT). An N-point rectangular window is applied to x[n] for 0, 1, … N−1, and X[k], for k = 0,
Let xc(t) be a real-valued, bandlimited signal whose Fourier transform xc(jΩ) is zero for |Ω| ≥ 2π (5000). The sequence x[n] is obtained by sampling xc(t) at 10 kHz. Assume that the sequence
Consider estimating the spectrum of a discrete-time signal x[n] using the DFT with a Hamming window for w[n]. A conservative rule of thumb for the frequency resolution of windowed DFT analysis is
Let x[n] be a discrete-time signal whose spectrum you wish to estimate using a windowed DFT. You are required to obtain a frequency resolution of at least π/25 and are also required to use a window
The following are three different signals xi[n] that are the sum of two sinusoids:x1[n] = cos (πn/4) + cos (17πn/64),x2[n] = cos (πn/4) + 0.8 cos (21πn/64),x3[n] = cos (πn/4) + 0.001 cos
Let x[n] be a discrete-time signal obtained by sampling a continuous-time signal xc(t) with some sampling period T so that x[n] = xc(nT). Assume xc(t) is bandlimited to 100 Hz, i.e, Xc(jΩ) = 0 for
Let x[n] be a 5000-point sequence obtained by sampling a continuous-time signal xc(t) at T = 50 μs. Suppose X[k] is the 8192-point DFT of x[n]. What is the equivalent frequency spacing in continuous
Assume that x[n] is a 1000-point sequence obtained by sampling a continuous-time signal xc(t) at 8 kHz and that Xc(jΩ) is sufficiently bandlimited to avoid aliasing. What is the minimum DFT length
Let Xr[k] be the time-dependent Fourier transform (TDFT) defined in Eq. (10.36). For this problem, consider the TDFT when both the DFT length N = 36 and the sampling interval R = 36. Let the window
Figure shows the magnitude |V[k]| of the 128-point DFT V[k] for a signal v[n]. The signal v[n] was obtained by multiplying x[n] by a 128-point rectangular window w[n]; i.e., v[n] = x[n]w[n]. Note
Figure shows the spectrogram of a chirp signal of the form? x[n] = sin (?0n + 1/2?n2). Note that the spectrogram is a representation of the magnitude of X[n, k], as defined in Eq. (??), where the
A continuous-time signal is sampled at a sampling rate of 10 kHz, and the DFT of 1024 samples is computed. Determine the continuous-time frequency spacing between spectral samples. Justify your
A signal x[n] is analyzed using the time-dependent Fourier transform Xr[k], as defined in Eq. (10.36). Initially, the analysis is performed with an N = 128 DFT using an L = 128-point Hamming window
Let x[n] be a signal with a single sinusoidal component. The signal x[n] is windowed with an L-point Hamming window w[n] to obtain v1[n] before computing V1(ejω). The signal is then windowed with an
Assume that you wish to estimate the spectrum of x[n] by applying a Kaiser window to the signal before computing the DTFT. You require that the side lobe of the window be 30 dB below the main lobe
It is desired to estimate the spectrum of x[n] by applying a 512-point Kaiser window to the signal before computing X(ejω).(a) The requirements for the frequency resolution of the system specify
Let x[n] = cos(2πn/5) and v[n] be the sequence obtained by applying a 32-point rectangular window to x[n] before computing V(ejω). Sketch |V(ejω) for –π ≤ ω ≤ π, labeling the frequencies
Sketch the spectrogram obtained by using a 256-point rectangular window and 256-point DFTs with no overlap (R = 256) on the signalx[n] cos [πn/4 + 1000sin (πn/8000)]for the interval 0 ≤ n ≤
Suppose that y[n] is the output of linear time-invariant FIR system with input x[n]; i.e., (a) Obtain a relationship between the time-dependent Fourier transform Y[n, ?) of the output of the linear
The periodogram I(?) if a discrete-time random signal x[n] was defined in Eq. (10.52) as? I(?) = 1/LU |V(ej?)|2, where V(ej?) is the discrete-time Fourier transform of the finite-length sequence v[n]

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