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engineering
telecommunication engineering

Questions and Answers of Telecommunication Engineering

Consider a finite-length sequence x[n] such that x[n] = 0 for n L. Define cxx[m] to be the aperiodic autocorrelation function of x[n]; i.e., (a) Determine the minimum value of N that can be used for
The symmetric Bartlett window, which arises in many aspects of power spectrum estimation, it is defined as? The Bartlett window is particularly attractive for obtaining estimates of the power
Consider a signal x[n] = [sin (?n/2)]2 u[n] whose time-dependent discrete Fourier transform is computed using the analysis window .
Show that the time-dependent Fourier transform, as defined by Eq. (10.18), has the following properties: (a) Linearity: If x[n] = ax1[n] + bx2[n], ? ? ? ??then? ? ? X[n, ?) = ?X1[n, ?) + bX2[n,
Suppose that xc(t) is a continuuous-time stationary random signal with autocorrelation function ?c(?) = ?{xc(t)xc(t + ?)} and power density spectrum Consider a discrete-time stationary random signal
In Section 10.6.5, we considered the estimation of the power spectrum of a sinusoid plus white noise. In this problem, we will determine the true powet spectrum of such a signal. Suppose that? x[n]
Consider a discrete-time signal x[n] of length N samples that was obtained by sampling a stationary, white, zero-mean continuous-time signal. It follows that? Suppose that we compute the DFT of the
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
Suppose that an estimate of the power spectrum of a signal is obtained by the method of avetraging periodograms, as discussed in Section 10.6.3 This is, the spectrum estimate is? where the K
Consider the computation of the autocorrelation estimate where x[n] is a real sequence. Since ?xx[?m] = ?xx[m], it is necessary onlu to evaluate Eq. (p10.38-1) for 0 ? m ? M ? 1 to obrain ?xx[m] for
In Section 10.3 we defined the time-dependent Fourier transform of the signal x[m] so that, for fixed n, it is equivalent to the regular discrete-time Fourier transform of the sequence x[n + m] w[m],
Consider a stable linear time-invariant system with a real input x[n], a real impulse response h[n], and output y[n]. Assume that the input x[n] is white noise with zero mean and variance σ2x. The
This problem illustrates the basis for an FFT-based procedure for interpolating the samples (obrained at a rate satisgying the Nyquist theorem) of a periodic continuous-time signal. Let be a
However, significant information can be gained from a windowed section of the sequence. In this problem, you will look at computing the Fourier transform of an infinite-duration signal x[n], given
Consider a sequence x[n] with descrete-time Fourier transform X(ejω). The sequence x[n] is real valued and causal, and Re{X(ejω)} = 2 – 2α cos ω.Determine Jm{X(ejω)}.
Consider a sequence x[n] and irs discrete-time Fourier transform X(ejω). The following is known:x[n] is real and causal,Re{X(ejω)} = 5/4 – cosω.Determine a sequence x[n] consistent with the
Consider a sequence x[n] and its discrete-time Fourier transform X(ejω). The following is known: x[n] is real,x[0] = 0,x[1] > 0, |X(ejω)|2 = 5/4 – cosω.Determine two distinct
Consider a complex sequence x[n] = xr[n] + jxi[n], where xr[n] and xi[n] are the real part and imaginary part, respectively. The z-transform X(z) of the sequence x[n] is zero on the bottom half of
Find the Hilbert transforms xi[n] = Hxr[n]} of the following sequences:(a) xr[n] = cos ω0n(b) xr[n] = sin ω0n(c) xr[n] = sin(ωcn)/πn
The imaginary part of X(ejω) for a casual, real sequence x[n] is X1(ejω) = 2 sin ω – 3 sin 4ω.Additionally, it is known that X(ejω)|ω=0 = 6. Find x[n].
(a) x[n] is a real, causal sequence with the imaginary part of its discrete-time Fourier transform X(ejω) given byJm{X(ejω)} = sin ω + 2 sin 2ω.(b) Is your answer to Part (a) unique? If so,
Consider a teal, causal sequence x[n] with discrete-time Fourier transform X(ej?) = XR(ej?) + jXI(ej?). The imaginary part of the discrete-time Fourier transform is? XI(ej?) = 3 sin(2?). Which of the
The following information is known about a real, causal sequence x[n] and its discrete-time Fourier transform X(ejω):Jm{X(ejω)} = 3 sin(ω) + sin(3ω),X(ejω)|ω=π = 3.Determine a sequence
Consider h[n], the real-valued impulse response of a stable, causal LTI system with frequency response H(ej?). The following is known: (i) The system has a stable, causal inverse. Determine h[n] in
Let x[n] = xr[n] + jxi[n] be a complex-valued sequence such that X(ej?) = 0 for ? ? ? ? Specify the real and imaginary parts of X(ej?).
h[n] is causal, real-valued sequence with h[0] nonzero and positive. The magnitude squared of the frequency response of h[n] is given by (a) Determine a choice for h[n]. (b) Is your answer to part
Let x[n] denote a causal, complex-valued sequence with Fourier transform X(ejω) = XR(ejω) + j XI(ejω).If XR(ejω) = 1 + cos(ω) + sin (ω) – sin(2ω), determine XI(ejω).
Consider a real, anticausal sequence x[n] with discrete-time Fourier transform X(ej?). The real part of X(ej?) is Find XI(ej?), the imaginary part of X(ej?). (Remember that a sequence is said to be
x[n] is a real, causal sequence with discrete-time Fouorier transform X(ej?). The imaginary part of X(ej?) is? jm{ X(ej?).} = sin ?, and it is also known that?
Consider a real, causal sequence x[n] with discrete-time Fourier transform X(ejω), where the following two facts are given about X(ejω):XR(ejω) = 2 – 4 cos(3ω),X(ejω)|ω=π = 7.Are these facts
Consider a real, causal, finite-length signal x[n] with length N = 2 and with a 2-point discrete-Fourier transform X[k] = XR[k] + j XI[k] for k = 0, 1. If XR[k] = 2δ[k] – 4δ[k–1], is it
Ler x[n] be a real-valued, causal, finite-length sequence with length N = 3. Find two choices for x[n] such that the real part of the discrete Fourier transform XR[k] matches that shown in Figure.
Consider a sequence x[n] that is real, causal, and of finite length with N = 6. The imaginary part of the 6-point discrete Fourier transform of this sequence is? Akkitionally, it is known that? Which
Let x[n] be a real, causal, finite-length sequence with length N = 4 that is also periodically causal. The real part of the 4-point discrete Fourier transform XR[k] for this sequence is shown in
Let x[n] be a real causal sequence for which |x[n]| < ∞. The z-transform of x[n] is which is a Taylor’s series in the variable z-1 and therefore converges to an analytic function everywhere
Show that the sequence of discrete Fourier series coefficients for the sequence Find the z-transform of the sequence? uN[n] = 2u[n] ? 2u[n ? N/2] ? ?[n] + ?[n ? N/2], and sample it to obtain U[k].
Consider a real-valued finite-duration sequence x[n] of length M. Specifically, x[n] = 0 for n < 0 and n > M – 1. Let X [k] denote the N-point DFT of x[n] with N ≥ M and N odd. The real
Consider a complex sequence h[n] = hr[n] + jhi[n], where hr[n] and hi[n] are both real sequences, and let H(ejω) = HR(ejω) + j HI(ejω) denote the Fourier transform of h[n], where HR(ejω) and
The ideal Hilbert transformer (90-degree phase shifter) has frequency response (over one period) Figure shows H(ej?), and Figure shows the frequency response of an ideal lowpass filter HIP(ej?) with
In Section 11.4.3, we discussed an efficient scheme for sampling a bandpass continuous-time signal with Fourier transform such that? Sc(j?) = 0? ? ? ??for |?| ? ?c and |?| ? ?c + ??. In that
Consider an LTI system with frequency response, The input x[n] to the system is restricted to be real valued and to have a Fourier transform (i.e., x[n] is absolutely summable). Determine whether or
Derive an integral expression for H(z) inside the unit circle in terms of Re{H(ejω)} when h[n] is a real, stable sequence such that h[n] = 0 for n > 0.
Let H{?} denote the (ideal) operation of Hilbert transformation; that is, Prove the following properties of the ideal Hilbert transform operator.
An ideal Hilbert transformer with impulse response has input xr[n] and output xi[n] = xr[n] * h[n], where xr[n] is a discrete-time random signal. (a) Find an expression for the autocorrelation
In Section 11.3, we mention that a causal complex cepstrum x[n] is equivalent to the minimum-phase condition of Section 5.4. Remember that x[n] is the inverse Fourier transform of X(ej?) as defined
Classify the following signals according to whether they are (1) one- or multi-dimensional; (2) single or multichannel, (3) continuous time or discrete time, and (4) analog or digital (in amplitude).
Determine which of the following sinusoids are periodic and computer their fundamental period.(a) Cos 0.01 πn(b) Cos (π 30n/105)(c) Cos 3πn(d) Sin 3n(e) Sin (π62n/10)
Determine whether or not each of the following signals is periodic. In case a signal is periodic, specify its fundamental period.(a) xa(t) = 3 cos(5t + π/6)(b) x(n) = 3 cos(5n + π/6)(c) x(n) = 2
(a) Show that the fundamental period Np of the signals sk (n) = ej2πkn/N. k = 0. 1. 2. . . .is given by Np = N/GCD(K,N), where GCD is the greatest common divisor of k and N. (b) What is
Consider the following analog sinusoidal signal:xa(t) = 3 sin (100πt)(a) Sketch the signal xa(t) for 0 ≤ t ≤ 30 ms.(b) The signal xa(t) is sampled with a sampling rate Fs = 300 samples/s.
A continuous-time sinusoid xa(t) with fundamental period Tp = 1/F0 is sampled at a rate Fs = 1/T to produce a discrete-time sinusoid x(n) = xa(nT).(a) Show that x(n) is periodic if T/Tp = k/N (i.e.,
An analog signal contains frequencies up to 10 kHz(a) What range of sampling frequencies allows exact reconstruction of this signal from its samples?(b) Suppose that we sample this signal with a
An analog electrocardiogram (ECG) signal contains useful frequencies up to 100 Hz.(a) What is the Nyquist rate for this signal?(b) Suppose that we sample this signal at a rate of 250samples/s, what
An analog signal xa(t) = sin(480πt) + 3sin(720πt) is sampled 600 times per second. (a) Determine the Nyquist sampling rate for xa(t). (b) Determine the folding frequency. (c) What are
A digital communication link carries binary-coded words representing samples of an input signal xa(t) = 3 cos 600πt + 2 cos 1800π.The link is operated at 10,000 bits/s and each input sample is
Consider the simple signal processing system shown in figure. The sampling periods of the A/D and D/A converters are T = 5 ms and T? = 1 ms. Respectively. Determine the output ya(t) of the system. If
(a) Derive the expression for the discrete-time signal x(n) in example using the periodicity properties of sinusoidal functions.(b) What is the analog signal we can obtain from x(n) if in the
The discrete-time signal x(n) = 6.35cos(π/10)n is quantized with a resolution(a) Δ= 0.1 of(b) Δ = 0.02. How many bits are required in the A/D converter in each case?
Determine the bit rate and the resolution in the sampling of a seismic signal with dynamic range of 1 volt if the sampling rate is Fs = 20 samples/s and we use an 8-bit A/D converter? What is the
Sampling of sinusoidal signal: aliasing consider the following continuous-time sinusoidal signal xa(t) = sin2?F01, - ???? t ??? ? Since xa (t) is described mathematically, its sampled version can be
Quantization error in A/D conversion of a sinusoidal signal?Let xq (n) be the signal obtained by quantizing the signals x(n) = sin 2?f0n.?The quantization error power Pq is defined by The
A discrete-time signal x(n) is defined as(a) Determine its values and sketch the signal x(n).(b) Sketch the signals that result if we:(1) First fold x(n) and then delay the resulting signal by four
A discrete-time signal x(n) us shown in figure. Sketch and label carefully each of the followingsignals.
Showthat
Show that any signal can be decomposed into an even and an odd component. Is the decomposition unique? Illustrate your arguments using thesignal
Show that the energy (power) of a real-valued energy (power) signal is equal to the sum of the energies (powders) of its even and odd components.
Consider the system y(n) = T [x(n) = x(n2)] (a)?Determine if the system is time invariant. (b) Clarify the result in part (a) assume that the signal is applied into the system. (1) Sketch the
Two discrete-time systems T1 and T2 are connected in cascade to form a new system T as shown in figure. Prove or disprove the following statements. (a) If T1 and T2 are linear, than T is linear
Let T be an LTI, relaxed, and BIBO stable system with input x(n) and output y(n) show that: (a) If x(n) is periodic with period N [i.e., x(n) = x(n + N) for all n ≥0], the output y(n) tends to
The following input-output pairs have been observed during the operation of a time-invariant system: Can you draw any conclusion regarding the linearity of the system? What is the impulse response
The following input-output pairs have been observed during the operation of a linear system: Can you draw any conclusions about the time invariance of this system?
The only available information about a system consist of N input-output pairs, of signals yi(n) = T[xi(n)], I = 1,2,..,N (a) What is the class of input signals for which we can determine the output,
How that the necessary and sufficient condition for a relaxed LTI system to be BIBO stable is for some constant Mn
Show that: (a) A relaxed linear system is causal if and only if for any input x(n) such that x(n) = 0 for n < n0 → y(n) = 0 for n< n0 (b) A relaxed LTI system is causal if and only if
(a) Show that for any real or complex constant a, and any finite integer numbers M and N, we have (b) Show that if |a|
Compute and plot the convolutions x(n) * h (n) and h(n) * x(n) for the pairs of signals shown infigure
Determine and sketch the convolution y(n) of the signals (a) Graphically (b) Analytically
Compute the convolution y(n) of thesignals
Consider the following three operations.(a) Multiply the integer number: 131 and 122.(b) Compare the convolution of signals: {1.3.1} * {1.2.2}.(c) Multiply the polynomials: 1 + 3z + z2 and 1 +
Compute the convolution y(n) = x(n) * h(n) of the following pairs of signals. (a) x(n) = axu(n), h(n) = bn?u(n) when a ? b and when a = b (b) ? ? (c) x(n) = u(n + 1) ? u(n - 4) - ?(n - 5), h(n) =
Let x(n) be the input signal to a discrete-time filter with impulse response hi(n) and let yi(n) be the corresponding out put. (a) Compute and sketch x(n) and yi(n) in the following cases. Using the
The discrete-time system y(n) = ny(n - 1) + x(n) n ≥ 0 is at rest [i.e., y(-1) = 0]. Check if the system is linear time invariant and BIBO stable.
Consider the signal ?(n) = an u(n), 0? (a) Show that any sequence x(n) can be decomposed as and express ck in terms of x(n). (b) Use the properties of linearity and time invariance to express the
Determine the zero-input response of the system described by the second-order difference equation x(n) – 3y(n - 1) – 4y(n-2) = 0
Determine the particular solution of the difference equation When the forcing function is x(n) =2nu(n)
Determine the response y(n), n ≥ 0, of the system describe by the second-order difference equation y(n) – 3y (n - 1) – 4y (n - 2) = x (n) + 2x (n - 1) to the input x(n) = 4n u(n).
Determine the impulse response of the following causal system: y(n) – 3y (n - 1) – 4y (n - 2) = x (n) + 2x (n - 1)
Let x(n), N1 ? n ? N2 and h (n), M1 ? n ? M2 be two finite- duration signals.(a) Determine the range L1 ? n ? L2 of their convolution, in terms of N1, N2, M1 and M2(b) Determine the limits of the
Determine the impulse response and the until step response of the system described by the differenceequation
Consider a system with impulse responseDetermine the input x(n) for 0 ? n ? 8 that will generate the output sequence
Consider the interconnection of LTI system as shown in figure (a) Express the overall impulse response in terms of h1 (n), h2 (n), h3 (n), and h4 (n). (b) Determine h(n) when (c) Determine the
Consider the system in figure with h (n) = anu(n), -1
Compute and sketch the step response of thesystem
Determine the range of values of the parameter a for which the linear time-invariant system with impulse response isstable.
Determine the response of the system with impulse response h(n) = anu(n)to the input signal x(n) = u(n) – u(n – 10)
Determine the response of the (relaxed) system characterized by the impulse responseh(n) = (1/2) n u(n)to the inputsignal
Determine the response of the (relaxed) system characterized by the impulse responseh(n) = (1/2)nu(n) to the input signals(a) x(n) = 2nu(n)(b) x(n) = u(-n)
Three system with impulse with impulse responses h1(n) = δ(n) – δ(n - 1), h2(n) = h(n), and h3 (n) = u (n), are connected in cascade. (a) What is the impulse response, h0(n), of the
(a) Prove and explain graphically the difference between the relations x(n)δ(n – n0) = x (n0)δ(n – n0) and x(n) * δ(n – n0) = x (n – n0) (b) Show that a discrete-time
Two signals s(n) and v(n) are related through the following difference equations s(n) + ais(n - 1) + . . . . + aN s(n - N) = b0v(n).Design the block diagram realization of: (a) The system that

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