Questions and Answers of Signals and Systems

Discuss the possibility of an “n-address” machine, where n > 3.
The instruction set of a certain processor does not have the JLE, JLT, JGE (jump less equal, less than, and greater or equal), and JGT instructions. Assume the process does not have all other
Describe the relationship between the main processor and coprocessor in a system with which you are familiar or one that you discover through Web research.
Should the instruction following the TEST instruction be interruptible? If so, what must the implicit BRANCH instruction (interrupt) do?
Microcoded computers tend to be superior to 1-, 2-, or 3- address computers with respect to overall performance. Why?
What is the difference between coprocessing and multiprocessing? What are the advantages and disadvantages of each?
GCP, Inc., has contracted you to analyze and develop commercial off-the-shelf (COTS) processors to be phased into their existing product lines. Your objective is to select four popular commercial
It seems that there are far fewer commercial implementations of SIMD, MIMD, and MISD architectures than there were just 10 years ago. This is probably due to a variety of factors, including high
What would be the appropriate operating system architecture for the:(a) Inertial measurement system(b) Nuclear monitoring system(c) Patient monitoring system(d) Airline reservations system(e) Pasta
Should a task be allowed to interrupt itself? If it does, what does this mean?
Discuss some of the advantages of EDF scheduling over RM scheduling and vice versa.
Show with an example that EDF is no longer an optimal scheduling policy if preemption is not allowed.
The following system of periodic tasks is scheduled and executed according to a cyclic schedule. Draw an execution trace (timeline) showing two occurances of each task. Ti ei Pi 1 8 T2 4 15 T3 3
Consider the following tasks with their resource requirements given as:a) τ1 = (10, 4, 1; [A; 1]), where the task executes for two time units, then requests the resource A.(b) τ2 = (7, 4, 2; [A;
For a machine you are familiar with, discuss whether the counting semaphore implementation given in this chapter has any critical-region problems. That is, can the semaphore itself be interrupted in
Why is it not wise to disable interrupts before the while statement in the binary semaphore, P(S)?
Discuss the problems that can arise if the test and set in the P(S) operation are not atomic. What could happen if the simple assignment statement in the V(S) operation were not atomic?
Rewrite the save and restore routines assuming that eight general register (R0–R7) and the program counter are to be saved on a stack. Do this for(a) 0-address machine(b) 1-address machine(c)
Rewrite the save and restore routines so that they save and restore to the head and tail of a ring buffer, respectively.
Rewrite the save and restore routines in 2-address code, assuming block move (BMOVE) and restore (BRESTORE) instructions are available. Make the necessary assumptions about the format of the
Rewrite the save and restore routines in the language of your choice so that they employ the push and pop procedures.
A real-time system has a fixed number of resources of types A, B, and C. There are five tasks in the system, and the maximum amount of resources A, B, and C needed for each task is known. Implement a
Show how priority inheritance can cause deadlock and also multiple blocking. For example, consider the following sequence (with τ1 τ2):τ1: Lock S1; Lock S2; Unlock S2; Unlock S1τ2: Lock S2; Lock
Modify the write procedure for the ring buffer to handle the overflow condition.
Consider a binary semaphore, counting semaphore, queues, and mailboxes. Any three can be implemented with the fourth. It was shown how binary semaphores can be used to implement counting semaphores
The TANDS instruction can be used in a multiprocessing system to prevent simultaneous access to a global semaphore by two processors. The instruction is made indivisible by the CPU refusing to issue
Obtain as much data as you can for as many of the existing commercial real-time systems as you can. Summarize your findings for each operating system, briefly, in narrative form.
Use your selection criteria and the information you have obtained to create a matrix of features by-products. In other words, present the findings you describe in step 2 more succinctly in tabular
For the following kinds of systems give your best recommendation as to the most likely commercial real-time operating system to use based on the selection criteria you developed(a) A controller
In filter design you will be asked to use hyper-bolic functions. In this problem we relate these functions to sinusoids and obtain a definition of these functions so that we can actually plot
Euler’s identity is very useful not only in obtaining the rectangular and polar forms of complex numbers, but in many other respects as we will explore in this problem.(a) Carefully plot x[n]
Consider the complex function x(t) = (1 + jt)2 for − ∞ < t < ∞.(a) Find the real and the imaginary parts of x(t) and carefully plot them with MATLAB. Try to make MATLAB plot x(t)
Consider complex numbers z = 1 +j, w = −1 + j, v = −1 − j, and u = 1 − j. You may use MATLAB compass to plot vectors corresponding to complex numbers to verify your analytic
The exponential x(t) = eat for t ≥ 0 and zero otherwise is a very common continuous-time signal. Likewise, y(n) = αn for integers n ≥ 0 and zero otherwise is a very common discrete-time signal.
Although sums behave like integrals, because of the discrete nature of sums one needs to be careful with the upper and lower limits more than in the integral case. To illustrate this consider the
Suppose you wish to find the area under a signal x(t) using sums. You will need the following result found above(a) Consider first x(t) = t, 0 ≤ t ≤ 1, and zero otherwise. The area under
Three laws in the computation of sums arefor any permutation p(k) of the set of integers kin the summation.(a) Explain why the above rules make sense when computing sums. To do that considerLet
Find the ordinary differential equation relating a current source is(t) = cos(?0t) with the current iL(t) in an inductor, with inductance L = 1 Henry, connected in parallel with a resistor of R = 1?
Another definition for the finite difference is the backward difference:∆1[x(nTs)] = x(nTs) − x((n − 1)Ts)(∆1[x(nTs)]/Ts approximates the derivative of x(t)).(a) Indicate how this new
Let y(t) = dx(t)/dt, where x(t) = 4cos(2πt), − ∞ < t < ∞. Find y(t) analytically and determine a value of Ts for which ∆[x(nTs)]/Ts = y(nTs)(consider as possible values Ts = 0.01
Consider a signal x(t) = 4cos(2πt) defined for − ∞ < t < ∞. For the following values of the sampling period Ts generate a discretetime signal x(n) = x(nTs) = x(t)∣t=nTs.(i) Ts =
The geometric series
To get an idea of the number of bits generated and processed by a digital system consider the following applications:(a) A compact disk (CD) is capable of storing 75 min of “CD quality”
A phasor can be thought of as a vector, representing a complex number, rotating around the polar plane at a certain frequency in radians/second. The projection of such a vector onto the real axis
Consider a function of z = 1 + j1, w = ez(a) Find (i) log(w), (ii) Re(w), (iii) Im(w)(b) What is w + w*, where w* is the complex conjugate of w?(c) Determine ∣w∣, ∠w, and ∣log(w)∣2?(d)
Consider the following problems related to computation with complexnumbers.(a) Find and plot all roots of(i) z3 = − 1, (ii) z2 = 1
Consider the calculation of roots of an equation zN = α where N ≥ 1 is an integer and α = ∣α∣ejφ a nonzero complex number.(a) First verify that there are exactly N roots for this
Use Euler’s identity to(a) Show the identities(i) cos(α + β) = cos(α) cos(β) − sin(α) sin(β)(ii) sin(α + β) = sin(α) cos(β) + cos(α) sin(β)(b) Find an expression for cos(α)
Use Euler?s identity in the following problems. (a)?Find trigonometric identities in terms of sin(?), sin(?), cos(?), cos(?) for (i) cos(? + ?) ? ? ?(ii) sin(? + ?) (b) Is it true that (c)?Is it
Use Euler?s identity to (a)?show that (i) cos(? ? ?/2) = sin(?), (ii) ? sin(? ? ?/2) = cos(?), (iii) cos(?) = sin(? + ?/2). (b) to find (i) cos(2πt) sin(2π t)dt, (ii) cos, (2πt)dt
Using the vectorial representation of complex numbers it is possible to get some interesting inequalities.(a) Is it true that for a complex number z = x + jy we have that ∣x∣ ≤ ∣z∣? Show it
Consider the following problems about trigonometric and polar forms.(a) Let z = 6ejπ/4 find (i) Re(z), (ii) Im(z)(b) If z = 8 + j3 and v = 9 − j2, is it true that(i) Re(z) = 0.5(z + z∗)?
Design an FIR low-pass filter with a cuttoff of π/3 and lengths N = 21 and N = 81,(a) Using a rectangular window(b) Using Hamming and Kaiser (β = 4.5) windows, and compare the magnitude of the
Given the signals x[n] = 2n (u[n] − u[n − 3]) and y[n] = 0.5n (u[n] − u[n − 3]) write a matrix equation to compute their circular convolution of lengths N = 3, 4, and 5. Call the
Consider an envelope detectorthat would be used to detect the message sent in an AM system. Consider the envelope detector as a system composed of the cascading of two sys-tems one which computes the
A discrete-time system has a unit impulse response h[n].(a) Let the input to the discrete-time system be a pulse x[n] = u[n] − u[n − 4] compute the output of the system in terms of the
An effect similar to multi-path in acoustics is echoing or reverberation. To see the effects of an echo in an acoustic signal consider the simulation of echoes on the handel.matsignal y[n]. Pretend
Suppose we sample the analog signal x(t) = e?2t u(t), using a sample period Ts = 1 (a)?Expressing the sampled signal as x(nTs) = x[n] = ?n u[n], what is the corresponding value of ?? Use stemto plot
The input of an LTI continuous-time system is x(t) = u(t) − u(t − 3.5). The system’s impulse response is h(t) = u(t) − u(t − 2.5).(a) Find the system’s output y(t) by graphically
Determine the impulse response h[n] of a LTI system represented by the difference equation y[n] = ?0.5y[n ? 1] + x[n], where x[n] is the input, y[n] is the output, and the initial conditions are
A two-bit quantizer has as input x(nTs) and as output x̂(nTs), ork∆ ≤ x(nTs) < (k + 1) ∆ → x̂(nTs) = k∆, k= − 2, − 1, 0, 1(a) Is this system time-invariant?
For a finite-support signal x[n] = r[n](u[n] − u[n − 11]) where r[n] is the discrete-time ramp function,(a) Find the energy of x[n].(b) Find the even xe[n] and the odd xo[n] components
Let x(t) = cos(? t)[u(t) ? u(t ? 2)] be the input of a zero-order hold sampler. The sampler samples every Ts sec starting at t = 0, and the impulse response of the zero-order hold is h(t) = u(t) ?
Suppose we would like to send the two messages mi(t), i = 1, 2, created in Problem 16 using the same bandwidth and to recover them separately. To implement this consider the QAM approach where the
The signal at the input of an AM receiver is u(t) = m1(t) cos(20t) + m2(t) cos(100t) where the messages mi(t), i = 1, 2 are the outputs of a low-pass Butterworth filter with inputs x1(t) = r(t) ?
Design an analog low-pass filter satisfying the following magnitude specifications:αmax = 0.5 dB αmin = 20 dBΩp
The specifications for a low-pass filter areα(0)=20 dBsΩp = 1500 rad/sec, α1 = 20.5 dBsΩs = 3500 rad/sec, α2 = 50
Consider the transmission of a sinusoid x(t) = cos(2πf0t) through a channel affected by multipath and Doppler. Let there be two paths, and assume the sinusoid is being sent from a moving object so
Control systems attempt to follow the reference signal at the input, but in many cases they cannot follow particular types of inputs. Let the system we are trying to control have a transfer function
Suppose you would like to obtain a feedback implementation of an all-pass system with transfer function (a)?Determine the feedforward transfer function G(s) and the feedback transfer function H(s)
An ideal operational amplifier circuit can be shown to be equivalent to a negative feedback system. (a)?Consider the inverting amplifier circuit and its two-port network equivalent shown in Figure
Consider a filter with frequency response or a sinc function in frequency. (a) Find the impulse response h(t) of this filter. Plot it and indicate whether this filter is a causal system or not. (b)
Let the signal x(t) = r(t + 1) − 2r(t) + r(t − 1) and y(t) = dx(t)/dt.(a) Plot x(t) and y(t)(b) Find X(Ω) and carefully plot its magnitude spectrum. Is X(Ω) real? Explain.(c) Find Y(Ω)(use
For signals with infinite support, their Fourier transforms cannot be derived from the Laplace transform unless they are absolutely integrable or the region of convergence of the Laplace transform
The Fourier transform of finite support signals, which are absolutely integrable or finite energy, can be obtained from their Laplace transform rather than doing the integral. Consider the following
Consider a signal x(t) = cos(t), 0 ≤ t ≤ 1(a) Find its Fourier transform X(Ω).(b) Let y(t) = x(2t), find Y(Ω), let z(t) = x(t/2), find Z(Ω).(c) Compare Y(Ω) and Z(Ω) with X(Ω).
The Fourier transforms of even and odd functions are very important. Let x(t) = e??t? and y(t) = e?t u(t) ? et u(?t). (a) Plot x(t) and y(t), and determine whether they are even or odd. (b) Show that
To understand the Fourier series consider a more general problem, where a periodic signal x(t), of period T0, is approximated as a finite sum of terms where {?k (t)} are ortho-normal functions. To
As you know, π is an irrational number that can only be approximated by a number with a finite number of decimals. How to compute this value recursively is a problem of theoretical interest. In this
The problem with thresholding the DCT coefficients of an image to compress it, is that the locations of the chosen coefficients are arbitrary and difficult to code. Consider then using a mask W(k,
An image can be blurred by means of a Gaussian filter which has an impulse response(a) If h[m, n] = h1[m]h1[n], i.e., separable determine h1[n]. Find the DFT of h[m, n] and plot its magnitude
Image filtering using 2D-FFT — Consider the linear filtering of an image using the 2D-FFT. Load the image clown and use three different filters to process it given in different formats.• Low-pass
To compute the 2D-DFT one can use 1D-DFT by separating the equation for the 2D-DFT as(a) Using the one-dimensional MATLAB function fft to implement the above result, and for the signalwith values M =
The convolution sum is a fast way to find the coefficients of the polynomial resulting from the multiplication of two polynomials.(a) Suppose x[n] = u[n] - u[n - 3] find its Z-transform X(z), a
A filter has a transfer function(a) Find the poles and zeros of H(z1; z2).(b) For what values of ẑ1 and ẑ2 is H(ẑ1, ẑ2) = 0/0?(c) Ignoring the numerator, i.e., lettingis this filter BIBO
Consider a system represented by the convolution sum(a) Obtain the BIBO stability condition for this system, assuming that the input x[m, n] is bounded.(b) Use the variable transformation
Use the binomial theoremto express the transfer functionasand determine the impulse response of the system. (z + y)* = E ()-"2 y"= =ng mi-n2 n2=0 H(21, 2) = 1/(1 – (2ī' + 2,')
Consider the separable two-dimensional Z-transformwhere p0 and p1 are poles of X(z1, z2).(a) Determine the poles and zeros of X(z1, z2).(b) Carefully indicate the ROC and the location of the unit
Detection of edges is a very important application in image processing. Taking the gradient of a two-dimensional function detects the changes the edges of an image. A filter than is commonly used in
Use the function imread, rgb2gray and double to read the color image peppers.png. Convert it into a gray level image I[m, n] with double precision. Add noise to it using the function randn (Gaussian
Let the inputand the impulse response of an FIR filter be(a) Use the two-dimensional convolution function conv2 to find the output y[m; n].(b) Is the impulse response separable, i.e., h[m; n] =
A discrete-time system has a unit impulse response h[n].(a) Let the input to the discrete-time system be a pulse x[n] = u[n] - [n - 4] compute the output of the system in terms of the impulse
Echoing in music — An effect similar to multi-path in acoustics is echoing or reverberation. To see the effects of an echo in an acoustic signal consider the simulation of echoes on the handel.mat
The impulse responsesatisfies the difference equationwith zero boundary conditions. By definition(a) Use the difference equation to show that(b) Show that for m ≥ 0; n > 0determine the values of
Consider the line impulseswhere -∞ < m < ∞ and -∞ < n < ∞.(a) Draw the line impulses x[m; n] and y[m; n]. Determine if they are separable.(b) Consider the product z[m; n] = x[m;
For the two-dimensional signals(a) Draw their supports, and express these domains in terms of u1[m; n].(b) Let z[m, n] = x[m, n] - y[m, n], and draw its support. r[m, n] = am+"u12[m, n] m+ru14[m, n]
The input of an LTI continuous-time system is x(t) = u(t) - u(t – 3.5). The system’s impulse response is h(t) = u(t) - u(t – 2.5).(a) Find the system’s output y(t) by graphically computing
Suppose we would like to send the two messages mi(t), i = 1; 2, created in Problem 12 using the same bandwidth and to recover them separately. To implement this consider the QAMapproach where the
The signal at the input of an AM receiver is u(t) = m1(t) cos(20t) + m2(t) cos(100t) where the messages mi(t), i = 1; 2 are the outputs of a low–pass Butterworth filter with inputsx1(t) = r(t) -
Consider the transmission of a sinusoid x(t) = cos(2f0t) through a channel affected by multi-path and Doppler. Let there be two paths, and assume the sinusoid is being sent from a moving object so

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