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Digital Signal Processing[ECEG-3171]-Ch1_L04Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L05Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L06Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
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Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
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EE402B Radio Systems and Personal Communication Networks-Formula sheetHaris Hassan
Programmes in which available:
Masters of Engineering - Electrical and Electronic
Engineering. Masters of Engineering - Electronic
Engineering and Computer Science. Master of Science -
Communication Systems and Wireless Networking.
Master of Science - Smart Telecom and Sensing
Networks. Master of Science - Photonic Integrated
Circuits, Sensors and Networks
To enable an extension of knowledge in fundamental data communications to radio communications and networks widely adopted
in modern telecommunications systems. To provide understanding of radio wave utilisation, channel loss properties, mobile
communication technologies and network protocol architecture applied to practical wireless systems
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Digital Signal Processing[ECEG-3171]-Ch1_L04Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L05Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L06Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
Digital Signal Processing[ECEG-3171]-Ch1_L02Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced
#Africa#Ethiopia
EE402B Radio Systems and Personal Communication Networks-Formula sheetHaris Hassan
Programmes in which available:
Masters of Engineering - Electrical and Electronic
Engineering. Masters of Engineering - Electronic
Engineering and Computer Science. Master of Science -
Communication Systems and Wireless Networking.
Master of Science - Smart Telecom and Sensing
Networks. Master of Science - Photonic Integrated
Circuits, Sensors and Networks
To enable an extension of knowledge in fundamental data communications to radio communications and networks widely adopted
in modern telecommunications systems. To provide understanding of radio wave utilisation, channel loss properties, mobile
communication technologies and network protocol architecture applied to practical wireless systems
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On The Fundamental Aspects of DemodulationCSCJournals
When the instantaneous amplitude, phase and frequency of a carrier wave are modulated with the information signal for transmission, it is known that the receiver works on the basis of the received signal and a knowledge of the carrier frequency. The question is: If the receiver does not have the a priori information about the carrier frequency, is it possible to carry out the demodulation process? This tutorial lecture answers this question by looking into the very fundamental process by which the modulated wave is generated. It critically looks into the energy separation algorithm for signal analysis and suggests modification for distortionless demodulation of an FM signal, and recovery of sub-carrier signals
MATLAB sessions: Laboratory 6
MAT 275 Laboratory 6
Forced Equations and Resonance
In this laboratory we take a deeper look at second-order nonhomogeneous equations. We will concentrate
on equations with a periodic harmonic forcing term. This will lead to a study of the phenomenon known
as resonance. The equation we consider has the form
d2y
dt2
+ c
dy
dt
+ ω20y = cosωt. (L6.1)
This equation models the movement of a mass-spring system similar to the one described in Laboratory
5. The forcing term on the right-hand side of (L6.1) models a vibration, with amplitude 1 and frequency
ω (in radians per second = 12π rotation per second =
60
2π rotations per minute, or RPM) of the plate
holding the mass-spring system. All physical constants are assumed to be positive.
Let ω1 =
√
ω20 − c2/4. When c < 2ω0 the general solution of (L6.1) is
y(t) = e−
1
2 ct(c1 cos(ω1t) + c2 sin(ω1t)) + C cos (ωt− α) (L6.2)
with
C =
1√
(ω20 − ω2)
2
+ c2ω2
, (L6.3)
α =
⎧
⎨
⎩
arctan
(
cω
ω20−ω2
)
if ω0 > ω
π + arctan
(
cω
ω20−ω2
)
if ω0 < ω
(L6.4)
and c1 and c2 determined by the initial conditions. The first term in (L6.2) represents the complementary
solution, that is, the general solution to the homogeneous equation (independent of ω), while the second
term represents a particular solution of the full ODE.
Note that when c > 0 the first term vanishes for large t due to the decreasing exponential factor.
The solution then settles into a (forced) oscillation with amplitude C given by (L6.3). The objectives of
this laboratory are then to understand
1. the effect of the forcing term on the behavior of the solution for different values of ω, in particular
on the amplitude of the solution.
2. the phenomena of resonance and beats in the absence of friction.
The Amplitude of Forced Oscillations
We assume here that ω0 = 2 and c = 1 are fixed. Initial conditions are set to 0. For each value of ω, the
amplitude C can be obtained numerically by taking half the difference between the highs and the lows
of the solution computed with a MATLAB ODE solver after a sufficiently large time, as follows: (note
that in the M-file below we set ω = 1.4).
1 function LAB06ex1
2 omega0 = 2; c = 1; omega = 1.4;
3 param = [omega0,c,omega];
4 t0 = 0; y0 = 0; v0 = 0; Y0 = [y0;v0]; tf = 50;
5 options = odeset(’AbsTol’,1e-10,’RelTol’,1e-10);
6 [t,Y] = ode45(@f,[t0,tf],Y0,options,param);
7 y = Y(:,1); v = Y(:,2);
8 figure(1)
9 plot(t,y,’b-’); ylabel(’y’); grid on;
c⃝2011 Stefania Tracogna, SoMSS, ASU 1
MATLAB sessions: Laboratory 6
10 t1 = 25; i = find(t>t1);
11 C = (max(Y(i,1))-min(Y(i,1)))/2;
12 disp([’computed amplitude of forced oscillation = ’ num2str(C)]);
13 Ctheory = 1/sqrt((omega0^2-omega^2)^2+(c*omega)^2);
14 disp([’theoretical amplitude = ’ num2str(Ctheory)]);
15 %----------------------------------------------------------------
16 function dYdt = f(t,Y,param)
17 y = Y(1); v = Y(2);
18 omega0 = param(1); c = param(2); omega = param(3);
19 dYdt = [ v ; cos(omega ...
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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2. P18.1
Consider the system in Figure P18.1-1 for discrete-time processing of a continuoustime
signal using sampling period T, where the C/D operation is as shown in Figure P18.1-2
and the D/C operation is as shown in Figure P18.1-3.
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3. The filter G(Q) is the lowpass filter shown in Figure P18.1-4.
The Fourier transform of xc(t), Xc(w) is given in Figure P18.1-5.
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4. The sampling frequency is 8 kHz. Sketch accurately the following transforms.
(a) X,(w)
(b) X(Q)
(c) Y(Q)
(d) Ye(w)
Consider the continuous-time frequency response in Figure P18.2.
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5. We want to implement this continuous-time filter using discrete-time processing.
(a) What is the maximum value of the sampling period T required?
(b) What is the required discrete-time filter G(Q) for T found in part (a)?
(c) Sketch the total system.
P18.3
The system in Figure P18.3 is similar to that demonstrated in the lecture. Note
that, as in the lecture, there is no anti-aliasing filter.
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6. P18.4 Suppose we want to design a variable-bandwidth, continuous-time filter
using the structure in Figure P18.4-1.
Find, in terms of wc, the value of the sampling period To and the corresponding
value co, such that the total continuous-time filter has the frequency response
shown in Figure P18.4-2.
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7. P18.5 Consider the system in Figure P18.5-1.
Let H(Q) be as given in Figure P18.5-2 and X(co) as given in Figure P18.5-3.
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8. (a) Sketch X(Q) and Y(Q).
(b) Suppose we replace the system in Figure P18.5-1 by the P18.5-4. Find
G(w) such that y[n] = z[n].
P18.6
Suppose we are given the system in Figure P18.6-1.
(a) Find the appropriate values of the sampling period To to avoid aliasing. Also
find the proper value for K so that the overall system has a gain of unity at w = 0
(i.e., no overall dc gain).
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9. (b) Suppose To is halved, but the anti-aliasing and reconstruction filters are not
modified.
(i) If X(w) is as given in Figure P18.6-2, find Y(Q).
(ii) If Y(Q) is as given in Figure P18.6-3, find Y(w).
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10. P18.7
Figure P18.7 shows a system that processes continuous-time signals using a
digital filter. The digital filter h[n] is linear and causal with difference equation
For input signals that are bandlimited so that Xe(w) = 0 for I l > ir/T, the system
is equivalent to a continuous-time LTI system. Determine the frequency
response He(w) of the equivalent overall system with input xc(t) and output
yc(t).
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11. P18.8 Figure P18.8-1 depicts a system for which the input and output are discrete-time
signals. The discrete-time input x[n] is converted to a continuous-time impulse train
x,(t). The continuous-time signal x,(t) is then filtered by an LTI system to produce the
output yc(t), which is then converted to the discrete-time signal y[n]. The LTI system
with input xc(t) and output yc(t) is causal and is characterized by the linear constant-
coefficient difference equation
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12. The overall system is equivalent to a causal discrete-time LTI system, as indicated
in Figure P18.8-2. Determine the frequency response H(Q) of the equivalent LTI
system.
P18.9
We wish to design a continuous-time sinusoidal signal generator that is capable of
producing sinusoidal signals at any frequency satisfying wi : W5 W2, where w,
and W2 are positive numbers.
Our design is to take the following form. We have stored a discrete-time
cosine wave of period N; that is, we have stored x[O], . . . , x[N - 1], where
Every T seconds we output an impulse weighted by a value of x[k], where we
proceed through the values of k = 0, 1, ... , N - 1 in a cyclic fashion. That is,
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13. (a) Show that by adjusting T we can adjust the frequency of the cosine signal
being sampled. Specifically, show that
where wo = 21r/NT. Determine a range of values for T so that y,(t) can
represent samples of a cosine signal with a frequency that is variable over the
full range
(b) Sketch Y,(w).
The overall system for generating a continuous-time sinusoid is depicted in
Figure P18.9-1. H(w) is an ideal lowpass filter with unity gain in its passband:
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14. The parameter we is to be determined such that y(t) is a continuous-time cosine signal
in the desired frequency band.
(c) Consider any value of T in the range determined in part (a). Determine the
minimum value of N and some value for w,such that y(t) is a cosine signal in the
range wi : O o02.
(d) The amplitude of y(t) will vary depending on the value of wchosen between wi
and W2. Determine the amplitude of y(t) as a function of w and as a function of N.
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15. P18.10
In many practical situations, a signal is recorded in the presence of an echo, which
we would like to remove by appropriate processing. For example, Figure P18.10-1
illustrates a system in which a receiver receives simultaneously a signal x(t) and an
echo represented by an attenuated delayed replication of x(t). Thus, the receiver
output is s(t) = x(t) + ax(t - TO), where jal < 1. The receiver output is to be processed
to recover x(t) by first converting to a sequence and using an appropriate digital filter
h[n] as indicated in Figure P18.10-2.
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16. Assume that x(t) is bandlimited, i.e., X(w) = 0 for IwI > wm, and that al < 1.
(a) If To < lr/M and the sampling period is taken equal to To (i.e., T = TO), determine the
difference equation for the digital filter h[n] so that yc(t) is proportional to x(t).
(b) With the assumptions of part (a), specify the gain A of the ideal lowpass filter so that
yc(t) = x(t).
(c) Now suppose that 2r/wm < To < 27r/WM. Determine a choice for the sampling period
T, the lowpass filter gain A, and the frequency response for the digital filter h[n] such
that yc(t) is equal to x(t).
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17. S18.1 (a) Since x,(t) = xc(t)p(t), then X,(w) is just a replication of Xe(w) centered at
multiples of the sampling frequency, namely 8 kHz or 27r8 X 10' rad/s. The sampling
period is T = 1/8000.
(b) X(Q) is just a rescaling of the frequency axis, where 21r8 X 103 becomes 2 1r.
X(Q) is shown in Figure S18.1-2.
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18. (c) Y(Q) is the product G(Q)X(Q). Therefore, Y(Q) appears as in Figure S18.1-3.
(d) Y,(w) is a frequency-scaled version of Y(w) but only in the range 0 = --w to 7,
as shown in Figure S18.1-4. Also note the gain of T.
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19. S18.2 (a) The maximum nonzero frequency component of H(w) is 5 00 w. Therefore,
this frequency can correspond to, at most, the maximum digital frequency before
folding, i.e., Q = x. From the relation wT = Q,we get
(b) Since w = 500w maps to Q = 7r, the discrete-time filter G(Q) is as shown in
Figure S18.2-1.
(c) The complete system is given by Figure S18.2-2. Note the need for an anti-
aliasing filter.
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21. Ye(w) is given by eq. (S18.3-2) and Figure S18.3-3.
Thus x(t) = y(t) in this case.
(b) Xc(w) is as given in Figure S18.3-4.
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22. We now use eq. (S18.3-1), shown in Figure S18.3-5.
Thus, in the range ± r, X(Q) = 20000 E",,U X,[20000(9 - 2irn)] is given as in
Figure S18.3-6.
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23. Using eq. (S18.3-2), we find Ye(w) as in Figure S18.3-7.
Note aliasing since 27000 Hz is above half the sampling rate of 20000 Hz.
(c) Xe(w) is as given in Figure S18.3-8.
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24. Again we use eq. (S18.3-1), shown in Figure S18.3-9.
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25. Thus X(Q) is given as in Figure S18.3-10.
Finally, from eq. (S18.3-2) we have Y,(w) shown in Figure S18.3-11.
S18.4 It is required that we sample at a rate such that the discrete-time frequency
7r/ 2 will correspond to c. The relation between 9, and cis Q, = ocTo. Thus, we
require
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26. As wc increases, demanding a wider filter, To decreases, and consequently the sampling
frequency must be increased. There are two ways to calculate Wa. First, since we are
sampling at a rate of
we need an anti-aliasing filter that will remove power at frequencies higher than half
the sampling rate; therefore wa = 2wc. Alternatively, we note that the "folding
frequency," or the frequency at which aliasing begins, is 0 = 7r. Since 9 = r/2
corresponds to wc, then r must correspond to 2w,.
S18.5 (a) We sketch X(Q) by stretching the frequency axis so that 2 7 corresponds to
the sampling frequency with a gain of 1/TO. We then repeat the spectrum, as shown
in Figure S18.5-1.
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27. After filtering, Y(Q) is given as in Figure S18.5-2.
(b) We see that Y(Q) looks like X(w) filtered and then sampled. The discrete-time
frequency is -/3. Again, 2-x corresponds to 27r/To, so 7/3 corresponds to ir/3T. Thus, if
x(t) is filtered by G(w) as given in Figure S18.5-3, then y[n] = z[n].
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28. Solutions to Optional Problems
S18.6 (a) Since we are allowing all frequencies less than 1007r through the anti-
aliasing filter, we need to sample at least twice 100r, or 200r. Thus, 2 007r = 2-
x/T or To = 10 ms. To find K, recall that impulse sampling introduces a gain of
1/TO. To account for this, K must equal To, or K = 0.01. (b) (i) Since X(w) is
bandlimited to 100-r, the anti-aliasing filter has no effect. The Fourier transform
of x,(t), the modulated pulse train, is given in Figure S18.6-1.
Since To = 0.005, the sampling frequency is 4007r. After conversion to a
discrete-time signal, X(Q) appears as in Figure S18.6-2.
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29. (ii) There are three effects to note in D/C conversion: (1) a gain of To, (2) a frequency
scaling by a factor of T,, and (3) the removal of repeated spectra. Thus, Y(w) is as
shown in Figure S18.6-4.
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30. S18.7 After the initial shock, you should realize that this problem is not as difficult
as it seems. If instead of h[n] we had been given the frequency response H(Q), then
He(w) would be just a scaled version of H(Q) bandlimited to ir/T. Let us find, then,
H(Q). Using properties of the Fourier transform, we have
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31. Therefore, the magnitude and phase of He(c) are as shown in Figure S18.7.
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32. From our previous study, we know that Xe(o) in the range ±ir/Tlooks just like X(Q) in
the range ±ir. Similarly, Y,(w) between -- r/T and +r/T looks like Y(Q) in the range -ir
to r. Although there is a factor of T, we can disregard it in analyzing this system
because it is accounted for in the H(w) filter. The transformation of xc(t) to yc(t) will
correspond to filtering x[n], yielding y[n]. In fact, the equivalent system will have a
system function H(Q) given by
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33. where He(w) is the Fourier transform of h(t). Thus, we need to find Hc(w). The relation
between yc(t) and xc(t) is governed by the following differential equation:
Using the properties of the Fourier transform, we have
S18.9
(a) It is instructive to sketch a typical y,(t), which we have done in Figure S18.9-1.
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34. Let us suppose that T is changed by being reduced. Then the envelope of y,(t)
seems to correspond to a higher-frequency cosine. At time kt,
where we use the sampling property of the impulse function. Thus,
(b) Recall that sampling with an impulse train repeats the spectrum with a period
of 2r/T and a gain factor of 1/T. Since 5([cos(2t/NT)J is as given by Figure S18.9-
2, Y,(w) is then given by Figure S18.9-3.
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35. (c) The minimum value of N is 2, corresponding to the impulses at o and (27r/T
wo) being superimposed at w/T. The lowpass filter cutoff frequency must be such
that the (superimposed) impulses at 7r/T are in the passband and those at 37r/T are
outside the passband. Consequently,
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36. (d) Comparing Y(w) and Y,(w) in Figures S18.9-2 and S18.9-3 respectively, we
see that for N > 2 the cosine output will have an amplitude of 1/T = w/21r. If N =
2, then the output amplitude will be 2/T = w/7r.
S18.10
(a) By sampling sc(t), we get
This is a first-order difference equation, so given s[n], we can find x[n]. Since x(t)
is appropriately bandlimited, we can then set
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37. (b) From part (a) we see that T = A will make y(t) = x(t).
(c) Since we do not want to alias, we still need T < 7r/wM. Now
Taking the continuousFourier transform, we see that
Thus, the continuous-time inverse system has frequency response
We want to implement this in discrete time. Therefore, using the relation, we obtain
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