assignment questions on signals and systems

Purdue University School of Electrical and Computer Engineering

Ee301 signals and systems spring 2024, exam 3: april 19 thru brightspace. coverage: chaps. 5-7, hmwks 8-9. executive summary: better versions of table 5.1 and 5.2: better version of tables 5.1 and 5.2, exam 2: march 22 thru brightspace. coverage: chaps. 3-4, hmwks 5,6,7. most emphasis on chap. 4 and hmwks 6,7. executive summary: my versions of table 4.1 and 4.2: my versions of tables 4.1 and 4.2 ; vip handout on sinc function products/sums: sinc function fourier transforms ; note on linearity of fourier transform linearity of fourier transform, a walk-through solution for exam 1 has been recorded and posted at brightspace under content as "lectures 17-18., class meets mwf 2:30-3:20 pm in lawson b155., textbook coverage order: chaps 1,2,3,4,7,5 + chap 10., purdue honors pledge, prof. michael d. zoltowski, main ta: arnav singh. email: [email protected], arnav's office hours: wednesday, 17:30-19:30 bhee 047 tables 1, 2 thursday, 16:00-18:00 bhee 047 table 3 friday, 16:00-18:00. bhee 047 table 1, 2, high-level course overview, time of due dates: hmwks and exams, vip course information, course outline, regular class time: mwf : 2:30-3:20 pm in laws b155., instructor office hours: prof zoltowski in lwsn b134: w-f at 1:30 pm and i can answer questions in b155 after class as well, instructor email: [email protected] or [email protected], catalog course description, blank pages, mini-tutorial on complex numbers: complex number overview, homework 1 on signals., hmwk 1 help. problems 1.26, 1.36, 1.54 ; mini-write-up on prob. 1.26(c). ; hmwk 1 soln, matlab code: rectangle function. ; matlab code: unit step function., m-file for function in problem 1.21 ; m-file for time-transformatons on hmwk 1 signal, homework 2 on systems. ; hmwk 2 help prob. 1.27 ; hmwk 2 help prob. 1.18, more hmwk 2 help: help for prob 1.20 setting prob 1.20 up prob. 1.20 set-up ; hmwk 2 soln, homework 3 on ct convolution., hmwk 3 help: prob. 2.22(a) ; problem 2.22 (e) ; problem 2.22 (c) note: h(t)= 2 ( u(t-1)-u(t-3)) ; problem 2.40, more hmwk 3 help: problem 2.11 ; problem 2.23 ; hmwk 3 soln, homework 4 on dt convolution., prob. 2.28 soln ; hmwk 4 help: probs 2.26_2.5_2.21 ; prob 2.24 help, homework 5 not turned in. see hmwk 5 help in chap. 3 section below, homework 6 on fourier transform due sat, mar 2, to gradescope by 8 am. use my versions of table 4.1 and 4.2: my versions of tables 4.1 and 4.2, hmwk 6 help: ft examples homework 6 help ; prob 4.1 help prob 4.1 help ; hmwk 6 soln, homework 7 on fourier transform due sat, mar 9, to gradescope by 8 am. use my versions of table 4.1 and 4.2: my versions of tables 4.1 and 4.2, hmwk 7 help: ft examples homework 7 help ; hmwk 7 soln, homework 8 on sampling theory due tues, apr 2, to gradescope by 9:00 am. hmwk 8 help homework 8 help ; probs 7.8-9-12 probs 7.8 7.9 7.12 help, homework 9 on dtft due tues., apr 9, to gradescope by 11:59 pm. help for prob 5.21 (hmwk 9) on chap. 5 dtft properties/pairs ; help for prob 5.35 (hmwk 9) on chap. 5 dtft all-pass filters, scanned-in hmwk problems from text, text chap 1 problem statements., text chap 2 problem statements., text chap 3 problem statements., text chap 4 problem statements., text chap 5 problem statements., text chap 7 problem statements., text scans plus supplementary instructor notes:, outline for course textbook., order that chapters will be covered: 1, 2, 3, 4, 7, 5, 10 (6, 8, 9 not covered)., chapter 1: signals, chapter 1 outline., text notes on basic signals., the notes below will be covered on jan. 8 and are helpful for hmwk 1:, notes on signal basics., the notes below were covered on jan. 10 and are helpful for hmwk 1:, notes on discrete-time signal basics., m-file: time-scaling examples ; m-file: dt sinewaves, chapter 1: systems, text notes on system properties., the notes below were covered on jan. 17 and are helpful for hmwk 2:, supplemental notes on system properties., system examples and properties., system example 1.16 from textbook., the notes below were covered on jan. 19 and are helpful for hmwk 2:, difference eqns are dt lti systems., chapter 2: lti systems: impulse response and convolution, text notes on convolution., text notes on lti system properties., the notes below were covered on jan. 22 on ct convolution. (sects. 2.2 and 2.3), ct convolution derivation, properties, and basic example ; ct convolution derivation: visuals, ct convolution: calculus analogy, the notes below were covered on jan. 24 on ct convolution. (sects. 2.2 and 2.3), convolution example: rectangular pulse with ramp-up or ramp-down line ;, note on the limits of the convolution integral, the notes below were covered on jan. 26 and are helpful for homework 3 (sect. 2.5):, vip *basic convolution results* vip compilation of notes for exam 1, impulse rssponse of basic ct lti systems, the notes below will be covered on jan. 29 on dt convolution (sect. 2.1), hmwk 3 help: prob. 2.22(a) ; problem 2.22 (e) ; problem 2.22 (c) ; problem 2.40, more hmwk 3 help: problem 2.11 ; problem 2.23, **vip** new: convolve"triangle" with exponential ;; m-file convolve triangle with exponential, the notes below will be covered on jan. 31 on dt convolution (sect. 2.1), dt convolution derivation, properties, and basic examples, the notes below will be covered on feb. 2 and are helpful for homework 4:, additional properties of dt convolution plus examples ; matlab code: dt convolution example., summary of key convolution properties, the notes below were covered on feb.5 and are helpful for exam 1:, approximate ct convolution via dt convolution ; m-file convolve 2 geometric sequences ; m-file convolve 2 truncated geometric sequences ; mfile convolve 2 gaussian ct signals ; m-file convolve 2 exponential ct signals ; ; mfile convolve rect and ramp ct signals, the notes below were covered on feb.7 and are helpful for exam 1:, impulse response of nonrecursive difference eqn, examples of simple first-order difference equations, inverse lti systems (sect. 2.3.5), stability and causality of dt & ct lti systems (sects. 2.3.6-2.3.7), summary of properties of lti systems based on impulse response. (sect. 2.3), chapter 3: fourier series for periodic signals, text notes on ct fourier series., text notes on dt fourier series., the notes below will be covered on feb.9 covering key points of ct fourier series:, basic and key points underlying ct fourier series ; vector space analogy, the notes below will be covered on feb. 12 covering key points of dt fourier series:, hmwk 5 help for chap 3 problems on fourier series, problem 3.22 on fourier series, basic and key points underlying dt fourier series, m-file to generate ct rectangular pulse train, m-file to generate dt rectangular pulse train, industry standard for how orthogonal sinewaves (ofdm) are used to transmit digital data see sect 5.6 on page 39., chap. 3: text notes on frequency selective filters., chapter 4: fourier transform, chap. 4, part i: text notes on ct fourier transform., chap. 4, part ii: text notes on convolution property of fourier transform., the notes below will be covered on feb. 21 on basic fourier transform theory., basic fourier transform theory fourier transform fundamentals, basic fourier transform theory: relationship to chap 3 on fourier series interpretation of inverse fourier transform, frequency ranges of biological, e&m, and other signals., the notes below will be covered on feb. 23 on basic fourier transform theory., matlab example of fourier transform properties: matlab demo of ft properties., symmetry properties of fourier transform: symmetry properties of fourier transform., vip: proofs of fourier transform properties plus examples., proofs for supplemental properties of the fourier transform, the notes below on fourier transform examples will be covered on feb. 26 and are helpful for hmwk. 6., fourier transform examples fourier transform examples(hmwk 6 help), revised table 4.1 of fourier transform properties : notation-revised table 4.1 plus additional properties ; addl table of fourier transform pairs/properties ; calculus analogy, the notes below on fourier transform examples will be covered on feb. 28, some fourier transform results involving two sinc functions., the notes below on fourier transform examples will be covered on mar. 1, application of fourier transform theory: multiplying signal by a sinewave ; frequency allocation chart compiled by fcc: frequency allocation chart, scan of chapter 8 of o&w textbook on communication theory: o&w textbook chap. 8., removing negative frequency content in transmitted signal., example for removing negative frequency content in transmitted signal., matlab demo of single sideband modulation., fourier transform of finite length sinewaves: fourier transform of finite length sinewaves, the ft notes below on multiplying a signal by a sinewave and removing negative frequency content to be covered on mar. 4, the notes below will be covered on mar. 6, vip handout for exam 2 on sinc function products: sinc function products, notes on response of lti system to sinewaves., hmwk 7 help: add'l ft examples homework 7 help, the notes below on further insights into the fourier transform will be covered on mar. 8 and are helpful for exam 2., interpretation of phase of fourier transform: special case of linear phase: interpretation of phase of fourier transform, exam 2 help: add'l ft example: text prob 4.25 text prob. 4.25, fourier transform of gaussian signal and related results vip info for exam 2, additional insights into fourier transform fourier transform insights, fourier transform tables where units for frequency variable is hertz: fourier transform tables, wikipedia on sinc function: the sinc function., chapter 7: sampling theory, chap. 7 text notes on sampling theory., the notes below on basic sampling theory will be covered on mar. 25 and use figures from the beginning of chapter 7 in the text., basic sampling theory basic sampling theory ; sample time invariance, m-file for illustrating sampling theory ; m-file for illustrating sampling theory for sinewave, the notes below on sampling theory will be covered on mar. 27, sampling theory reprise: ideal reconstruction of bandlimited signal from its samples ideal signal reconstruction, homework 8 help., below is vip information on the benefits of digital over analog in terms of storage, transmission, and flexibility/programmability. covered on mar. 29, illustration of 3-bit encoding notes : quantization and binary encoding, digital upsampling for practical d/a conversion: digitalupsampling, advantages of digital over analog, part i: digital communications and regeneration, advantages of digital over analog, part ii error correcting codes, matlab example of ldpc code: 1 bit per symbol ldpceg.m ; matlab example of ldpc code: 2 bits per symbol ldpcqameg.m, matlab example of digital upsampling and zero-order-hold dac zoheg2.m, illustration of quantization and binary encoding ; additional material including real-world signal bandwidths, specs for ti dac chip for digital audio with 96 khz sampling and 24 bits/sample, wikipedia on how information is stored and read off a cd-rom/dvd. add'l link: how binary bit are stored and read off a cd-rom/dvd., chapter 5: discrete-time fourier transform, chap. 5: text notes on dt fourier transform., chap. 5: text notes on dt fourier transform -- second half.., the notes below on the discrete-time fourier transform (dtft) were covered on apr.1 and also relate the dtft to the ct fourier transform:, discrete-time fourier transform (dtft)basics dtft basics, the notes below related to the dtft and helpful for hmwk 9 will be covered on apr. 3, dtft properties and examples dtft properties/examples revised table 5.1 of dt fourier transform properties : notation-revised table 5.1 and table 5.2, help for prob 5.21 (hmwk 9) on chap. 5 dtft properties/pairs, help for prob 5.35 (hmwk 9) on chap. 5 dtft all-pass filters, better version of tables 5.1 and 5.2, exam 3 help problem, exam 3 big picture, revised table 5.1 of dt fourier transform properties : notation-revised table 5.1, ctft-dtft relationship for a sampled sinewave., ctft-dtft relationship plus example for exam 3., notes on relationship of dtft to ctft., supplemental notes on chap.5 related to frequency response of lti systems described by difference equations. chap. 5 supplement, chapter 10: z-transform, the notes below related to the z-transform and will be covered on apr. 27-29, chap. 10: text notes on z-transform -- first half., chap. 10: text notes on z-transform -- second half., vip "big picture" notes tying together chapters 4, 5, 7, and 10: relationship between dt and ct fourier transform, the notes below related to the z-transform and will be covered on apr. 29, vip last week notes laplace transform - z tranform, z-transform basics. z-transform basics, handout on chap.10 material: z-transform. z-transform: main handout, handout on graphical frequency response. graphicalfrequencyresponse.pdf, supplemental notes on all-pass filters. all-pass filters, material for lecture on cat scans and the theory of 2-d fourier transforms: 2-d fourier transform and cat scan theory, notes on theory of two-dimensional signals and 2-d fourier transform 2-d signals, systems, and transforms, reference for cat scan theory, and 2-d fourier transform: section_6.4 ; supplementary notes: theory of ct scans, exam 1: coverage: chaps. 1-2, hmwks 1-4. more emphasis on chap. 2. open-book plus one 8.5 x11 crib sheet. exam 1 cover sheet: exam 1 cover sheet executive summary: executive summary for exam 1 ;, old exam 1's:, example problem for exam 1 from spring 2019: example: convolution trick, exam 1 from spring 2020: exam 1 ; exam 1 soln: exam 1 solution exam 1 stats: exam 1 stats, exam 1 from spring 2019: exam 1 ; exam 1 soln: exam 1 solution exam 1 stats: exam 1 stats, exam 1 from spring 2018: exam 1 ; exam 1 soln: exam 1 solution, exam 1: exam 1 ; exam 1 soln: exam 1 solution ; exam 1 stats: exam 1 stats, exam 1 from spring 2016: exam 1 ; exam 1 soln: exam 1 soln ; ; exam 1 stats: exam 1 stats ;, exam 1: thurs., feb. 25, in me 1061 4:30-5:45 pm. exam 1 ;; condensed exam 1, exam 1 from spring 2015: exam 1 and solution to exam 1., exam 1 from spring 2014: exam 1 and solution to exam 1., exam 1 from spring 2013: exam 1 and solution to exam 1. ; histogram of scores ; histogram and stats in pdf, exam 1 from spring 2012: exam 1 and solution to exam 1. , and b&w solution with point breakdown. , histogram of scores and statistics, exam 1 from spring 2011: exam 1 and solution to exam 1. , note on integrators with finite limits, exam 1 from spring 2010: exam 1 and solution to exam 1., exam 1 from spring 2009: exam 1 and solution to exam 1., exam 1 from spring 2008: exam 1 and solution to exam 1., exam 1 from spring 2007: exam 1 and solution to exam 1., exam 1 from spring 2005: exam 1 and solution to exam 1., exam 1 from spring 2004: spring 2004 exam 1 and solutions, fall 2002 exam 1. solutions, fall 1998 exam 1. solutions, spring 1999 exam 1. solutions, old exam 2's :, exam 2 coverage: primarily text chapter 4, homeworks 6-7. the primary topic covered is the fourier transform, properties and pairs. only a single handwritten two-sided crib sheet is allowed for this exam, in addition to the textbook (open book) no calculators. a table of fourier transform properties will be attached to the exam, with the j notation removed and some additions (e.g. duality property) relative to table 4.1 in the textbook., exam 2 from spring 2019: exam 2 ; unverified soln: exam 2 solution ; exam 2 stats: exam 2 statistics, exam 2 from spring 2018: exam 2 ; exam 2 soln: exam 2 solution ; exam 2 stats: exam 2 statistics, exam 2 from spring 2017: exam 2 ; exam 2 soln: exam 2 solution ; exam 2 stats: exam 2 statistics, exam 2 from spring 2016: exam 2 sp 2016 and solution solution to exam 2., exam 2 from spring 2015: exam 2 sp 2015 and solution solution to exam 2., exam 2 from spring 2014: exam 2 sp 2014 and solution solution to exam 2., exam 2 from spring 2013: exam 2 sp 2013 and solution solution to exam 2., exam 2 from spring 2012: exam 2 sp 2012 and solution solution to exam 2. , exam 2 statistics. ; example problem 2 solution., exam 2 from spring 2011: exam 2 sp 2011 and tables and block diagrams and extra handout and solution to exam 2. , exam 2 statistics., exam 2 from spring 2010: exam 2 sp 2010 and solution to 2010 exam 2. and block diagrams, exam 2 from spring 2009: exam 2 sp 2009 and solution to 2009 exam 2. and block diagrams, exam 2 from spring 2008: exam 2 sp 2008 and solution to 2008 exam 2., exam 2 from spring 2007: exam 2 sp 2007 and solution to 2007 exam 2., exam 2 from spring 2005: exam 2 sp 2005 and solution to 2005 exam 2., exam 2 from spring 2004: exam 2 sp 2004 and solution to 2004 exam 2., old exam 3's:, exam 3 coverage: primarily text chapters 7 and 5, homeworks 8-9. primary coverage topics: sampling theory and discrete-time fourier transform, properties and pairs. also, relationship between dtft and fourier transform when the dt signal was obtained by sampling a ct (analog) signal. also, frequency response of dt lti systems described by difference equations., exam 3 sp 2019 ; exam 3 solution ; supplemental exam 3 solution ;, exam 3 from spring 2018: exam 3 ; exam 3 soln: exam 3 solution ; exam 3 stats: exam 3 statistics, exam 3 spring 2017: exam 3 ; exam 3 soln: exam 3 solution ; exam 3 stats: exam 3 statistics, exam 3 spring 2016: exam 3 ; exam 3 soln: exam 3 soln ; ; exam 3 stats: exam 3 stats ;, exam 3 from spring 2015: solution: exam 3 solution ; exam 3 statistics: exam 3 stats, exam 3 from spring 2014 exam 3 and solution to exam 3., exam 3 from spring 2013 exam 3 and solution to exam 3. ; histogram of scores ; histogram and stats in pdf, exam 3 from spring 2012: exam 3 sp 2012 and solution solution to exam 3. ; exam 3 statistics. ; exam 3 statistics., exam 3 from spring 2011: exam 3 sp 2011 and tables and extra handout and solution to 2011 exam 3. and exam 3 statistics., exam 3 from spring 2010: exam 3 sp 2010 and solution to 2010 exam 3. matlab help ctft exam3soln.m ; matlab help dtft exam3solnrads.m ;, exam 3 from spring 2009: exam 3 sp 2009 and solution to 2009 exam 3., exam 3 from spring 2008: exam 3 sp 2008 and solution to 2008 exam 3., exam 3 from spring 2007: exam 3 sp 2007 and solution to 2007 exam 3., exam 3 from spring 2005: exam 3 sp 2005 and solution to 2005 exam 3., exam 3 from spring 2004: exam 3 sp 2004 and solution to 2004 exam 3., old final exams:, final exam from spring 2019: final exam sp 2019 ; solution: final exam sp2019 solution ; better solution: better final sp2019 solution, final exam from spring 2018: final exam sp 2018 ; solution to problem 3., final exam from spring 2017: final exam sp 2017 ; solution: final exam sp2017 solution, final exam from spring 2016 final exam sp 2016 ; final exam sp 2016 solution, final exam from spring 2015 final exam sp 2015 solution: final exam sp 2015 solution, final exam from spring 2014 final exam and soln to spring 2014 final exam., final exam from spring 2013 final exam and solution to final exam., final exam from spring 2012: final exam sp 2012 and solution to 2012 final exam., final exam from spring 2011: final exam sp 2011 and solution to 2011 final exam., final exam from spring 2010: final exam sp 2010 ., final exam from spring 2009: final exam sp 2009 ., final exam from spring 2008: final exam sp 2008 ., final exam from spring 2007: final exam sp 2007 ., final exam from spring 2005: final exam sp 2005 ., final exam from spring 2004: final exam sp 2004 ., excellent tutorial article on error control coding, illustration of quantization and binary encoding, powerpoint tutorial on digital communications, pdf version of tutorial on digital communications, wikipedia on how information is stored and read off a cd-rom/dvd., current frequency allocation table by federal communications commission, vip link for frequency allocation, medical school used seinfeld as educational tool, seinfeld: george and marine biology seinfeld: marine biologist, seinfeld: kramer vs mail fraud seinfeld: kramer vs the mail., seinfeld_kramer challenged, seinfeld: kramer assman seinfeld: kramer assman, seinfeld: kramer gets a job seinfeld: kramer gets a job, seinfeld: kramer chicken roaster seinfeld: kramer chicken roasters., seinfeld: elaine and dancing seinfeld: elaine dancing., seinfeld: kramer stops talking seinfeld: kramer stops talking, seinfeld: george the bad boy seinfeld: george the bad boy., seinfeld: george bootlegger blooper seinfeld: george bootlegger blooper., seinfeld: george and jerk-store comeback seinfeld: george and jerk-store comeback., seinfeld: george "it's not you, it's me" seinfeld: george "it's not you, it's me"., seinfeld: the lip reader seinfeld: the lip reader., seinfeld: kramer pipe night seinfeld: kramer pipe night., seinfeld: pipe night blooper seinfeld: pipe night blooper., seinfeld: kramer and jerry's apartment seinfeld: kramer and jerry's apartment., seinfeld: kramer and merv griffin show seinfeld: kramer and merv griffin show., seinfeld: george's opposite religion seinfeld: george opposite religion, seinfeld: kramer is batman seinfeld: kramer is batman, seinfeld: the lopper seinfeld: the lopper, seinfeld: kramerica industries seinfeld: kramerica industries., seinfeld: kramer and mail fraud seinfeld: kramer and mail fraud., seinfeld: kramer and novacaine seinfeld: kramer and novacaine., seinfeld: kramer karate sessions seinfeld: kramer and karate., seinfeld: black market shower heads seinfeld: low-flow shower heads., seinfeld: keith hernandez part 1 seinfeld: keith hernandez part 1, seinfeld: keith hernandez part 1 seinfeld: keith hernandez part 2, seinfeld: george and fire at children's party seinfeld: george and fire at children's party, seinfeld: kramer's license plates seinfeld: kramer's license plates, seinfeld: george and blind date seinfeld: george and blind date, seinfeld: nothing higher than architect seinfeld: george and the architect., seinfeld: kramer/newman and bottle deposit refund seinfeld: kramer/newman and bottle deposit refund, seinfeld: mechanic becomes attached to jerry's car seinfeld: mechanic becomes attached to jerry's car., seinfeld: kramer/newman rickshaw idea seinfeld: kramer/newman rickshaw idea., seinfeld: kramer & jerry's apt seinfeld: kramer & jerry's apt, seinfeld: not that there's anything wrong with that seinfeld: not that there's anything wrong with that, reference: prof bouman's lecture notes., previous class offering (inc. exams and hmwks plus solutions.), supplementary notes on fourier series and linear algebra ., ken fischer's web supplements ( ct convolution ; dt convolution )..

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2.7: Signals and Systems Problems

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Learning Objectives

• Signals and Systems practice problems.

Complex Number Arithmetic

Find the real part, imaginary part, the magnitude and angle of the complex numbers given by the following expressions.

• \[\frac{1+\sqrt{3}i}{2} \nonumber \]
• \[1+i+e^{i\tfrac{\pi }{2}} \nonumber \]
• \[e^{i\tfrac{\pi }{3}}+e^{i\pi }+e^-({i\tfrac{\pi }{3}}) \nonumber \]

Discovering Roots

Complex numbers expose all the roots of real (and complex) numbers. For example, there should be two square-roots, three cube-roots, etc. of any number. Find the following roots.

• What are the cube-roots of 27? In other words, what is 27 1/3 ?
• What are the fifth roots of 3(3 1/5 )?
• What are the fourth roots of one?

Cool Exponentials

• \[i^{i} \nonumber \]
• \[i^{2i} \nonumber \]
• \[i^{i^{-1}} \nonumber \]

Complex-valued Signals

Complex numbers and phasors play a very important role in electrical engineering. Solving systems for complex exponentials is much easier than for sinusoids, and linear systems analysis is particularly easy.

• \[3\sin (24t) \nonumber \]
• \[\sqrt{2} \cos \left ( 2\pi 60t + \frac{\pi }{4} \right ) \nonumber \]
• \[2 \cos \left (t + \frac{\pi }{6} \right ) + 4 \sin \left ( t - \frac{\pi }{3} \right ) \nonumber \]
• Show that for linear systems having real-valued outputs for real inputs, that when the input is the real part of a complex exponential, the output is the real part of the system's output to the complex exponential (see figure below).

\[S\left ( \Re (Ae^{i2\pi ft}) \right ) = \Re \left (S (Ae^{i2\pi ft}) \right ) \nonumber \]

• \[v(t) = \cos (5t) \nonumber \]
• \[v(t) = \sin \left ( 8t+\frac{\pi }{4} \right ) \nonumber \]
• \[v(t) = e^{-t} \nonumber \]
• \[v(t) = e^{-(3t)}\sin \left ( 4t+\frac{3\pi }{4} \right ) \nonumber \]
• \[v(t) = 5e^{(2t)}\sin (8t + 2\pi ) \nonumber \]
• \[v(t) = -2 \nonumber \]
• \[v(t) = 4\sin (2t) + 3\cos (2t) \nonumber \]
• \[v(t) = 2\cos \left ( 100\pi t + \frac{\pi }{6} \right )- \sqrt{3}\sin \left ( 100\pi t + \frac{\pi }{2} \right ) \nonumber \]

Linear, Time-Invariant Systems

When the input to a linear, time-invariant system is the signal x(t), the output is the signal y(t) ,

• Find and sketch this system's output when the input is a unit step.

Linear Systems

The depicted input x(t) to a linear, time-invariant system yields the output y(t) .

• What is the system's output to a unit step input u(t) ?

Communication Channel

A particularly interesting communication channel can be modeled as a linear, time-invariant system. When the transmitted signal x(t) is a pulse, the received signal r(t) is as shown:

• What will be the received signal when the transmitter sends the pulse sequence x 1 (t) ?
• What will be the received signal when the transmitter sends the pulse sequence x 2 (t) that has half the duration as the original?

Analog Computers

So-called analog computers use circuits to solve mathematical problems, particularly when they involve differential equations. Suppose we are given the following differential equation to solve.

\[\frac{\mathrm{d\: y(t)} }{\mathrm{d} t} + ay(t) = x(t) \nonumber \]

In this equation, a is a constant.

• When the input is a unit step \[(x(t) = u(t)) \nonumber \] the output is given by \[y(t) = (1-e^{-(at)})u(t) \nonumber \] What is the total energy expended by the input?
• Instead of a unit step, suppose the input is a unit pulse (unit-amplitude, unit-duration) delivered to the circuit at time t =10 , what is the output voltage in this case? Sketch the waveform.

Signals and Systems (EENG226 / INFE226)

Topic outline.

Signals and Systems (EENG226)

Lecturer:  Prof. Dr. Hasan AMCA

email:  [email protected]

ALL EENG226 STUDENTS

PLEASE NOTE THAT THERE WILL BE A 30 MINUTES QUIZ ON CHAPTER 1 ON THURSDAY DURING THE LECTURE HOUR. YOU WILL BE RESPONSIBLE FOR ALL OF CHAPTER 1, WHICH COVERS THE FOLLOWING SUBJECTS.

Classification of Signals

Basic Operations on Signals

Elementary Signals

Systems Viewed as Interconnections of Operations

Properties of Systems

Introduction to Signals and Systems

What is a signal: Continuous and Discrete Signals, Analog or Digital Signals...

What is a system: Linear Time Invariant Systems

System properties, convolution sum  and the convolution integral representation, system properties, LTI systems  described by differential and difference equations.

Introduction to LTI Systems

Convolution theorem.

Assistants solve the TUTORIAL questions on Discrete and Continuous Time Convolution...

Assistants: Solve tutorial question employing intermediate function

Interconnections of LTl Systems

Fourier series.

Representation of periodic continuous-time and discrete-time signals and  filtering.

Introduction to Fourier Series

Fourier representations for four classes of signals.

Assistants solve tutorial questions on Fourier Representation

Properties of Fourier Representations

Applications of fourier representations to mixed signal classes.

Time and  frequency shifting, conjugation, differentiation and integration, scaling,  convolution, and the Parseval’s relation

Sample Exam Questions

Tutorials and suplementary materials.

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Bryan Huang

khalid nasir

Solution 2.1 x(n) is periodic if x(n) = x(n + N) for some integer value of N. For the sequence in (a), x (n + N) = A cos (27 n + N-) x(n + N) = x(n) if 7 N is an integer multiple of 27. The smallest value of N for which this is true is N = 14. Therefore the sequence in (a) is periodic with period 14. For the sequence in (b), (+ N_-7) x(n + N) = e nT)jN jN = ej 8 e 8 =x(n) e N The factor ej8-is unity for (N/8) an integer multiple of 2rr. This requires that N-= 2TrR 8 where N and R are both integers. This is not possible since 7 is an irrational number. Therefore this sequence is not periodic. Solution 2. 2 x(n) =-26(n + 3)-6(n) + 36(n-1) + 26(n-3) Solution 2. 3 Each of the systems given can be tested against the definitions of linearity and time invariance. For example, for (a), T[x (n)] = 2x (n) + 3 T[x 2 (n)] = 2x 2 (n) + 3 Since T(ax 1 (n) + bx2 (n)] = 2[ax 1 (n) + bx2 (n)] + 3 and aT(x 1 (n)] + bT (x 2 (n)) = 2axl(n) + 2bx 2 (n) + 3(a + b) The system is not linear. The system is, however, shift-invariant since T[x(n-n 0)J = 2x(n-n) + 3 = y(n-n)

ANDREE FRANKLIN SALAZAR ROJAS

System F input u(t) output y(t) System F input u(t) System G output y(t) 2 LINEAR SYSTEMS 2 2 LINEAR SYSTEMS We will discuss what we mean by a linear time-invariant system, and then consider several useful transforms. 2.1 Definition of a System In short, a system is any process or entity that has one or more well-defined inputs and one or more well-defined outputs. Examples of systems include a simple physical object obeying Newtonian mechanics, and the US economy! Systems can be physical, or we may talk about a mathematical description of a system. The point of modeling is to capture in a mathematical representation the behavior of a physical system. As we will see, such representation lends itself to analysis and design, and certain restrictions such as linearity and time-invariance open a huge set of available tools. We often use a block diagram form to describe systems, and in particular their interconnec­ tions: In the second case shown, y(t) = G[F [u(t)]]. Looking at structure now and starting with the most abstracted and general case, we may write a system as a function relating the input to the output; as written these are both functions of time: y(t) = F [u(t)] The system captured in F can be a multiplication by some constant factor-an example of a static system, or a hopelessly complex set of differential equations-an example of a dynamic system. If we are talking about a dynamical system, then by definition the mapping from u(t) to y(t) is such that the current value of the output y(t) depends on the past history of u(t). Several examples are: � t y(t) = u 2 (t 1)dt 1 , t−3 N y(t) = u(t) + � u(t − nδt). n=1 In the second case, δt is a constant time step, and hence y(t) has embedded in it the current input plus a set of N delayed versions of the input.

Sung min Kim

bugaje aminu

For the single-sided spectra, write the signal in terms of cosines: x(t) = 10cos(4πt + π/8) + 6 sin(8πt + 3π/4) = 10cos(4πt + π/8) + 6 cos(8πt + 3π/4 − π/2) = 10cos(4πt + π/8) + 6 cos(8πt + π/4) For the double-sided spectra, write the signal in terms of complex exponentials using Euler's theorem: x(t) = 5exp[(4πt + π/8)] + 5 exp[−j(4πt + π/8)] +3 exp[j(8πt + 3π/4)] + 3 exp[−j(8πt + 3π/4)] The two sets of spectra are plotted in Figures 2.1 and 2.2. Problem 2.2 The result is x(t) = 4e j(8πt+π/2) + 4e −j(8πt+π/2) + 2e j(4πt−π/4) + 2e −j(4πt−π/4) = 8cos(8πt + π/2) + 4 cos (4πt − π/4) = −8 sin (8πt) + 4 cos (4πt − π/4)

Yaw Ntiamoah

Souleyman Hassan

This chapter is about LTI system and the inpulse response.

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EE344 - Signals and Systems I - Fall 2006

  lecture-by-lecture overview.

EEE2047S: Signals and Systems I

Course presentation, teaching assistance, lecture notes and textbook.

• Introduction ( 2up ).
• Signals ( 2up ).
• Systems ( 2up ).
• Fourier series ( 2up ).
• Fourier transform ( 2up ).
• Filtering ( 2up ).
• Laplace transform ( 2up ).
• None at present.
• MATLAB convolution demo: cconvdemo on this page.
• Fourier series demo: demo_fourierseries.m .
• Fourier transform from Fourier series demo: demo_fseries2ftform.m .
• Fourier reconstruction demo: demo_fourierreconstruct.m .
• System frequency response demo: demo_freqresp1.m .
• Frequency response for periodic input demo: demo_freqresp2.m .

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• Assignment --> (and --> solutions --> . Hand-in Friday 4 April, at start of --> .

Tutorials and Additional Questions

Class tests, exam information, current class record.

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Signals and Systems

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

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Prof. Kushal K. Shah

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Gabriela Salinas and Jeeva Somasundaram

Roula Khalaf, Editor of the FT, selects her favourite stories in this weekly newsletter.

In April this year, Hein Schumacher, chief executive of Unilever, announced that the company was entering a “new era for sustainability leadership”, and signalled a shift from the central priority promoted under his predecessor , Alan Jope.

While Jope saw lack of social purpose or environmental sustainability as the way to prune brands from the portfolio, Schumacher has adopted a more balanced approach between purpose and profit. He stresses that Unilever should deliver on both sustainability commitments and financial goals. This approach, which we dub “realistic sustainability”, aims to balance long- and short-term environmental goals, ambition, and delivery.

As a result, Unilever’s refreshed sustainability agenda focuses harder on fewer commitments that the company says remain “very stretching”. In practice, this entails extending deadlines for taking action as well as reducing the scale of its targets for environmental, social and governance measures.

Such backpedalling is becoming widespread — with many companies retracting their commitments to climate targets , for example. According to FactSet, a US financial data and software provider, the number of US companies in the S&P 500 index mentioning “ESG” on their earnings calls has declined sharply : from a peak of 155 in the fourth quarter 2021 to just 29 two years later. This trend towards playing down a company’s ESG efforts, from fear of greater scrutiny or of accusations of empty claims, even has a name: “greenhushing”.

Test yourself

This is the fourth in a series of monthly business school-style teaching case studies devoted to the responsible business dilemmas faced by organisations. Read the piece and FT articles suggested at the end before considering the questions raised.

About the authors: Gabriela Salinas is an adjunct professor of marketing at IE University; Jeeva Somasundaram is an assistant professor of decision sciences in operations and technology at IE University.

The series forms part of a wider collection of FT ‘instant teaching case studies ’, featured across our Business Education publications, that explore management challenges.

The change in approach is not limited to regulatory compliance and corporate reporting; it also affects consumer communications. While Jope believed that brands sold more when “guided by a purpose”, Schumacher argues that “we don’t want to force fit [purpose] on brands unnecessarily”.

His more nuanced view aligns with evidence that consumers’ responses to the sustainability and purpose communication attached to brand names depend on two key variables: the type of industry in which the brand operates; and the specific aspect of sustainability being communicated.

In terms of the sustainability message, research in the Journal of Business Ethics found consumers can be less interested when product functionality is key. Furthermore, a UK survey in 2022 found that about 15 per cent of consumers believed brands should support social causes, but nearly 60 per cent said they would rather see brand owners pay taxes and treat people fairly.

Among investors, too, “anti-purpose” and “anti-ESG” sentiment is growing. One (unnamed) leading bond fund manager even suggested to the FT that “ESG will be dead in five years”.

Media reports on the adverse impact of ESG controversies on investment are certainly now more frequent. For example, while Jope was still at the helm, the FT reported criticism of Unilever by influential fund manager Terry Smith for displaying sustainability credentials at the expense of managing the business.

Yet some executives feel under pressure to take a stand on environmental and social issues — in many cases believing they are morally obliged to do so or through a desire to improve their own reputations. This pressure may lead to a conflict with shareholders if sustainability becomes a promotional tool for managers, or for their personal social responsibility agenda, rather than creating business value .

Such opportunistic behaviours may lead to a perception that corporate sustainability policies are pursued only because of public image concerns.

Alison Taylor, at NYU Stern School of Business, recently described Unilever’s old materiality map — a visual representation of how companies assess which social and environmental factors matter most to them — to Sustainability magazine. She depicted it as an example of “baggy, vague, overambitious goals and self-aggrandising commitments that make little sense and falsely suggest a mayonnaise and soap company can solve intractable societal problems”.

In contrast, the “realism” approach of Schumacher is being promulgated as both more honest and more feasible. Former investment banker Alex Edmans, at London Business School, has coined the term “rational sustainability” to describe an approach that integrates financial principles into decision-making, and avoids using sustainability primarily for enhancing social image and reputation.

Such “rational sustainability” encompasses any business activity that creates long-term value — including product innovation, productivity enhancements, or corporate culture initiatives, regardless of whether they fall under the traditional ESG framework.

Similarly, Schumacher’s approach aims for fewer targets with greater impact, all while keeping financial objectives in sight.

Complex objectives, such as having a positive impact on the world, may be best achieved indirectly, as expounded by economist John Kay in his book, Obliquity . Schumacher’s “realistic sustainability” approach means focusing on long-term value creation, placing customers and investors to the fore. Saving the planet begins with meaningfully helping a company’s consumers and investors. Without their support, broader sustainability efforts risk failure.

Questions for discussion

Read: Unilever has ‘lost the plot’ by fixating on sustainability, says Terry Smith

Companies take step back from making climate target promises

The real impact of the ESG backlash

Unilever’s new chief says corporate purpose can be ‘unwelcome distraction ’

Unilever says new laxer environmental targets aim for ‘realism’

How should business executives incorporate ESG criteria in their commercial, investor, internal, and external communications? How can they strike a balance between purpose and profits?

How does purpose affect business and brand value? Under what circumstances or conditions can the impact of purpose be positive, neutral, or negative?

Are brands vehicles by which to drive social or environmental change? Is this the primary role of brands in the 21st century or do profits and clients’ needs come first?

Which categories or sectors might benefit most from strongly articulating and communicating a corporate purpose? Are there instances in which it might backfire?

In your opinion, is it necessary for brands to take a stance on social issues? Why or why not, and when?

Climate Capital

Where climate change meets business, markets and politics. Explore the FT’s coverage here .

Are you curious about the FT’s environmental sustainability commitments? Find out more about our science-based targets here

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Course info.

• Prof. Dennis Freeman

Departments

• Electrical Engineering and Computer Science

As Taught In

• Digital Systems
• Signal Processing

Learning Resource Types

Signals and systems.

Exams from previous years (Spring 2010 and Fall 2009) are provided for review purposes; Quiz 3 from Spring 2010 covers different topics and therefore has been omitted.

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