FE Exam Tips: Answering Difficult Engineering Questions Pt. 1

Taking the Fundamentals of Engineering (FE) exam can be a daunting task for many engineering students and graduates. The breadth of knowledge required to pass this 6-hour-long exam covering a variety of engineering disciplines is vast, leading many examinees to spend months in preparation through reviewing engineering concepts, solving practice problems, and taking practice exams. However, even with thorough preparation, the FE exam can present unexpected challenges on test day. Difficult exam questions can catch students off guard, which can cause anxiety and stress during the testing period.

Having a structured game plan in approaching difficult questions can help ease nerves and optimize performance when encountering unclear or tricky concepts on the FE. This is the first in a two-part series of blogs providing tips and techniques for students to use when answering difficult questions during the FE exam. We will cover strategies such as flagging questions for later, logically working through multi-step problems, leveraging your strengths across knowledge domains, and maintaining a positive mindset. Read on to learn how to conquer tough exam questions on test day.

In this Blog,

1. Assess the Question

2. Work Through the Fundamentals

3. Skip and Flag Questions

4. Leverage Your Strengths

5. Stay Positive

Assess the Question

When presented with a difficult question on the FE exam, before delving in too quickly, take a step back to fully assess what is being asked. Read the question prompt carefully and identify exactly what needs to be determined or calculated. Underline or highlight key parts of the question or any data that is provided. Then, take a few seconds to think - what engineering concept or formula will help me solve this? Trying to isolate the fundamental theory or relationship at hand is crucial, rather than jumping right into calculations. This aligns your thought process in the right direction from the start.

For multi-step quantitative problems, visualizing the scenario can assist with this assessment phase. Quickly sketch or draw the components involved, labeling any information given. This puts the abstract problem into a concrete perspective, clarifying the goal and known factors. Once the question is fully parsed apart, determining the proper formula or method will be more apparent. Rushing into solving before understanding all aspects of a difficult question often leads to mistakes or wasted time from improper approaches. So, take a breath, analyze what’s being asked, and set yourself up for success.

Work Through the Fundamentals

Encountering difficult questions in subject areas that you are less confident about, lean on the fundamentals to push through to the solution. Break down complex problems into simpler, more familiar components that you can grasp. For example, electromagnetism problems can be simplified using basic electromagnetic principles of oppositely charged particles, conductors, electric fields, etc. Reduce the problem to the basics within these core concepts. Identify any simplifying assumptions you can make to remove unnecessary complexity. Isolating these to walk through methodically will often help unlock the proper problem-solving path.

It can also help you think back to the most basic conceptual questions or example problems covered in your study resources for the topic. Difficult exam problems usually test the same principles but in more complex scenarios. If you are completely unsure of how to proceed, go back to the simple case and expand from there. The way core engineering principles interact and build upon each other will guide you through even nasty, difficult questions. Stick to the fundamentals and avoid getting overwhelmed by complexities.

Skip and Flag Questions

Be alert for any difficult questions that you are completely unsure of how to approach after a quick assessment. These types of questions are time sinks, distracting mental energy and motivating a random guessing approach. For these, it is often best to immediately skip them and flag for later return. Quickly flag it in your exam software, write a physical mark on your note board, or even note it on a written list of flagged questions you want to revisit. This will allow you to stay focused on questions you can solve confidently.

Getting mired down on a few difficult questions can derail your progress and timing for the exam. Make sure to budget your time across all knowledge areas accordingly, flagging questions that are too difficult to come back to later. Doing this allows you to maximize your score potential across the board. After returning to other questions, with more time and cleared thoughts, flagged questions may even seem less daunting. Just be sure to keep a close track of flags to circle back before the end of your allotted exam time.

Leverage Your Strengths

On the FE exam, answers to difficult questions in one topic area are sometimes obscured in the relationships to other topics. For example, a challenging dynamics problem may require understanding electrical circuit analysis to unlock the solution. When stuck on a question, think broadly across engineering disciplines and leverage your strengths. What other knowledge domains might offer clues to help solve this? The FE tests a breadth of interrelated engineering concepts, so find the connections.

Also, consider strengths in problem-solving approaches that could be applied. For challenging quantitative questions, is there an easier method you have experience with? Could a scalar-based shortcut help rather than brute-force vector calculations? Your strengths in certain simplification techniques or approaches to specific problem archetypes can enable breakthroughs.

Finally, is there another way to frame the question or check your work? Trying to come at the problem from a different perspective can provide surprisingly simple solutions to seemingly difficult questions. Play to your advantages in mental modeling ability, conceptual understanding, and problem-solving creativity.

Stay Positive

Maintaining a positive mindset when encountering difficult questions is critical for FE exam performance. Getting derailed mentally by tricky concepts can quickly snowball into a downward spiral of self-doubt and rushed guessing. Be ready to take a few deep breaths when prompted with confusing questions to clear your head. Remember that you studied and prepared for this and have the engineering knowledge to reason through it. Trust your mental process, isolate the fundamentals, and be open to creative approaches.

Also, be sure to pat yourself on the back for small successes throughout the day. Give yourself credit for everything you know you answered correctly so far, rather than dwelling on uncertainties. Have confidence in all the knowledge you possess. Motivate yourself to push through the final stages of the mentally taxing FE exam. Keep spirits high, knowing this is a milestone accomplishment in your engineering journey.

With the right problem-solving mindset and strategies, difficult questions on the FE exam can be overcome for success. Assess the core of the question, stick to the engineering fundamentals, strategically skip confusing questions, leverage your strengths, and stay positive. Put these tips into practice during your FE exam preparation and on test day. Trust yourself, your knowledge, and your abilities. You’ve got this!

Did you know School of PE offers tutoring if you need help mastering certain concepts on your FE exam? Our tutors are subject-matter and industry experts who teach only in their respective fields of expertise. They are experienced educators dedicated to helping our students succeed. Check out our tutoring options today and gain the confidence you need to pass on your first try!

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

FE Exam Week Survival Guide

The Fundamentals of Engineering (FE) exam is a pivotal milestone for any engineer or engineering student's career. Passing this exam is the first step to earning the Professional Engineer (PE) license, and it opens doors to enhanced career opportunities and higher earning potential. With so much riding on this one test, it's crucial to go into it fully prepared through diligent studying and review.

The week leading up to the FE exam is an anxious but exciting time. Ideally, you will have already put in months of hard work studying, and you can consider these final days as your last chance to brush up on concepts, work on your test-taking strategies, and get yourself into the right mindset. Structuring this critical week properly can give you the edge you need to pass. In this blog post, I'll share my tips on how to make the most of the week before the FE exam.

FE Exam Week Survival Guide

1. Get Organized

2. Review Key Topics

3. Practice Memorizing Formulas

4. Take Practice Exams

5. Simulate Test Conditions

6. Brush Up on Mathematics

7. Review Weak Topics

8. Learn Testing Strategies

9. Maintain Physical and Mental Health

Get Organized

In the week before the FE exam, your study habits don’t need to kick into overdrive. In most cases, cramming everything in at the last-minute leads to stress and confusion. That's why staying organized is so important.

Make a study calendar that maps out which topics you'll review each day. Block out specific chunks of time for studying, breaks, meals, exercise, etc. Having a plan prevents you from wasting time figuring out what to do next. It also holds you accountable to ensure you don’t procrastinate.

Stock up on snacks, frozen meals, and anything else that will minimize chores and errands in your final week. Get all your FE exam supplies together in one place - calculator, pencils, passport, snacks, etc. Gather up all your FE study materials and notes and organize them by topic. The more structured you are, the easier it'll be to maximize your remaining prep time.

Review Key Topics

In your first pass through the FE exam content, you likely focused on understanding the main concepts and formulas. Now is the time to consolidate that knowledge. Go through each topic again with an eye for the types of problems and situations that are most likely to show up on the actual exam.

Make flashcards of must-know equations and definitions that you have struggled with. Work through practice problems without references to simulate exam conditions. Identify the topics you're shakiest on and spend extra time rehearsing those. The goal is to reinforce the key principles and relationships, so they become second nature.

Practice Memorizing Formulas

You won't get every formula memorized, but there are certain foundational ones you absolutely should know by heart. These include equations for dynamics, strength of materials, fluid mechanics, thermodynamics, and mathematics. Committing them to memory will save you precious minutes flipping through your reference manual on test day.

Make flashcards for the essential formulas and drill yourself on them throughout the week. Come up with mnemonic devices and other tricks to aid recall. Recite the formulas aloud regularly to strengthen the pathways in your brain. By exam day, you want these critical equations to roll off your tongue on command.

Take Practice Exams

Full-length practice exams are your best gauge of FE exam readiness. Try to take at least two to three practice tests under simulated conditions during your final week of preparation. Grade yourself honestly to reveal lingering weak areas. Review the problems you got wrong in-depth to understand where your knowledge is lacking.

Taking practice exams also familiarizes you with the exam format, timing, and question styles. Build your test-taking endurance so you're prepared to power through the grueling 5-hour and 30-minute exam. Having exposure to the real thing boosts confidence on exam day when those first questions pop up on the screen.

Simulate Test Conditions

To get the most accurate results from practice tests, you need to recreate exam conditions as closely as possible. Take the practice test in a quiet room free of distractions. Time yourself rigorously, and don't go over the allotted time limit. Avoid referencing notes or textbooks except for the reference manual. Program your calculator fully before starting. Sitting at a desk or table for the length of the exam gets you used to the physical demands.

Mimicking the real testing environment prevents nasty surprises about the pace, pressure, or computer interface. It also engrains the mindset and stamina required on exam day. Think of each practice test as a dress rehearsal to get your mental game sharper.

Brush Up on Mathematics

Rusty math skills are one of the biggest pitfalls for FE exam takers. Math topics like algebra, geometry, trigonometry, and calculus form the bedrock principles for solving engineering problems. Don't let careless mathematical mistakes trip you up.

Use your practice exams to identify where your math foundations need strengthening. Target those areas for extra review. Work through math problems without relying on calculators to sharpen your skills. Brush up on techniques for rearranging equations, handling exponents and radicals, graphing functions, and solving integrals. The faster you can handle the math portion, the more time you'll have for the complex engineering questions.

Review Weak Topics

After taking practice exams, analyze your wrong answers to reveal weak spots in your knowledge. Were there particular subjects or problem types that you consistently struggled with? Go back and thoroughly review those shaky topics.

Study the theory and work out plenty of practice problems to build mastery. Check textbook examples and YouTube videos to clarify concepts you're foggy on. Strengthening your weak areas can earn you more points on test day. Don't ignore identified deficiencies and hope they won't be on the exam. Shore up those gaps now before it's too late.

Learn Testing Strategies

Cracking the FE exam takes more than just content knowledge. You also need test-taking strategies tailored to this exam's unique format and constraints.

For multiple-choice questions, learn techniques like elimination and educated guessing. Brush up on strategies for solving word problems efficiently. Master time management strategies to pace yourself properly over the 5-hour and 30-minute endurance test.

On test day, you'll also need to navigate the computer interface confidently while managing anxiety. Getting familiar with the exam's logistics and psychological demands ensures they don't trip you up.

Maintain Physical and Mental Health

Don't neglect your physical and mental well-being amidst the exam stress. Lack of sleep, exercise, healthy food, and relaxation can nuke your focus, retention, and stamina. Here are some tips for taking care of yourself that crucial week:

• Stick to a regular sleep schedule, turning in early if needed. Being well-rested boosts productivity.
• Eat healthy brain foods like berries, nuts, whole grains, and fatty fish. Stay hydrated and limit caffeine.
• Exercise daily to reduce anxiety, elevate mood, and boost thinking. Aim for 30-60 minutes a day.
• Give your brain breaks by listening to music, going outside, or talking to friends. Don't burn yourself out.
• Reduce social media and TV to limit distractions and mental clutter.
• Turn to relaxation techniques like meditation, yoga, and deep breathing to calm nerves.
• Get pep talks from loved ones to stay motivated.
• Trust your preparation. Having confidence in yourself will help you succeed.

Staying balanced physically and emotionally keeps you firing on all cylinders for the final exam push. Don't neglect self-care in pursuit of perfection.

Conclusion

Preparing for the FE exam is a months-long endeavor. However, how you spend the final week before the test can make or break your chances of success. Follow these tips to make sure you cross the finish line strongly:

Get hyper-organized to maximize study time. Drill essential formulas until they're second nature. Take mock exams under real conditions. Review deficient areas. Maintain fitness and reduce stress. With smart, focused preparation, you'll walk into that test center ready to conquer the FE exam.

One of the best ways to gain confidence for your FE exam is to take an exam review course. School of PE’s comprehensive courses combine structured lectures with exclusive study materials to bring you the best chance of success on your exam. Register for a course today!

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

How to Pass the FE Exam on Your First Try

A crucial first step to becoming a licensed Professional Engineer (PE) in the United States is to pass the Fundamentals of Engineering (FE) exam and receive your Engineer in Training (EIT) certification. As someone who has successfully accomplished the first milestone, allow me to provide my personal strategies on how to adequately prepare and set yourself up for success.

In this Blog,

1. Understanding the FE Exam Structure

2. Creating a Winning Study Strategy

3. Your First Month

4. Practice, Practice, Practice

5. Stress Management

Understanding the FE Exam Structure

First things first, let us familiarize ourselves with the structure of the FE; the exam is a 6-hour long computer-based test, organized into 5 major knowledge areas - mathematics, engineering economics, ethics, probability & statistics, engineering sciences, and the specific engineering discipline of your choice. Composed of 110 multiple-choice questions in total, with the exact breakdown of topics varying slightly for each specific engineering discipline. For example, the Chemical FE exam places more emphasis on material science and thermodynamics, while the Civil FE exam focuses more on structural analysis and construction. Keep this in mind when creating your study plan.

Creating a Winning Study Strategy

Now we know how the exam will be structured, step one is to start preparing early - at least 2-3 months in advance if possible. Don't wait until the last minute to start studying. For example, if you plan on taking the FE exam in mid-April, you should begin studying in early January. Obtain the official FE Reference Handbook from the NCEES website and spend time familiarizing yourself with the equations, tables, and graphs provided. This reference handbook is the single approved material you are allowed to use during the actual exam, so it is vital to know the contents well. You should also purchase an FE exam prep book that contains sample questions and practice exams. I recommend an NCEES practice exam and prep books from proven exam prep providers like School of PE.

Your First Month

In your first month of studying, focus on thoroughly reviewing the reference handbook, refreshing yourself on mathematical concepts like calculus and linear algebra, and taking an initial practice exam to benchmark your skills. This will identify weak areas for you to focus on. For example, I struggled with the engineering economics questions, so I spent extra time re-learning topics like the time value of money, depreciation, and inflation. In the second month, work through your prep book methodically, reading chapters and completing the practice problems related to each knowledge area. Really focus on practice problems similar to actual exam questions. In the final weeks before the exam, your prep should increase to solving 4-5 hours of practice problems per night. Take multiple timed, simulated exams to improve your pacing and stamina. I recommend taking at least two full-length practice exams.

Practice, Practice, Practice

In addition to individual subject review, practicing full-length exams is essential preparation. Simulated exams help build the mental endurance required for the 6-hour test. On exam day, you will receive an online agreement form and have 8 minutes of tutorial time to get accustomed to the format before the clock starts. The exam allows for one scheduled 50-minute break, which I recommend taking to refresh your brain before diving into the next section. In addition to the scheduled 50-minute break, you will have the flexibility to take unscheduled breaks as needed for restroom usage or grabbing a snack/water. However, it is prudent to be judicious when taking excessive unscheduled breaks, as the exam clock continues counting down during any time away from your computer. While essential personal needs should always be met, aim to limit unscheduled breaks to just 5-10 minutes so as not to squander valuable exam time. Make use of the restroom just before beginning a new section. With thoughtful time budgeting, you can satisfy necessary breaks without compromising your ability to complete this time-sensitive exam. Keep this timed exam structure in mind as you work practice tests at home so you are prepared for the pacing required on the actual test day.

Stress Management

Managing anxiety and stress is also hugely important when preparing for and taking the FE exam. Have confidence in your diligent preparation and utilize relaxation techniques leading up to exam day. Get adequate sleep, eat a nutritious breakfast, and do breathing exercises on the morning of the test. Arrive early at the testing center so you do not feel rushed. I brought a bottle of water, a snack, and earplugs to aid my comfort and focus during the exam. When you begin taking the test, quickly skim through the whole thing first to allocate your time appropriately before diving into the details. Flag questions you do not know immediately and come back to them later so you don't get bogged down. With thorough preparation and remaining calm on exam day, you will be primed for success.

Conclusion

While the FE exam is certainly difficult, passing it simply requires diligent preparation over 2-3 months, learning from practice problems and simulated exams, and not losing hope. If you don't pass on your first try, thoroughly review areas needing improvement per your diagnostic report. Then, sign up and study again using what you learned. Many examinees pass their second or third attempts with improved knowledge of their weak points. With smart preparation strategies tailored to your needs and perseverance, you will be an FE exam success story before you know it.

Are you ready to pass your FE exam on your first try? Enroll in one of our comprehensive FE exam prep courses today.

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

From Pyramids to Skyscrapers: The History of Engineering

Engineering has been a fundamental driver of human progress for thousands of years. From ancient civilizations to the technological marvels of today, engineers have pushed boundaries to imagine, design, and create remarkable structures, machines, systems, and processes that have shaped the modern world.

In this Blog,

1. Early Engineering
i. Egypt and Mesopotamia
ii. Ancient Rome
iii. Europe and the Renaissance

2. Pivot Point: Industrial Revolution

3. Engineering’s Effects on Industry
i. Automobile
ii. Air Travel
iii. Electronics
iv. Nuclear

4. Pivot Point: Information Age

5. What Does the Future Hold?

Early Engineering

Egypt and Mesopotamia

The foundations of engineering can be traced back to the great public works and military achievements of antiquity. The construction of the pyramids of ancient Egypt, spanning almost a century, represented a tremendous engineering feat. The massive blocks, precisely cut and positioned to astronomical alignments, still leave archaeologists perplexed as to the methods used. In Mesopotamia, irrigation systems (Figure 1) enabled large-scale agriculture through the controlled distribution of water (Crabben, 2023). This knowledge spread to early civilizations around the world.

Figure 1: Mesopotamia Irrigation Canals

Ancient Rome

The ancient Romans developed civil engineering on an epic scale. They built over 50,000 miles of roads (International Museum of the Horse, n.d.) and aqueducts like the Pont du Gard (Figure 2) that transported water hundreds of miles (Desbordes & Lescure, n.d.). The iconic Colosseum and Pantheon employed arches and concrete to allow huge enclosed public spaces. Fortifications and siege weapons showcased Roman military engineering prowess.

Figure 2: Pont Du Gard (Roman Aqueduct)

Europe and the Renaissance

Medieval engineering advanced with the construction of elaborate cathedrals and fortresses in Europe. Building ever-higher and larger cathedrals required innovations in columns, arched entries, flying buttresses, and intricate stonework. Technologies like water wheels and windmills were adopted for early industry. Late Medieval polymaths like Leonardo da Vinci embodied the guild artisans of the time period, displaying engineering creativity through detailed drawings and models.

The Renaissance accelerated engineering progress through the application of science and mathematical principles. Artists and architects carefully studied proportions and perspective, creating some of the great masterpieces of the era. Movable type printing using metal blocks revolutionized mass communication. Highly varied technical treatises were widely published. Exquisite mechanical devices and automatons were built to demonstrate principles of physics and mechanics.

Pivot Point: Industrial Revolution

The Industrial Revolution marked the transformation into modern engineering. Steam power opened entirely new possibilities for manufacturing, transportation, and large-scale machinery. Civil engineers designed roads, railways, bridges, tunnels, ports, and other infrastructure to support travel and commerce. The telegraph and telephone connected distant parts of the world with communication networks. Electrical engineering soon emerged to capitalize on electricity for lighting, motors, and appliances.

Twentieth-century engineering produced an astounding array of technologies that changed everyday life in the automobile/air travel, electronic, and nuclear industries.

Engineering’s Effects on Industry

Automobile

Automobiles and air travel became ubiquitous. In the early 20th century, cars and planes were novelties accessible only to the wealthy. However, mass production techniques pioneered by Henry Ford's Model T made automobiles affordable for the average American. By 1930, 23 million cars were registered in the U.S., with one car for every five Americans (National Museum of American History Behring Center, n.d.). The growing roadway infrastructure allowed driving to become a part of everyday life.

Figure 3: Ford with his early Model T

Air Travel

Similarly, air travel transformed from daredevil barnstormers to comfortable and common public transportation. The Douglas DC-3, introduced in 1936, could carry 21 passengers on a transcontinental flight across the U.S. within 15 hours (The Geography of Transport Systems, n.d.). By 1958, over 60 million Americans had traveled by air. Cars and planes had evolved from playthings of the rich to ubiquitous technologies relied upon by the masses for business and pleasure.

Figure 4: Douglas DC-3

Electronics

Radio, television, and later computers provided entertainment and information globally. Huge dams and power grids delivered electricity across nations. Electronics amplified communications, computing, and control systems. The invention of the vacuum tube in the early 20th century enabled a breakthrough in electronics (Smil, 2019). Radios using vacuum tubes became extremely popular, allowing live broadcasting of speech, music, and news into homes across the nation. Later, the development of television brought both sound and dynamic visuals delivered over airwaves. Telephone networks also transitioned from operator-assisted switchboards to automated systems, enabling more conversations simultaneously through electronic amplification and switching. In World War II, advanced electronics like radar and analog computers provided tactical advantages. After the war, electronics research produced the transistor, allowing computing power to advance from room-sized mainframes to minicomputers and, eventually, microprocessors (Computer History Museum, n.d.). The space race drove further miniaturization of electronics. By the end of the 20th century, digital communications networks, computers, and consumer devices had electronics integrated into nearly every aspect of life.

Nuclear

Nuclear engineering emerged from physics into a practicable option for power generation as well as destructive weapons. In the 1930s, physicists discovered that the fission of uranium could yield tremendous amounts of energy. This was demonstrated practically in 1942 with the creation of the first nuclear reactor at the University of Chicago (U.S. Department of Energy, n.d.). Only a few years later, this principle was applied to develop the atomic bombs (Manhattan Project) dropped on Hiroshima and Nagasaki, demonstrating the immense destructive potential of nuclear physics.

Figure 5: Manhattan Project during testing

After the war, nuclear technology was applied to peaceful purposes. In 1951, the first nuclear power plant was opened in Idaho, and it was able to provide electricity to the local region. Over the next decades, nuclear power expanded to become a major source of energy worldwide. Nuclear naval propulsion also emerged from physics research, with the USS Nautilus launching in 1954 as the first nuclear-powered submarine (Department of Energy, n.d.). So, within just two decades, nuclear engineering had branched from physics theories into both powerful weapons and practical generators of electricity and propulsion. The principles of physics were turned into real-world applications that would change energy and warfare.

Pivot Point: Information Age

The Information Age has accelerated the integration of all engineering disciplines. Software engineering made computers flexible and accessible to the masses. The internet networked the world like never before. Biomedical engineering harnessed technology to improve human health and enhance lives. Environmental engineering works to sustainably manage the planet’s resources. Nanotechnology manipulates matter at molecular scales with vast potential.

What Does the Future Hold?

Looking forward, the possibilities seem endless. But even with infinite computing power, engineering innovation requires human creativity, problem-solving, and perseverance. The grand challenges of the future, whether climate change, accessible healthcare, cybersecurity, or space exploration, will rely on engineers pushing boundaries to make the seemingly impossible - possible. The enduring legacy of engineering history lies in working to uplift humanity through technology.

References

Computer History Museum. (n.d.). 1947: Invention of the Point-Contact Transistor | The Silicon Engine. Computer History Museum. Retrieved September 14, 2023, from https://www.computerhistory.org/siliconengine/invention-of-the-point-contact-transistor/

Crabben, J. v. d. (2023, March 22). Agriculture in the Fertile Crescent & Mesopotamia. World History Encyclopedia. Retrieved September 14, 2023, from https://www.worldhistory.org/article/9/agriculture-in-the-fertile-crescent--mesopotamia/

Department of Energy. (n.d.). Timeline of Events: 1951 to 1970. Department of Energy. Retrieved September 14, 2023, from https://www.energy.gov/lm/timeline-events-1951-1970

Desbordes, M., & Lescure, M. (n.d.). 40 centuries of history. Le Pont du Gard. Retrieved September 14, 2023, from https://pontdugard.fr/en/discover/history

The Geography of Transport Systems. (n.d.). Selected Transcontinental DC-3 Routes, Late 1930s. The Geography of Transport Systems. Retrieved September 14, 2023, from https://transportgeography.org/contents/chapter5/air-transport/continental-dc3-routes-1930s/

International Museum of the Horse. (n.d.). Roman Roads. International Museum of the Horse. Retrieved September 14, 2023, from https://imh.org/exhibits/past/legacy-of-the-horse/roman-roads/

National Museum of American History Behring Center. (n.d.). Americans Adopt the Auto | National Museum of American History. National Museum of American History. Retrieved September 14, 2023, from https://americanhistory.si.edu/visitor-guides/america-move/americans-adopt-auto

Smil, V. (2019, January 24). During the 20th Century, Vacuum Tubes Improved in a Moore's Law-Like Way. IEEE Spectrum. Retrieved September 14, 2023, from https://spectrum.ieee.org/during-the-20th-century-vacuum-tubes-improved-in-a-moores-lawlike-way

U.S. Department of Energy. (n.d.). Manhattan Project: Fission Comes to America, 1939. OSTI.GOV. Retrieved September 14, 2023, from https://www.osti.gov/opennet/manhattan-project-history/Events/1890s-1939/fission_america.htm

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

Does a Master's Degree Give You an Advantage in Your FE Exam Preparation?

Passing the Fundamentals of Engineering (FE) exam and gaining an Engineering in Training (EIT) certification is the first step in the journey of becoming a licensed professional engineer in the United States. The exam is designed to test an individual's knowledge of fundamental engineering principles and their ability to apply that knowledge in practical ways. Passing the FE exam is required before an engineer can take the Professional Engineering (PE) exam and obtain their engineering license. This begs the question, would having a master's degree give you an advantage when taking the FE exam? There are a few key factors to consider when considering this question.

Does a Master's Degree Give You an Advantage in Your FE Exam Preparation?

1. Academic Knowledge

2. Problem-Solving Abilities

3. Studying and Test-Taking Strategies

4. Self-Motivation

Academic Knowledge

A master's degree will provide you with advanced technical knowledge and a deeper understanding of engineering concepts, which can certainly help when taking the FE exam. The FE will test your knowledge of subjects like mathematics, statistics, chemistry, physics, engineering economics, ethics, and more. All of these general topics will be covered in greater depth and technical detail at the graduate level compared to an undergraduate engineering degree. The advanced coursework and research conducted for a master's program expand your base of engineering knowledge, which you can leverage for the FE exam.

However, it's important to note that the FE exam is focused on only testing fundamentals rather than highly advanced concepts. Much of the exam covers engineering principles and skills that would be introduced and developed in an undergraduate program. So, while additional technical knowledge from a master's can be useful, it's not necessary nor required to pass the FE if you already have a solid grasp of the core fundamentals from an accredited undergraduate engineering program. The FE aims to set a minimum level of competency for all engineers at the early stages of their careers.

Problem-Solving Abilities

More so than just technical knowledge, the FE exam emphasizes critical thinking and problem-solving skills. You have to be able to analyze engineering situations and apply foundational concepts to solve quantitative problems. A master's degree may provide an advantage in that sense, considering several of their courses are aimed at developing strong analytical and problem-solving abilities.

Graduate-level coursework and research require a student to investigate open-ended problems, evaluate alternatives, and think through solutions rigorously. Master's programs also emphasize higher-order skills like critical thinking, data analysis, and logical reasoning. Dealing with complex technical concepts and assignments hones your ability to parse information, extract key insights, and make calculations—abilities all very relevant to the FE exam. While undergraduate studies certainly teach problem-solving as well, the more advanced skills developed in a master's program can further boost your proficiency.

Studying and Test-Taking Strategies

Pursuing a master's degree also helps develop general studying and test-taking abilities that could easily be applied to your FE exam preparation. A master's coursework requires managing a heavy workload, synthesizing large amounts of information, and applying knowledge across multiple domains. Conducting research develops skills like information gathering, statistical analysis, and technical writing. Master's students have to stick to deadlines, manage stress, and get comfortable taking exams in advanced subjects. These skills earned through graduate school prepare examinees for detailed studying, time management, and test-taking, which are invaluable when preparing for and taking the rigorous 6-hour FE exam.

There have been many who have excellent study habits and many who have certainly prepared well for the FE with just an undergraduate background. The experience of pursuing a master's degree provides an additional level of preparation for those who may need it. The demands of graduate-level academics serve as good training for both the content and process of taking the FE exam.

Self-Motivation

Finally, the drive and self-motivation required to pursue a master's degree beyond your undergraduate education also translate into a greater focus on FE exam preparation. Succeeding in a master's program requires disciplined studying, passion for your field, and the ability to self-motivate over an extended period of time. These soft skills around dedication and focus are important when taking on the challenge of the FE exam. If you have the motivation to consistently study and improve over the 1-2 years of a master's program, you will likely also have the right mindset to stay committed to your FE exam preparation.

While someone without a master's can certainly be highly motivated as well, and there have been many who have done so, the determination demonstrated by pursuing (and completing) a graduate degree shows you have the kind of work ethic and persistence needed to pass the FE. If you have conquered the rigors of a master's program, you have also likely developed the self-motivation needed to fully dedicate yourself and put in the hours required to study and be ready for the FE.

In summary, a master's degree can provide some advantages when preparing for the FE exam, such as:

• Deeper academic knowledge, especially in advanced math/science topics
• Enhanced critical thinking and problem-solving abilities
• Improved study and test-taking strategies
• Higher levels of self-motivation and dedication

However, a master's degree is not required to pass the FE exam; plenty of individuals without a master's degree succeed in passing the exam each year. With proper preparation using undergraduate education and FE exam-specific study materials, you can develop the fundamental engineering knowledge and skills needed to do well. The FE aims to test basic competency and is fully feasible without a master's degree. However, the advanced engineering proficiency and academic maturity gained in graduate studies can provide an incremental benefit.

Conclusion

Ultimately, the FE exam tests your comprehension of core engineering principles and your ability to solve problems efficiently. The pursuit or completion of a master's degree would only strengthen both comprehension and problem-solving through advanced technical knowledge and overall academic development. But, for many, it is not mandatory. With diligent preparation, both master's graduates and undergraduates can perform well on the FE exam and take the first step to professional engineering licensure. The degree to which a master's degree helps mostly depends on the quality of the program and your effort to leverage the experience for your FE exam preparation.

Whether you have a master’s degree or not, we can help you pass your FE exam on your first try! Join the tens of thousands of engineers who’ve passed their FE exam with School of PE. Register now!

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

Ultimate Guide to Balancing Study Time for the FE Exam

Studying for the Fundamentals of Engineering (FE) exam can feel overwhelming. With so much material to cover, it's crucial to have an effective study plan to manage your time wisely. As a former engineering student who successfully passed the FE exam, I've discovered some time management solutions that have helped me tremendously to maximize my preparation. In this blog post, I'll share the best tips I've learned for developing an FE exam study schedule, sticking to it, and retaining what you learn.

In this Blog,

1. Create a Study Calendar

2. Take Practice Exams

3. Follow the NCEES Practice Exam Format

4. Use Memory Techniques

5. Study in Groups

6. Try Timers for Focused Sessions

7. Review Key Concepts Before Bed

8. Eat Brain Foods

9. Review, Don't Cram

Create a Study Calendar

The first step is to make a realistic calendar that maps out the weeks and months leading up to your exam date. Outline how many hours you can devote to studying each day and week. The general rule of thumb is to study 200-300 hours total over several months. Calculate what pace you need to maintain to reach your target hours.

I recommend studying 10-15 hours per week if you have 3-4 months until test day. Break that down into 3-4 study sessions per week. Studying every day for short bursts of time is more effective than cramming on weekends. Your brain does better at absorbing and retaining information through consistent repetition than through an impulse of learning.

Once you've created your FE exam study calendar, do your best to stick to it! Treat study sessions like important appointments that can't be rescheduled. However, build in some wiggle room for inevitable busy weeks or emergency interruptions. Life happens; adjust your schedule accordingly, but don’t forget to get right back on track the following week.

Take Practice Exams

Taking timed practice exams is one of the most valuable ways to prepare for the FE exam. Full-length practice tests closely simulate the actual testing environment, giving you a realistic assessment of your knowledge. Here are some tips for using practice tests effectively:

• Take your first practice test 2-3 months before the exam to identify weak areas to focus on. Take another one 1-2 months out, and your final test just before exam day.
• Strictly follow the 5-hour 20-minute time limit to build your test-taking endurance.
• Grade each test thoroughly and note the concepts/types of problems you missed. Emphasize reviewing trouble topics in your following study sessions.
• Pay attention to how you felt physically and mentally while taking the practice exam. Adjust your schedule to prevent any onset of burnout.
• Purchase at least 2 different practice exams. Repeating the same test won't help as much. Varying your practice gives a well-rounded experience.

Follow the NCEES Practice Exam Format

The National Council of Examiners for Engineering and Surveying (NCEES) designs the FE exam and regularly updates the exam specs. It is best to follow their test specifications closely when you prepare for your exam. Look for a practice exam in the format of the real exam, like the ones School of PE offers.

The exam covers 110 questions and is 5 hours and 20 minutes long. All the questions will be either multiple choice or alternative item types such as:

• Multiple correct options (allows multiple choices to be correct)
• Point and click (requires examinees to click on part of a graphic to answer)
• Drag and drop (requires examinees to click on and drag items to match, sort, rank, or label
• Fill in the blank (provides a space for examinees to enter a numerical response to the question)

Study with the NCEES breakdown in mind.

Use Memory Techniques

With so many equations, constants, conversions, and formulas to remember, using memory techniques is crucial for recall on test day. Here are two effective strategies I use:

Chunking - Group concepts or steps that go together into "chunks". Then, memorize the chunks as a whole. This prevents your brain from getting overloaded.

Mnemonics - Create acronyms, rhymes, songs, or mental images to represent concepts. Associating information with visuals or music makes it stick better.

Study in Groups

Studying with other FE exam takers has many benefits beyond moral support. Group members can quiz each other, explain tricky concepts, and share study resources. Having to verbalize your knowledge out loud helps strengthen your understanding. Just be sure to stay on task and take breaks to avoid mental fatigue.

If your college doesn't have an FE review group, check social media for local study partners. There are also online FE exam preparation groups where you can connect virtually. Studying together keeps you accountable and motivated!

Try Timers for Focused Sessions

With so much material to cover, maintaining focus is a challenge. Timers and productivity apps are useful for keeping study sessions targeted. Here are some you can try:

• Pomodoro Technique - Study intensely for 25 mins, take a 5 min break. After 4 rounds, take a 30-minute break.
• Forest - This app lets you set a time limit and grow a virtual tree while you stay off your phone. If you switch apps before the timer ends, the tree dies.
• Be Focused - Customize sessions with timed intervals of working, short breaks, and long breaks.
• Focus Keeper - Alternate between focused study and rest periods. Track your daily progress.

I like to switch up my timer style every few sessions to keep my brain engaged. The forced breaks help processing and retention.

Review Key Concepts Before Bed

In the last hour before bedtime, review your highlighted notes or flashcards on foundational concepts. Repetition right before sleep helps strengthen and consolidate memory. This practice really works - I can often recall entire equations the next morning!

Make your "before bed review" quick, about 10 minutes. Don't do any intense problem-solving that will keep your mind too active. The key is falling asleep promptly so memory integration can occur overnight.

Eat Brain Foods

You'll retain so much more studying while properly nourished. Complex carbs provide steady energy, protein aids concentration, and antioxidants boost brain function. Some of my go-to study snacks include:

• Oatmeal with berries
• Greek yogurt with mixed nuts
• Apple with peanut butter
• Carrots with hummus
• Trail mix
• Green tea

Drink plenty of water to stay hydrated as well. Proper nutrition gives both your body and mind the resources to perform at your peak.

Review, Don't Cram

If the exam is just a week or two away, resist the urge to panic cram! Review key concepts methodically instead. Re-work any practice problems that you have struggled with. Skim through flashcards. But don't try digesting new material. Stick to strengthening your existing knowledge.

Cramming overload causes mental fatigue faster. To simulate the endurance needed on exam day, treat the last week as a practice run. Maintain your regular study hours and methods. Get plenty of rest the night before the test. You'll feel confident and prepared walking in.

While studying for a big exam like the FE can feel overwhelming at times, staying organized and managing your time effectively makes reaching your goal totally doable. Implementing just a few of these tips can optimize your study plan. Maintain self-care, stick to your schedule, and keep chipping away consistently. I hope these strategies for balancing FE exam prep with daily life can help maximize your study time. With determination and commitment, you've got this!

If you’re interested in a comprehensive course designed to help you pass your FE exam on your first try, check out School of PE! School of PE's #1-rated engineering exam review courses have provided tens of thousands of professionals with the tools to pass their exams.

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

FE Exam or Engineering Experience: Which Comes First?

Many engineering students wonder if they need to pass the Fundamentals of Engineering (FE) exam before beginning their engineering careers and gaining industry experience. The short answer is no - you do not have to pass the FE exam before beginning your engineering career. However, there are several good reasons to take and pass the exam as early as possible. Here is an in-depth look at the FE exam and whether you need to pass it before gaining experience.

In this Blog,

1. Do You Have to Pass the FE Before Gaining Experience?

2. How Long to Study for the FE Exam?

3. Should You Work Before Taking the FE Exam?

4. How to Prepare for the FE Exam

5. What's Next After Passing the FE Exam?

6. Key Takeaways

Do You Have to Pass the FE Before Gaining Experience?

Nothing is stopping you from working as an engineer before taking or passing the exam. The FE is not a prerequisite to begin working after graduation.

However, there are a few reasons why you may want to take the FE exam as soon as you can:

1. The concepts are fresh in your mind.

Engineering students typically take the FE exam right after graduation, as the test material is still fresh in their minds. The FE tests your comprehension of basic engineering principles, which you should have learned during your undergrad.

Engineering knowledge is freshest after graduation, making the FE exam a bit easier. Your undergraduate coursework should have prepared you with broad engineering and scientific fundamentals to handle the breadth of the FE.

Studying and passing the exam only gets more difficult and time-consuming the longer you are out of school. Preliminary review suggests new graduates have higher FE exam pass rates than those further into their careers.

The further you get from your college courses, the harder it will be to recall conceptual topics that you may not use regularly in your job. Reviewing and relearning concepts you’re rusty on will require extra time and effort.

2. You must take it before taking the PE exam and earning your PE license.

To get your PE license, the first step is to pass the FE exam and gain relevant work experience, usually four years. If your goal is to become a licensed Professional Engineer, you’ll want to pass the FE exam as soon as possible.

Gaining the 4+ years of experience required to take the PE exam and your PE license takes time. Knocking out the FE exam requirement early lets you get the licensure clock ticking as soon as possible.

The FE and PE exams are spaced apart by design. Passing the FE as soon as you can after graduation clears a major hurdle on the path to licensure. The PE license application process can't formally start until you pass the FE. Waiting to take the test delays earning your PE credential.

3. Some employers require it.

While not all engineering jobs require the FE or PE, there are always some exceptions to the rule. Certain employers make passing the FE an official condition of hiring and continued employment for engineering positions.

For example, civil engineers working on public construction projects may need licensure to comply with contracts. Their employer likely requires passing the FE.

If your target companies do not require the FE upfront, you still gain an edge over other applicants by passing the exam early. It shows commitment to professional development.

4. It will boost your resume.

While not strictly required for non-licensure careers, passing the FE exam still looks impressive on a resume. It demonstrates you have acquired a body of engineering knowledge and possess analytical problem-solving skills.

The FE shows initiative to go above and beyond and master your foundational engineering education. Some companies may offer promotions and raises to engineers who pass the FE or pursue licensure.

How Long to Study for the FE Exam?

Studying for the FE exam takes dedication. The test covers a wide range of engineering topics, so thorough prep is essential.

On average, examinees spend 80-100 hours studying to pass the FE exam. Preparation time depends on your engineering discipline and academic background.

Some accelerated FE exam prep courses claim you can cram in just 5 weeks. However, most examinees study for 2-3 months to truly cement knowledge and practice working speed. Don’t underestimate the exam - take time to prepare properly.

The FE exam recently transitioned to a computer-based format, allowing year-round testing. This gives flexibility to take the FE when you feel ready rather than waiting for specific dates. Just be sure to meet registration deadlines.

Earning an engineering degree shows you have the knowledge. Passing the FE proves you can apply that knowledge efficiently. Dedicate the time to refresh concepts and practice working on exam problems at a brisk pace.

Should You Work Before Taking the FE Exam?

There are no rules saying you can't gain engineering experience before passing the FE exam. Working for a few years before taking the test is fairly common.

However, don't wait too long after graduation. Some pros of gaining experience before attempting the FE include:

• Earning money to finance FE prep materials and expenses
• Gaining some field experience to inform exam studying
• Having more "think like an engineer" problem-solving practice

Yet, spending too long outside of school can work against you. Engineering concepts won't get reinforced and will become rustier. Plus, you may find it harder to find the time and motivation for rigorous FE studying after starting your career.

Ideally, take the FE exam within two years of finishing your engineering degree. This strikes a balance between allowing some field experience while concepts are still fresh.

How to Prepare for the FE Exam

Whatever your experience level, disciplined preparation helps boost FE exam success:

• Use the NCEES exam specifications to focus studying on the topics covered. Review college textbooks and notes.
• Work through many practice problems under timed conditions. Practice builds speed and accuracy.
• Learn the FE exam format and how to strategically tackle questions. Don't waste time on difficult problems.
• Master your calculator to efficiently handle mathematical computations. Bring spare batteries.
• Brush up on fundamentals you haven’t used recently, like chemistry, physics, or advanced math.
• Take a review course with a top-ranked exam prep provider like School of PE.
• Keep taking practice exams to evaluate preparedness until consistently scoring over 70%.
• Rest up the day before. Have what you need ready so you're not rushing around.

What's Next After Passing the FE Exam?

Congratulations, you've passed the FE exam! Now you can:

• Update your resume and LinkedIn profile to highlight your accomplishments.
• Join engineering associations and networks like the National Society for Professional Engineers.
• Formally start accruing the professional engineering experience needed for PE licensure. Track hours.
• Identify a PE mentor at your company who can provide guidance.
• Continue developing technical skills in your field. Seek opportunities to gain experience in areas covered on the PE exam.
• Maintain knowledge by attending engineering seminars and events or taking continuing education courses. Stay up to date on regulations.
• Save FE exam references and resources in case you need to brush up again for the PE exam.

Passing the FE opens opportunities in your engineering career. For those pursuing licensure, it's the critical first step to becoming a Professional Engineer. For others, it strengthens your foundation to grow professionally even without the PE.

Key Takeaways

• You do NOT have to pass the FE exam before starting your engineering career or gaining experience. Nothing prevents working as an engineer before or without FE certification.
• However, it is wise to take the test as soon as possible after finishing your accredited engineering degree. Knowledge is freshest, and the exam is often easier right out of college.
• Passing the FE exam early starts the licensure clock if your goal is to become a licensed Professional Engineer. It also boosts your resume and competitiveness.
• Disciplined studying is key to succeeding on the broad FE exam, which covers many engineering topics. On average, examinees spend 80-100 hours preparing.
• The FE exam is your chance to prove your comprehension and ability to apply foundational engineering concepts. Take advantage while the material is still fresh after graduation.

Ready to pass your FE exam on the first try? Don't leave it to chance. Choose School of PE, the top-rated engineering exam prep provider. Our expert-led courses and comprehensive study materials will equip you with the knowledge and confidence to succeed. Check out our courses here!

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.

Mastering Digital Signal Processing: Your Essential Guide for the FE Electrical Exam

The FE requires examinees for the FE Electrical exam to be able to solve DSP equations. Here, we will provide a brief introduction to the topic and several useful equations that will help prepare you for your upcoming exam.

In this Blog,

1. Sampling
a. Nyquist Theorem
b. Aliasing

2. Digital Filters
a. Difference Equation
b. Z-Transform
c. Finite Impulse Response (FIR)
d. Infinite Impulse Response (IIR)

Sampling

The sampling theorem states that in order to accurately reconstruct a continuous-time signal from its samples, the sampling rate must be at least twice the highest frequency component in the signal. This minimum sampling rate is known as the Nyquist rate. If the sampling rate is below the Nyquist rate, aliasing occurs, leading to distorted or misleading reconstructions.

1. Nyquist Theorem

A signal can be represented in its sample and can be recovered back when using the sampling frequency (f_s) or sampling angular frequency (ω_s=2πf_s) is greater or equal to twice the max frequency component of the signal (McClellan et al., 2021, 111) where:

f_s≥2f_max or ω_s≥2ω_max   (1)

For instance, considering the constraints of human auditory perception, the upper limit of discernible frequencies for human hearing is approximately 20 kHz (D et al., 2001). Consequently, to accurately capture and reproduce audio signals while adhering to the Nyquist Theorem, widely adopted media formats such as compact discs (CDs) and numerous digital audio files employ a sampling frequency of 44.1 kHz.

During the sampling process, it is customary to adhere to the Nyquist rate to ensure a faithful representation of the original signal. However, it is worth noting that alternate approaches, such as oversampling or undersampling, can also be employed to capture signals with distinct characteristics. 

2. Aliasing

Aliasing occurs when the sampling rate is less than the Nyquist rate, causing overlapping of the frequency components of the continuous-time signal. Aliasing leads to the creation of false frequency components, making it difficult to accurately reconstruct the original signal.

However, the signal can be reconstructed by filtering out frequencies above ¹/₂ f_s. To ensure accurate sampling, it is crucial that the signal to be sampled possesses a frequency content that is limited to half the sampling frequency. Failure to adhere to this requirement leads to aliasing, an undesirable phenomenon where different frequency components overlap and distort the reconstructed signal.

When the highest frequency in the signal is greater than half of the sampling frequency, then aliasing occurs. Through graphical analysis, it can be observed that the frequency (f) that surpasses half the sampling frequency will result in a signal reconstruction with aliasing at the difference between the sampling frequency (f_s) and f, denoted as f_s - f.

For 0.5f_s < f <1.5f_s, the aliasing will appear at |f_s-f|.

For 1.5f_s < f <2.5f_s, the aliasing will appear at |2f_s-f|.

For 2.5f_s< f <3.5f_s, the aliasing will appear at |3f_s-f| and so on (Hasegawa, 2021, 23).

Example: 

What is the frequency (Hz) of the artifact in the reconstructed signal if the function is Acos(640πt+Φ) and the signal is sampled at 120 Hz?

Solution:

1. Since the question is asking for frequency (Hz), we need to identify the max frequency and sampling frequency:

f_max=640π/2π=320 (Hz)

f_s=120 (Hz)

2. Check the criteria of f_max compare to f_s. You will need to keep testing until the right criteria is reached.

2.5f_s< f <3.5f_s

2.5(120)<320<3.5(120)

300<320<420

3. Find the artifact frequency via |3f_s-f|

f_artifact=|3(120)-320|=40 (Hz)

Digital Filters

Based on the FE Electrical and Computer topic list, the subtopics covered in the Digital Filters are Difference Equation and Z-Transform. However, Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) shall not be neglected even though they were not listed. 

1. Difference Equation

A difference equation is a mathematical equation that represents the relationship between the input and output signals of a discrete-time system. It describes how the current and past input values, as well as the current and past output values, are combined to compute the current output value (McClellan et al., 2021, 147).

A general form of a difference equation is:

y[n]=a₀ x[n]+a₁ x[n-1]+...+b₀ y[n-1]+b₁ y[n-2]+...   (x)

Where, 

y[n] is the current output at discrete time n.

x[n] is the current input at discrete time n.

x[n-1],... is the past input values.

y[n-1],... is the past output values.

a₀, … and b₀,... are the coefficients of the difference equation.  

Example: 

y[n]=4x[n]+2y[n-1]+2y[n-2]; y[n]=0 for n<0

What is the impulse response of a system described by the difference equation? 

Solution: 

1. For impulse response, x[n]=δ[n] and δ[0]=1 and δ[n≥1]=0

. n=0

y[0]=4δ[0]+2y[0-1]+2y[0-2]=4(1)+y[-1]+y[-2]=4+0+0=4

. n=1

y[1]=4δ[1]+2y[1-1]+2y[1-2]=4(0)+2y[0]+2y[-1]=0+2(4)+2(0)=8

. n=2

y[2]=4δ[2]+2y[2-1]+2y[2-2]=4(0)+2y[1]+2y[0]=4+2(8)+2(4)=28

You can keep computing for n approaches ∞ but the solution right is h[n]={4.8,28,...}

2. Z-Transform

The z-transform provides a way to represent discrete-time signals or sequences as functions of a complex variable, denoted as "z." It is defined as the sum of the sequence multiplied by powers of z, where z can be any complex number (McClellan et al., 2021, 348). The z-transform of a discrete-time sequence x[n] is given by:

∞
X(z)=∑	x[n]z-n(x)
n = 0

In the exam, a table of z transform paris is provided in the Reference Handbook p.372. 

Example: 

Find the z-transform X(z) of the following signal: 

x[n]=δ[n]+1/5 δ[n-2]-3/4 δ[n-4]

Solution: 

1. Use the summation definition (equation x) to solve for the z transform:

∞
X(z)=∑	x[n]z-n
n = 0

2. x[n] is zero for all values of n except when n=0,2,and 4.

x[n]={1,0,1/5,0,-3/4}

. n=0, z transform is

x[0]z^-0=1⋅1=1

. n=2, z transform is

x[2]z^-2=1/5⋅z^-2

. n=4, z transform is

x[4]z^-2=-3/4⋅z^-4

So, X(z)=1+1/5 z^-2-3/4 z^-4

3. Finite Impulse Response (FIR)

The FIR system, or, as we often refer to them, FIR filter, is a type of digital filter whose output is determined by a weighted sum of past and present input samples (McClellan et al., 2021, 147).

The output of the FIR filter can be calculated as:

∞
Y(n)=∑	bnx[n-k]
k = 0

4. Infinite Impulse Response (IIR)

In contrast to FIR filter, the output of IIR filter is the sum of the values of past inputs plus past outputs (McClellan et al., 2021, 394).

The output of the IIR filter can be calculated as: 

y[n] = ∑_l=0^∞ b_l x[n-l] + ∑_k=0^∞ a_k y[n-k]

Example: 

With the given filter below

y[n]=x[n]+1/6 x[n-1]

What type of filter is it?

Solution: 

1. By looking at the function, the output contains the sum of the current and past inputs. Therefore, it is an FIR filter. 

2. Do a z-transform to determine if it is a low-pass, high-pass, band-pass, or band-reject filter. 

y[n]=x[n]-1/6 x[n-1]

Use the z-transform pair from Reference Handbook p.372, we have 

Y(z)=X(z)-1/6 X(z)z^(-1)

H(z)=(Y(z))/(X(z))=1-1/6 z^(-1)=(z-1/6)/z

Conclusion

This blog aims to serve as a valuable point of reference, offering concise explanations and practical examples for each topic addressed in the exam. It is important to note, however, that while the blog provides a helpful overview, it may not encompass all aspects related to digital signal processing. For a more thorough and comprehensive study plan, I recommend visiting School of PE’s FE Electrical exam review course FE Electrical exam review course, which offers an extensive and detailed curriculum to ensure a comprehensive understanding of the subject matter.

References

D, P., GJ, A., & D, F. (2001). Neuroscience 2nd edition. Sunderland (MA): Sinauer Associates. https://www.ncbi.nlm.nih.gov/books/NBK10924/

Hasegawa, M. (2021). Lecture 6: Sampling and Aliasing. Retrieved June 6, 2023, from https://courses.engr.illinois.edu/ece401/fa2021/lectures/lec06.pdf

McClellan, J. H., Schafer, R. W., & Yoder, M. A. (2021). DSP First Second Edition. Pearson.

About the Author: Khoa Tran

Khoa Tran is an electrical engineer working at the Los Angeles Department of Water and Power and is currently pursuing his master's in electrical Power from the University of Southern California. He is fluent in both Vietnamese and English and is interested in outdoor activities and exploring new things.