Introduction: Today, I received two students who answered questions in the course. Through discussions with them, I not only answered their questions in the learning process, but also further enriched my understanding of the questions of middle school students in the course content.
关键词: Signals and Systems , Q&A
§ 01 Course Q&A
1.1 Q&A time in the course
Today (2022-04-01) evening is the time to answer questions for the students of the signal and system course. Since the beginning of the school year, there have been no students to answer questions.
The time for answering questions is decided by class voting with the students in the class, and it is defined from 8:00 to 10:00 pm every Friday.
For the past few weeks, Q&A time has always been me alone in the office waiting.
It was just after 8:30 today. Today, two little beauties in the class finally came to ask questions about the course.
This time they asked a lot of interesting questions.
The first question they asked was very interesting, and actually caught me from the very beginning.
▲ 图1.1.1 翻看着笔记，询问问题
I don't know which of the two of them came up with this question. It feels like they were prepared and premeditated to make things difficult for their teachers.
This is a question about the scale properties of the Fourier transform. They set the Fourier transform corresponding to the 1/t function as F(omiga), and then perform scale transformation to expand the 1/t signal by a factor of two to obtain a spectrum of 2F(2omiga).
▲ 图1.1.2 第一个问题
However, the scale of 1/t expands on both sides, which is actually equivalent to 2 divided by t, then according to the linear characteristic of Fourier transform, the corresponding result should be 2 times F(omiga).
▲ 图1.1.3 提出的问题
Therefore, there is a contradiction in the middle. Why do the same signals have different corresponding spectral functions?
▲ 图1.1.4 出现了矛盾
This question can be really confusing. No student has asked me this question before. But soon, we realized that the problem occurs in the 1/t signal, which does not meet the Dirichlett condition of the Fourier transform.
▲ 图1.1.5 提问的学生
The Dirichlet of the Fourier transform is a sufficient and unnecessary condition of the Fourier transform. One of them is to require the signal to be absolutely integrable. And 1/t does not satisfy this condition. Therefore, to obtain its Fourier transform, it needs to be obtained according to the generalized Fourier transform, or the properties of the Fourier transform.
▲ 图1.1.6 傅里叶变换的Dirichlett条件
To find the Fourier transform of 1/t, you can use the dual characteristics of FT to get the corresponding result, which is a sign function of pi times. This result can also be obtained from the Hilbert transform formula. Since the Fourier transform of 1/t is in the form of a sign function,
▲ 图1.1.7 Hilbert变换
And any scale change of the sign function is actually the same sign function.
▲ 图1.1.8 符号函数
So back to the student's question, for F(omiga), since it is in symbolic functional form, F(omiga) and F(2omiga), which look different in functional form, are actually the same function.
▲ 图1.1.9 频谱相等没有关系了
I didn't actually give the above answer during the discussion, and my brain circuit was short-circuited, so I didn't find the answer right away. Now supplement the answer with this video.
▲ 图1.1.10 询问问题的学生
Later, they also asked a lot of interesting questions in the homework and in the course, which were the main points that were not discussed carefully in the class.
▲ 图1.1.11 询问问题
Discussing with them allows me to see where students may be confused during the learning process, and to improve in later classes.
▲ 图1.1.12 询问问题
More importantly, their questions can make classroom teachers feel job satisfaction.
Finally, the two classmates also told me the software they used to take notes on the PAD, which helped me to make better use of the PAD to organize teaching materials. Thank them.
▲ 图1.1.13 答疑的同学们
※ Summary ※
Today, I received two students who asked questions about the course. Through discussions with them, I not only answered their questions in the learning process, but also further enriched my understanding of the questions of middle school students in the course content.
● Links to related charts:
- Figure 1.1.1 Flip through notes and ask questions
- Figure 1.1.2 The first question
- Figure 1.1.3 Questions raised
- Figure 1.1.4 There is a contradiction
- Figure 1.1.5 Students asking questions
- Figure 1.1.6 Dirichlett condition for Fourier transform
- Figure 1.1.7 Hilbert transform
- Figure 1.1.8 Symbolic function
- Figure 1.1.9 Spectrum equality doesn't matter anymore
- Figure 1.1.10 Students asking questions
- Figure 1.1.11 Ask a question
- Figure 1.1.12 Ask a question
- Figure 1.1.13 Students answering questions