-- Daniel introduces the signal and system

 

Reprinted-Daniel introduced "Signals and Systems"Source: Zhangke's log of Xi Qin people

Lesson 1 What is convolution? What is the use of convolution? Fu Liye transformation? What is Laplace transformation? 

Introduction
Many of my friends, like me, have learned a bunch of signal courses for electronics majors in Engineering. I have not learned anything. I took the test on the formula and then graduated.

Let's talk about the question "What is the use of convolution. (Someone replied that "convolution" exists in the subsequent sections of the course "signal and system. I yelled at him and took him out !)

Tell a story:
John has just been hired as a tester for an electronic product company. He has never taken the "signal and system" course. One day, he got a product. The developer told him that the product has an input end and an output end. A limited input signal only produces a limited output.
Then, the manager asked Michael to test what kind of waveform The product outputs when the sin (t) (T <1 second) signal is input (with a signal generator. Three photos were taken and a waveform chart was taken.
"Good! "Said the manager. Then the manager gave a heap of A4 paper: "There are thousands of signals. The formula shows that the duration of the input signal is also determined. Test the Output Waveforms of the following products respectively! "

Now, Michael is in his mind. "God, help me. How can I draw these waveforms? "
So God came up: "James, you only need to perform a test and draw the Output Waveforms corresponding to all input waveforms in a mathematical way ".
God went on to say, "To give the product a pulse signal, the energy is one second, and the output waveform is shown! "
James took care of it. "What then? "
God said, "For an Input Waveform, you imagine dividing it into countless small pulses and inputting it to the product. The result of the superposition is your output waveform. You can imagine that these small pulses lined up into your product. Every time you generate a small output, when you draw a timing diagram, the waveform of the input signal seems to be in turn into the system. "
James learned: "Oh, the output results are all points! Thank God. What is the name of this method? "
God said, "convolution! "

From then on, James's work was much easier. Every time the manager asks him to test some signal output results, James only needs to make calculus on A4 paper to submit the task!
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James worked happily until one day, and his peaceful life was broken.
The manager took a small electronic device and connected it to the oscilloscope and said to James, "Look, the waveforms produced by this small device cannot be described by a simple function. Moreover, it continuously sends signals! But fortunately, this continuous signal repeats every other time. John, let's test the following output waveforms when connected to our devices! "
Zhang San waved his hand: "the input signal has no time limit. Do I have to test the infinite time to obtain a stable and repeated waveform output? "
The manager is angry: "You can fix it for me, or you may have fired the squid! "
Zhang sanzhi thought: "this time, the input signal is connected to the formula, and a chaotic waveform. The time is infinitely long, And the convolution cannot work. What should I do? "
In time, God appeared again: "ing chaotic time domain signals to another mathematical domain, and ing back after computation is complete"
"Every atom in the universe is rotating and oscillating. You can regard the time signal as a combination of several vibrations, that is, something that can be determined and has a fixed frequency. "
"I will give you a mathematical function f, with infinite input signals in the time domain limited in the f domain. Chaotic input signals in the time domain are neat and easy to see in the f domain. In this way, you can calculate"
"At the same time, the convolution of the time domain in the f domain is a simple multiplication relationship. I can prove it to you"
"After a limited program is computed, you get an output waveform after the F (-1) inversion is returned to the time domain, and the rest is your mathematical computation! "
James thanked God and kept his job. Later, he learned that the F-domain transformation has a name called Fu Liye. What is it ......
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Later, the company developed a new electronic product with an infinite length of output signal. This time, James started learning Laplace ......

Postscript:

It's not because the teaching materials are poor or the teachers are not good at learning.
I really appreciate Google's interview questions: I used three sentences, like the old lady, to explain what a database is. Such a proposition is very good, because we do not have a deep understanding of a proposition, and do not think carefully about the design philosophy of a thing, we will fall into the details of the mud: Back formulas, mathematical derivation, points, answer the question, but there is no time to answer "why ". What a university teacher cannot do is to "read thick books and thin books". He can't tell the philosophical truth, simply endorse and turn over PPT, and make boring mathematical proofs, then I blame "the current generation of students is not like a generation". What is the significance?


In the second lesson, what is frequency and system? 

In this article, I will explain the Fourier transformation F. Note that the name of Fourier transformation F can represent the concept of frequency (freqence), or any other concept, because it is just a conceptual model, it is constructed to solve the computing problem (for example, how to obtain the output signal for an infinitely long input signal in the time domain ). Let's look at the Fourier transform into a function in C language, and the output of the signal is regarded as an I/O problem, then, you can use X-> F (x)-> F-1 (x)-> Y to solve any difficult X-> Y problems.

1. What is frequency?
A basic assumption: All information has the characteristics of frequency, the sound level of the audio signal, the spectrum of light, the cycle of the electronic shock, and so on. We abstract a concept of harmonic vibration, the mathematical name is called frequency. Imagine an atom on the x-y plane moving around the origin with a radius of 1 at a uniform circular speed, and imagining the X axis as time, the projection of the circular motion on the Y axis is a sin (t) waveform. I believe that middle school students can understand this.
Then, different frequency models correspond to different circular motion speeds. The faster the circular motion, the narrower the waveform of sin (t. Two scaling modes are available for frequency scaling.
(A) old-fashioned radios use tapes as the media of music. When we are fast-moving, we will feel the sound of singing becomes strange and the tone is very high, that is because the speed of "circular motion" doubles, and the sin (t) output of each sound Component Changes to sin (NT ).
(B) fast or full on the CD/computer. The singer is singing or singing slowly, so there will be no higher tones: because the time-domain sampling method is adopted in fast release, some waveforms are discarded, but the output waveforms carrying information do not change in width or width. When the full-space mode is used, the time-domain signal fill is stretched.

2. Does F transformation result in negative/plural values?
Explanation: F transformation is a mathematical tool and does not have a direct physical significance. The existence of negative or plural numbers is only for the integrity of computation.

3. What are the basic things of Signals and Systems?
For communication and electronics students, in many cases, our job is to design or the physical layer technology in the OSI 7-layer model. The complexity of this technology lies in that you must first establish the electrical characteristics of the transmission media, generally, different transmission media have different processing capabilities for signals in different frequency segments. The ethernet cable processes the baseband signal, and the wide area network light emits high-frequency modulation signals. Mobile Communication, 2G and 3G require different carrier frequencies. Can these media (air, wire, optical fiber, etc.) input at a certain frequency be basically unchanged after a certain distance is transmitted? Then we need to establish a mathematical model of the frequency of the media. At the same time, knowing the frequency characteristics of the media, how can we design the signal transmitted on it to reach the theoretical maximum transmission rate? ---- This is the world that the course of Signals and Systems leads us.
Of course, signals and systems are not only used in this way, but are linked to Shannon's information theory. They can also be used in information processing (sound, image), pattern recognition, intelligent control, and other fields. If the computer science curriculum is the logic model of data expression, then the signal and system are built at a lower level, representing a mathematical model of physical significance. Data Structure Knowledge can solve the encoding and Error Correction of logical information, and signal knowledge can help us design the physical carrier of the code stream (if the received signal waveform is chaotic, so what is the basis for determining whether this is 1 or 0? Logical error correction is meaningless ). In the field of industrial control, the premise of computer application is a variety of digital-to-analog conversion, so the continuous analog signal generated by various physical phenomena (temperature, resistance, size, pressure, speed, etc)
To convert a specific device to a meaningful digital signal, we must first design an available mathematical conversion model.

4. How to design the system?
Design physical system functions (continuous or discrete States) with inputs and outputs. The intermediate processing process is related to the specific physical implementation, not the focus of this course (Electronic Circuit Design ?). In the final analysis, Signals and Systems design a system function for specific needs. The premise of designing a system function is to use the function to represent the input and output (such as sin (t )). The analysis method is to break down a complex signal into several simple signals and accumulate them. The specific process is a lot of calculus. The specific mathematical operation is not the central idea of this course.
What types of systems are there?
(A) Classification by function: modulation and demodulation (signal sampling and Reconstruction), superposition, filtering, power amplifier, phase adjustment, signal clock synchronization, negative feedback PLL, and a more complex system composed of several subsystems-you can draw a system flowchart. Is it very close to the logic flowchart of programming? There is no difference in the space of symbols. There is also digital signal processing in discrete state (subsequent courses ).
(B) Classification by system category, stateless systems, finite state machines, and linear systems. Physical layer continuous system functions are a complex linear system.

5. What are the best teaching materials?
The core of a symbolic system is set theory, not calculus. it is meaningless to implement the calculus used without the set theory. You don't even know what to do after half a day. One of the best teaching materials for learning Signals and Systems from a computer perspective is<Structure
And interpretation of Signals and Systems>The author is Edward A. Lee and Pravin varaiya of UC Berkeley-first defined and implemented, in line with human thinking habits. All the textbooks in China are mathematical derivation, that is, they refuse to say for what purpose, for what to get, for what to build, and for what to prevent; they are not discussed in terms of epistemology and needs, the whole article is a method that does not show purpose, and the last word is an inverse.

Lesson 3 What is the sampling theorem? 

1. for example, when a telephone call is made, the signal sent by the telephone is Pam pulse amplitude modulation. The telephone line uploads not the voice, but the pulse sequence after the voice is converted through channel encoding, recover the Voice Waveform at the receiving end. So how can we convert continuous speaker speech signals into certain columns of pulses to ensure basic non-distortion and transmission? Obviously, we think of sampling. We can sample the electrical signal amplitude once every m milliseconds, convert the amplitude to pulse encoding, transmit it, and regenerate the language at the receiver according to certain rules.
So the question is: How small is m enough to sample data every m milliseconds? How can the language waveform be restored at the receiving end?
For the first problem, we consider that the voice signal is a time frequency signal (so the corresponding F transformation indicates the time frequency) the speech signal is decomposed into several single-audio mixture of different frequencies (the frequency of the frequency function is expanded by the compound-benefit leaf series, and the non-cyclic interval function can be regarded as the cyclic signal expansion after completion, with the same effect ), for the highest-frequency signal component, if the sampling method can guarantee the restoration of this component, other low-frequency components can be saved by sampling. If the human voice's high frequency is limited to 3000Hz, the high frequency component is regarded as sin (3000 t). This sin function needs to save the information through sampling, which can be seen as: for a period, we can use the sampling signal to represent the original analog continuous signal without loss. The two signals correspond one by one and are equivalent to each other.
For the second question, how can we recover analog continuous signals from the pulse sequence (carding waveform) at the receiving end? First of all, we have confirmed that the pulse sequence above the frequency domain already contains all information, but the original information only exists below a certain frequency. How can this problem be solved? Let's let the input pulse signal I through a device X, the output signal is the original voice O, then I (*) x = O, here (*) indicates convolution. If the characteristics of the time domain are not well analyzed, then the f (I) * f (x) = f (o) multiplication relationship in the frequency domain is obvious, as long as f (x) it is an ideal low-pass filter (a box is drawn in the f domain), which is a clock function in the time domain (because it contains a negative part of the timeline, so it does not actually exist). To make such a signal processing device, we can use the input pulse sequence to obtain almost ideal original speech. In practical application, our sampling frequency is usually a bit more than the nequest frequency. The sampling standard is 8 k Hz for the voice signal of 3 k Hz.
2. For a digital image, the sampling theorem corresponds to the resolution of the image-the higher the sampling density, the higher the resolution of the image. If our sampling frequency is not enough, the information will be mixed-there is an image on the Internet. Einstein is seen in myopia glasses, and Monroe is taken out of the eyes-because there is no eye, if the resolution is not enough (the sampling frequency is too low), high-frequency component distortion is mixed into low-frequency components, which leads to a visual trap. Here, F changes in the image correspond to the spatial frequency.
After all, isn't it good to upload the original voice signal directly on the channel? The simulated signal has no anti-interference capability, and no error correction capability. The sampled signal has the digital characteristics, and the transmission performance is better.
What signals cannot be properly sampled? The time domain has a hop, and the frequency domain has an infinite width, such as a square wave signal. If a sampling signal with limited bandwidth is used to represent it, it is equivalent to taking the partial part of the compound leaf series. When the original signal is restored, there will be burrs on the non-bootable points, also known as the garbage box phenomenon.
3. Why did Fourier come up with such a series? This is derived from the basic idea of Western philosophy and science: orthogonal analysis method. For example, to study a three-dimensional shape, we use three orthogonal axes X, Y, and Z: the projection of any axis on other axes is 0. In this way, the three views of an object can fully express its shape. Similarly, how can signals be decomposed and analyzed? The infinite Sum of trigonometric component orthogonal to each other: This is the contribution of Fourier.

Lesson 4 complex wavelet of Fourier Transformation 

Broadly speaking, "plural" is a "concept" rather than an objective existence.
What is "concept "? How many faces does a piece of paper have? Two, Here "face" is a concept, a subjective perception of objective existence, just like the concepts of "big" and "small", only meaningful to people's consciousness, there is no significance for objective existence itself (Kant: a pure rational criticism ). Turn the two sides of the paper into a "Mobius circle", and there is only one "face" left. The concept is the processing of the objective world, reflecting things in consciousness.
The concept of number is promoted as follows: What number X makes x ^ 2 =-1? The real number axis obviously does not work. (-1) * (-1) = 1. So if there is an abstract space that includes both the real number of the real world and the imaginary x ^ 2 =-1, then we call this space "complex field ". The algorithm of real numbers is a special case of the complex number field. Why 1 * (-1) =-1? +-The symbol represents the direction in the complex field.-1 indicates "backward, turn! "For such a command, a 1 becomes-1 after the circular motion of 180 degrees. Here, the number axis and the circular rotation of the straight line are unified in the space of the plural.
Therefore, (-1) * (-1) = 1 can be interpreted as "backward" + "backward" = return to the original place. How does the complex field represent x ^ 2 =-1? Very simple. "Turn left" and "Turn left" are equivalent to "turn backward ". Because the single-axis real number field (straight line) does not contain such elements, the complex number field must be represented by two orthogonal number axes-the plane. Obviously, we can obtain a feature of the multiplication of the complex number field, that is, the absolute value of the result is the multiplication of the absolute values of two plural numbers, and the rotation angle = the Rotation Angle of the two plural numbers. In the high school era, we learned the dimover theorem. Why is there such a multiplication? It is not because the complex field has such a multiplication property (nature determines understanding), but because the person who invented the complex field developed such a complex field based on such requirements (Understanding determines the nature ), it is a research method of subjective idealism. To construct x ^ 2 =-1, we must consider multiplication as a set composed of two elements:
Product and angle rotation.
As trigonometric functions can be seen as a projection of circular motion, in the complex field, trigonometric functions and multiplication operations (exponent) are unified. Starting with the Fourier series in the real number field, we can immediately obtain the Fourier complex series in a simpler form. Because the complex number field is simple in form, it is easy to study ---- although there is no complex number in nature, but because of the one-to-one correspondence with the series in the real number field, we can obtain physical results by reflecting the ing.
So what is the meaning of the difficult-to-understand Fourier transformation formula? Let's take a look at its relationship with the Fourier series in the complex field. What is calculus, that is, first differentiation, then integration, and the Fourier series have already produced infinite differentiation, corresponding to the sum of the shock signals of countless discrete frequency components. Fourier transform solves the problem of non-cyclic signal analysis. Imagine that this non-cyclic signal is also a periodic signal: only the cycle is infinite, and each frequency component is infinitely small (otherwise the integral result is infinite ). Then we can see that, in the process of solving each component constant of the Fourier series, the integral interval is changed from t to positive and negative infinity. Because the constant of each frequency component is infinitely small, let each component be divided by F to get the number of values-so the Fourier transformation of the periodic function corresponds to a bunch of pulse functions. Similarly, each frequency component is infinitely close, because f is very small, F, 2f, and 3f in the series are almost close together, and eventually get together, just like convolution, the sum of the series in the complex frequency space can eventually become an integral formula: the Fourier series is transformed into a Fourier transformation. Note that there is a conceptual change: discrete frequency. Each frequency has a "weight" value, while the weighted value of each frequency in the continuous F field is infinitely small (Area = 0 ), only the "spectrum" within a frequency range corresponds to certain energy points. The frequency point is the line of the spectrum.

Therefore, what is obtained by Fourier transformation is a continuous function that can draw an image above the complex frequency field? What is 2pai? It is only used to ensure that the signal remains unchanged after the positive transformation is reversed. We can divide the positive transformation by 2 and the reverse transformation by Pi. Slow down, how does one have a "negative" part, or that sentence, is the rotation of the direction of the number axis corresponding to the plural axis, or the phase component corresponding to the trigonometric function, so it is easy to understand. What are the benefits? We ignore the phase and only study the "Amplitude" factor to see the frequency characteristics in the real frequency domain.
From the real number (trigonometric decomposition)-> the complex number (E and PI)-> the complex number Transformation (f)-> the complex number inverse transformation (F-1)-> the complex number (take the amplitude component) -> real numbers seem complicated, but this tool makes it possible to solve frequency analysis problems that cannot be solved by a single real number field. The relationship between the two is: The frequency amplitude component in the Fourier series is a1-an, b1-bn, these discrete numbers represent the frequency characteristics, each number is the result of integral. The result of Fourier transformation is a continuous function: For each value point a1-aN (n = infinite) in the f domain, its value is the original time domain function and a trigonometric function (expressed as a complex number) the result of the integral-this solution is the same as the expression of the series. However, N discrete integral sub-statements are unified into a general and continuous integral sub-statements.

In the frequency-complex mode, we can't draw a picture, but let me draw it! Because not a graph can be clearly expressed. In Chinese only:
1. Draw a plane composed of X and Y axes and draw a circle centered on the origin (r = 1 ). Draw another vertical line (linear equation x = 2) and regard it as a baffle plate.
2. Imagine that there is an atom starting from () and moving along the circle at a counter-clockwise speed. Imagine that the sunlight emits a positive direction from the plural direction of the X axis, and the projection of the Atomic Motion on the baffle (x = 2) is a coseismic reduction.
3. let's modify it. x = 2 does not correspond to a baffle plate, but to a printer's paper exit port. Then, the atomic movement will draw a continuous sin (t) on the White Paper) curve!
What are the three items described above? The trigonometric function corresponds to the circular motion one by one. If I want sin (t + x), or cos (t), I just need to change the starting position of the atom: the vector of level coordinates, the radius remains unchanged and the phase changes.
The expanded form of the Fourier series. Each frequency component is represented as ancos (NT) + bnsin (NT). We can prove that, this formula can be changed to a single trigonometric form such as sqr (an ^ 2 + BN ^ 2) sin (nt + x). So: The real value pair (an, bn ), it corresponds to a point on the two-dimensional plane. Phase X corresponds to the phase of the point. The one-to-one correspondence between real numbers and complex numbers is established. Therefore, the real number frequency uniquely corresponds to a complex number frequency, so we can use the complex numbers to conveniently study the calculation of real numbers: convert triangle operations into exponential and multiplication addition operations.
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However, F transformation is still restricted (the input function must meet the diyheri condition ), in order to more widely use the "Domain" transformation idea to represent a "generalized" frequency information, we have invented the Laplace transformation. Its continuous form corresponds to the F transformation, the discrete form is a z transformation. What about discrete signals? The F series of discrete periodic functions. The number of items is limited. The number of discrete non-periodic functions is still a discrete periodic function after the periodic extension. The number of discrete F series is still limited. Discrete F transformations are easy to understand-continuous signals are multiplied by a periodic sampling filter, that is, a frequency domain and a heap of pulses. Time-Domain sampling corresponds to frequency-domain periodic extension. Why? In turn, it is easy to understand that the periodic continuation of the time domain corresponds to a bunch of pulses in the frequency domain.
Difference between the two: FT [F (t)] = from negative infinity to positive infinity pair [F (t) exp (-JWT)] points lt [F (t)] = from zero to positive infinity for [F (t) exp (-St)] integral (due to actual application, only one-side Laplace transformation is usually performed, that is, the integral starts from zero, in the Fourier integral transformation, the multiplication factor is exp (-JWT). Here,-JWT is obviously a pure virtual number. In the Laplace transformation, the multiplication factor is exp (-St ), S is a complex number: S = d + JW, JW is a virtual part, equivalent to JWT In the Fourier transformation, and D is a real part, as the attenuation factor, in this way, many functions that cannot be used for Fourier Transformation (such as exp (AT), a> 0) can be used for domain transformation.
In short, the ztransform refers to the Laplace transformation of discrete signals (also called sequences), which can be derived from the Laplace transformation of Sampling Signals. ZT [F (n)] = sums [F (n) Z ^ (-N)] From n to negative infinity. Physical significance of the Z domain: because the values are discrete, the input/output process and the physical time consumed have no inevitable relationship (T is only meaningful to the continuous signal ), therefore, the frequency field investigation becomes simple. We regard the basic sequence (1,-1) as the sequence with the highest digital frequency, the digital frequency is 1Hz (Digital angle frequency 2 pi). The frequencies of other digital sequences are all 1Hz in n. The result of frequency decomposition is a set of several values in the 0-2 pi angle frequency, it is also a heap of discrete numbers. Because the time frequency is discrete, you do not need to write the impact function factor during the transformation.
Discrete Fourier transformation to Fast Fourier transformation ---- since the number of discrete Fourier transformation is O (n ^ 2), we consider decomposing the discrete sequence into a group of two for Discrete Fourier transformation, the computational complexity of the transformation is reduced to O (nlogn), and the calculation result is accumulated O (n), which greatly reduces the computational complexity.
Another high-level topic: wavelet. In practical engineering applications, most of the transformations mentioned above have been replaced by wavelet transformations.
What is wavelet? First, let's talk about what is a wave: The component in the Fourier series. The Sin/COS function is a wave, and sin (T)/cos (t) is scaled down by amplitude and the frequency is tightened, it becomes the sum of a series of waves, and uniformly converges to the original function. Note that the convergence of Fourier series summation is strict for the entire number axis. However, as we mentioned earlier, when applying FFT, we only need to pay attention to the Fourier transformation of some signals and then find a sum. Then, for part of the component of the function, we only need to ensure that this is used to act as the "wave function" of bricks. In a certain interval (filtered by a window function), it is enough to meet the definitions that can be integrated and converged, therefore, the Fourier transform "wave" factor can use a series of function families constructed from some basic functions instead of trigonometric functions, as long as the basic function meets the convergence and orthogonal conditions. How to construct such basic functions? After sin (t) is added with a square window, ing to the frequency domain is a bunch of infinite hashes, so trigonometric functions cannot be used. We need to obtain a function family with good frequency convergence, which can cover the low-end part of the frequency domain. Far from that, if the wavelet transform of digital signals is used, the basic wavelet must ensure that the digital angle frequency is the largest.
2PI. The method of using wavelet for off-spectrum analysis is not to find all the frequency components like the Fourier series, nor to look at the spectrum characteristics like the Fourier transformation, but to perform some filtering, let's see what the peak value of a certain digital angle frequency is. You can obtain a sequence of numbers as needed.
We use a multiplier relationship such as (0, F), (F, 2f), (2f, 4f) to evaluate the frequency characteristics of the function family, then, the corresponding time waveform is a series of Function Families with multiples Scaling (and modulation included. The frequency domain is the basic function of the window function, and the time domain is the bell function. Of course, other types of wavelet, although the frequency domain is not a window function, are still available: because the transformation obtained from the wavelet integral is a value, such as the total value contained in (0, F, (F, 2f) contains the total value. Therefore, even if the division in the frequency domain is not a rectangle But other images, the results will not be affected. At the same time, the resolution density of the value in this frequency field conflicts with the time resolution of the time-domain wavelet basis function (time-domain closeness in frequency-domain width and time-domain closeness ), therefore, the design was subject to the uncertainty principle of the heenburg test. Since the time frequency is local and the transformation result is a numerical point rather than a vector, the computation complexity is reduced from the O (nlgn) of the FFT to O (n ), excellent performance.
After speaking so much in Chinese, the basic idea has been clearly expressed. For the convenience of research, we have expanded from the real Fourier series to the Fourier series created in the complex field, then, the Fourier transformation is extended to the pull transformation, and the z transformation is simplified to the case where both the time and frequency are discrete. All of them are linked by a main line.