Signal and System

Daniel is a very popular introduction to the "Signal and system"

Lesson one what is convolution convolution what's the use of Fourier leaf transform what is Laplace transform

Introduction

Many friends and I like, engineering electronics major, learned a bunch of signal aspects of the class, nothing learned, back the formula test, and then graduated.

"What is the use of convolution"? (Some people answer, "convolution" is to learn the "signal and system" course of the following chapters exist.) I yelled and dragged him out to shoot! )

?? tell a story:

?? Zhang San just applied to an electronics company as a tester, and he did not learn the "signal and system" course. One day, he got a product, the developer told him that the product has an input, there is an output, a limited input signal will only produce a limited output.

?? Then, the manager Jeng Changsan tests when the input sin (t) (t<1 second) signal (with a signal generator), the product output what waveform. Zhang San, took a waveform diagram.

? ?" Very good! The manager said. Then the manager gave Zhang San a stack of A4 paper: "There are thousands of kinds of signals, all in the formula, the input signal duration is also determined." What do you test the output waveform of our products separately? "

?? This Zhang San, he was thinking "God, help me, how do I draw these waveforms?"

?? Then God appeared: "Zhang San, you just have to do a test, you can use the mathematical method to draw all the input waveform corresponding output waveform."

?? God went on: "Give the product a pulse signal, the energy is 1 joules, the output waveform drawing out!" "

?? Zhang San obeyed, "then?"

?? god added, "for an input waveform, you imagine that it is a differential into countless small pulses that are input to the product, and the result of the overlay is your output waveform." You can imagine these little pulses lined up into your product, each producing a small output, and when you draw the timing diagram, the waveform of the input signal appears to go into the system in turn. "

?? Zhang San realized: "Oh, the output of the results of the points out!" Thank God. What's the name of this method? "

God said, "It's called convolution!" "

Since then, Zhang San's work has been much easier. Every time the manager asked him to test the output of some of the signals, Zhang San only needed to do calculus on A4 paper and submit the task!

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?? Zhang San worked happily until one day, the quiet life was broken.

The manager brought a small electronic device to the oscilloscope and said to Zhang San, "Look, the waveform generated by this little device simply can't be explained by a simple function, and it's continuously signaled!" Fortunately, this continuous signal repeats every once in a while. Zhang San, you come to test the following, connected to our device, will produce what output waveform! "

?? Zhang San waved: "The input signal is infinite time, I want to test the infinite time to get a stable, repetitive waveform output?"

The manager angered: "Anyway, you fix it, or you get fired!" "

? Zhang San thought: "This time the input signal even formulas are given out, a very chaotic waveform, time is infinitely long, convolution also die, how to do?"

?? in time, God appeared again: "Map chaotic time domain signals to another math domain, and then map back after the calculation is complete."

? ?" Every atom in the universe is spinning and vibrating, and you can think of a time signal as a combination of several shocks, which is something that can be determined, with fixed frequency characteristics. "

? ?" I'll give you a mathematical function f, the time domain infinite input signal in the F domain is limited. The time domain waveform chaotic input signal in the F field is neat and easy to see clearly. So you can calculate the "

? ?" Also, the time domain convolution in the F field is a simple multiplication relationship, and I can prove it to you to see "

? ?" After calculating the finite program, take the F (-1) Back to the time domain, you get an output waveform, and the rest is your mathematical calculation! "

?? Zhang Sanxie over God and saved his job. Then he knew that the transformation of the F-Domain had a name, called the Fourier leaf, what what ...

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?? later, the company developed a new electronic product, the output signal is an infinite length of time. This time, Zhang San began to learn Laplace ...

Postscript:

It is not that we learn bad, because the textbook is not good, the teacher is not good at speaking.

? I really appreciate Google's interview question: Use 3 words like an old lady to tell what a database is. Such a proposition is very good, because without an in-depth understanding of a proposition, without careful consideration of the design philosophy of a thing, we fall into the mire of detail: the back formula, the mathematical deduction, the integral, the problem, and the lack of time to answer "why." Do the university teacher's do not "the thick book reads thin" This point, cannot speak the philosophical level truth, blindly endorse and to turn the PPT, does the dull mathematics proof, then blames "the current student generation is inferior to the generation", has what significance?

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Lesson two in the end what is the frequency what is the system?

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? ? In this article, I unfold the Fourier transform F. Note that the Fourier transform's name F can represent the concept of frequency (freqence), or any other concept, because it is just a conceptual model that is constructed to solve computational problems (such as an infinite time domain input signal, how to get the output signal). We look at the Fourier transform as a function of the C language, the output and output of the signal as an IO problem, and then any x->y problems that are difficult to solve can be obtained with x->f (x)->f-1 (x)->y.

1. What is frequency?

A basic hypothesis: any information has the characteristics of frequency, the sound level of the audio signal, the spectrum of the light, the period of the electronic oscillation, and so on, we abstract a concept of harmonic vibration, the mathematical name is called frequency. Imagine an atom on the X-y plane with a radius of 1 uniform circular motions around the origin, and an x-axis as time, then the projection of that circular motion on the y-axis is the waveform of sin (t). I believe that middle school students can understand this.

?? So, different frequency models actually correspond to different circumferential motion velocities. The faster the circular motion, the narrower the waveform of sin (t). There are two modes of frequency scaling

(a) old-fashioned radios are tape-used as music media, and when we put them on, we feel the sound of singing become strange and high-toned, because the speed of "circular motion" is multiplied, and the sin (t) output of each sound component becomes sin (NT).

(b) on the cd/computer on the fast or full feel singers singing or slow singing, will not appear the phenomenon of high pitch: Because the time when the time domain sampling method, discarded some waveforms, but the output waveform bearer information will not have the width of the change; when full, the time-domain signal is filled to elongate.

2. What is the physical significance of the F-transform resulting in a negative/plural part?

?? Explanation: The F-transform is a mathematical tool that does not have a direct physical meaning, and the existence of negative/complex numbers is only for the sake of computational integrity.

3. Signals and systems what is the basic thrust of this lesson?

For the communications and electronics students, in many cases our work is the design or the OSI seven layer model of the physical layer technology, the complexity of the first is that you must establish the transmission medium electrical characteristics, usually different transmission media for different frequency segments of the signal has different processing power. Ethernet wire processing baseband signal, WAN light out of high-frequency modulation signals, mobile communications, 2G and 3G respectively need to have different carrier frequency characteristics. So these media (air, wire, fiber, etc.) for the input of a certain frequency can be transmitted at a certain distance after the basic constant input? Then we need to establish the corresponding mathematical model of the frequency of the medium. At the same time, knowing the frequency characteristics of the medium, how to design the signal transmitted above it can be large to the theoretical maximum transmission rate?----This is the signal and the system this lesson led us into a world.

? Of course, the application of signals and systems is more than that, and Shannon's information theory is linked, it can also be used in information processing (sound, image), pattern recognition, intelligent control and other fields. If the course of computer science is the logical model of data expression, then the signal and system are built on the lower level, which represents a mathematical model of some physical meaning. The knowledge of the data structure can solve the logic information coding and error correction, and the knowledge of the signal can help us to design the physical carrier of the code stream (if the received signal waveform is chaotic, then I based on what to judge whether this is 1 or 0? Logical error correction loses its meaning). In the field of industrial control, the premise of computer application is a variety of digital-analog conversion, then the various physical phenomena produced by the continuous analog signal (temperature, resistance, size, pressure, speed, etc.) how to be a specific device to convert to a meaningful signal, first of all we need to design a usable mathematical transformation model.

4. How to design the system?

? Design physical system functions (continuous or discrete state), with inputs, outputs, and intermediate processes and specific physical implementations, not the focus of this lesson (electronic circuit design?). In the final analysis, signals and systems design a system function for specific requirements. The prerequisite for designing a system function is to represent the inputs and outputs in functions (such as sin (t)). The method of analysis is to decompose a complex signal into a number of simple signal accumulation, the specific process is a lot of calculus, the specific mathematical operation is not the central idea of this course.

What kind of system do you have?

(a) Classification by function: modulation and demodulation (signal sampling and reconstruction), superposition, filtering, 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 not very close to writing a program logic flowchart? They do not differ in the symbolic space. There is also the discrete state of digital signal processing (follow-up course).

(b) Classification by system, stateless systems, finite state machines, linear systems, etc. and the continuous system function of the physical layer is a complex linear system.

5. What are the best textbooks?

? ? The core of the symbology is the set theory, not the calculus, the system that does not have the set theory constructs, the realization uses the calculus to be meaningless----you do not even know what to do for a half day. To learn the signal and system from the computer's point of view, one of the best textbooks is <>, the author is the UC Berkeley Edward A.lee and Pravin Varaiya----first defined and realized, in line with human thinking habits. The domestic textbooks are mathematical deduction, is not to say that these derivations are for what purpose to do, what to get, what to build, to prevent anything, not to discuss from the epistemology and demand, the entire article is not see the purpose of the methodology, the cart before the horse.

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The third lesson The sampling theorem is what?

1. For example, when the telephone call, the signal is Pam pulse amplitude modulation, in the phone line is not the voice, but voice through the channel code conversion pulse sequence, in the receiver to restore the voice waveform. Then for the continuous speaker voice signal, how to convert into some column pulse to ensure that the basic non-distortion, can be transmitted? Obviously, we think of sampling, every m milliseconds to the voice sample to see the amplitude of the signal, the amplitude into pulse encoding, transmission out, at the end of a rule to regenerate the language.

? ? So, the question is, how small is it enough to sample every m milliseconds? How can I restore the language waveform at the end of the collection?

? ? For the first problem, we consider that the voice signal is a time-frequency signal (so the corresponding F-transform represents a time frequency) to decompose the speech signal into several different tones of the mono-mixed body (periodic function of the compound Leaf series expansion, non-periodic interval function, can be regarded as the completion of the periodic signal expansion, the same effect), For the highest frequency signal component, if the sampling method can guarantee the recovery of this component, then the other low-frequency components can be sampled to enable the information to be saved. If a person's voice is limited to 3000Hz, then the high-frequency component we see as sin (3000t), the SIN function to save information by sampling, can be seen as: for a period, the peak sample once, the trough sampling once, that is, the sampling frequency is the highest frequency component twice times (Nyquist sampling theorem), We can represent the original analog continuous signal without compromising the sample signal. The two signals one by one correspond to each other and are equivalent.

? ? For the second question, at the receiver, how to restore the analog continuous signal from the pulse sequence (comb-loaded waveform)? First, we have affirmed that the pulse sequence above the frequency domain already contains all the information, but the original information only exists at a certain frequency, how to do? We let the input pulse signal I through a device X, the output signal for the original voice O, then I (*) x=o, here (*) represents the convolution. When the characteristics of the time domain is not analyzed, then in the frequency domain F (I) *f (X) =f (O) Multiply the relationship, this is very obvious, as long as F (X) is an ideal, low-pass filter can be (in the F field is drawn out is a box), it is a clock type function in the time domain So the actual does not exist), to make such a signal processing device, we can get the almost ideal original speech by the input pulse sequence. In practice, our sampling frequency is usually a bit more than the Nyquist frequency, 3k Hz voice signal, sampling standard is 8k Hz.

2. Another example, 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, the clearer it is. If our sampling frequency is not enough, the information will occur aliasing----online there is a picture, myopia wearing glasses to see Einstein, take off the eyes see is Monroe----because without eyes, resolution is not enough (sampling frequency is too low), high-frequency component distortion is mixed with low-frequency components, only to create a visual trap. Here, the F-change of the image corresponds to the spatial frequency.

? ? In other words, is it not good to upload the original voice signal directly in the channel? Analog signal has no anti-jamming ability, no error correction ability, sampled signals, with digital characteristics, better transmission performance.

? ? What signals are not ideal for sampling? The time domain is hopping and the frequency domain is infinitely wide, such as a square wave signal. If it is represented by a sampling signal of limited bandwidth, it is equivalent to a partial sum of compound-interest-leaf series, and this part and when recovering the original signal, there will be burrs on the non-conductive point, also known as Gibbs phenomenon.

3. Why did Fourier come up with such a progression? This is the basic idea of Western philosophy and Science: Orthogonal analysis method. For example, to study a stereoscopic shape, we use X, Y, Z, three orthogonal axes: any one axis has a projection of 0 above the other axes. In this case, the 3 view of an object can fully express its shape. Similarly, how does the signal decompose and analyze? The infinity of the triangular function components that are orthogonal to each other: this is the contribution of Fourier.

Introductory Lesson IV Fourier transform complex wavelets

? ? Broadly speaking, "plural" is a "concept", not an objective existence.

? ? What is "concept"? How many sides does a piece of paper have? Two, here "face" is a concept, a subjective understanding of the objective existence, just like the concept of "big" and "small", only to the human consciousness has meaning, to the objective existence itself has no meaning (Kant: The Critique of pure rationality). Turn the note on both sides of the connection, into a "Möbius circle", this note is only one "face". The concept is the processing of the objective world, reflected in the consciousness of things.

? ? The concept of number is so popularized: what number x makes X^2=-1? The real axis is obviously not, (-1) * (-1) = 1. So if there is an abstract space that includes both real-world real numbers and imaginary x^2=-1, then we call this imaginary space "plural domain". Then the algorithm of the real number is a special case of the plural domain. Why 1* (-1) =-1? +-symbol in the plural field to represent the direction, -1 is "backwards, go!" Such a command, a 1 in the circular motion after 180 degrees into-1, here, the line axis and circular rotation, in the complex space is unified.

? ? Therefore, (-1) * (-1) =1 can be interpreted as "turn backward" + "go backwards" = go back to the same place. So how does the plural field represent x^2=-1? Very simple, "turn left", "turn left" two times equivalent to "turn backwards". Because a single-axis real field (line) does not contain such an element, the complex field must be represented by two orthogonal axes-the plane. Obviously, we can get a characteristic of the multiplication of complex fields, that is, the absolute value of the result is multiplied by the absolute value of two, and the angle of rotation = Two of the complex number. In high school, we learned the Dimover theorem. Why is there such a multiplicative nature? It is not because the plural domain happens to have such multiplicative nature (the nature determines the cognition), but the person who invents the plural domain is to make such a complex domain according to such demand (the cognition decides the nature), is a kind of subjective idealism research method. To construct the X^2=-1, we must consider multiplication as a set of two elements: product and angular rotation.

? ? Because trigonometric functions can be seen as a projection of circular motion, trigonometric functions and multiplication (exponents) are unified in complex fields. We start with the Fourier series of the real field, and we can get the Fourier Ye Shi number of the form simpler, the plural domain and the real field one by one. Because of the simple form of complex numbers, it is convenient to study----although there is no complex number in nature, but because it corresponds to the number of series one by one in the real field, we can get a result with physical meaning by reflecting the fire.

? ? So what is the meaning of the Fourier transform, the incomprehensible conversion formula? We can look at the relationship between it and the Fourier series of the plural domain. What is calculus, first differential, then integral, Fourier series has been made infinite differential, corresponding to countless discrete frequency component impulse signal and. Fourier transform to solve the problem of non-periodic signal analysis, imagine this aperiodic signal is also a periodic signal: Only the period is infinite, each frequency component is infinitely small (otherwise the result of the integration is infinite). Then we see Fourier series, each component constant of the solution process, the interval of the integral is from T to positive and negative infinity. And because the constants of each frequency component are infinitely small, so that each component is divided by F, it gets the number of values----so the Fourier transform of the periodic function corresponds to a bunch of pulse functions. Similarly, each frequency component is infinitely close, because F is small, the f,2f,3f in the series are almost next to each other, and, as with convolution, the sum of the series of the complex frequency space can eventually become an integral formula: The Fourier series becomes the Fourier transform. Note that there is a conceptual change: discrete frequency, each frequency has a "weight" value, and the continuous f-domain, each frequency of the weighted value is infinitesimal (area =0), only one frequency range of "spectrum" corresponding to a certain energy integral. The frequency point becomes the line of the spectrum.

? ? So the Fourier transform is a usually a continuous function, is the complex frequency domain above can draw the image of things? What is that square root 2Pai? It is just to ensure that the signal is not changed when the reverse transformation is returned. We can divide the positive transformation by 2 and let the inverse transform divide by pi. Slow, how there is a "negative" part, or that sentence, is the direction of the axis of the number of axes of the rotation, or corresponding to the triangular function of the phase component, so that is very good understanding. What are the benefits? We ignore the phase and only study the "amplitude" factor, so we can see the frequency characteristic in the real frequency domain.

? ? We look very complicated from the real numbers, complex numbers ((F), Complex inverse Transformations (F-1), complex (E and pi), and complex (c) and complex (magnitude)----(trigonometric decomposition)-----------this tool makes it possible to analyze the problem from a frequency that cannot be solved by the real field can be solved. 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, and each number is the result of the integral. The result of the Fourier transform is a continuous function: For each value point A1-an (n= Infinity) of the F domain, its value is the result of the original time domain function and a trigonometric (represented as complex) integral----the solution and the representation of the series are the same. But it is to unify the N discrete integral formula for a universal, continuous integral equation.

? ? Complex frequency domain, we all say that cannot draw, but I come to draw! Because not a diagram can be expressed clearly. I use pure Chinese to say:

1. Draw a plane consisting of an x, Y axis, and draw a circle at the center of the origin (R=1). Draw a vertical line: (linear equation x=2), think of it as a baffle.

2. Imagine that there is an atom, starting from the point of (1,0), along this circle counterclockwise uniform circular motions. Imagine the sunlight from the x-axis in the plural direction of the x-axis positive direction, then this atom motion on the bezel (x=2) above the projection, is a simple co-vibration.

3. Again, x=2 corresponds to not a baffle, but a printer's exit, then the process of atomic motion on the white Paper has drawn a continuous sin (t) curve!

? ? What does the above 3 show? The trigonometric and circumferential motions correspond to one by one. If I want sin (t+x), or cos (t) in this form, I just need to change the starting position of the atom: that is, the vector of the level coordinates, the radius is constant, the phase changes.

? ? The real expansion form of Fourier series, each frequency component is expressed as Ancos (NT) +bnsin (NT), we can prove that this equation can become sqr (an^2+bn^2) sin (nt+x) as a single trigonometric form, then: real value pairs (an,bn), Corresponds to a point above the two-dimensional plane, and phase x corresponds to the phase of the point. The one by one correspondence between the real number and the complex number is established, so the real frequency uniquely corresponds to a certain complex frequency, and we can use complex numbers to conveniently study the operations of real numbers: the trigonometric operations into exponential and multiplication addition operations.

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? ? However, the F-transform is still limited (the representation of the input function must satisfy the Diehe condition, etc.), in order to use the idea of "domain" transformation to represent a "generalized" frequency information, we invented the Laplace transform, its continuous form corresponds to the F-transform, and the discrete form becomes the Z-transform. What about discrete signals? The F-series of discrete periodic functions, finite number of items, discrete non-periodic functions (which are still discrete periodic functions after the periodic continuation), and the discrete F-series, are still limited in number. The discrete f-transform is easy to understand----continuous signals are multiplied by a periodic sampling filter, which is the frequency domain and a bunch of pulses. Time domain sampling corresponds to the frequency domain periodic continuation. Why? In turn, it is easy to understand that the periodic extension of the time domain corresponds to a bunch of pulses in the frequency domain.

? ? The difference between the two: ft= from negative infinity to positive infinity to integral lt= from zero to positive infinity (due to practical application, usually only unilateral Laplace transformation, that is, the integration from the zero) specifically, in the Fourier integration transformation, the multiplicative factor is exp (-JWT), here,- The JWT is obviously a pure imaginary number, whereas in the Laplace transformation, the multiplicative factor is exp (-st), where S is a complex number: S=D+JW,JW is the imaginary part, which is equivalent to the JWT in the Fourier transform, and D is the real part, as the attenuation factor, which can Functions that make Fourier transformations (such as exp (at), a>0) do domain transformations.

? ? The z-transform, simply said, is a Laplace transformation of discrete signals (also called sequences), which can be derived from the Laplace transformation of sampled signals. zt= the sum of negative infinity to positive infinity from N. The physical meaning of Z-domain: Because the value is discrete, so the process of input and output and the physical time spent has no inevitable relationship (T is only meaningful to the continuous signal), so the frequency domain investigation becomes and simple, we regard (1,-1,1,-1,1,-1) such a basic sequence as the highest number of series , his digital frequency is 1Hz (digital angular frequency 2Pi), the other digital sequence frequency is n per 1Hz, the result of frequency decomposition is a set of several values in the 0-2pi angular frequency, is a bunch of discrete numbers. Since the time-frequency is discrete, it is not necessary to write the factor of the impact function when doing the transformation

? ? Discrete Fourier transform to fast Fourier transform----because the number of discrete Fourier transforms is O (n^2), so we consider the discrete sequence decomposition into 221 groups for the discrete Fourier transform, the computational complexity of the transformation down to O (Nlogn), and then the results of the calculation of O (N), This greatly reduces the computational complexity.

? ? Another high-level topic: wavelets. In practical engineering applications, most of these transformations have been replaced by Wavelet transforms.

What is wavelet? First say what is wave: The component inside the Fourier series, the Sin/cos function is the wave, sin (t)/cos (t) through the amplitude of the contraction and the tightening of the frequency, into a series of sums of waves, uniformly converge to the original function. Note that the convergence of the Fourier series summation is strictly for the whole axis. But as we said earlier, when we actually apply the FFT, we just need to focus on the Fourier transform of some of the signals and then find a whole and then we can, then for the part of the function, we just need to ensure that this is used as a brick "wave function", in a certain interval (filter by window function) in accordance with The few integrals and convergent definitions are possible, so the Fourier transform's "wave" factor, instead of using trigonometric functions, uses a series of function families constructed from some basic functions, as long as the basic function conforms to those conditions of convergence and orthogonality. How do you construct such a basic function? When sin (t) is added to a square window, mapping to the frequency domain is a bunch of infinite hash pulses, so it is no longer possible to use trigonometric functions. We want to get a family of functions with good convergence in the frequency domain, which can cover the lower part of the frequency domain. Say a little farther, if it is to take the digital signal of the wavelet transform, then the basic wavelet to ensure that the digital angular frequency is the largest 2Pi. The method of spectrum analysis using wavelets is not to find out all the frequency components like Fourier series, nor to look at the spectral characteristics as Fourier transforms, but to do some sort of filtering to see what the crest value of a certain digital angular frequency is probably. A sequence of numbers, such as a dry one, can be obtained according to actual needs.

? ? We used (0,f), (f,2f), (2f,4f) Such a multiplier relationship to investigate the frequency characteristics of the function family, then the corresponding time waveform is a multiple expansion (and contains modulation---so that the spectrum is moved) a series of function families. The frequency domain is the basic function of the window function, and the time domain is the bell-shaped function. Of course other types of wavelets, although the frequency domain is not a window function, but still available: because the wavelet integral of the transformation, is a value, for example (0,F) contains the total energy value, (F,2F) contains the total energy value. So even if the segmentation of the frequency domain is not rectangular but other graphs, it has little effect on the result. At the same time, the value of this frequency domain, its resolution density and temporal resolution of the wavelet function is a conflict (time domain tight frequency domain width, time domain tight), so the design time by the Heisenberg uncertainty principle constraints. Jpeg2000 compression is the wavelet: because the time-frequency is local, the transformation result is a numerical point rather than a vector, so the computational complexity from the FFT o (nlgn) down to O (N), the performance is very good.

In Chinese, the basic thought has been expressed clearly, in order to "research convenience", from the real Fourier series to create a complex number of Fourier series expansion, and then to the Fourier transform, then extended to the pull-type transformation, and then for the time-frequency are discrete conditions simplified to Z-transform, all with a main line linked up.

Signal and System