Understanding of time domain and frequency domain

Understanding of time domain and frequency domain

  The time domain is to describe the relationship between mathematical function or physical signal and time。

  For example, the time-domain waveform of a signal can express the change of semaphore with time. If discrete-time is considered, the function or signal in the time domain is known at each discrete time point. If continuous time is considered, the value of function or signal at any time is known. In the study of time-domain signs, the oscilloscope is often used to convert the signal into its time-domain waveform.

The time domain is the real world and the only entire domain. Because our experiences are developed and verified in the time domain, we are used to chronological order events. When evaluating digital products' performance, it is usually analyzed in the time domain because products' performance is ultimately measured in the time domain. The clock waveform is shown in Figure 1 below.

  That can see from Figure 1 that two essential parameters of the clock waveform are clock period and rise time. The figure shows the clock period and 10% - 90% rise time of the 1GHz clock signal. The falling time is generally shorter than the rising time, and sometimes more noise will appear. The clock cycle is the time interval of clock cycle repetition, usually measured by NS. The clock frequency clock, that is, the number of clock cycles in one second, is the clock cycle clock's reciprocal.

   The rise time is related to the time of signal jumping from low level to high level. There are usually two definitions. One is the rise time of 10% - 90%, which refers to when the signal jumps from 10% of the final value to 90%. Usually, a default expression that they can read directly from the waveform's time-domain graph. The second definition is 20% - 80% rise time, which refers to 20% to 80% of the final value.

There is also a corresponding value for the time domain waveform descent time. According to the logic series, the downtime is usually shorter than the uptime, caused by a typical CMOS output driver's design. In a specific output driver, the p-tube and n-tube are connected in series between the power track VCC and VSS, and the output is joined in the middle of the two tubes. At any one time, only one transistor is on, and which transistor is on depends on the high or low state of the output.

The frequency-domain takes frequency as the primary variable when analyzing problems.

The frequency domain is a coordinate system used to describe the frequency characteristics of signals.

The description needs to be carried out from many aspects, and each element only provides part of the information for us to understand this thing. For example, there is a car in front of me. I can describe it in this way: 1. Color, length, height. Aspect 2: displacement, brand, price. For a signal, it has many characteristics. For example, the change rule of signal strength with time (time-domain features), frequency-domain elements single frequency signals are synthesized. 

Frequency domain analysis

Frequency domain (frequency domain) - the independent variable is the frequency, i.e., the horizontal axis is the frequency, and the vertical axis is the amplitude of the frequency signal, i.e., the frequency spectrum. The spectrum chart describes the signal's frequency structure and the relationship between the frequency and frequency signal amplitude.

When analyzing signals in the time domain, sometimes the time domain parameters of some signals are the same, but it does not mean that the signals are the same. Because the signal changes with time and relates to frequency, phase, and other information, it is necessary to analyze the frequency structure of the signal further and describe the signal in the frequency domain. The transformation of dynamic signal from a time domain to frequency domain is mainly realized by Fourier series and Fourier transform. The periodic signal depends on the Fourier series, and the aperiodic signal depends on the Fourier transform.

A simple example of frequency domain analysis can be illustrated in Figure 2: a simple linear process for children's toys. The linear system consists of a weight suspended by a spring mounted on a handle. The child controls the position of importance by moving the handle up and down.

Anyone who has played kind of game knows that if the handle is moved more or less in a sine wave, then the weight will start to oscillate at the same frequency, although the vibration of the importance and the movement of the handle are not synchronous at this time. Only when the spring can not be fully extended will the weight and spring move synchronously and act at a relatively low frequency.

   As the frequency gets higher and higher, the mass oscillation phase may be more ahead of the handling stage and may also be lagging. At the natural frequency point of the process object, the mass oscillation height will be the highest. The process object's natural frequency is determined by the mass of the weight and the spring's strength coefficient.

   When the input frequency is more extensive than the natural frequency of the process object, the amplitude of the mass oscillation will reduce. The phase will be more and more lagging (in other words, the amplitude of the mass oscillation will be less and less, and its phase lag will be larger and larger). In the case of very high frequency, the weight moves only slightly, which is the opposite direction of the handle.

Bode figure

All linear process objects exhibit similar properties. These process objects convert the sine wave's input to the sine wave's output with the same frequency. The difference is that the amplitude and phase of the production and the information are changed. The magnitude of amplitude and phase variation depends on the phase lag and gain of the process object. The growth can be defined as "the ratio coefficient between the amplitude of the output sine wave and the phase lag can be defined as "the degree of the output signal lag when comparing the output sine wave with the input sine wave."

   Unlike the steady-state gain K, the process object's gain and phase lag will change. Unlike the steady-state gain K, the process object's gain and phase lag vary with the input sine wave signal frequency. The spring counterweight object does not substantially change the low-frequency sine wave input signal's amplitude in the above example. That is to say, the item has only one low-frequency gain coefficient. When the signal frequency is close to the process object's natural frequency, because the amplitude of the output signal is larger than the amplitude of the input signal, the gain coefficient is larger than the coefficient under the above low frequency. When the toy in the above example is shaken rapidly, that can consider the object's high-frequency gain as zero because the heavy object can hardly vibrate.

   The phase lag of the process object is a prime factor. In the above example, the phase lag starts from the low-frequency input signal close to zero because the weight and handle oscillate synchronously when the handle moves very slowly. In a high-frequency input signal, the phase lag is "- 180 degrees", that is, the weight and handle move in the opposite direction (therefore, we often use "lag 180 degrees" to describe the reverse movement of these two kinds of situations).

   The Bode spectrum shows the complete spectrum of the system gain and phase lag in the frequency range of 0.01-100 radians/second for the spring weight object. An example of the Bode map was a graphical analysis tool invented by Hendrick's Bode of Bell Laboratories in the 1940s. Using this tool, we can judge the vibration amplitude and phase of the corresponding output signal when the process object is driven by the sine wave input signal of a specific frequency. To obtain the amplitude of the output signal, multiply the input signal's amplitude by the "gain coefficient corresponding to this frequency in the Bode. 

Fourier theorem

The gain coefficient and phase lag value displayed in the Bode diagram of the process object reflect the system's very definite characteristics. For an experienced control engineer, the chart accurately tells him all the process object attributes that he needs to know. Using this tool, the control engineer can predict the system response caused by the control action of a sine wave in the future and see the system response caused by any control action.

The Fourier theorem makes the above analysis possible. It shows that any time series or signal measured continuously can be expressed as an infinite superposition of sine wave signals with different frequencies. In 1822, the mathematician Fourier proved this famous theorem and created a well-known algorithm called Fourier transform. The algorithm uses the original signal measured directly to calculate the frequency, amplitude, and phase of different sine wave signals in an additive way.

In theory, Fourier can use bode graphs to predict linear process objects' responses when they are affected by the time sequence of control action. See the following:

1) using the mathematical method of Fourier transform, the control function provided to the process object is theoretically decomposed into different sine wave signal composition or frequency spectrum.

  2) the Bode diagram can be used to judge each sine wave signal's changes when passing through the object. In other words, they can find the difference in amplitude and phase of the sine wave at each frequency on the graph.

  3) on the contrary, by using the inverse Fourier transform method, each sine wave signal, which is changed separately, can be converted into a signal.

Since the inverse Fourier transform is essentially an additive processing, the linear characteristics of the process object will ensure that the set of "theoretical sine waves calculated in the first step" has a different effect, which should be equivalent to the impact of "the cumulative set of different sine waves." Therefore, the total signal calculated in the third step can represent "the actual value of the process object when the provided control effect is input to the process object."

Note that none of the above steps is made up of a single sine wave generated by the controller drawn on the graph. All these analysis techniques in the frequency domain are conceptual. This is a convenient mathematical method, which uses Fourier transform (or closely related Laplace transform) to convert time-domain signals into frequency-domain signals, then uses Bode diagram or other frequency-domain analysis tools to solve some problems at hand, and finally uses inverse Fourier transform to convert frequency-domain signals into time-domain signals.

Most of the control design problems that can be solved by this method can also be solved by direct control in the time domain, but for calculation, the way using the frequency domain is usually more straightforward. In the above example, multiplication and subtraction are used to calculate the spectrum of the process's actual value. The process's real value is obtained through the Fourier transform of the given control effect and then compared with the Bode diagram analysis.

If all the sine waves are correctly accumulated, the shape predicted by the Fourier transform will be produced. When this phenomenon is not intuitive at times, for example, it may be helpful to understand.

Think again about the child's weight in the example above - the spring toy, the seesaw on the playground, and the boat on the outer ocean. Suppose that the ship rises and falls on the sea in the form of a sine wave with a frequency of W and amplitude of A. at the same time, we assume that the seesaw also oscillates in the form of a sine wave with a frequency of 3W and amplitude of a / 3 and that the child shakes the toy in the form of a sine wave with a frequency of 5W and amplitude of a / 5. "Three separate sine wave waveforms" have shown that if we observe three different sine wave motions separately, each sine wave motion will reflect the form.

Now suppose the child sits on the seesaw, which in turn is fixed on the deck of the ship. If the three individual sine waves are arranged correctly, then the toy's overall motion is about a square wave - as shown in Figure 4: the sine wave of the three combinations.

The above is not a very clear practical example. Still, the sum of the fundamental frequency sine wave, the three times frequency harmonic with the amplitude of one third, and the five times frequency harmonic with the amplitude of one fifth are approximately equal to the square wave with the frequency of W and the amplitude of a. Even if we add the seven times frequency harmonic with the amplitude of one-seventh and the nine times frequency harmonic with the amplitude of one-ninth, the total waveform will be more like a square wave. The Fourier theorem has long shown that when sine waves of different frequencies are added infinitely in infinite series, the resulting total superimposed signal is a square wave with an amplitude of an in a strict sense. that can also use Fourier theorem to decompose aperiodic signals into sine wave signals.

It is beneficial to analyze the time-domain performance by solving differential equations, but it is more troublesome for more complex systems. Because the work of translating the differential equation will increase with the increase of the order of the differential equation, besides, when the equation has been solved, and the response of the system can not meet the technical requirements, it is not easy to determine how to adjust the procedure to obtain the expected results. From the engineering point of view, we hope to find a way to predict the system's performance without solving the differential equation. At the same time, it can also point out how to adjust the performance of the system. The frequency-domain analysis method has the above characteristics. It is a classical method to study the control system and an engineering method to evaluate the system's performance using the frequency domain's graphic analysis method. This method takes the input signal frequency as the variable to study the system's performance in the frequency domain. The frequency characteristics can be obtained by differential equation or transfer function and measured by experiment. The frequency-domain analysis method does not need to solve the system's differential equation directly but indirectly reveals its time-domain performance. It can easily show the influence of system parameters on the system performance and further indicate how to design and calibrate. This analysis method is conducive to system design and can estimate the impact Frequency range of system performance. When there are some components in the system that are difficult to describe by a mathematical model, the system's frequency characteristics can be obtained by the experimental method to accurately and effectively analyze the design and components.

Signal frequency domain analysis

The signal frequency domain analysis uses the Fourier transform to transform the time domain signal x (T) into the frequency domain signal x (f) to help people understand the signal's characteristics from another angle. The signal spectrum X (f) represents the size of signal components in different frequency components, providing more intuitive and rich information than the signal waveform in the time domain.

In 1822, French mathematician J. Fourier (1768-1830) published the "theory of heat analysis" when he studied heat conduction theory. He proposed and proved the principle of expanding periodic function into sine series, which laid the Fourier series's theoretical foundation.

Poisson, Guass, and so on applied this achievement to electricity, and that widely used it.

At the end of the 19th century, people made capacitors for engineering practice.

After entering the 20th century, a series of specific problems such as resonance circuit, filter, and sinusoidal oscillator has opened up a broad prospect for further applying the sinusoidal function and Fourier analysis.

In the theoretical research and practical application of communication and control system, the Fourier transform method has many advantages.

FFT gives new vitality to Fourier analysis.

The frequency-domain analysis takes the frequency of the input signal as the variable. The frequency-domain studies the relationship between the structural parameters and the system's performance, thus derive the essential concepts of the signal spectrum, bandwidth, filtering, modulation, and frequency division multiplexing.

Advantages of frequency domain analysis

Frequency domain analysis has obvious advantages: it does not need to solve differential equations, graphical (frequency characteristic diagram) method, indirectly reveals the system performance, points out the direction of improving performance and is easy to experimental analysis. It can be extended to some nonlinear systems (such as systems with delay links) and can be efficiently designed to suppress noise effectively.

Frequency domain analysis includes

1. Frequency response refers to the system's steady-state response to the sinusoidal input signal.

2. Frequency characteristic: The element that the system's steady-state output changes with the frequency (ω changes from 0 to ∞) when the sinusoidal signal input of different frequencies.

3. Amplitude frequency characteristic and phase-frequency characteristic constitute the frequency characteristic of the system.

4. Amplitude frequency characteristic refers to the change element of|g (J ω) |, recorded as a (ω) when ω changes from 0 to ∞.

5. Phase frequency characteristic: it refers to that when ω changes from 0 to ∞, the change aspect of ∠ g (J ω) is called phase-frequency characteristic, which is recorded as ϕ (ω).

——转自《电子工程专辑》