Signal and Spectrum

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Author: vamei Source: http://www.cnblogs.com/vamei welcome reprint, please also keep this statement. Thank you!

 

Introduction to singal

We often encounter signals in our lives. For example, stock trend chart and heartbeat pulse chart. In the communication field, no matter GPS, mobile phone voice, radio, or Internet communication, we send and receive signals. Recently, the communication system of Shenzhen Metro is suspected to be in conflict with the Wi-Fi signal, that is, the subway antenna receives the Wi-Fi signal, and mistakenly regards the signal as the subway communication signal. Our social informatization is based on signals.

Signal: Shanghai Stock Index in the last three years

 

Signal isA sequence that changes with time or space. In signal processing, we often use "signals" to referOne-Dimensional SignalThat is, a sequence that only changes with a single temporal or spatial dimension. Such signals can be expressedF (t)OrF (x)Such a function.What corresponds to the formation of one-dimensional signals isMultidimensional SignalFor example, an image is a two-dimensional signal. It is representedF (x, y). In the absence of special declarations, "signals" are used to represent one-dimensional signals.

Although the signal is widely used, there is no mysterious point in the mathematical sense of the signal, just a common sequence (function ). The signal processing method is acceptable.GeneralSignal in any field (communication, finance, or other fields ).Signal Processing.

 

The simplest signal: Simple Harmonic)

A sine wave and a cosine wave are collectively referred to as a harmonic. Sine waves can be written as functions:

The preceding expression has three parameters,Amplitude(A, ampl ),Frequency(F, frequency ),Phase(PHI, phase), which respectively control different features of the sine wave. Replace the letter with a real value, and we get a sine wave:

 

A sine signal (Y axis is the amount indicated by the signal, such as the stock price)

 

Amplitude, frequency, and phase control different characteristics of the sine wave respectively. By adjusting them, we can obtain different sine wave signals.

 

Upper left: lower left: 2 times frequency upper right: 2 times amplitude lower right: Phase Movement

(My friend said that I am "using volume to control the tone": Singing should change the frequency but change the amplitude .)

The form of the cosine wave function is similar to that of the sine wave, but the sin is changed to Cos. In fact, we can change the sine wave to obtain the cosine wave from the sine wave. For example, right is a cosine wave of a = 1, F = 1, Phi = 0.

 

Fourier transform)

Simple Harmonic is simple, but it is of great significance for signal processing. Fourier is an engineer who found that any signal can be actually obtained through the sum of simple harmonic. That isFourier Theorem(Fourier Inversion Theorem):

Any signal can be obtained by adding a simplified harmonic.

Therefore, complex signals canDecompositionIt becomes a simple harmonic. A signal is obtained by adding the harmonic of multiple frequencies. A simple harmonic that constitutes a signal. It is calledComponent(Component ).

 

For example, it shows how we use the superposition of Simple Harmonic to constantly approach the blue signal:

 

From Wikipedia

 

Fourier transformIs a set of fixed computing methods used to calculate each component of the signal (that is, the preceding an, BN). During signal processing, the signal can be transformed into a combination of simple harmonic. By separately controlling the harmonic components at various frequencies, we can perform signal processing more effectively. For example, we can use high-frequency harmonic signals during communication. However, the antenna receiving the signal may receive interference signals at other frequencies. At this time, we can integrate the received hybrid signal for Fourier transformation, and only extract the high-frequency components of the target. This is a common method to reduce signal noise. The process of Fourier transformation is somewhat complicated, but there are already a large numberProgramIt can help you. All you need isInput SignalThe computer will help you calculate its components.

 

 

For example, if the signal f (x) is periodic, we can convert it:

Parameters:

They represent the strength (and phase) of the harmonic components of signals at various frequencies ). By bringing the f (x) signal into the above formula, we can calculate the harmonic at each frequency.

(The complete discussion of Fourier transform involves the complex space. The above shows a more intuitive series form)

 

Spectrum (frequency spectrum)

Through Fourier transformation, we can obtain the strength (amplitude) of the component of a signal f (t) At each frequency ). The spectrum is obtained by drawing such information. For example, the spectrum obtained from the daily precipitation sequence:

There is an obvious peak of the harmonic component in the cycle of one year, that is, the component has a relatively large amplitude. This is understandable, because precipitation always changes regularly throughout the year. PassSignal> Fourier transform> SpectrumWe can understand complex signals from the perspective of simplified harmonic motion.

 

Image Processing)

Fourier TheoremMultidimensional SignalAlso true. We can also use the multi-dimensional Fourier transformation to obtain the multi-dimensional signal spectrum:

 

 

Lenna and her Spectrum

The left side is a two-dimensional signal (image, f (x, y )). We use black and white to indicate the signal strength. The right side is the spectrum of a two-dimensional image. The X axis represents the frequency in the X direction, the Y axis represents the frequency in the Y direction, and the black and white represent the amplitude strength of different frequency components. In the following line, lenna is intentionally added with noise, and the spectrum changes accordingly.
The center of the spectrum represents the amplitude of the low frequency signal, and the far from the center represents the amplitude of the high frequency signal. As you can see, if noise is added, the high-frequency signal (non-center part) is significantly enhanced. That is to say, noise is concentrated in the high-frequency section in the spectrum. If you can filter high-frequency signals, it can suppress noise.

(If you know something about image processing, you will surely know the name of lenna. She is a Playboy Girl, but also a goddess in the image processing field. You can search "lenna full image" to find the full image. Lenna is now an old lady. She "witnessed" the development of image processing .)

 

Summary

Signals can be decomposed into simple harmonic waves of different frequencies, which helps us better understand complex signals. Fourier transform is a basic tool for Signal Processing (and image processing. Through Fourier transformation, we can obtain the signal spectrum.

The spectrum provides us with another perspective of understanding the signal. The original signal is based on the time (or space) domain, while the spectrum is based on the frequency domain. Sometimes it is difficult to analyze or mix signals (such as the original signal and noise) in the time and space fields, and then split the spectrum to better understand and process the signals.