How does the Fourier transform apply to a real world physical signal?

The signal is the closest, that is, from the perspective of the filter, for example, the Fourier transform of the signal x with a frequency range of 0-k and a length of 1024 points can be understood as using a 1024-segment filter bank to filter the signal, and then because the frequency range of each segment is only k 1024, the separated signal of each segment can be resampled with 1 1024 times the sampling rate of the original signal x (if you learn the filter group, it must be easy to understand) Then each output signal can be resampled with the original sampling rate of 1 1024, because the total length of the original signal is 1024, exactly one sampling point for each segment of the decomposition frequency. From this point of view, the amplitude spectrum of the Fourier transform is actually the amplitude of each segment of the signal after being filtered by a 1024-segment filter, which shows that the physical meaning of the Fourier transform is the output of a bunch of filters. From this point of view, it is also possible to explain the short-time Fourier transform of various types of windowing--- i.e., each segment is the output of the filter bank.

The short-time Fourier transform for each of the different window functions passes through the output of a different set of filters. <>

First, let's look at it from the perspective of the characteristic signals of the physical system. We know that many phenomena in nature can be abstracted into a linear time-invariant system, whether you describe it in differential equations, transfer functions, or state spaces.

A linear time-invariant system can be understood in such a way that the input and output signals satisfy a linear relationship, and the system parameters do not change with time. For many systems in nature, after a sinusoidal signal is input, the output is still sinusoidal, only the amplitude and phase may change, but the frequency and wave shape are still the same.

That is, the sinusoidal signal is the eigenvector of the system! Of course, exponential signals are also eigenvectors of the system, representing the decay or accumulation of energy. Most of the attenuation or diffusion phenomena in nature are exponential, or have both fluctuation and exponential decay (complex exponential form), so the characteristic basis function changes from a trigonometric function to a complex exponential function.

However, if the input is a square wave, a triangle wave, or some other waveform, the output may not necessarily look like it. Therefore, waveforms other than exponential and sinusoidal signals are not characteristic signals. What do you mean by the names eigenvectors and eigensignals?

In fact, this is ** in linear algebra: we know that matrix a acts on an eigenvector x can be described in mathematical language in this way: then the system acts on a eigensignal in mathematical language.

The formal structure is the same, except that it is a vector of finite length and the other is a signal of infinite length. Since it is an eigenvector, we wonder if we can use eigenvectors to represent signals and a physical system in nature. The advantage of this is that we know the input, and we can easily write the output.

Let's look at a practical example, the percussion instrument – the piano. When the keys are struck with a small hammer, a sound is produced. <>

In the actual circuit, the signal is really Fourier transform, some circuits decompose (Fourier positive transform circuit), some circuits achieve superposition (Fourier inverse transform circuit), and only the interaction of these two circuits can realize the function of the filter. There are a few problems. However, this is because there is no actual contact with the hardware system, stupidly think about it, the purpose of the Fourier transform is to remove clutter and noise, not a simple transformation (this is only meaningful to mathematics, not to engineering), since it is for filtering, then it is necessary to perform a Fourier positive and negative transformation of the input signal, (why do you do two transformations?).

The purpose of the forward transformation is to break up unwanted clutter and noise and then filter them out, and the purpose of the reverse transformation is to restore the remaining useful signal back to the next level of circuitry).

The Fourier transform isDigital Signal ProcessingA very important algorithm in the fieldTo understand the significance of the Fourier transform algorithm, we must first understand the significance of the Fourier principle.

The Fourier principle shows that any continuously measured time series or letter number can be expressed as a sine wave of different frequencies.

Infinite stacking of signals. The Fourier transform algorithm based on this principle uses the directly measured raw signal to calculate the frequency and amplitude of different sine wave signals in the signal in an additive way.

and phase. <>

The Fourier transform is proposed:

with a sinusoidal curve.

to replace the original curve instead of a square wave.

The reason for this is that there are infinite ways to decompose a signal, but the purpose of decomposing a signal is to process the original signal more simply. With sine and cosine.

It would be simpler to represent the original signal, because the sine and Cos have a property that the original signal does not have: sinusoidal fidelity.

After a sinusoidal signal is input, the output is still sinusoidal, only the amplitude and phase may change, but the frequency and wave shape are still the same. And only sinusoidal curves have such properties, which is why we don't use square or triangular waves to represent them.

Fourier transform is an important algorithm in the field of digital signal processing.

To understand the significance of the Fourier transform algorithm, we must first understand the significance of the Fourier principle. The Fourier principle states that any continuously measured timing or signal can be expressed as an infinite superposition of sine wave signals of different frequencies.

The Fourier transform algorithm based on this principle uses the directly measured raw signal to calculate the frequency, amplitude and phase of different sine wave signals in the signal in an additive manner.

The counterpart to the Fourier transform algorithm is the inverse Fourier transform algorithm. This inverse transformation is also essentially an additive process, which converts a separately changed sine wave signal into a single signal.

The role of the Fourier transform

The purpose of the Fourier transform is to transform a non-sine and cosine (note that it must be a periodic function) function into an infinite number of regular sine and cosine functions. After becoming a regular function, although there are infinite terms, it is enough to take the first few items of precision in the project. Regular functions are good for calculation.

Transform functions that are difficult or impossible to calculate into functions that can be calculated.

For example, the function of the approximate rectangle at the front is composed of infinite terms of the color at the back. It is to use the Fourier function to decompose into an infinite number of regular sine and cosine functions at the end.

Fourier series of periodic signals.

is the amplitude of the signal at each discrete frequency component.

The Fourier transform of a non-periodic signal can be understood as periodic infinity.

The Fourier series of the periodic signal. At this point, the discrete frequencies gradually become continuous frequencies, and the spectral density value at a certain point frequency is meaningless, like a probability density function.

It only makes sense to find the area value formed by the spectral density function in a short frequency near that point, which represents the amplitude of the signal at that frequency point.

Derivation: f m = n = 0 n 1 f n e 2 i m n n f n = 1 n m = 0 n 1 f m e 2 i m n n f m = sum f ne leftrightarrow f n= frac sum f me fm=n=0 n 1fne 2 imn n fn=n1m=0 n 1fme2 imn n

f(jw)=[w-w0)-πw+w0)]/j。

Find the Fourier transform of f(x)=sinw0t.

w0 in order to distinguish it from w).

According to Euler's formula.

get sinw0t=(e jw0t-e (-jw0t) (2j).

Because the Fourier transform of DC signal 1 is 2 δ(w).

And E jw0t is the frequency shift of the Fourier transform of the DC signal.

So the Fourier transform of e jw0t is 2 δ (w-w0), and the Fourier transform of e (-jw0) is 2 δ (w+w0).

So f(jw)=[w-w0)- w+w0)] j。

Fourier transform:

Fourier transform or transformée de fourier has multiple Chinese translations, the common ones are "Fourier transform" and "Fu Liye".

Transformation", "Fourier Transformation", "Fourier Transformation", "Fourier Transformation", "Fourier Transformation", etc.

Fourier transform.

It is a method of analyzing a signal that analyzes the components of the signal and can also synthesize the signal from those components. Many waveforms can be used as components of a signal, such as a sine wave.

Square waves, sawtooth waves, etc., the Fourier transform uses a sine wave as a component of the signal.

f(jw)=[w-w0)-πw+w0)]/j。

Find the Fourier transform of f(x)=sinw0t (w0 to distinguish from w).

According to Euler's formula, sinw0t=(e jw0t-e (-jw0t) (2j).

Because the Fourier of the DC signal 1 is changed to 2 δ (w).

And E jw0t is the frequency shift of the Fourier transform of the DC signal.

So the Fourier transform of e jw0t is 2 δ (w-w0), and the Fourier transform of e (-jw0) is 2 δ (w+w0).

So f(jw)=[w-w0)- w+w0)] j。 Carry liquids.

Fourier transform:

Fourier transform or transformée de fourier has multiple Chinese translations, the common ones are "Fourier transform", "Fu Liye transform", "Fourier transform", "Fourier transform", "Fourier transform", "Fourier transform", and so on.

The Fourier transform is a method of analyzing a signal by analyzing the components of a signal and synthesizing a signal from those components. Many waveforms can be used as components of signals, such as sine waves, square waves, sawtooth waves, etc., and the Fourier transform uses sine waves as components of signals.

1. Gate function f(w)=2w w sin=sa() w.

2. The exponential function (unilateral) f(t)=e-atu(t) f(w)=1, which is actually a low-pass filter a+jw.

3. The unit impulse function f(w)=1, the frequency band is infinitely wide, and it is a uniform spectrum.

4. Constant 1 Constant 1 is a DC signal, so of course its spectrum only has a value when w=0, which is reflected as (w). f(w)=2(w) can be obtained from the symmetry of the Fourier transform.

5. The sinusoidal function f(ejw0t)=2(w-w0), which is equivalent to the displacement of the DC signal. f(sinw0t)=f((ejw0t-e-jw0t)/2)=(w-w0)-(w+w0))f(sinw0t)=f((e。

6. Unit impact sequence jw0t-e-jw0t) 2j)=j((w-w0)-(w+w0)) t(t)=(t-tn) - This is a periodic function, every t there is an impact, the Fourier transform of the periodic function is discrete f(t(t))=w0(w-nw0)=w0, w0(w) n=- The Fourier transform of the unit impact sequence is still a periodic series, and the period is w0=2t.

Fourier transform.

A Fourier transform is an integral that expresses a function that satisfies certain conditions as a trigonometric function. The Fourier transform is produced in the study of Fourier series. In different fields of study, the Fourier transform has different effects.

When analyzing the signal, the frequency, amplitude, and phase are mainly considered.

The function of the Fourier transform is mainly to transform the function into the form of multiple sinusoidal combinations (or e-exponents), in essence, the signal is still the original signal after the transformation, but it is just a different way of expression, so that the frequency, amplitude, and phase components of a function can be analyzed more intuitively.

Therefore, the frequency, phase, and amplitude components of a complex signal can be easily determined by simply going through the Fourier transform.

According to Euler's formula, cos 0t = [exp(j 0t)+exp(-j 0t)] 2.

The Fourier transform of a DC signal is 2 δ (

According to the property of the frequency shift, the Fourier transform of exp(j 0t) is 2 δ(0).

According to the linear nature, it can be obtained.

The Fourier transform of cos 0t=[exp(j 0t)+exp(-j 0t)] 2 is δ(0)+0). Contained.