Fourier analysis Basis (i)--fourier series

Objective

The role of Fourier analysis is to transform a function into a form of triangular functions, i.e. decomposition. First, the Fourier series is introduced, the function of the Fourier series is to change the functions into the sum of infinite trigonometric functions, and the frequency of these trigonometric functions is an integer multiple of a certain base frequency. If the base frequency is approaching 0 infinitely, then the parameter (frequency) of the function is continuous in the case of limit, and the transformation of continuous time domain function to continuous frequency domain function is the standard Fourier transform.

Because most of the signals in the project are discrete, the time domain discretization is not possible after the continuous frequency domain function, so in the frequency domain is also not continuous, this discrete time domain sequence to the discrete frequency domain sequence of the transformation called the discrete Fourier transform (DFT), and then someone opened a fast calculation of fast Fourier transform (FFT).

Each of the Fourier transformations described above has its inverse transformation. Fourier series

Consider the assumption that there are 2 of sequences and that there is a number.

Now there is a function that changes on the time domain. This function can be expressed in the following form:

This is the most basic form of Fourier series, Fourier transform. It may not be intuitive to have a neutral form, and it might be intuitive to draw a portion of it.

, is the 1rad/s~10rad/s of the sine signal from the circle frequency.

Fourier series expansion in space

Considering the Fourier series, we first consider the expansion of the periodic function, but before the expansion, several calculations and proofs are needed.

Calculation (not calculated, this is the same).

The integration process is slightly, obtained.

So we can find the norm on the

Then the orthogonal base may use the composition.

However, I do not know whether this set is orthogonal to the base, need to prove Ah!

The following points prove that the following integral relationship is established:

Among them, it is easy to get: The correctness of 2,4,5, the key lies in the proof of 1,3,6.

The above six equations are proved to be equivalent to a set of orthogonal bases of space.

First we all build:

Spicy? Two add a bit to get:

You get it by reducing it:

, the above two points on the goods are 0 can be seen with the naked eye.

I can't see the ... Oh

The final proof of the 1-type, first hypothesis, it is easy to prove that, then, the function is an odd function, so the relative origin of the symmetric interval on the integral is 0.

Finish the card.

Now we've got a set of orthogonal bases:

The following will use it to decompose the function, that is, to calculate these coefficients.

This is an averaging operation that should be well understood.

The above two formulas should also not be difficult to understand, but there are two details, the first, why the coefficients are used in the same way as the DC (or average), rather than the use, because this is essentially the case with the time when the base does the inner product.

Secondly, why the preceding coefficients are, in fact, to match the form:

Our trigonometric function bases are all bands, and if need not, then the coefficients are multiplied and the coefficients are simply:

The expression must be written as:

If we let the new coefficients become coefficients, then we can get the concise form of the direct multiplication of sin and cos, this time the integral becomes:

The two multiply and you get it. Fourier series expansion in space

What happens if we pan the area above that width?

A: Yarn will not happen.

So what do we need to change?

A: Change the upper and lower limit of the points can be. For example, the original expression needs to be changed to:

The rest and so on are the same, without any change.

The reason for this is because the right and wrong of the six equations that derive the orthogonal basis are not changed by the addition of this change, and the lemma naturally does not change. Fourier series expansion in space

The change seems to be asking: what if the interval stretches?

The definition field (interval) of the function is stretched, so we stretch the base together to be OK.

So, here's the question: How do you stretch it?

Very simple, the first straight stretch:

Then make sure the norm is 1.

The calculation of the norm is almost the same as the calculation of the above norm, so I would like to briefly say the result here:

Space to meet:

The formula for the parameter is as follows:

The final free form of the Fourier series: progression in Space

With the above two deduction, it is now possible to formally draw a more general expansion:

At this time, the length of the interval is equal to just, then the transformation from the parameter to the function is shown above.

This time the parameter is calculated as follows:

So it's over, and we've got a way to expand the function in any finite interval, so we can develop something more exotic, such as piecewise fitting a function using a Fourier base.

Trick:

When the function of human Fourier series expansion, if the interval about the origin of symmetry, the first observation of the form of the function, if it is an odd function, then the part of the COS (including DC components) can not look: there must be no ~

If it is even function, then the part of sin will not be seen. Boulevard to Jane: the complex exponential form of the Fourier series

By the well-known Euler formula:

Among them, on many occasions generally use J instead of I, but in fact generally defined, but because of the results of the two conjugate, so the above relationship is generally not used, but in order to make some explanations to facilitate the use of J instead of-I.

We may as well set up, then we can get the Fourier of the complex index form:

From three formulas into a formula, sure enough, the body and mind are happy a lot, but why is the transformation effective?

The following proofs take two steps, the first step, to prove why the Euler formula shape is so bizarre but is right. Second, why you can use the plural sin and cos together.

Proof of the Euler formula:

First, the exponential function and the two trigonometric functions are expanded by Taylor:

Then make it possible to push and export the following formula:

It's such a happy proof.

So why would a complex number be used to blend two functions together?

Because on the complex plane, the real and imaginary axes are perpendicular.

Moreover, we have a trigonometric 0, and the derivative is greater than 0 of the point is the phase of 0 points, then the phase of sin is 0, and the phase of the Cos is 90°, that is.

So, if the phase of sin is -90°, then the phase of the Cos is 0°, which is a relative relationship, and there is no absolute. We continue to analyze, in fact, a spiral of three-dimensional space:

In addition to having a good property that can directly represent the phase in the complex plane, or a qualified complex periodic function ~ Then we may as well abandon from the old version of Fourier to the new version of Fourier mapping, reconstruct the entire proof!

Now we create a new canonical orthogonal base: We can prove it is canonical orthogonal.

According to the definition of inner product, but under the plural it is wrong!

The plural below is actually:

In this way, the part of the imaginary number can be eliminated by conjugation, and the obtained norm is true.

Say the exponential function is really a professional against the derivation of the 500-year function, its integration is easier, here does not prove that the direct conclusion: its norm is, so the above-mentioned orthogonal basis is normative.

So I don't know if quadrature is orthogonal (but I'm anxious to get him to standardize the orthogonal base).

In the time can get the function integral is 0, if the time is abnormal, but not difficult to solve, the direct constant of the integral who will, get, is the above conclusion. (To be careful or to prove something you don't want to write)

In fact, this time Fourier series has been reborn, fundamentally changed a set of bases, but also from the original field of real numbers to expand into a more broad complex field, but the form is more concise.

Fourier analysis Basis (i)--fourier series