Classical Algorithm Research Series: 10. A thorough understanding of Fourier transform algorithms from start to end

Author: July, dznlong February 20, 2011

Recommended reading:The Scientist and Engineers Guide to Digital Signal Processing, By Steven W. Smith, Ph. D.Book address:Http://www.dspguide.com/pdfbook.htm.

Author's note: I. The Discrete Fourier transformation method described in this article is based on this book: The Scientist and Engineers Guide to Digital Signal Processing, By Steven W. smith, Ph. d. translated from:Http://www.dspguide.com/pdfbook.htm. II. At the same time, a considerable part of the content was edited from dznlong's blog and posted its blog address to pay tribute to the original author:Http://blog.csdn.net/dznlong. In these years, there are very few people who really calm down and write original articles.
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A thorough understanding of the Fourier transform algorithm
Preface
Part 1, DFT
Chapter 1: Evolution of Fourier Transformation
Chapter 2 Real number form discrete Fourier transform (Real DFT)

Chapter 3, plural
Chapter 4. complex form discrete Fourier Transformation

 

Preface:
"The description of Fourier transformation can be easily found in books or on the Internet, but most of them are original and abstract articles, A list of seemingly daunting formulas makes it hard for people to understand their senses. "--- dznlong,

So what is the Fourier Transform Algorithm column? What are the complex columns involved in Fourier transformation?
Fourier transform(Fourier transform) is a linear integral transformation. Because his basic idea was first proposed by French scholar Fourier systematically, his name was used as a souvenir.

Oh, Fourier transformation is just a transformation, but this transformation is a change from time to frequency. Now, you know, Fourier is a transformation, and what is a transformation column? It is a change or mutual conversion from time to frequency.

OK. Let's take a general look at Fourier transformation, which gives you a general impression. Let's also look at the formula involved in Fourier transformation. How complicated is it:
The following are four variants of Fourier transform (from Wikipedia)
Continuous Fourier Transformation
Generally, if the word "Fourier transformation" does not contain any limitation, it refers to "continuous Fourier transformation ". The continuous Fourier transformation expresses the square product function f (t) as the integral or series form of the complex exponent function.

This is the integral form of the F (t) function of the frequency field.

Inverse Fourier transform is:

That is, the function f (t) in the time field is expressed as the credits of the function F (ω) in the frequency field.

Generally referred to as a functionF (t)It is called the original function.F (ω)This is a fourier transform function. The original function and the image function form a fourier transform pair (transform pair ).

In addition, there are other types of conversion pairs, the following two types are also often used. In terms of communication or signal processing, it is often used for replacement to form a new transformation pair:

Or a new transformation pair is obtained due to the redistribution of coefficients:

The extension of continuous Fourier transformation is Fractional Fourier Transform ). Fractional Fourier transform (FRFT) refers to the generalization of Fourier transform (FT.
The physical meaning of Fractional Fourier transformation is to perform Fourier transformation a times, where a does not have to be an integer. After the fractional Fourier transformation, the signal or input function appears in the fractional domain between the time domain and the frequency domain ).

When f (t) is an even function (or an odd function), its sine (or cosine) component will die, but it can be called the transformation to the cosine transformation (cosine transform) or sine transform ).

Another noteworthy property is that when f (t) is a pure real function, F (−ω) = F * (ω) is true.

Fourier Series
The continuous form of Fourier transformation is actually a promotion of Fourier series, because the integral is actually a limit form of summation operator. For cyclic functions, the Fourier series exists:

Fn indicates the amplitude. For a real-value function, the Fourier series of the function can be written as follows:

Where an and bn are the amplitude of the real frequency component.

Discrete Time Domain Fourier Transformation
Discrete Fourier transformation is a special case of Discrete Time Fourier Transformation (DTFT) (sometimes used as the approximation of the latter ). DTFT is discrete in the time domain and periodic in the frequency domain. DTFT can be considered as the inverse transformation of Fourier series.

Discrete Fourier Transformation
Discrete Fourier Transform (DFT) is a form of continuous Fourier transform that is discrete in both the time and frequency domains. It transforms the sampling of time domain signals into sampling in the Discrete Time Fourier transform (DTFT) frequency domain. In form, the sequences at both ends of the Transformation (in the time domain and in the frequency domain) are finite lengths. In fact, these two sequences should be considered as the primary value sequences of discrete periodic signals. Even if a finite-length discrete signal is used as DFT, it should be considered as a periodic signal after periodic extension and then transformed. In practical applications, Fast Fourier transformation is usually used to calculate DFT efficiently.

In order to use computers for Fourier transformation in scientific computation and digital signal processing, the xn function must be defined in a discrete point rather than a continuous domain and must meet finite or periodic conditions. In this case, discrete Fourier transform (DFT) is used to represent the xn function as the following summation form:

Xk is the Fourier amplitude. The computation complexity calculated using this formula is O (n * n), while the Fast Fourier Transform (FFT) can improve the complexity to O (n * lgn ). (We will explain in detail how FFT reduces complexity to O (n * lgn .) The reduction of computing complexity and the development of digital circuit computing capabilities make DFT a very practical and important method in the field of signal processing.

Next, compare the four variants of the Fourier transform,

As shown above, it is easy to find that the discretization of a function in the time (frequency) Field corresponds to the periodicity of its image function in the frequency (frequency) field. Otherwise, the signal in the corresponding domain is non-cyclical. That is to say, the discretization in time corresponds to the periodicity in frequency. At the same time, note that the Discrete Time Fourier transformation, time discretization, and frequency are not discrete, and it is still continuous in the frequency domain.
If you read this, you don't understand it. It doesn't matter. You don't have to worry about the above four variants. Continue to look at it and you will be very open. (If you have any questions, please submit them or criticize and correct them)

OK, this article, next, starts with Fourier transformation, and then focuses on Discrete Fourier transformation and Fast Fourier algorithm, and finally implements the FFT algorithm completely. The whole article strives to be easy to understand and read smoothly, teaches you to thoroughly understand the Fourier Transform Algorithm from start to end.Because of the Fourier transformation, also known as the Fourier transformation, which is referred to below as the Fourier transformation, the same transformation, different names, readers do not have to feel strange.

Style = "FONT-SIZE: large"> Part 1, DFT
Chapter 1: Evolution of Fourier Transformation
To understand Fourier transformation, we must first know how Fourier transformation is transformed. Of course, we also need a certain foundation for advanced mathematics. The most basic thing is series transformation, fourier series transformation is the basic formula of Fourier transformation.
 
I. Proposal of Fourier Transformation

Fourier is a French mathematician and physicist, formerly Jean Baptiste Joseph Fourier (1768-1830). In 1807, Fourier published a paper at the French Scientific Conference, the paper describes the use of sine curves to describe the temperature distribution. There was a controversial decision in the paper: any continuous cycle signal can be combined by a set of appropriate sine curves.

At that time, he reviewed the paper, and he resolutely opposed the publication of the paper. Then, in the last 50 years, he insisted that the Fourier method could not represent a signal with edges and corners, for example, the slope of a non-continuous change occurs in a square wave. It was not until 15 years after his death that the paper was published.
Who is right? Returns the right result from the use of the Laplace curve. The Sine Curve cannot be combined into a signal with edges and corners. However, we can use a sine curve to represent it very closely. There is no energy difference between the two Representation Methods. Based on this, Fourier is correct.

Why should we replace the original curve with a sine curve? For example, we can also use square waves or triangular waves instead. The signal decomposition method is infinite, but the signal decomposition aims to process the original signal more simply.
It is easier to use the positive cosine to represent the original signal, because the positive cosine has the properties that the original signal does not have: sine curve fidelity. After a positive cosine curve signal is input, the output is still a positive cosine curve. Only the amplitude and phase may change, but the frequency and wave shape are still the same. And only the positive cosine curve has this property, so we don't need a square or triangular wave to represent it.

Ii. Fourier transform Classification