Fourier transform 2

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Fourier transform: Any continuous periodic signal can be combined by a set of appropriate sine curves

There are a series of problems when we touch Fourier transform, we learn the Fourier transform from these problems slowly and deeply.

question one : Why use a sinusoidal curve instead of the original

A: It is simpler to replace the original signal with a sine-cosine curve, because the sine-cosine curve has the properties that the original signal does not have: a positive cosine curve fidelity, a sine-cosine curve signal input, the output is still the positive cosine curve, but the amplitude and phase may change, but the frequency and wave shape is still the same. And only the sine-cosine curve has such properties.

question two : There are several types of Fourier changes

1, non-cyclical continuous signal This is the Fourier change FT

2, non-cyclical continuous signal Fourier series The proof of the high number of basic knowledge in the upper section FS

3. Discrete time domain Fourier transform DTFT discrete signal with non-periodicity

4. Discrete Fourier variation of periodic discrete signal may be expanded into series form DFT according to Dilichlie theorem

question three : These four kinds of Fourier transform are for positive infinity or negative infinity signal, that is, the length of the signal is infinite. We know that this is not possible for computer processing. so is there a finite Fourier variation for the length?

A: No, because the cosine signal is defined as from negative infinity to positive infinity, we cannot synthesize a signal with a finite length of signal.

Methods: The signal of limited length is expressed as an infinite signal, that is, the limited signal is infinitely extended from left to right, and the extended part is expressed in the form of zero, so that the signal can be considered as a non-periodic discrete signal can be converted into a third class for research and processing. Because of the continuous situation of computer science is not considered.

question four : is the Fourier transform the same as the function transformation

A: Different, although all are mathematical transformation, but with function transformation, function transformation conforms to the principle of one-to-one mapping. The Fourier transform described here is a mathematical transformation tool that combines abstract irregular signals with a signal form known as a familiar comparative law. is to expose an abstract irregular signal to another form of processing so it can be more than just a form, not a one by one mapping. It's simply a way to convert a bunch of data into another heap of data.

question five : real Fourier transform and imaginary Fourier transform how to differentiate between individual transformation forms

A: First of all, each Fourier transform is divided into methods: real Fourier transform and imaginary Fourier transform. But the Fourier transform of imaginary numbers is more complicated than the Fourier transform of real number. We first do the Fourier transform of real numbers to learn and understand. In this case, we should not dwell on the relationship of imaginary real Fourier transform and the respective realization form.

Here's an example of a real-number discrete Fourier transform look for the feeling:

is the original signal:

The length of the signal is 16, so the signal can be decomposed into 9 cosine signals and 9 sinusoidal signals, why? (A signal of length n can be decomposed into a n/2 sine cosine signal) look at the decomposition graph below

Cosine signal:

Sinusoidal signal:

How to change it: The simple is to put 9 positive cosine curve in the computer by a fixed arrangement of the combination of the form. Just need to understand that he is a combination just fine.

Fourier transform 2