Fourier transform: the most detailed analysis

Fourier transform is one of the basic concepts in signal processing and data analysis. But what does it mean? Geometric interpretation.

Take the classic task – working with sound. Now add specifics.

Your friend brings a recording of his live performance. And this is a very good performance. But! Although the recording was done on a good microphone, there is still noise in it. A friend asks for help to remove it or at least reduce it.

This is where the knowledge of the Fourier transform comes in handy.

What is sound in a mathematical sense?

A single note is a harmonic signal with a specific frequency and amplitude.

As a rule, a melody, speech or other sound signal can be represented as a sum of harmonic signals. In this case, noise is the term that corresponds to any unwanted sound.

The Fourier transform allows you to decompose the original signal into harmonic components, which is required for the selection of noise.

We write the definition:

Here g (t) is the original signal (in our case, a friend’s record). In the context of the Fourier transform, it is called the original. G (f) is the Fourier image, and the parameter f is the frequency.

You may already know this definition. But do you know how this transformation happens? If you saw it for the first time, would you understand how to analyze the original signal with it?

Geometric interpretation of the Fourier transform

Grant Sanderson offers a geometric analogue of  the Fourier transform. For several graphic transitions from the original signal to the image, each of the components of the definition acquires meaning, and the transformation itself receives a new geometric reading.

In the following discussion it is assumed that you are familiar with vectors, integration, and the concept of a complex number . If you still do not have enough knowledge, check out the materials from our selection on university mathematics .

1. reel signal

Let’s start with the simplest case. Consider a harmonic signal , making 3 oscillations per second (f 0 = 3s -1 ) :

g (t) = 1 + cos (6πt).

Map g (t) onto the complex plane. To do this, we introduce a radius vector that rotates evenly clockwise. Its length at each time instant is equal to the magnitude of the signal value, and the rotation frequency is chosen arbitrarily.

Now we will build the trajectory of the end of the vector making a complete revolution in two seconds, or, in other words, with the rotation frequency V = 0.5 rev / s .

It looks like we wound the original signal to the origin. At the signal minima, the resulting “winding” merges with the origin, and when approaching the maxima it deviates.

While not very informative, is it?

And now we increase the frequency of winding.

At first, the graph is distributed quite symmetrically relative to the origin of coordinates up to the rotation frequency B = 3 rev / s . Then the maxima dramatically shift to the right half-plane, and the winding ceases to resemble the spirograph pattern.

2. We are looking for the center of mass

Let’s take a closer look at what is happening. As a characteristic of the winding, we take the averaged value of all its points – the center of mass (note it in orange).

We build the dependence of the position of the center of mass on the winding frequency. Now it is enough for us to consider the x-coordinate, but in the future to determine the Fourier transform we need both coordinates.

We see two peaks: at points B = 0 rev / s and B = 3 rev / s . Based on this behavior of the center of mass, it is already possible to judge the frequency of the original signal (it varies with f = 3s -1 ).

Then what does splash at low frequencies mean?

3. Analyzing the effect of bias.

You may have noticed that the signal we are considering is shifted by one. The shift was introduced for clarity, but it was he who leads to the complication of the center of mass behavior.

At zero frequency, the entire display of the signal on the complex plane is located on the x-axis. At low frequencies, the winding is still grouped in the right half-plane.

As soon as we remove the shift, that is, we take a signal of the form g (t) = cos (6πt) , the winding at low frequencies is shifted to the left along the x -axis.

The construction of the radius vector remains the same. Its length is equal to the magnitude of the signal value, the direction of rotation is positive. But when the sign of g (t) is changed, the direction of the vector is reversed.

Now you will see how the winding and x-coordinate of the center of mass of the unbiased signal change.

Thus, there is only one sharp jump on the chart.

This is an important point when using the Fourier transform: linear trend and displacement occur at low frequencies, because they are excluded from the original signal.

4. Select the frequency of the polyharmonic signal

Now consider the sum of two harmonic signals with an oscillation frequency 1 = 2 s -1 and 2 = 3 s -1. Let’s do the same operations with it – we “wind it up” near the origin of coordinates, and, changing the rotation frequency, we will plot the x-coordinates of the center of mass.

We observe two peaks at points B = 2 rev / s and B = 3 rev / s , which corresponds to the frequency composition of the initial sum.

We note another interesting fact that is true for both the x-coordinate and the Fourier transform. The conversion for the sum of the signals and the sum of the signal conversions are the same. That is, the Fourier transform is linear.

Thus, this approach allows determining the frequency of oscillations of both mono- and polyharmonic signals. It remains to describe mathematically the procedure for calculating the center of mass of the winding.

Fourier transform output

At the very beginning of the review, we mapped the source signal to the complex plane. This choice is not accidental – it allows you to consider the points on the plane as complex numbers and use the Euler formula to describe the winding:

 = cos (φ) + i · sin (φ).

Geometrically, this relation means that for any φ the point  on the complex plane lies on the unit circle.

Construct the radius vector  for different values ​​of φ .

When φ changes by 2π, the vector goes full counterclockwise, since  is the length of the unit circle. To set the rotational speed of the vector, the exponent is multiplied by ft , and to change the direction of rotation – by -1 .

Then the winding of the signal g (t) is described as g (t) e -2π ift .

Now we calculate the center of mass. To do this, we note N arbitrary points on the winding graph and calculate the average:

Center of mass estimate

If we increase the number of points in question, we come to the limiting case:

Center of mass

where 1 and 2 are the limits of the interval on which the signal is considered.

The expression in front of the integral is a scaling factor, but does not reflect the behavior of the center of mass. Because it can be discarded.

The resulting expression will be a Fourier transform with the difference that, in general, the integration is given in the interval from -∞ to + ∞ .

Such a transition to an infinite interval means that we do not impose any restrictions on the duration of the signal under consideration.

Applying Fourier Transform to Filter

Now, speaking of the Fourier transform, you can represent its geometric interpretation – winding the signal onto the complex plane and calculating the center of mass.

In this case, the winding frequency f becomes the input parameter for the Fourier image. The center of mass is an estimate of how well the parameter f is correlated (correlated) with the frequencies present in the signal.

After you find all the frequency components in the brought record, you will only have to subtract them from the image and apply the inverse Fourier transform.

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