Fourier series representation of periodic signals. Examples of Fourier series expansion

General Descriptions

The French mathematician Fourier (J. B. J. Fourier 1768-1830) proposed a hypothesis that was quite bold for its time. According to this hypothesis, there is no function that cannot be expanded into a trigonometric series. However, unfortunately, such an idea was not taken seriously at that time. And this is natural. Fourier himself could not provide convincing evidence, and it is very difficult to intuitively believe in Fourier’s hypothesis. It is especially difficult to imagine the fact that when adding simple functions, similar to trigonometric ones, functions that are completely different from them are reproduced. But if we assume that the Fourier hypothesis is correct, then a periodic signal of any shape can be decomposed into sinusoids of different frequencies, or vice versa, by means of the appropriate addition of sinusoids with different frequencies it is possible to synthesize a signal of any shape. Therefore, if this theory is correct, then its role in signal processing can be very large. In this chapter, we will first try to illustrate the correctness of the Fourier hypothesis.

Consider the function

f(t)= 2sin t – sin 2t

Simple trigonometric series

The function is the sum trigonometric functions, in other words, is presented as a trigonometric series of two terms. Let's add one term and create new row of three members

By adding a few terms again, we get a new trigonometric series of ten terms:

We denote the coefficients of this trigonometric series as b k , where k - whole numbers. If you look closely at the last ratio, you will see that the coefficients can be described by the following expression:

Then the function f(t) can be represented as follows:

Odds b k - these are the amplitudes of sinusoids with angular frequency To. In other words, they set the magnitude of the frequency components.

Considering the case where the superscript To equals 10, i.e. M= 10. By increasing the value M up to 100, we get the function f(t).

This function, being a trigonometric series, is close in shape to a sawtooth signal. And it seems that Fourier's hypothesis is completely correct in relation to physical signals with which we are dealing. In addition, in this example the waveform is not smooth, but includes break points. And the fact that the function is reproduced even at breakpoints looks promising.

There are indeed many phenomena in the physical world that can be represented as sums of oscillations of various frequencies. A typical example of these phenomena is light. It is the sum of electromagnetic waves with a wavelength from 8000 to 4000 angstroms (from red to violet). Of course you know that if white light pass through a prism, a spectrum of seven pure colors appears. This occurs because the refractive index of the glass from which the prism is made changes depending on the length of the electromagnetic wave. This is precisely proof that white light is the sum of light waves of different lengths. So, by passing light through a prism and obtaining its spectrum, we can analyze the properties of light by examining color combinations. Likewise, by decomposing the received signal into its various frequency components, we can find out how the original signal originated, what path it followed, or, finally, what external influence it was subjected to. In short, we can obtain information to find out the origin of the signal.

This method of analysis is called spectral analysis or Fourier analysis.

Consider the following system of orthonormal functions:

Function f(t) can be expanded over this system of functions on the interval [-π, π] as follows:

Coefficients α k,β k, as shown earlier, can be expressed through scalar products:

IN general view function f(t) can be represented as follows:

Coefficients α 0 , α k,β k is called Fourier coefficients, and such a representation of a function is called expansion in a Fourier series. Sometimes this representation is called valid expansion in a Fourier series, and the coefficients are real Fourier coefficients. The term “real” is introduced to distinguish the presented expansion from the Fourier series expansion in complex form.

As mentioned earlier, an arbitrary function can be expanded into a system of orthogonal functions, even if the functions from this system are not represented as a trigonometric series. Usually, by Fourier series expansion we mean expansion into a trigonometric series. If the Fourier coefficients are expressed in terms of α 0 , α k,β k we get:

Since at k = 0 cost= 1, then the constant a 0 /2 expresses the general form of the coefficient and k at k= 0.

In relation (5.1), the oscillation of the longest period, represented by the sum cos t and sin t is called the fundamental frequency oscillation or first harmonic. An oscillation with a period equal to half the main period is called the second harmonic. An oscillation with a period equal to 1/3 of the main period is called third harmonic etc. As can be seen from relation (5.1) a 0 is a constant value expressing the average value of the function f(t). If the function f(t) represents electrical signal, That a 0 represents its constant component. Consequently, all other Fourier coefficients express its variable components.

In Fig. Figure 5.2 shows the signal and its expansion into a Fourier series: into a constant component and harmonics of various frequencies. In the time domain, where time is the variable, the signal is expressed by the function f(t), and in the frequency domain, where the variable is frequency, the signal is represented by Fourier coefficients (a k, b k).

The first harmonic is a periodic function with a period 2 π. Other harmonics also have a period that is a multiple of 2 π . Based on this, when generating a signal from the components of the Fourier series, we will naturally obtain a periodic function with a period 2 π. And if this is so, then the Fourier series expansion is, strictly speaking, a way of representing periodic functions.

Let us expand a signal of a frequently occurring type into a Fourier series. For example, consider the sawtooth curve mentioned earlier (Figure 5.3). A signal of this shape on a segment - π < t < π i is expressed by the function f( t)= t, so the Fourier coefficients can be expressed as follows:

Example 1.

Fourier series expansion of a sawtooth signal

f(t) = t,

A) Subsequence rectangular pulses .

Figure 2. Sequence of rectangular pulses.

This signal is an even function and it is convenient to use sine-cosine form Fourier series:

. (17)

The duration of the pulses and their repetition period are included in the resulting formula in the form of a ratio, which is called duty cycle of the pulse sequence :.

. (18)

The value of the constant term of the series, taking into account corresponds to:

.

Representation of a sequence of rectangular pulses in the form of a Fourier series has the form:

. (19)

The graph of the function has a lobe pattern. The horizontal axis is graduated in harmonic numbers and frequencies.

Fig 3. Representation of a sequence of rectangular pulses

in the form of a Fourier series.

Petal width, measured in the number of harmonics, is equal to the duty cycle (at , we have , if ). This implies an important property of the spectrum of a sequence of rectangular pulses - in it there are no harmonics with numbers that are multiples of the duty cycle . The frequency distance between adjacent harmonics is equal to the pulse repetition frequency. The width of the lobes, measured in frequency units, is equal to, i.e. is inversely proportional to the duration of the signal. We can conclude: the shorter the pulse, the wider the spectrum .

b) Ramp signal .

Fig 4. Ramp wave.

A sawtooth signal within a period is described by linear function

, . (20)

This signal is an odd function, therefore its Fourier series in sine-cosine form contains only sine components:

The Fourier series of the sawtooth signal has the form:

For the spectra of rectangular and sawtooth signals, it is characteristic that the amplitudes of the harmonics with increasing numbers decrease proportionally .

V) Triangular pulse sequence .

The Fourier series has the form:

Figure 5. Sequence of triangular pulses.

As we can see, in contrast to a sequence of rectangular and sawtooth pulses, for a triangular periodic signal the amplitudes of the harmonics decrease in proportion to the second power of the harmonic numbers. This is due to the fact that the rate of spectrum decay depends on degree of signal smoothness.

Lecture No. 3. Fourier transform.

Properties of the Fourier transform.

1.3 Draw general conclusions.

Part 2

Purpose of the work: deepening the theoretical knowledge gained during the study of the Fourier transform(Fourier Transform)

Necessary theoretical information.

Changing the period T and pulse duration as shown in Fig. 7, you can change the signal spectrum. As the period increases, the harmonics move closer together without changing the shape of the envelope.

Fig. 7 – Spectrum change

Let's simulate a single rectangular pulse, a periodic sequence of pulses with a period T And 10T .

Let's carry out a spectral analysis of the received signals. Non-periodic processes - these are information signals, single pulses, chaotic oscillations(noises) - have solid or continuous spectrum. Intuitively, this conclusion can be reached by representing a single pulse as part of a periodic sequence, the period of which increases indefinitely. Indeed, as the interval between pulses increases, the harmonics in the spectral diagrams of periodic sequences of pulses come closer together: the less frequently the pulses follow, the smaller the distance between neighboring harmonics (it is equal to 1/ T). The spectrum of a single pulse (the limiting case of increasing the period) becomes continuous, and it is introduced not in rows, but Fourier integrals.

Fourier transform(Fourier transform) is a spectral analysis tool non-periodic signals.

The functions described below implement a special Fast Fourier Transform (FFT) method - Fast Fourier Transform (FFT), which allows you to sharply reduce the number of arithmetic operations during the above transformations. The method is especially effective if the number of processed elements (samples) is 2n, where n is a positive integer. IN MatLab The following functions are used:

fft(X) - returns a discrete Fourier transform for the vector X, using the fast Fourier transform algorithm if possible. If X is a matrix, the fft function returns the Fourier transform for each column of the matrix;

fft(X.n)- returns the n-point Fourier transform. If the length of the vector X is less than n, then the missing elements are filled with zeros. If the length of X is greater than n, then the extra elements are removed. When X is a matrix, the lengths of the columns are adjusted similarly;

ft(X,[ Ldirn) and fft(X,n,dim)- apply the Fourier transform to one of the array dimensions depending on the parameter value dim.

A one-dimensional inverse Fourier transform is possible, implemented by the following functions:

ifft(F)- returns the result of the discrete inverse Fourier transform of a vector F . If F - matrix, then ifft returns the inverse Fourier transform for each column of this matrix;

ifft(F.n)- returns the result of the n-point discrete inverse Fourier transform of the vector F ;

ifft(F.,dim) иу = ifft(X,n,dim)- return the result of the inverse discrete Fourier transform of the array F by rows or columns depending on the value of the scalar dim .

For anyone X the result of sequential execution of direct and inverse Fourier transforms ifft(fft(x)) equals X up to rounding error. If X - an array of real numbers, ifft(fft(x)) may have small imaginary parts.

Let us obtain the spectra of the simulated signals.

Let's call the program SPTool (Signal Processing Tool). Let's import the simulated signals and calculate the signal spectrum. To do this, select the signal in the list of signals and press the button Create, located under the list of spectra. In the window Spectrum Viewer in the field Parameters you need to specify the spectral analysis method. We specify the DFT method (the fast Fourier transform FFT is used). Having specified the method, click on the button Apply. The power spectral density graph will be displayed. It is possible to display spectra on a linear or logarithmic scale (menu Options).

The spectrum is continuous (solid) chaotic(noise) fluctuations. In this case, the spectral characteristic, as a function of frequency, is also chaotic(random) process, the statistical parameters of which are determined by the specifics of a particular random time process. Let's generate a signal containing regular components with frequencies of 50 Hz and 120 Hz and a random additive component with zero average.

TASK 2

Examples of Fourier series expansion.

A) Rectangular pulse sequence .

Figure 2. Sequence of rectangular pulses.

This signal is an even function and it is convenient to use sine-cosine form Fourier series:

. (17)

The duration of the pulses and their repetition period are included in the resulting formula in the form of a ratio, which is usually called duty cycle of the pulse sequence :.

. (18)

The value of the constant term of the series, taking into account corresponds to:

.

Representation of a sequence of rectangular pulses in the form of a Fourier series has the form:

. (19)

The graph of the function has a lobe pattern.
Posted on ref.rf
The horizontal axis is graduated in harmonic numbers and frequencies.

Fig 3. Representation of a sequence of rectangular pulses

in the form of a Fourier series.

Petal width, measured in the number of harmonics, is equal to the duty cycle (at , we have , in case ). This implies an important property of the spectrum of a sequence of rectangular pulses - in it there are no harmonics with numbers that are multiples of the duty cycle . The frequency distance between adjacent harmonics is equal to the pulse repetition frequency. The width of the lobes, measured in frequency units, is , ᴛ.ᴇ. is inversely proportional to the duration of the signal. We can conclude: the shorter the pulse, the wider the spectrum .

b) Ramp signal .

Fig 4. Ramp wave.

A sawtooth signal within a period is described by a linear function

, . (20)

This signal is an odd function, and therefore its Fourier series in sine-cosine form contains only sine components:

The Fourier series of the sawtooth signal has the form:

It is important to note that for the spectra of rectangular and sawtooth signals it is characteristic that the amplitudes of the harmonics with increasing numbers decrease proportionally .

V) Triangular pulse sequence .

The Fourier series has the form:

Figure 5. Sequence of triangular pulses.

As we can see, in contrast to a sequence of rectangular and sawtooth pulses, for a triangular periodic signal the amplitudes of the harmonics decrease in proportion to the second power of the harmonic numbers. This is due to the fact that the rate of spectrum decay depends on degree of signal smoothness.

Lecture No. 3. Fourier transform.

Properties of the Fourier transform.

Examples of Fourier series expansion. - concept and types. Classification and features of the category "Examples of Fourier series expansion." 2017, 2018.

Forms of recording the Fourier series. The signal is called periodic, if its shape repeats cyclically in time Periodic signal u(t) in general it is written like this:

u(t)=u(t+mT), m=0, ±1,±2,…

Here is the T-period of the signal. Periodic signals can be either simple or complex.

For the mathematical representation of periodic signals with a period T series (2.2) is often used, in which harmonic (sine and cosine) oscillations of multiple frequencies are chosen as basis functions

y 0 (t)=1; y 1 (t)=sinw 1 t; y 2 (t)=cosw 1 t;

y 3 (t)=sin2w 1 t; y 4 (t)=cos2w 1 t; ..., (2.3)

where w 1 =2p/T is the main angular frequency of the sequence

functions. For harmonic basis functions, from series (2.2) we obtain the Fourier series (Jean Fourier - French mathematician and physicist of the 19th century).

Harmonic functions of the form (2.3) in the Fourier series have the following advantages: 1) simple mathematical description; 2) invariance to linear transformations, i.e. if at the input linear circuit If a harmonic oscillation operates, then at its output there will also be a harmonic oscillation, differing from the input only in amplitude and initial phase; 3) like a signal, harmonic functions are periodic and have infinite duration; 4) the technique for generating harmonic functions is quite simple.

It is known from a mathematics course that to expand a periodic signal into a series in harmonic functions(2.3) the Dirichlet conditions must be satisfied. But all real periodic signals satisfy these conditions and can be represented in the form of a Fourier series, which can be written in one of the following forms:

u(t)=A 0 /2+ (A’ mn cosnw 1 t+A” mn nw 1 t), (2.4)

where are the coefficients

A 0 =

A mn ”= (2.5)

u(t)=A 0 /2+ (2.6)

A mn = (2.7)

or in complex form

u(t)= (2.8)

Cn= (2.9)

From (2.4) - (2.9) it follows that in the general case the periodic signal u(t) contains a constant component A 0 /2 and a set of harmonic oscillations of the fundamental frequency w 1 =2pf 1 and its harmonics with frequencies w n =nw 1, n=2 ,3,4,… Each of the harmonic

Fourier series oscillations are characterized by amplitude and initial phase y n .nn

Spectral diagram and spectrum of a periodic signal. If any signal is presented as a sum of harmonic oscillations with different frequencies, then it is said that spectral decomposition signal.

Spectral diagram signal is usually called a graphical representation of the Fourier series coefficients of this signal. There are amplitude and phase diagrams. In Fig. 2.6, on a certain scale, the values ​​of harmonic frequencies are plotted along the horizontal axis, and their amplitudes A mn and phases y n are plotted along the vertical axis. Moreover, the harmonic amplitudes can take only positive values, the phases can take both positive and negative values ​​in the interval -p£y n £p

Signal spectrum- this is a set of harmonic components with specific values ​​of frequencies, amplitudes and initial phases, which together form a signal. In technical applications, in practice, spectral diagrams are called more briefly - amplitude spectrum, phase spectrum. Most often people are interested in the amplitude spectral diagram. It can be used to estimate the percentage of harmonics in the spectrum.

Example 2.3. Expand a periodic sequence of rectangular video pulses into a Fourier series With known parameters (U m , T, t z), even "Relative to point t=0. Construct a spectral diagram of amplitudes and phases at U m =2B, T=20ms, S=T/t and =2 and 8.

A given periodic signal on an interval of one period can be written as

u(t) =

To represent this signal, we will use the Fourier series form V form (2.4). Since the signal is even, only cosine components will remain in the expansion.

Rice. 2.6. Spectral diagrams of a periodic signal:

a - amplitude; b- phase

The integral of an odd function over a period is equal to zero. Using formulas (2.5) we find the coefficients

allowing us to write the Fourier series:

To construct spectral diagrams for specific numerical data, we set i=0, 1, 2, 3, ... and calculate the harmonic coefficients. The results of calculating the first eight components of the spectrum are summarized in table. 2.1. In a series (2.4) A" mn =0 and according to (2.7) A mn =|A’ mn |, the main frequency f 1 =1/T= 1/20-10 -3 =50 Hz, w 1 =2pf 1 =2p*50=314 rad/s. The amplitude spectrum in Fig.

2.7 is built for these n, at which And mn more than 5% of the maximum value.

From the given example 2.3 it follows that with increasing duty cycle, the number of spectral components increases and their amplitudes decrease. Such a signal is said to have a rich spectrum. It should be noted that for many practically used signals there is no need to calculate the amplitudes and phases of harmonics using the previously given formulas.

Table 2.1. Amplitudes of the Fourier series components of a periodic sequence of rectangular pulses

Rice. 2.7. Spectral diagrams of a periodic pulse sequence: A- with duty cycle S-2; - b-with duty cycle S=8

In mathematical reference books there are tables of expansions of signals in a Fourier series. One of these tables is given in the Appendix (Table A.2).

The question often arises: how many spectral components (harmonics) should be taken to represent real signal near Fourier? After all, the series is, strictly speaking, endless. A definite answer cannot be given here. It all depends on the shape of the signal and the accuracy of its representation by the Fourier series. Smoother signal change - less harmonics required. If the signal has jumps (discontinuities), then it is necessary to sum larger number harmonics to achieve the same error. However, in many cases, for example in telegraphy, it is believed that three harmonics are sufficient for transmitting rectangular pulses with steep fronts.