Fourier transforms. Fourier transform Fourier integral complex form of the integral Fourier transform cosine and sine transforms amplitude and phase spectra application properties

One of the powerful tools for studying problems in mathematical physics is the method of integral transformations. Let the function f(x) be given on an interval (a, 6), finite or infinite. An integral transformation of a function f(x) is a function where K(x, w) is a function fixed for a given transformation, called the kernel of the transformation (it is assumed that the integral (*) exists in its proper or improper sense). §1. Fourier integral Any function f(x), which on the interval [-f, I] satisfies the conditions of expansion into a Fourier series, can be represented on this interval by a trigonometric series. Coefficients a*, and 6„ of series (1) are determined by the Euler-Fourier formulas : FOURIER TRANSFORM Fourier integral Complex form of the integral Fourier transform Cosine and sine transformation Amplitude and phase spectra Properties Applications The series on the right side of equality (1) can be written in a different form. For this purpose, we enter into it from formulas (2) the values of the coefficients a" and op, put cos ^ x and sin x under the signs of the integrals (which is possible, since the integration variable is m) O) and use the formula for the cosine of the difference. We will have If the function /(x) was initially defined on an interval of the numerical axis greater than the segment [-1,1] (for example, on the entire axis), then expansion (3) will reproduce the values of this function only on the segment [-1, 1] and will continue to the entire numerical axis as a periodic function with a period of 21 (Fig. 1). Therefore, if the function f(x) (generally speaking, non-periodic) is defined on the entire number line, in formula (3) one can try to go to the limit at I +oo. In this case, it is natural to require that the following conditions be met: 1. f(x) satisfies the conditions of expansion into a Fourier series on any finite segment of the Ox axis\ 2. the function f(x) is absolutely integrable on the entire real number line. If condition 2 is satisfied, the first term on the right side of the equality (3) as I -* +oo tends to zero. In fact, Let us try to establish what the sum on the right side of (3) turns into in the limit at I +oo. Let us assume that Then the sum on the right side of (3) takes the form Due to the absolute convergence of the integral, this sum for large I differs little from the expression which resembles the integral sum for a function of the variable £ compiled for the interval (0, +oo) of change. Therefore, it is natural to expect, that for sum (5) goes into the integral. On the other hand, for fixed) it follows from formula (3) that we also obtain the equality. The sufficient condition for the validity of formula (7) is expressed by the following theorem. Theorem 1. If the function f(x) is absolutely integrable on the entire real number line and has, together with its derivative, a finite number of discontinuity points of the first kind on any interval [a, 6], then the equality holds: Moreover, at any point xq that is a discontinuity point 1 function f(x) of the th kind, the value of the integral on the right side of (7) is equal to Formula (7) is called the Fourier integral formula, and the integral on its right side is called the Fourier integral. If we use the formula for the cosine of the difference, then formula (7) can be written in the form The functions a(ξ), b(ζ) are analogues of the corresponding Fourier coefficients an and bn of a 2m-periodic function, but the latter are defined for discrete values n, while a(0> HO are defined for continuous values £ G (-oo, +oo). Complex form of the Fourier integral Assuming /(x) is absolutely integrable on the entire Ox axis, consider the integral This integral converges uniformly for, since and therefore, it is a continuous and, obviously, odd function of But then, on the other hand, the integral is an even function of the variable so that Therefore, the Fourier integral formula can be written as follows: Let us multiply the equality by the imaginary unit i and add it to equality (10). by Euler's formula we have This is the complex form of the Fourier integral. Here the outer integration over £ is understood in the sense of the Cauchy principal value: §2. Fourier transform Cosine and sine transforms Let the function f(x) be piecewise smooth on any. finite segment of the Ox axis and is absolutely integrable on the entire axis. Definition: The function from which, by virtue of Euler’s formula, is called the Fourier transform of the function /(r) (spectral function). This is the integral transformation of the function /(r) on the interval (-oo). ,+oo) with kernel Using the Fourier integral formula we obtain This is the so-called inverse Fourier transform, which gives the transition from F(ξ) to f(x). Sometimes the direct Fourier transform is defined as follows: Then the inverse Fourier transform is defined by the formula The Fourier transform of the function /(x) is also defined as follows: FOURIER TRANSFORM Fourier integral Complex form of the integral Fourier transform Cosine and sine transforms Amplitude and phase spectra Properties Applications Then, in turn, In this case, the position of the factor ^ is quite arbitrary: it can be included either in formula (1") or in formula (2"). Example 1. Find the Fourier transform of the function -4 We have This equality allows differentiation with respect to £ under the integral sign (the integral obtained after differentiation converges uniformly when ( belongs to any finite segment): Integrating by parts, we will have The out-of-integral term vanishes, and we we obtain from where (C is the constant of integration). Setting £ = 0 in (4), we find C = F(0). By virtue of (3) we have It is known that In particular, for) we obtain that Example 2 (discharge of the codemsetor through the copropylene ). Let us consider the function 4 For the spectra of the function F(ξ), we obtain Hence (Fig. 2). The condition for the absolute integrability of the function f(x) on the entire number line is very strict. It excludes, for example, such elementary functions , as) = cos x, f(x) = e1, for which the Fourier transform (in the classical form considered here) does not exist. Only those functions that quickly tend to zero as |x| have a Fourier transform. -+ +oo (as in examples 1 and 2). 2.1. Cosine and sine Fourier transforms Using the cosine and difference formula, we rewrite the Fourier integral formula in the following form: Let f(x) be an even function. Then we have equality (5). In the case of odd f(x), we similarly obtain If f(x) is given only on (0, -foo), then formula (6) extends f(x) to the entire Ox axis in an even manner, and formula (7) - odd. (7) Definition. The function is called the Fourier cosine transform of f(x). From (6) it follows that for an even function f(x) This means that f(x), in turn, is a cosine transform for Fc(£). In other words, the functions / and Fc are mutual cosine transformations. Definition. The function is called the Fourier sine transform of f(x). From (7) we obtain that for an odd function f(x), i.e. f and Fs are mutual sine transformations. Example 3 (rectangular pulse). Let f(t) be an even function defined as follows: (Fig. 3). linear operator . Denoting it by we will write. That is, the operation of multiplying f(x) by the argument x goes after the Fourier transform into the operation t. If, together with the function f(x), the functions are absolutely integrable on the entire Ox axis, then the differentiation process can be continued. We obtain that the function has derivatives up to order m inclusive, and Thus, the faster the function f(x) decreases, the smoother the function becomes. Theorem 2 (about the drill). Let be the Fourier transforms of the functions f,(x) and f2(x), respectively. Then where the double integral on the right side converges absolutely. Let's put - x. Then we will have or, changing the order of integration, The function is called the convolution of functions and is denoted by the symbol Formula (1) can now be written as follows: This shows that the Fourier transform of the convolution of the functions f\(x) and f2(x) is equal to y/2x multiplied by product of Fourier transforms of convolvable functions. Remark. It is not difficult to establish the following properties of convolution: 1) linearity: 2) commutativity: §4. Applications of the Fourier transform 1. Let P(^) be a linear differential operator of order m with constant coefficients. Using the formula for the Fourier transform of derivatives of the function y(x), we find " Consider the differential equation where P is the differential operator introduced above. Assume that the desired solution y(x) has the Fourier transform y (O. and the function f(x) has the transform /(£) Applying the Fourier transform to equation (1), we obtain instead of a differential algebraic equation on the axis relative to where so that formally where the symbol denotes the inverse Fourier transform The main limitation of the applicability of this method is due to the following fact. differential equation with constant coefficients contains functions of the form eL*, eaz cos fix, eax sin рх. They are not absolutely integrable on the -oo axis< х < 4-оо, и преобразование Фурье для них не определено, так что, строго говоря, применятьданный метод нельзя. Это ограничение можно обойти, если ввести в рассмотрение так называемые обобщенные функции. Однако в ряде случаев преобразование Фурье все же применимо в своей классической форме. Пример. Найти решение а = а(х, t) уравнения (а = const), при начальных условиях Это - задача о свободных колебаниях бесконечной однородной струны, когда задано начальное отклонение <р(х) точек сгруны, а начальные скорости отсутствуют. 4 Поскольку пространственная переменная х изменяется в пределах от -оо до +оо, подвергнем уравнение и начальные условия преобразованию Фурье по переменной х. Будем предполагать, что 1) функции и(х, t) и

Fourier transform is a family of mathematical methods based on the decomposition of an initial continuous function of time into a set of basic harmonic functions (which are sinusoidal functions) of various frequencies, amplitudes and phases. From the definition it is clear that the main idea of the transformation is that any function can be represented as an infinite sum of sinusoids, each of which will be characterized by its amplitude, frequency and initial phase.

The Fourier transform is the founder of spectral analysis. Spectral analysis is a signal processing method that allows you to characterize the frequency composition of the measured signal. Depending on how the signal is represented, different Fourier transforms are used. There are several types of Fourier transform:

– Continuous Fourier Transform (in English literature Continue Time Fourier Transform – CTFT or, for short, F.T.);

– Discrete Fourier Transform (in English literature Discrete Fourier Transform – DFT);

– Fast Fourier transform (in English literature Fast Fourier transform – FFT).

Continuous Fourier transform

The Fourier transform is a mathematical tool used in various scientific fields. In some cases, it can be used as a means of solving complex equations that describe dynamic processes that arise under the influence of electrical, thermal or light energy. In other cases, it allows one to isolate the regular components in a complex vibrational signal, which makes it possible to correctly interpret experimental observations in astronomy, medicine and chemistry. The continuous transformation is actually a generalization of Fourier series, provided that the period of the expanded function tends to infinity. Thus, the classical Fourier transform deals with the spectrum of the signal taken over the entire range of existence of the variable.

There are several types of recording of the continuous Fourier transform, differing from each other in the value of the coefficient in front of the integral (two forms of recording):

where and is the Fourier transform of a function or the frequency spectrum of a function;

- circular frequency.

It should be noted that different types of recording are found in different fields of science and technology. The normalization factor is necessary for correct scaling of the signal from the frequency domain to the time domain. The normalization factor reduces the signal amplitude at the output of the inverse conversion so that it matches the amplitude of the original signal. In the mathematical literature, the direct and inverse Fourier transforms are multiplied by a factor , while in physics most often the direct transformation does not include a factor, but the inverse transform uses a factor . If you sequentially calculate the direct Fourier transform of a certain signal, and then take the inverse Fourier transform, then the result of the inverse transform must completely coincide with the original signal.

If the function is odd on the interval (−∞, +∞), then the Fourier transform can be represented through the sine function:

If the function is even on the interval (−∞, +∞), then the Fourier transform can be represented through the cosine function:

Thus, the continuous Fourier transform allows us to represent a non-periodic function in the form of an integral of a function that represents at each point the coefficient of the Fourier series for the non-periodic function.

The Fourier transform is invertible, that is, if its Fourier transform was calculated from a function, then the original function can be uniquely restored from the Fourier transform. By inverse Fourier transform we mean an integral of the form (two forms of notation):

where is the Fourier transform of a function or the frequency spectrum of a function;

- circular frequency.

If the function is odd on the interval (−∞, +∞), then the inverse Fourier transform can be represented through the sine function:

If the function is even on the interval (−∞, +∞), then the inverse Fourier transform can be represented through the cosine function:

As an example, consider the following function . The graph of the exponential function under study is presented below.

Since the function is an even function, the continuous Fourier transform will be defined as follows:

As a result, we obtained the dependence of the change in the exponential function under study on the frequency interval (see below).

The continuous Fourier transform is used, as a rule, in theory when considering signals that change in accordance with given functions, but in practice they usually deal with measurement results that represent discrete data. The measurement results are recorded at regular intervals with a certain sampling frequency, for example, 16000 Hz or 22000 Hz. However, in the general case, discrete readings can be uneven, but this complicates the mathematical apparatus of analysis, and therefore is not usually used in practice.

There is an important theorem of Kotelnikov (in foreign literature the name “Nyquist-Shannon theorem”, “sampling theorem” is found), which states that an analog periodic signal having a finite (limited in width) spectrum (0...fmax) can be uniquely restored without distortions and losses in their discrete samples taken with a frequency greater than or equal to twice the upper frequency of the spectrum - sampling frequency (fsample >= 2*fmax). In other words, at a sampling rate of 1000 Hz, a signal with a frequency of up to 500 Hz can be reconstructed from an analog periodic signal. It should be noted that discretization of a function by time leads to periodization of its spectrum, and discretization of the spectrum by frequency leads to periodization of the function.

This is one of the Fourier transforms widely used in digital signal processing algorithms.

The direct discrete Fourier transform associates a time function, which is defined by N-measurement points on a given time interval, with another function, which is defined on a frequency interval. It should be noted that the function on the time domain is specified using N-samples, and the function on the frequency domain is specified using the K-fold spectrum.

k ˗ frequency index.

The frequency of the kth signal is determined by the expression

where T is the period of time during which the input data was taken.

The direct discrete transformation can be rewritten in terms of real and imaginary components. The real component is an array containing the values of the cosine components, and the imaginary component is an array containing the values of the sine components.

From the last expressions it is clear that the transformation decomposes the signal into sinusoidal components (which are called harmonics) with frequencies from one oscillation per period to N oscillations per period.

The discrete Fourier transform has a special feature, since a discrete sequence can be obtained by a sum of functions with different compositions of the harmonic signal. In other words, a discrete sequence is decomposed into harmonic variables - ambiguous. Therefore, when expanding a discrete function using a discrete Fourier transform, high-frequency components appear in the second half of the spectrum that were not in the original signal. This high-frequency spectrum is a mirror image of the first part of the spectrum (in terms of frequency, phase and amplitude). Typically, the second half of the spectrum is not considered, and the signal amplitudes of the first part of the spectrum are doubled.

It should be noted that the decomposition of a continuous function does not lead to the appearance of a mirror effect, since a continuous function is uniquely decomposed into harmonic variables.

The amplitude of the DC component is the average value of the function over a selected period of time and is determined as follows:

The amplitudes and phases of the frequency components of the signal are determined by the following relationships:

The resulting amplitude and phase values are called polar notation. The resulting signal vector will be determined as follows:

Let's consider an algorithm for transforming a discretely given function on a given interval (on a given period) with the number of initial points

D sparkle Fourier transform

As a result of the transformation, we obtain the real and imaginary value of the function, which is defined on the frequency range.

The inverse discrete Fourier transform associates a frequency function, which is defined by the K-fold spectrum on the frequency interval, with another function, which is defined on the time interval.

N ˗ number of signal values measured over a period, as well as the frequency spectrum multiplicity;

k ˗ frequency index.

As already mentioned, the discrete Fourier transform associates N-points of a discrete signal with N-complex spectral samples of the signal. To calculate one spectral sample, N complex multiplication and addition operations are required. Thus, the computational complexity of the discrete Fourier transform algorithm is quadratic, in other words, complex multiplication and addition operations are required.

I believe that everyone is generally aware of the existence of such a wonderful mathematical tool as the Fourier transform. However, for some reason it is taught so poorly in universities that relatively few people understand how this transformation works and how it should be used correctly. Meanwhile, the mathematics of this transformation is surprisingly beautiful, simple and elegant. I invite everyone to learn a little more about the Fourier transform and the related topic of how analog signals can be effectively converted into digital signals for computational processing.

Without using complex formulas and Matlab, I will try to answer the following questions:

FT, DTF, DTFT - what are the differences and how do seemingly completely different formulas give such conceptually similar results?
How to Correctly Interpret Fast Fourier Transform (FFT) Results
What to do if you are given a signal of 179 samples and the FFT requires an input sequence of length equal to a power of two
Why, when trying to obtain the spectrum of a sinusoid using Fourier, instead of the expected single “stick”, a strange squiggle appears on the graph and what can be done about it
Why are analog filters placed before the ADC and after the DAC?
Is it possible to digitize an ADC signal with a frequency higher than half the sampling frequency (the school answer is incorrect, the correct answer is possible)
How to restore the original signal using a digital sequence

I will proceed from the assumption that the reader understands what an integral is, a complex number (as well as its modulus and argument), convolution of functions, plus at least a “hands-on” idea of what the Dirac delta function is. If you don’t know, no problem, read the above links. Throughout this text, by “product of functions” I will mean “pointwise multiplication”

We should probably start with the fact that the usual Fourier transform is some kind of thing that, as you can guess from the name, transforms one function into another, that is, it associates each function of a real variable x(t) with its spectrum or Fourier image y (w):

If we give analogies, then an example of a transformation similar in meaning can be, for example, differentiation, turning a function into its derivative. That is, the Fourier transform is essentially the same operation as taking the derivative, and it is often denoted in a similar way by drawing a triangular “cap” over the function. Only in contrast to differentiation, which can also be defined for real numbers, the Fourier transform always “works” with more general complex numbers. Because of this, problems constantly arise with displaying the results of this transformation, since complex numbers are determined not by one, but by two coordinates on a graph operating with real numbers. The most convenient way, as a rule, is to represent complex numbers in the form of a modulus and an argument and draw them separately as two separate graphs:

The graph of the argument of the complex value is often called in this case the “phase spectrum”, and the graph of the modulus is often called the “amplitude spectrum”. The amplitude spectrum is usually of much greater interest, and therefore the “phase” part of the spectrum is often skipped. In this article we will also focus on “amplitude” things, but we should not forget about the existence of the missing phase part of the graph. In addition, instead of the usual module of a complex value, its decimal logarithm multiplied by 10 is often drawn. The result is a logarithmic graph, the values of which are displayed in decibels (dB).

Please note that not very much negative numbers logarithmic graph (-20 dB or less) in this case correspond to almost zero numbers on the “normal” graph. Therefore, the long and wide “tails” of various spectra on such graphs, when displayed in “ordinary” coordinates, as a rule, practically disappear. The convenience of such a strange at first glance representation arises from the fact that Fourier images various functions often it is necessary to multiply among themselves. With such pointwise multiplication of complex-valued Fourier images, their phase spectra are added, and their amplitude spectra are multiplied. The first is easy to do, while the second is relatively difficult. However, the logarithms of the amplitude add up when multiplying the amplitudes, so logarithmic amplitude graphs can, like phase graphs, simply be added pointwise. In addition, in practical problems it is often more convenient to operate not with the “amplitude” of the signal, but with its “power” (the square of the amplitude). On logarithmic scale both graphs (amplitude and power) look identical and differ only in the coefficient - all values on the power graph are exactly twice as large as on the amplitude scale. Accordingly, to plot a graph of power distribution by frequency (in decibels), you can not square anything, but calculate the decimal logarithm and multiply it by 20.

Are you bored? Just wait a little longer, we'll be done with the boring part of the article explaining how to interpret graphs soon :). But before that, you should understand one extremely important thing: Although all of the above spectrum plots were drawn for some limited ranges of values (positive numbers in particular), all of these plots actually continue to plus and minus infinity. The graphs simply depict some “most meaningful” part of the graph, which is usually mirrored for negative values parameter and is often repeated periodically with a certain step when considered on a larger scale.

Having decided what is drawn on the graphs, let's return to the Fourier transform itself and its properties. There are several different ways how to determine this transformation, differing in small details (different normalizations). For example, in our universities, for some reason, they often use the normalization of the Fourier transform, which defines the spectrum in terms of angular frequency (radians per second). I will use a more convenient Western formulation that defines the spectrum in terms of ordinary frequency (hertz). The direct and inverse Fourier transforms in this case are determined by the formulas on the left, and some properties of this transformation that we will need are determined by a list of seven points on the right:

The first of these properties is linearity. If we take some linear combination of functions, then the Fourier transform of this combination will be the same linear combination of the Fourier images of these functions. This property allows you to reduce complex functions and their Fourier transforms to simpler ones. For example, the Fourier transform of a sinusoidal function with frequency f and amplitude a is a combination of two delta functions located at points f and -f and with coefficient a/2:

If we take a function consisting of the sum of a set of sinusoids with different frequencies, then according to the property of linearity, the Fourier transform of this function will consist of a corresponding set of delta functions. This allows us to give a naive but visual interpretation of the spectrum according to the principle “if in the spectrum of a function frequency f corresponds to amplitude a, then the original function can be represented as a sum of sinusoids, one of which will be a sinusoid with frequency f and amplitude 2a.” Strictly speaking, this interpretation is incorrect, since the delta function and the point on the graph are completely different things, but as we will see later, for discrete Fourier transforms it will not be so far from the truth.

The second property of the Fourier transform is the independence of the amplitude spectrum from the time shift of the signal. If we move a function to the left or right along the x-axis, then only its phase spectrum will change.

The third property is that stretching (compressing) the original function along the time axis (x) proportionally compresses (stretches) its Fourier image along the frequency scale (w). In particular, the spectrum of a signal of finite duration is always infinitely wide and, conversely, the spectrum of finite width always corresponds to a signal of unlimited duration.

The fourth and fifth properties are perhaps the most useful of all. They make it possible to reduce the convolution of functions to a pointwise multiplication of their Fourier images, and vice versa - the pointwise multiplication of functions to the convolution of their Fourier images. A little further I will show how convenient this is.

The sixth property speaks of the symmetry of Fourier images. In particular, from this property it follows that in the Fourier transform of a real-valued function (i.e., any “real” signal), the amplitude spectrum is always an even function, and the phase spectrum (if brought to the range -pi...pi) is an odd one . It is for this reason that the negative part of the spectrum is almost never drawn on spectrum graphs - for real-valued signals it does not give any new information(but, I repeat, it is not zero either).

Finally, the last, seventh property, says that the Fourier transform preserves the “energy” of the signal. It is meaningful only for signals of finite duration, the energy of which is finite, and suggests that the spectrum of such signals at infinity quickly approaches zero. It is precisely because of this property that spectrum graphs usually depict only the “main” part of the signal, which carries the lion’s share of the energy - the rest of the graph simply tends to zero (but, again, is not zero).

Armed with these 7 properties, let's look at the mathematics of signal “digitization”, which allows you to convert a continuous signal into a sequence of numbers. To do this, we need to take a function known as the “Dirac comb”:

A Dirac comb is simply a periodic sequence of delta functions with unity coefficient, starting at zero and proceeding with step T. To digitize signals, T is chosen as small a number as possible, T<<1. Фурье-образ этой функции - тоже гребенка Дирака, только с гораздо большим шагом 1/T и несколько меньшим коэффициентом (1/T). С математической точки зрения, дискретизация сигнала по времени - это просто поточечное умножение исходного сигнала на гребенку Дирака. Значение 1/T при этом называют частотой дискретизации:

Instead of a continuous function, after such multiplication, a sequence of delta pulses of a certain height is obtained. Moreover, according to property 5 of the Fourier transform, the spectrum of the resulting discrete signal is a convolution of the original spectrum with the corresponding Dirac comb. It is easy to understand that, based on the properties of convolution, the spectrum of the original signal is “copied” an infinite number of times along the frequency axis with a step of 1/T, and then summed.

Note that if the original spectrum had a finite width and we used a sufficiently high sampling frequency, then the copies of the original spectrum will not overlap, and therefore will not sum with each other. It is easy to understand that from such a “collapsed” spectrum it will be easy to restore the original one - it will be enough to simply take the spectrum component in the region of zero, “cutting off” the extra copies going to infinity. The simplest way to do this is to multiply the spectrum by a rectangular function equal to T in the range -1/2T...1/2T and zero outside this range. Such a Fourier transform corresponds to the function sinc(Tx) and according to property 4, such a multiplication is equivalent to the convolution of the original sequence of delta functions with the function sinc(Tx)

That is, using the Fourier transform, we have a way to easily reconstruct the original signal from a time-sampled one, working provided that we use a sampling frequency that is at least twice (due to the presence of negative frequencies in the spectrum) higher than the maximum frequency present in the original signal. This result is widely known and is called the “Kotelnikov/Shannon-Nyquist theorem”. However, as it is easy to notice now (understanding the proof), this result, contrary to the widespread misconception, determines sufficient, but not necessary condition for restoring the original signal. All we need is to ensure that the part of the spectrum that interests us after sampling the signal does not overlap each other, and if the signal is sufficiently narrowband (has a small “width” of the non-zero part of the spectrum), then this result can often be achieved at a sampling frequency much lower than twice the maximum frequency of the signal. This technique is called “undersampling” (subsampling, bandpass sampling) and is quite widely used in processing all kinds of radio signals. For example, if we take an FM radio operating in the frequency band from 88 to 108 MHz, then to digitize it we can use an ADC with a frequency of only 43.5 MHz instead of the 216 MHz assumed by Kotelnikov’s theorem. In this case, however, you will need a high-quality ADC and a good filter.

Let me note that “duplication” of high frequencies with frequencies of lower orders (aliasing) is an immediate property of signal sampling that irreversibly “spoils” the result. Therefore, if the signal can, in principle, contain high-order frequencies (that is, almost always), an analog filter is placed in front of the ADC, “cutting off” everything unnecessary directly in the original signal (since after sampling it will be too late to do this). The characteristics of these filters, as analog devices, are not ideal, so some “damage” to the signal still occurs, and in practice it follows that the highest frequencies in the spectrum are, as a rule, unreliable. To reduce this problem, the signal is often oversampled, setting the input analog filter to a lower bandwidth and using only the lower part of the theoretically available frequency range of the ADC.

Another common misconception, by the way, is when the signal at the DAC output is drawn in “steps”. “Steps” correspond to the convolution of a sampled signal sequence with a rectangular function of width T and height 1:

The signal spectrum with this transformation is multiplied by the Fourier image of this rectangular function, and for a similar rectangular function it is again sinc(w), “stretched” the more, the smaller the width of the corresponding rectangle. The spectrum of the sampled signal with such a “DAC” is multiplied point by point by this spectrum. In this case, unnecessary high frequencies with “extra copies” of the spectrum are not completely cut off, but the upper part of the “useful” part of the spectrum, on the contrary, is attenuated.

In practice, of course, no one does this. There are many different approaches to constructing a DAC, but even in the closest weighting-type DACs, the rectangular pulses in the DAC, on the contrary, are chosen to be as short as possible (approaching the real sequence of delta functions) in order to avoid excessive suppression of the useful part of the spectrum. “Extra” frequencies in the resulting broadband signal are almost always canceled out by passing the signal through an analog low-pass filter, so that there are no “digital steps” either “inside” the converter, or, especially, at its output.

However, let's go back to the Fourier transform. The Fourier transform described above applied to a pre-sampled signal sequence is called the Discrete Time Fourier Transform (DTFT). The spectrum obtained by such a transformation is always 1/T-periodic, therefore the DTFT spectrum is completely determined by its values on the segment )