# Amplitude modulation of an arbitrary signal

As you know, AM is a type of modulation in which the amplitude of the carrier signal changes according to the law of the modulating (informational) signal. There are many sources with a theoretical and practical description of AM. The description is given, first of all, in order to show the frequency composition of the AM signal. A single-tone signal is usually considered as a modulating signal. This signal is given by a simple sine function. People always asked me, and I wondered how to describe AM in case there is an arbitrary signal as a modulating signal. It is an arbitrary signal, the frequency spectrum of which consists of many components, is of interest, since AM is used in broadcasting to transmit sound.

Let us try to describe AM for the above case, taking into account that the modulating signal can be represented as a continuous sum of simple single-tone signals of different frequencies with different amplitudes and phases. Without going into the subtleties of mathematical analysis, this signal can be written as a continuous sum (integral) of Fourier:

The integrand of this formula is a so-called. trigonometric convolution in the amplitude-phase view of the Fourier series term into which the signal can be expanded. The integral in (1) can be called the Fourier integral, since, in fact, it is a continuous sum, i.e. continuous Fourier series in which the original signal is expanded. The decomposition of the signal in a similar series gives an idea of the frequency composition of this signal. Thus, the original modulating signal is represented as a continuous sum of sinusoids (in this case, for convenience,$$c o s ) different frequencies$$f from$$0 to$$m , each of them has its own amplitude$$A ( f ) phase shift$$φ ( f ) . Function$$A ( f ) is the frequency spectrum of the original signal$$S ( t ) .

It should be noted that the signal is considered for a limited period of time.$$t ∈ [ 0 ; t 0 ] . Generally speaking, if we are talking about a sound signal, then, as a rule, the frequency spectrum has a practical meaning to consider for very short signal fragments. Obviously, the longer the duration of the signal, the more low-frequency (approaching zero) components will appear in the spectral composition, which cannot be compared with sound frequencies in the audible range.

In addition to the modulating signal, there is a tone signal, which is a carrier wave with a frequency$$f c amplitude$$C and zero initial phase:

We now turn directly to the process of amplitude modulation.

It is known that the AM signal$$S A M is the result of multiplying the carrier signal and the modulating signal previously shifted and “indexed” by the modulation index$$k , i.e.

Substitute the source data (1) and (2) into expression (3), open the brackets, fill in the integral independent of the integration variable$$f some factors:

Continuing the equality, we divide the integral of the resulting sum by the sum of two integrals, open the brackets and put out the necessary factors in the function arguments:

We will use this replacement:

Thus, new piecewise set functions (4) and (5) were introduced, describing the change in amplitude and phase as a function of frequency. Looking at the components of the function (4), you can see that the third component is obtained by parallel transfer of the function$$ on $$, and the first - also with a preliminary mirror turn. Factors-constants in front of the functions that reduce the amplitude, I do not take into account. That is, in the spectrum of the AM signal there are three components: carrier, upper side and lower side, which was reflected in (4).

In conclusion, it is worth noting that AM can be described using a more complex approach based on complex signals and complex numbers. The usual signal, which was discussed in this article, has no imaginary component. Taking into account the representation using vector diagrams on the complex plane, a signal without an imaginary component is made up of two complex signals with both components. This is obvious if the single-tone signal is represented as a sum of two vectors that rotate in opposite directions symmetrically about the x axis (Re). The rotational speed of these vectors is equivalent to the signal frequency, and the direction is the sign of the frequency (positive or negative). It follows from this that the frequency spectrum of a signal without an imaginary component has not only a positive, but also a negative component. And of course it is symmetrical about zero. It is with this representation that it can be stated that in the process of amplitude modulation, the spectrum of the modulating signal is transferred on the frequency scale to the right from zero to the carrier frequency (and to the left too). In this case, the “lower side” does not occur, it already exists in the original modulating signal, although it is located in the negative frequency range. It sounds strange at first glance, since it would seem that there are no negative frequencies in nature. But math presents a lot of surprises. the truth is in the negative frequency range. It sounds strange at first glance, since it would seem that there are no negative frequencies in nature. But math presents a lot of surprises. the truth is in the negative frequency range. It sounds strange at first glance, since it would seem that there are no negative frequencies in nature. But math presents a lot of surprises.

Let us try to describe AM for the above case, taking into account that the modulating signal can be represented as a continuous sum of simple single-tone signals of different frequencies with different amplitudes and phases. Without going into the subtleties of mathematical analysis, this signal can be written as a continuous sum (integral) of Fourier:

$$S ( t ) = m ∫ 0 A ( f ) cos ( 2 π f t + φ ( f ) ) d f ,( 1 )

Where $$m is the upper limit of the signal frequency (baseband signal),$$f is the integration variable responsible for the frequency, and$$f ∈ ( 0 ; m ] . Functions$$A ( f ) and$$φ ( f ) - amplitude and phase of the signal component at the frequency$$f .The integrand of this formula is a so-called. trigonometric convolution in the amplitude-phase view of the Fourier series term into which the signal can be expanded. The integral in (1) can be called the Fourier integral, since, in fact, it is a continuous sum, i.e. continuous Fourier series in which the original signal is expanded. The decomposition of the signal in a similar series gives an idea of the frequency composition of this signal. Thus, the original modulating signal is represented as a continuous sum of sinusoids (in this case, for convenience,$$c o s ) different frequencies$$f from$$0 to$$m , each of them has its own amplitude$$A ( f ) phase shift$$φ ( f ) . Function$$A ( f ) is the frequency spectrum of the original signal$$S ( t ) .

It should be noted that the signal is considered for a limited period of time.$$t ∈ [ 0 ; t 0 ] . Generally speaking, if we are talking about a sound signal, then, as a rule, the frequency spectrum has a practical meaning to consider for very short signal fragments. Obviously, the longer the duration of the signal, the more low-frequency (approaching zero) components will appear in the spectral composition, which cannot be compared with sound frequencies in the audible range.

In addition to the modulating signal, there is a tone signal, which is a carrier wave with a frequency$$f c amplitude$$C and zero initial phase:

$$S c ( t ) = C sin ( 2 π f c t ) ,( 2 )

and $$f c ≫ m . Indeed, in broadcasting, the carrier frequency is many times greater than the band of the transmitted signal.We now turn directly to the process of amplitude modulation.

It is known that the AM signal$$S A M is the result of multiplying the carrier signal and the modulating signal previously shifted and “indexed” by the modulation index$$k , i.e.

$$S A M ( t ) = S c ( t ) ( 1 + k S ( t ) ) .( 3 )

In order to avoid the so-called overmodulation $$k ∈ ( 0 ; 1 ) .Substitute the source data (1) and (2) into expression (3), open the brackets, fill in the integral independent of the integration variable$$f some factors:

$$S A M ( t ) = C sin ( 2 π f c t ) ( 1 + k m ∫ 0 A ( f ) cos ( 2 π f t + φ ( f ) ) d f ) == C sin ( 2 π f c t ) + C sin ( 2 π f c t ) k m ∫ 0 A ( f ) cos ( 2 π f t + φ ( f ) ) d f == C sin ( 2 π f c t ) + k C m ∫ 0 A ( f ) sin ( 2 π f c t ) cos ( 2 π f t + φ ( f ) ) d f .

Let us apply the well-known school trigonometric formula for transforming a product for integrands:$$

This formula is key in AM and emphasizes these very “two side” in the spectral composition of the AM signal.Continuing the equality, we divide the integral of the resulting sum by the sum of two integrals, open the brackets and put out the necessary factors in the function arguments:

$$

The three resulting terms, respectively, are, as can be seen from equality, the carrier signal, the signals of the "lower" and "upper" side. Before giving a specific explanation, we continue the equality by applying the variable replacement method in the following configuration:$$

We will use this replacement:

$$

By swapping the integration limits in the first integral (as a result of which the sign before the integral changes to the opposite), we can combine the two integrals into one. Moreover, there can also be made the first term describing the carrier signal. In this case, naturally, the integrand functions of amplitude and phase need to be generalized. This is all done conditionally and for more detailed clarity, without going into the subtleties of mathematical analysis. Thus, it turns out:$$

Where$$

and$$

Thus, new piecewise set functions (4) and (5) were introduced, describing the change in amplitude and phase as a function of frequency. Looking at the components of the function (4), you can see that the third component is obtained by parallel transfer of the function$$ on $$, and the first - also with a preliminary mirror turn. Factors-constants in front of the functions that reduce the amplitude, I do not take into account. That is, in the spectrum of the AM signal there are three components: carrier, upper side and lower side, which was reflected in (4).

In conclusion, it is worth noting that AM can be described using a more complex approach based on complex signals and complex numbers. The usual signal, which was discussed in this article, has no imaginary component. Taking into account the representation using vector diagrams on the complex plane, a signal without an imaginary component is made up of two complex signals with both components. This is obvious if the single-tone signal is represented as a sum of two vectors that rotate in opposite directions symmetrically about the x axis (Re). The rotational speed of these vectors is equivalent to the signal frequency, and the direction is the sign of the frequency (positive or negative). It follows from this that the frequency spectrum of a signal without an imaginary component has not only a positive, but also a negative component. And of course it is symmetrical about zero. It is with this representation that it can be stated that in the process of amplitude modulation, the spectrum of the modulating signal is transferred on the frequency scale to the right from zero to the carrier frequency (and to the left too). In this case, the “lower side” does not occur, it already exists in the original modulating signal, although it is located in the negative frequency range. It sounds strange at first glance, since it would seem that there are no negative frequencies in nature. But math presents a lot of surprises. the truth is in the negative frequency range. It sounds strange at first glance, since it would seem that there are no negative frequencies in nature. But math presents a lot of surprises. the truth is in the negative frequency range. It sounds strange at first glance, since it would seem that there are no negative frequencies in nature. But math presents a lot of surprises.