Fourier amplitude spectrum. Spectra of amplitudes and phases of periodic signals

SPECTRAL ANALYSIS OF SIGNALS

In radio engineering (communications, navigation, television, radar), signals of complex shapes are widely used when transmitting information. To analyze the passage of such signals through a circuit, they act in this way: they represent a complex signal as a sum of harmonic oscillations and, using a known method (for example, the method of complex amplitudes), analyze the passage of each harmonic through the circuit. In accordance with the principle of superposition, the shape of the original signal is determined as the sum of the original harmonics.

The representation of a complex signal in the form of harmonic oscillations is explained by the fact that a harmonic signal is the only signal that does not change its shape when passing through a circuit. Only its amplitude and initial phase change, which greatly simplifies the analysis of the passage of complex signals.

The spectrum of a signal is the set of harmonic oscillations that make up the signal itself.

More strictly speaking, there are two main types of spectra: amplitude-frequency (amplitude) and phase-frequency (phase) spectrum.

Amplitude spectrum is called the distribution of amplitudes of harmonic components by frequency.

Phase spectrum is called the distribution of the initial phases of harmonic components by frequency.

Amplitude and phase spectrum image

Amplitude spectrum

The amplitude spectrum is always positive. The phase spectrum can be either positive or negative.

Spectrum of periodic signals

For the spectral representation of periodic oscillations, the expansion of these oscillations into a trigonometric Fourier series is used:

- period of the periodic signal.

Spectrum of a periodic sequence of rectangular video pulses

According to the picture the function
is even. Then in the trigonometric form of writing the series, only cosine terms remain, because the coefficients are equal to zero.

Let us determine the magnitude of the constant component and the amplitude of the harmonics

- duty cycle. Thus

Amplitude spectrum

P
Since the main part of the pulse energy is concentrated in the region of the main lobe, the width of the main lobe is taken as the spectrum width

Theoretically, the spectrum extends to infinity.

Phase spectrum

Spectrum of a non-periodic signal

Consider a non-periodic signal
, specified as some function that differs from zero in the interval
. Let's add the signal to a periodic one as shown in the figure.

Let's select an arbitrary period of time T , which includes the interval, we will represent the signal assignments in the form of a complex Fourier series

,

Where
Odds are determined by the expression

To go to a single pulse, you need to go to the limit at
.

If , then

As a result we get

Direct conversion

Elichina

called spectral density .

Physically, spectral density characterizes the total amplitude of oscillations of a single frequency range of the signal spectrum, and the value
characterizes the total amplitude of oscillations of the frequency range
.

The spectrum of a non-periodic signal is continuous.

Knowing the spectral density, you can find the signal shape

Inverse Fourier transform

In this way

Spectral Density Properties

Between a signal and its spectrum
there is a one-to-one correspondence, which is expressed by a number of properties.

1. The modulus of the spectral density is an even function of frequency, and the argument is an odd one:

2. The relationship between the spectra of periodic and non-periodic signals.

Let us have a signal and the corresponding spectral density
. When following pulses with a period the interval between adjacent harmonics is . Amplitude The th harmonic is correspondingly equal to

Spectral density of a non-periodic signal

From here we find

Conclusion. The modulus of the spectral density of a non-periodic signal (single pulse) and the envelope of the line spectrum of a periodic signal (pulse sequence) coincide in shape and differ only in scale.

3. Property of linearity. Based on the fact that the Fourier transform is linear, when adding signals
and which have spectra
And
, total signal
will have a spectrum
.

4. Time shift of signals(delay theorem). Signal
has a spectral density of arbitrary shape
.

When this signal is delayed for a time t 0 (while maintaining its form) we get a new function
. Let's determine the spectral density signal

Let's introduce a new variable
. Then we get

Thus, the time shift of the function by leads to a change in the phase characteristic of the spectrum by the amount
. The spectral density modulus does not depend on the position of the signal on the time axis.

5. Changing the time scale (scale theorem). Signal
duration  and can be compressed in time. New compressed signal

Signal duration
V times less than and equals
. Let's determine the spectral density of the compressed signal

Let's introduce a new variable
Then

.

When the signal is temporarily compressed, its spectrum expands by the same amount.

6
.Shifting the signal spectrum (shift theorem). Let's write down the spectral density for the product of signals
And
.
.

Thus

IN
conclusion Multiplying a function by an oscillation
equivalent to spectrum splitting
into two parts, which are shifted respectively by
And
.

This theorem allows us to find the spectrum of a radio signal (that is, a signal with high-frequency filling) from the spectrum of a video signal.

It follows from the figure that at a significant filling frequency of the radio pulse  0 in the region of positive frequencies (there are no negative frequencies) we can neglect the term (1/2) ( + 0) and determine the spectral density using the approximate formula

7. Energy distribution in the spectrum of non-periodic oscillations

The energy of the pulse as it passes through the resistance is equal to

Parseval's equality

Conclusion: the square of the spectral density modulus has the meaning of energy density, that is, the energy per unit frequency band [J/Hz].

8. Convolutionsignals. Let the spectral density correspond to the signals
. That is . Then the product of two spectra
signal convolution will respond:

Spectral densities of typical pulses

1. Exponential impulse:

A pulse of this shape occurs during lightning discharges, in car ignition systems. Wherever there are rubbing contacts.

2. Step function (Heaviside function):

The spectrum is found from the spectrum of the exponential pulse at
:

3. Rectangular video pulse:

Let's use the formula

Most of the pulse energy is concentrated in the main lobe region (more than 90%). Therefore, the width of the main lobe in the positive frequency range is taken as the spectrum width:

4. Spectrum of a single pulse (spectrum of the Dirac function)

Dirac function

The Dirac function represents the limit of a sequence of rectangular video pulses, provided that the area
and the duration
.

Physically, the Dirac function is a pulse of finite energy with a very short duration and a very large amplitude. With the help of this impulse, short-term strong influences (impacts) are described.

Thus

Conclusion: the spectrum of a single pulse is constant and extends to infinity.

5. Bell-shaped pulse spectrum:

The peculiarity of this pulse is that its shape coincides with the shape of the spectrum.

6. Spectrum of a rectangular radio pulse:

To determine the spectrum we will use

displacement theorem

Spectral density of a rectangular video pulse

Hence

7. Spectrum of a periodic sequence of rectangular radio pulses

To find the spectrum, we use the relationship between the spectra of a single radio pulse and a periodic sequence

Some pulses used in special communication systems

8. Spectrum of a triangular pulse

9. Spectrum of a trapezoidal pulse

Signal spectral density

Since the trapezoidal pulse is the result of pulse integration, its spectral density is equal to

From here we find the spectral density modulus

At
spectral density equal to the area of ​​the trapezoid

.

Qualitative view of the spectral density at positive frequencies:

The number of side lobes is determined by the ratio between And
. The smaller , that is, the steeper the tipulse fronts, the greater the number of side lobes in the region from 0 to . In the limit when the steepness of the fronts tends to infinity
the spectrum of the trapezoid transforms into the spectrum of a rectangular pulse.

1
0. Spectrum of cosine-square pulse

To determine the spectral density, we use the Laplace transform. To do this, we introduce two functions:
And
.

Let
. Then, in accordance with the delay theorem

Since the cosine square momentum is

, then its spectral density

Using the formula

,

This original corresponds to the Laplace image

Believing
, we find the complex spectral density

Considering that

We finally find

Spectral density module

This follows from the direct Fourier transform.

The qualitative appearance of the spectral density will be the same as that of a rectangular video pulse. Only the level of the side lobes will be significantly lower.

1
1. Spectrum of a cosine pulse

Determination of spectrum width and pulse duration

Since the signal has a limited duration, theoretically its spectrum is always infinite. Therefore, in practice, the width of the signal spectrum is determined based on the frequency region in which most of the pulse energy is concentrated (90%, 95%, 99%).

In the general case, the spectrum width and pulse duration are determined from Parseval’s equality

Spectrum width
and pulse duration (it is assumed that the pulse starts from zero time) are found from the conditions

Magnitude

Spectrum width
and the rate of decrease of the side lobes
various impulses

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The concept of "signal" can be interpreted in different ways. This is a code or sign transmitted into space, an information carrier, a physical process. The nature of the alerts and their relationship to noise influences its design. Signal spectra can be classified in several ways, but one of the most fundamental is their variation over time (constant and variable). The second main classification category is frequencies. If we consider in more detail in the time domain, among them we can distinguish: static, quasi-static, periodic, repeating, transitional, random and chaotic. Each of these signals has certain properties that can influence related design decisions.

Signal types

Static by definition is unchanged over a very long period of time. Quasi-static is determined by the DC level, so it must be handled in low-drift amplifier circuits. This type of signal does not occur at radio frequencies because some such circuits can produce a constant voltage level. For example, a continuous wave alert with constant amplitude.

The term "quasi-static" means "almost unchanged" and therefore refers to a signal that changes unusually slowly over a long period of time. It has characteristics more similar to static alerts (persistent) than dynamic ones.

Periodic signals

These are the ones that are accurately repeated on a regular basis. Examples of periodic waveforms include sine, square, sawtooth, triangle, etc. The nature of a periodic waveform indicates that it is identical at similar points along a time line. In other words, if the time line advances exactly one period (T), then the voltage, polarity, and direction of the waveform change will repeat. For the voltage waveform, this can be expressed by the formula: V (t) = V (t + T).

Repeated signals

They are quasi-periodic in nature, so they have some similarities to a periodic waveform. The main difference between them is found by comparing the signal at f(t) and f(t+T), where T is the warning period. Unlike periodic alerts, in repeating sounds these points may not be identical, although they will be very similar, as will the overall waveform. The alert in question may contain either temporary or stable symptoms, which vary.

Transient signals and pulse signals

Both types are either a one-time event or a periodic event in which the duration is very short compared to the period of the waveform. This means that t1<<< t2. Если бы эти сигналы были переходными процессами, то в радиочастотных схемах намеренно генерировались бы в виде импульсов или переходного режима шума. Таким образом, из вышеизложенной информации можно сделать вывод, что фазовый спектр сигнала обеспечивает колебания во времени, которые могут быть постоянными или периодическими.

Fourier series

All continuous periodic signals can be represented by a fundamental sine wave of frequency and a set of cosine harmonics that sum linearly. These vibrations contain swell forms. An elementary sine wave is described by the formula: v = Vm sin(_t), where:

• v - instantaneous amplitude.
• Vm - peak amplitude.
• "_" - angular frequency.
• t - time in seconds.

The period is the time between repetitions of identical events or T = 2 _ / _ = 1 / F, where F is the frequency in cycles.

The Fourier series that makes up the waveform can be found if a given quantity is decomposed into its component frequencies, either by a bank of frequency-selective filters or by a digital signal processing algorithm called fast transform. The method of building from scratch can also be used. The Fourier series for any waveform can be expressed by the formula: f(t) = a o/2+ _ n -1.

9. Properties of the Fourier transform. Properties of linearity, changes in time scale, others. Derivative spectrum theorem. Theorem on the spectrum of the integral.

10. Discrete Fourier transform. Interference with radio reception. Classification of interference.

Discrete Fourier transform can be obtained directly from the integral transformation of the argument discretizations (t k = kt, f n = nf):

S(f) =s(t) exp(-j2ft) dt, S(f n) = ts(t k) exp(-j2f n kt), (6.1.1)

s(t) =S(f) exp(j2ft) df, s(t k) = fS(f n) exp(j2nft k). (6.1.2)

Let us recall 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. We should also not forget that the values ​​(6.1.1) of the number series S(f n) are discretizations of the continuous function S"(f) of the spectrum of the discrete function s(t k), as well as the values ​​(6.1.2) of the number series s(t k) are a discretization of the continuous function s"(t), and when reconstructing these continuous functions S"(f) and s"(t) from their discrete samples, the correspondence S"(f) = S(f) and s"(t) = s (t) is guaranteed only if the Kotelnikov-Shannon theorem is satisfied.

For discrete transformations s(kt)  S(nf), both the function and its spectrum are discrete and periodic, and the numerical arrays of their representation correspond to the task on the main periods T = Nt (from 0 to T or from - Т/2 to Т/2), and 2f N = Nf (from -f N to f N), where N is the number of samples, in this case:

f = 1/T = 1/(Nt), t = 1/2f N = 1/(Nf), tf = 1/N, N = 2Tf N . (6.1.3)

Relations (6.1.3) are conditions for informational equivalence of dynamic and frequency forms of representation of discrete signals. In other words: the number of samples of the function and its spectrum must be the same. But each sample of the complex spectrum is represented by two real numbers and, accordingly, the number of samples of the complex spectrum is 2 times greater than the function samples? This is true. However, the representation of the spectrum in complex form is nothing more than a convenient mathematical representation of the spectral function, the real samples of which are formed by adding two conjugate complex samples, and complete information about the spectrum of the function in complex form is contained only in one half of it - the samples of the real and imaginary parts of complex numbers in frequency interval from 0 to f N, because the information of the second half of the range from 0 to -f N is conjugate with the first half and does not carry any additional information.

When representing signals discretely, the argument t k is usually filled with sample numbers k (by default t = 1, k = 0.1,...N-1), and Fourier transforms are performed on the argument n (frequency step number) at the main periods. For values ​​of N that are multiples of 2:

S(f n)  S n = s k exp(-j2kn/N), n = -N/2,…,0,…,N/2. (6.1.4)

s(t k)  s k = (1/N)S n exp(j2kn/N), k = 0.1,…,N-1. (6.1.5)

The main period of the spectrum in (6.1.4) for cyclic frequencies from -0.5 to 0.5, for angular frequencies from - to . For an odd value of N, the boundaries of the main period in frequency (values ​​f N) are half the frequency step behind the samples (N/2) and, accordingly, the upper summation limit in (6.1.5) is set equal to N/2.

In computing operations on a computer, to eliminate negative frequency arguments (negative values ​​of numbers n) and use identical algorithms for direct and inverse Fourier transforms, the main period of the spectrum is usually taken in the range from 0 to 2f N (0  n  N), and the summation in (6.1 .5) is produced accordingly from 0 to N-1. It should be taken into account that the complex conjugate samples S n * of the interval (-N,0) of the two-sided spectrum in the interval 0-2f N correspond to the samples S N+1- n (i.e., the conjugate samples in the interval 0-2f N are the samples S n and S N+1- n).

Example: On the interval T= ,N=100, a discrete signal is given s(k) =(k-i) - a rectangular pulse with unit values ​​at points k from 3 to 8. The shape of the signal and the modulus of its spectrum in the main frequency range, calculated by the formula S(n) = s(k)exp(-j2kn/100) with numbering from -50 to +50 with frequency steps, respectively, =2/100, are shown in Fig. 6.1.1.

Rice. 6.1.1. Discrete signal and module of its spectrum.

In Fig. 6.1.2 shows the envelope of values ​​of another form of representation of the main range of the spectrum. Regardless of the form of representation, the spectrum is periodic, which is easy to verify if you calculate the spectrum values ​​for a larger interval of the argument n while maintaining the same frequency step, as shown in Fig. 6.1.3 for the envelope of spectrum values.

Rice. 6.1.2. Spectrum module. Rice. 6.1.3. Spectrum module.

In Fig. 6.1.4. shows the inverse Fourier transform for a discrete spectrum, performed using the formula s"(k) =(1/100)S(n)exp(j2kn/100), which shows the periodization of the original function s(k), but the main periodk=( 0.99) of this function completely coincides with the original signal s(k).

Rice. 6.1.4. Inverse Fourier transform.

Transformations (6.1.4-6.1.5) are called discrete Fourier transforms (DFT). For DFT, in principle, all properties of integral Fourier transforms are valid, but the periodicity of discrete functions and spectra should be taken into account. The product of the spectra of two discrete functions (when performing any operations when processing signals in frequency representation, such as filtering signals directly in frequency form) will correspond to the convolution of periodized functions in time representation (and vice versa). Such a convolution is called cyclic (see Section 6.4) and its results at the end sections of information intervals can differ significantly from the convolution of finite discrete functions (linear convolution).

From the DFT expressions one can see that to calculate each harmonic, N complex multiplication and addition operations are needed and, accordingly, N 2 operations are required to complete the DFT. With large volumes of data, this can lead to significant time costs. Acceleration of calculations is achieved by using the fast Fourier transform.

Interference is usually called extraneous electrical disturbances that interfere with the transmitted signal and make it difficult to receive. If the interference intensity is high, reception becomes almost impossible.

Interference classification:

a) interference from neighboring radio transmitters (stations);

b) interference from industrial installations;

c) atmospheric interference (thunderstorms, precipitation);

d) interference caused by the passage of electromagnetic waves through layers of the atmosphere: the troposphere, ionosphere;

e) thermal and shot noise in elements of radio circuits, caused by the thermal movement of electrons.

Mathematically, the signal at the receiver input can be represented either as the sum of the transmitted signal and interference, and then the interference is called additive, or just noise, or in the form of a product of the transmitted signal and interference, and then such interference is called multiplicative. This interference leads to significant changes in the signal intensity at the receiver input and explains such phenomena as fading.

The presence of interference makes it difficult to receive signals when the intensity of interference is high, signal recognition can become almost impossible. The ability of a system to withstand the interfering effects of noise is called noise immunity.

External natural active interference is noise resulting from radio emission from the earth's surface and space objects, and the operation of other radio-electronic equipment. A set of measures aimed at reducing the influence of mutual interference between electronic zones is called electromagnetic compatibility. This complex includes both technical measures to improve radio equipment, the choice of signal shape and method of processing it, and organizational measures: frequency regulation, distribution of radio electronic zones in space, standardization of the level of out-of-band and spurious emissions, etc.

11. Sampling of continuous signals. Kotelnikov's theorem (samples). The concept of Nyquist frequency. The concept of sampling interval.

Sampling of analog signals. Kotelnikov series

Any continuous message s(t) occupies a finite time interval T With, can be transmitted with sufficient accuracy by a finite number N samples (samples) s(nT), i.e. a sequence of short pulses separated by a pause.

Time sampling of messages is a procedure consisting of replacing an uncountable set of instantaneous signal values ​​with their countable (discrete) set, which contains information about the values ​​of a continuous signal at certain points in time.

With a discrete method of transmitting a continuous message, it is possible to reduce the time during which the communication channel is busy transmitting this message, with T With to , where is the duration of the pulse used to transmit the sample; It is possible to simultaneously transmit several messages over a communication channel (time multiplexing of signals).

The simplest is the discretization method, based on the theorem of V.A. Kotelnikov, formulated for signals with a limited spectrum (sampling theorem):

if the highest frequency in the spectrum of the function s(t) is less than F m , then the function s(t) is completely determined by the sequence of its values ​​at moments separated from each other by no more than seconds and can be represented side by side:

.

Here the value denotes the interval between readings on the time axis, and

Sampling time - signal value at the time of counting.

Series (1) is called the Kotelnikov series, and samples (samples) of the signal ( s(nT)) is sometimes called the time spectrum of the signal.

has the following properties:

a) at a point t=nT the function is equal to 1, because at this point the argument of the function is 0, and its value is 1;

b) at points t=kT, function, because the argument of the sine at these points is equal, and the sine itself is equal to zero;

c) spectral density function u n (nT) uniform across the frequency band and equal. This conclusion is made on the basis of the reciprocity theorem of frequency and time of a pair of Fourier transforms. The phase response characteristic of the spectral density is linear and equal to (in accordance with the signal shift theorem). Thus,

.

Time and frequency representations of a function u n (t) are given in Fig. 3.

A graphical interpretation of the Kotelnikov series is presented in Fig. 4.

The Kotelnikov series (1) has all the properties of a generalized Fourier series with basis functions u n (nT), and therefore defines the function s(t) not only at reference points, but also at any point in time.

Function orthogonality interval u n equals infinity. Norm square

The coefficients of the series, determined by the general formula for the Fourier series, are equal (using Parseval’s equality):

hence

When the signal spectrum is limited to the finite highest frequency, series (1) converges to the function s(t) at any value t.

If we take the interval T between samples is less than , then the width of the spectrum of the basis function will be greater than the width of the spectrum of the signal, therefore the accuracy of signal reproduction will be higher, especially in cases where the signal spectrum is not limited in frequency and the highest frequency F m one has to choose from energy or information considerations, leaving the “tails” of the signal spectrum unaccounted for.

As the distance between samples () increases, the spectrum of the basis function becomes narrower than the signal spectrum, the coefficients C n will be samples of another function s 1 (t), the spectrum of which is limited by frequency.

If the signal duration T c is finite, then its frequency band is strictly infinity, because the conditions of finite duration and band are incompatible. However, it is almost always possible to select the highest frequency so that the “tails” contain either a small fraction of energy or have little effect on the shape of the analog signal. With this assumption, the number of samples N on time T With will be equal T With /T, i.e. N=2F m T c. Series (1) in this case has limits 0 , N.

Number N sometimes called the number of degrees of freedom of the signal, or base signal. As the base increases, the accuracy of reconstructing an analog signal from a discrete one increases.

12. Time and frequency characteristics of linear radio circuits. The concept of impulse response. The concept of transient response. The concept of input and transfer frequency characteristics.

When considering radio signals, it was found that the signal can be presented both in the time (dynamic representation) and in the frequency (spectral representation) domains. Obviously, when analyzing signal conversion processes, the circuits must also have appropriate descriptions by time or frequency characteristics.

Let's start by considering the timing characteristics of linear circuits with constant parameters. If a linear circuit carries out a transformation in accordance with the operator and a signal is applied to the input of the circuit in the form of a delta function (in practice a very short pulse), then the output signal (reaction of the circuit)

called impulse response chains. Impulse response forms the basis of one of the methods for analyzing signal conversion, which will be discussed below.

If a signal is received at the input of a linear circuit, i.e. signal of the form “single drop”, then the output signal of the circuit

called step response.

There is an unambiguous relationship between the impulse and the transient response. Since the delta function (see subsection 1.3):

,

then substituting this expression into (5.5), we obtain:

In turn, the transition characteristic

. (5.8)

Let's move on to consider the frequency characteristics of linear circuits. Apply the direct Fourier transform to the input and output signals

The ratio of the complex spectrum of the output signal to the complex spectrum of the input signal is called complex transfer coefficient

(5.9)

It follows that

Thus, operator When converting a signal by a linear circuit in the frequency domain, the complex transmission coefficient is used.

Let us represent the complex transmission coefficient in the form

where and are the modulus and argument of the complex function, respectively. The modulus of the complex transfer coefficient as a function of frequency is called amplitude-frequency characteristic (AFC), and the argument is phase-frequency characteristic (PFC). The amplitude-frequency response is even, and the phase-frequency characteristic is odd function of frequency.

The time and frequency characteristics of linear circuits are interconnected by the Fourier transform

which is understandable, since they describe the same object - a linear chain.

13. Analysis of the impact of deterministic signals on linear circuits with constant parameters. Time, frequency, operator methods.