Amplitude and phase spectra of signals. Spectra of amplitudes and phases of periodic signals Amplitude Fourier spectrum
) we are familiar with the concept harmonic (sinusoidal) functions. Are there non-harmonic functions and signals and how to work with them? This is what we have to figure out today 🙂
Harmonic and non-harmonic signals.
And to begin with, let's take a closer look at how signals are classified. First of all, we are interested in harmonic signals, the shape of which repeats after a certain time interval, called a period. Periodic signals, in turn, are divided into two large classes - harmonic and non-harmonic. A harmonic signal is a signal that can be described by the following function:
Here, is the signal amplitude, is the cyclic frequency, and is the initial phase. You ask - what about the sine? Isn't a sinusoidal signal harmonic? Of course, it is, the fact is that, that is, the signals differ in the initial phase, respectively, the sinusoidal signal does not contradict the definition that we gave for harmonic oscillations 🙂
The second subclass of periodic signals are non-harmonic vibrations. Here is an example of a non-harmonic signal:
As you can see, despite the “non-standard” shape, the signal remains periodic, that is, its shape repeats after a time interval equal to the period.
To work with such signals and study them, there is a certain technique, which consists in decomposing the signal into Fourier series. The essence of the technique is that a non-harmonic periodic signal (under certain conditions) can be represented as a sum of harmonic oscillations with certain amplitudes, frequencies and initial phases. An important nuance is that all harmonic oscillations that are involved in the summation must have frequencies that are multiples of the frequency of the original non-harmonic signal. Perhaps this is not entirely clear yet, so let's look at a practical example and understand a little more 🙂 For example, we use the signal that is shown in the figure just above. It can be represented as follows:
Let's plot all these signals on one chart:
Functions are called harmonics signal, and the one whose period is equal to the period of the non-harmonic signal is called first or fundamental harmonic. In this case, the first harmonic is a function (its frequency is equal to the frequency of the studied non-harmonic signal, respectively, and their periods are equal). And the function is nothing more than the second harmonic of the signal (its frequency is twice as high). In general, a non-harmonic signal is decomposed into an infinite number of harmonics:
In this formula, is the amplitude, and is the initial phase of the kth harmonic. As we mentioned a little earlier, the frequencies of all harmonics are multiples of the frequency of the first harmonic, in fact, this is what we see in this formula 🙂 - this is the zero harmonic, its frequency is 0, it is equal to the average value of the function over the period. Why average? Look - the average value of the sine function for the period is 0, which means that when averaging in this formula, all terms, except for them, will be equal to 0.
The totality of all harmonic components of a non-harmonic signal is called spectrum this signal. There are phase and amplitude spectrum of the signal:
- phase spectrum of the signal - a set of initial phases of all harmonics
- amplitude spectrum of the signal - the amplitudes of all harmonics that make up the non-harmonic signal
Let's look at the amplitude spectrum in more detail. For a visual image of the spectrum, diagrams are used, which are a set of vertical lines of a certain length (the length depends on the amplitude of the signals). Harmonic frequencies are plotted on the horizontal axis of the diagram:
On the horizontal axis, both frequencies in Hz, and simply numbers of harmonics, as in this case, can be plotted. And along the vertical axis - the amplitudes of the harmonics, everything is clear here :). Let's build the amplitude spectrum of the signal for the non-harmonic oscillation, which we considered as an example at the very beginning of the article. I remind you that its expansion in a Fourier series is as follows:
We have two harmonics whose amplitudes are 2 and 1.5, respectively. Therefore, there are two lines in the diagram, the lengths of which correspond to the amplitudes of harmonic oscillations.
The phase spectrum of the signal is constructed in a similar way, with the only difference that the initial phases of the harmonics are used, and not the amplitudes.
So, we figured out the construction and analysis of the amplitude spectrum of the signal, let's move on to the next topic of today's article - the concept of amplitude-frequency response.
Amplitude-frequency characteristic (AFC).
Frequency response is the most important characteristic of many circuits and devices - filters, sound amplifiers, etc. Even simple headphones have their own frequency response. What does she show?
Frequency response is the dependence of the amplitude of the output signal on the frequency of the input signal.
As we found out in the first part of the article, a non-harmonic periodic signal can be expanded into a Fourier series. But now we are primarily interested in the audio signal, and it looks like this: 
As you can see, there is no question of any periodicity here 🙂 But, fortunately, there are special algorithms that allow you to represent an audio signal in the form of a spectrum of frequencies included in it. We will not analyze these algorithms in detail now, this is a topic for a separate article, we will simply accept the fact that they allow us to perform such a conversion with an audio signal 🙂
Accordingly, we can plot the amplitude spectrum of the audio signal. And passing through any circuit (for example, through headphones when playing sound), the signal will be changed. So the amplitude-frequency characteristic just shows what changes the input signal will undergo when passing through a particular circuit. Let's discuss this point a little more...
So, at the input we have a number of harmonics. The amplitude-frequency characteristic shows how the amplitude of a particular harmonic will change when passing through the circuit. Consider an example of frequency response:
Let's figure it out step by step what is shown here ... Let's start with the axes of the frequency response graph. On the y-axis, we plot the value of the output voltage (or gain, as in this figure). We set the gain factor in dB, respectively, a value equal to 0 dB corresponds to a gain of 1, that is, the signal amplitude remains unchanged. The x-axis plots the frequencies of the input signal. Thus, in the case under consideration, for all harmonics whose frequencies lie in the range from 100 to 10000 Hz, the amplitude will not change. And the signals of all other harmonics will be attenuated.
Frequencies and are separately marked on the graph - their distinctive feature is that the signal of harmonics of these frequencies will be attenuated by 1.41 times (3 dB) in voltage, which corresponds to a 2-fold decrease in power. The frequency band between and is called the passband. It turns out the following situation - the signals of all harmonics, the frequencies of which lie within the bandwidth of the device / circuit, will be attenuated by less than 2 times in power.
The frequency range of audio devices is usually broken down into low, medium and high frequencies. Approximately it looks like this:
- 20 Hz - 160 Hz - low frequency region
- 160 Hz - 1.28 kHz - midrange
- 1.28 kHz - 20.5 kHz - high frequency region
It is this terminology that can usually be found in various equalizer programs used to adjust the sound. Now you know that beautiful graphs from such programs are precisely the amplitude-frequency characteristics that we met in today's article 🙂
At the end of the article, let's look at a couple of frequency responses obtained in a software equalizer:
Here we can see the frequency response of the amplifier. Moreover, the mid-range frequencies will be predominantly amplified.
But here the situation is completely different - low and high frequencies are amplified, and in the mid-frequency region for harmonics with a frequency of 500 Hz, we observe a significant attenuation.
And here only low frequencies are amplified. Audio equipment with such a frequency response will have a high level of bass 🙂
This concludes our today's article, thank you for your attention and we are waiting for you on our website again!
2.1. Spectra of Periodic Signals
A periodic signal (current or voltage) is called such a type of influence when the waveform repeats after a certain time interval T which is called the period. The simplest form of a periodic signal is a harmonic signal or a sine wave, which is characterized by amplitude, period, and initial phase. All other signals will inharmonious or non-sinusoidal. It can be shown, and practice proves it, that if the input signal of the power supply is periodic, then all other currents and voltages in each branch (output signals) will also be periodic. In this case, the waveforms in different branches will differ from each other.
There is a general technique for studying periodic non-harmonic signals (input actions and their reactions) in an electrical circuit, which is based on the decomposition of signals into a Fourier series. This technique consists in the fact that it is always possible to select a number of harmonic (i.e. sinusoidal) signals with such amplitudes, frequencies and initial phases, the algebraic sum of the ordinates of which at any time is equal to the ordinate of the studied non-sinusoidal signal. So, for example, the voltage u in fig. 2.1. can be replaced by the sum of stresses and , since at any time the identical equality takes place: 
. Each of the terms is a sinusoid, the oscillation frequency of which is related to the period T integer ratios.
For the example under consideration, we have the period of the first harmonic coinciding with the period of the non-harmonic signalT 1 = T, and the period of the second harmonic is two times smallerT 2 = T/2, i.e. instantaneous values of harmonics should be written as:
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Here, the amplitudes of harmonic oscillations are equal to each other ( 
), and the initial phases are equal to zero.
Rice. 2.1. Example of addition of the first and second harmonics
non-harmonic signal
In electrical engineering, a harmonic component whose period is equal to the period of a non-harmonic signal is called first or basic signal harmonics. All other components are called higher harmonic components. A harmonic whose frequency is k times greater than the first harmonic (and the period, respectively, k times less) is called
k - th harmonic. Allocate also the average value of the function for the period, which is called null harmonica. In the general case, the Fourier series is written as the sum of an infinite number of harmonic components of different frequencies:
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(2.1) |
where k is the harmonic number; - angular frequency of the k - th harmonic;
ω 1 \u003d ω \u003d 2 π / T- angular frequency of the first harmonic; - zero harmonic.
For commonly occurring waveforms, a Fourier series expansion can be found in the specialized literature. Table 2 shows the expansions for eight waveforms. It should be noted that the expansions given in Table 2 will take place if the origin of the coordinate system is chosen as indicated in the figures on the left; when changing the origin of time t the initial phases of the harmonics will change, while the amplitudes of the harmonics will remain the same. Depending on the type of signal under study, V should be understood as either a value measured in volts if it is a voltage signal, or a value measured in amperes if it is a current signal.
Fourier series expansion of periodic functions
table 2
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Schedule f(t) |
Fourier series of functionsf(t) |
Note |
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k=1,3,5,... |
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k=1,3,5,... |
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k=1,3,5,... |
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k=1,2,3,4,5 |
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k=1,3,5,... |
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k=1,2,3,4,5 |
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S=1,2,3,4,.. |
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k=1,2,4,6,.. |
Signals 7 and 8 are generated from a sinusoid by gate circuits.
The set of harmonic components that form a non-sinusoidal signal is called the spectrum of this non-harmonic signal. From this set of harmonics, they distinguish and distinguish amplitude And phase range. The amplitude spectrum is a set of amplitudes of all harmonics, which is usually represented by a diagram in the form of a set of vertical lines, the lengths of which are proportional (in the chosen scale) to the amplitude values of the harmonic components, and the place on the horizontal axis is determined by the frequency (harmonic number) of this component. Similarly, phase spectra are considered as a set of initial phases of all harmonics; they are also shown to scale as a set of vertical lines.
It should be noted that it is customary to measure the initial phases in electrical engineering in the range from -180 0 to +180 0. Spectra consisting of individual lines are called lined or discrete. Spectral lines are at a distance f apart, where f- frequency interval equal to the frequency of the first harmonic f.Thus, the discrete spectra of periodic signals have spectral components with multiple frequencies - f, 2f, 3f, 4f, 5f etc.
Example 2.1. Find the amplitude and phase spectrum for a rectangular signal, when the durations of the positive and negative signals are equal, and the average value of the function over the period is zero
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u(t) = Vat 0<t<T/2 |
u(t) = -Vat T/2<t<T |
For signals of simple, frequently used forms, it is advisable to find a solution using tables.
Rice. 2.2. Linear amplitude spectrum of a rectangular signal
From the Fourier expansion of a rectangular signal (see Tables 2 - 1), it follows that the harmonic series contains only odd harmonics, while the amplitudes of the harmonics decrease in proportion to the number of the harmonic. The amplitude line spectrum of harmonics is shown in fig. 2.2. When constructing, it is assumed that the amplitude of the first harmonic (here voltage) is equal to one volt: B; then the amplitude of the third harmonic will be equal to B, the fifth - B, etc. The initial phases of all harmonics of the signal are equal to zero, therefore, the phase spectrum has only zero values of the ordinates.
Problem solved.
Example 2.2.Find the amplitude and phase spectrum for a voltage that varies according to the law: at - T/4<t<T/4; u(t) = 0 for T/4<t<3/4T. Such a signal is formed from a sinusoid by eliminating (by circuitry using valve elements) the negative part of the harmonic signal.
a) b)
Rice. 2.3. The line spectrum of a half-wave rectification signal: a) amplitude; b) phase
For a half-wave rectification signal of a sinusoidal voltage (see Tables 2 - 8), the Fourier series contains a constant component (zero harmonic), the first harmonic, and then a set of only even harmonics, the amplitudes of which rapidly decrease with increasing harmonic number. If, for example, we put the value V = 100 B, then, multiplying each term by the common factor 2V/π , we find(2.2)
The amplitude and phase spectra of this signal are shown in Fig. 2.3a,b.
Problem solved.
In accordance with the theory of Fourier series, the exact equality of a non-harmonic signal to the sum of harmonics takes place only for an infinitely large number of harmonics. The calculation of harmonic components on a computer allows you to analyze any number of harmonics, which is determined by the purpose of the calculation, the accuracy and form of non-harmonic effects. If the duration of the signalt regardless of its shape, much less period T, then the amplitudes of the harmonics will decrease slowly, and for a more complete description of the signal, it is necessary to take into account a large number of terms in the series. This feature can be traced for the signals presented in Tables 2 - 5 and 6, provided that the condition τ <<T. If a non-harmonic signal is close to a sinusoid in shape (for example, signals 2 and 3 in Table 2), then the harmonics decrease rapidly, and for an accurate description of the signal, it is enough to limit ourselves to three to five harmonics of the series.
The set of harmonics that form the Fourier series (4.10) in trigonometric form is called spectrum of a periodic signal, and the sets of amplitudes Umk and initial phases of these harmonics - spectra amplitudes and phases. Each harmonica:
can be displayed with two vertical lines. To do this, on one frequency axis, it is necessary to plot the value of the frequency of this harmonic and draw a vertical line with a height equal to the amplitude of the harmonic, then on the other frequency axis at the frequency of the same harmonic, draw a second vertical line equal in height to the initial phase of the harmonic.
The Fourier series (4.3) can be rewritten as
Given that the cosine function is periodic with a period 2 = 360°, i.e. its values repeat through 360°, you can subtract an integer number of periods from the phase of the harmonic components. Then we obtain one more form of the series (4.3):
These series can be represented graphically. The harmonics of this signal, which are included in formula (4.3), are shown in the timing diagrams in fig. 4.1, b-d. Another way to graphically represent the components of the Fourier series for the signal in Fig. 4.1, but is shown in fig. 4.5, A–V. The amplitudes of the harmonics decrease according to the law , Where P is the number of the harmonic, and the phases of the harmonics change according to the law n where is the phase of the first harmonic.
For a periodic sequence of rectangular pulses shifted by a quarter of the period (Fig. 4.3, A) the formula of the Fourier series (4.6) can be modified if we remember that the minus sign in front of the harmonic oscillation means the rotation of the oscillation in phase by 180 °:
Rice. 4.5. Amplitudes and phases of signal harmonics (4.12) and (4.13)
The initial phases of oscillations in the series (4.14) alternately take on the values 0 and 180°. The graphic representation of series (4.14) is given in fig. 4.5, a and b.
The vertical lines in fig. 4.5 and 4.6 were named spectral lines, and the sets of these lines, or, what is the same, the sets of amplitudes of the phase harmonics in (4.10), form amplitude and phase spectra this signal.
Rice. 4.6. Amplitudes and phases of signal harmonics (4.14)
Radio engineers are familiar with devices - spectrum analyzers that respond to each harmonic that is part of a signal of complex shape and allow them to be measured.
Thus, the amplitude spectrum is a set of harmonic amplitudes , , , ... (including the constant and the main components), included in the Fourier series written in trigonometric form (4.10), and the phase spectrum is a set of initial phases,, ... of these harmonics. The complex amplitudes from (4.12) form the complex spectrum of the signal u(t).
Analysis of the spectral (harmonic) composition of periodic signals is the calculation of the amplitudes and initial phases of the harmonic components of the Fourier series. Usually, to calculate these quantities, the form of the Fourier series (4.2) is used:
Let us show that the notation (4.15) is equivalent to notation (4.7).
It follows from the above reasoning that, in order to analyze the spectral composition of a signal, it suffices to know how to calculate the quantities , U" mn And U’ mn in expression (4.15).
From formulas (4.2) we know that the constant component of the series is calculated as the average value of the function:
Odds U" mk And U"" mk calculated as weighted averages with weights cos k and sin, respectively:
Because the, 
That
Applying the Euler formula
we finally obtain the expression for the complex spectrum of the signal:
The spectrum of the signal is affected not only by the shape of the signal, but also by its parameters. It is best to consider this effect on a specific example, and the easiest way is on the example of a periodic sequence of rectangular pulses. In a fairly general case, this sequence is shown in Fig. 4.7, A. The pulse repetition period is indicated T", and the ratio of the period to the duration of the pulses "is called the duty cycle and denote.
The calculation of the coefficients of the Fourier series in trigonometric form using formulas (4.16) - (4.18) leads us to write (see Table 4.1)
Where U 0 =U/ q And
Rice. 4.7. Periodic sequence of rectangular pulses with duty cycle q= 3 and its spectrum
The amplitude spectrum of such a periodic sequence with a duty cycle q= 3 is shown in fig. 4.7, b.
For values k, multiples of duty cycle q pulse sequence, the function takes zero values and harmonics with these numbers have zero amplitudes (in our example with k= 3.6, 9, ...). The frequency of the first harmonic is determined by the formula
For harmonicas with numbers k, for which the amplitude is positive, the phase angle is equal to zero; for harmonicas same with numbers k, for which the value turns out to be negative, the phase angle takes on the value of 180 ° (Fig. 4.7, c).
Let us consider the influence on the spectrum of a sequence of rectangular pulses of such parameters as the period and duration of the pulse.
First of all, the frequency of the fundamental harmonic depends on the value of the period, i.e. its location on the spectrum. If we, for example, increase the period of the pulse sequence (Fig. 4.7, A), then the frequency of the first harmonic will decrease.
This will lead to a thickening of the spectral lines (Fig. 4.8, b And V). The duty cycle of the pulses will also increase with increasing period (in our example q= 5), therefore, harmonics with higher numbers that are multiples of q (k= 5, 10, 15, ...). The amplitudes of all harmonics will decrease.
Rice. 4.8. A sequence of rectangular pulses with a duty cycle q= 5 and its spectrum
On the other hand, if the sequence period is left unchanged (for example, ), and the pulse duration, say, is reduced (for example, to the value , as in fig. 4.9 A), then the first harmonic will not change its location in the signal spectrum. With an increase in the duty cycle, as before, harmonics with numbers that are multiples of q (in Fig. 4.8, b at k= 5,10,15,… ).
Rice. 4.9. Influence of pulse duration on the signal spectrum
Rice. 4.10. Influence of pulse duration and their repetition period on the signal spectrum
On fig. 4.10, the case is shown when both the period and the duration of the pulse have been changed. We invite readers to analyze this situation for themselves. Examples of solving problems for calculating periodic signals are also given in.
Although we have analyzed rather particular examples, the characteristic behavior of the spectrum is also observed for other types of periodic pulse sequences. It consists in the following:
As the sequence period increases T the frequency of the first harmonic decreases and the spectral lines thicken; on the contrary, as the period decreases, the frequency of the first harmonic increases and the spectral lines become rarer;
The shorter the pulses in the sequence, the slower they decrease with increasing number P amplitudes of harmonics; on the contrary, the wider the pulses, the faster the amplitudes of higher harmonics decrease.
The main provisions of the materials set out in clause 4.2.
The signal is called periodical, if its form is cyclically repeated in time. A periodic signal is generally written as follows:
Here is the period of the signal. Periodic signals can be both simple and complex.
For the mathematical representation of periodic signals with a period, this series is often used, in which harmonic (sinusoidal and cosine) oscillations of multiple frequencies are chosen as the basis functions:
Where . - fundamental angular frequency of the sequence of functions. With harmonic basis functions, from this series we obtain a Fourier series, which in the simplest case can be written in the following form:
where coefficients
It can be seen from the Fourier series that, in the general case, a periodic signal contains a constant component and a set of harmonic oscillations of the fundamental frequency and its harmonics with frequencies . Each harmonic oscillation of the Fourier series is characterized by amplitude and initial phase.
Spectral diagram and spectrum of a periodic signal.
If any signal is presented as a sum of harmonic oscillations with different frequencies, then this means that spectral decomposition signal.
Spectral diagram signal is called a graphical representation of the coefficients of the Fourier series of this signal. There are amplitude and phase diagrams. To build these diagrams, on a certain scale, harmonic frequencies are plotted along the horizontal axis, and their amplitudes and phases are plotted along the vertical axis. Moreover, the amplitudes of the harmonics can take only positive values, the phases - both positive and negative values in the interval .
Spectral diagrams of a periodic signal:
a) - amplitude; b) - phase.
Signal spectrum- this is a set of harmonic components with specific values of frequencies, amplitudes and initial phases, forming a signal in total. In practice, spectral diagrams are called more briefly - amplitude spectrum, phase spectrum. The greatest interest is shown to the amplitude spectral diagram. It can be used to estimate the percentage of harmonics in the spectrum.
Spectral characteristics in telecommunication technology play an important role. Knowing the spectrum of the signal, you can correctly calculate and set the bandwidth of amplifiers, filters, cables and other nodes of communication channels. Knowledge of signal spectra is necessary for building multichannel systems with frequency division of channels. Without knowing the interference spectrum, it is difficult to take measures to suppress it.
From this we can conclude that the spectrum must be known for the implementation of undistorted signal transmission over a communication channel, to ensure signal separation and interference mitigation.
To observe the spectra of signals, there are devices called spectrum analyzers. They allow you to observe and measure the parameters of individual components of the spectrum of a periodic signal, as well as measure the spectral density of a continuous signal.
In this section, we will consider the description of two-dimensional time series in the frequency domain. It will be shown that the sample cross-covariance function discussed in the previous section has a Fourier transform called the sample cross spectrum. This spectrum is a complex-valued function that can be written as a product of a real function, called the sample cross-amplitude spectrum, and a complex-valued function, called the sample phase spectrum. Similarly, the Fourier transform of a theoretical cross covariance function is called the cross spectrum. It can be represented as a product of the mutual amplitude and phase spectra. The cross amplitude spectrum shows how large the amplitudes of the coupled frequency components are in two rows at a particular frequency. Similarly, the phase spectrum shows how far behind or ahead in phase such a component in one of the rows is the corresponding component in the other row for a given frequency. The following section gives examples of cross amplitude and phase spectra obtained from the cross spectrum of the two-dimensional linear process (8.1.14). Then a somewhat more useful concept than the mutual amplitude spectrum is introduced, namely the coherence spectrum. We will show that the coherence spectrum and the phase spectrum give a complete description of a two-dimensional normal random process.
8.3.1. Applying Fourier Analysis to Bivariate Time Series
Fourier analysis can be applied to two-dimensional time series in the same way as to one-dimensional ones. Suppose, for example, that - two cosine waves of the same frequency but with different amplitudes and phases, i.e.
If the length of the available records is equal to T, then using (2.2.11) we obtain the Fourier transform
Hence, the sample spectra (6.1.6) of these two signals are
These expressions tend to
Thus, the dispersion, or average power of a cosine wave, is distributed as -functions at frequencies
Suppose now that we want to describe the covariance of two cosine waves. In this case, it is natural to use the selective cross power spectrum, or, in short, the selective cross spectrum
where the asterisk denotes complex conjugation. Substituting (8.3.2) into (8.3.3), we find that the sample cross spectrum of two cosine waves is
that when tends to
Definition (8.3.3) is natural, since it contains all the information about the dependence of two signals. In the particular case of cosine waves, equation (8.3.5) shows that this information consists of a phase difference showing how much one of the cosine waves leads the other, and a mutual amplitude showing how large the corresponding amplitudes are in the two signals at a given frequency.
Selective phase and selective mutual amplitude spectra. More generally, assume that arbitrary real signals with Fourier transforms, respectively. These transformations give the amplitude and phase distribution of signals, i.e.
where is a non-negative even function and is an odd function. According to (8.3.3), the sample cross spectrum in this case will be equal to
which can also be written in the form
Therefore, the covariance of two series can be described using the sample phase spectrum
and selective mutual amplitude spectrum
The sampled phase spectrum shows whether the frequency component of one series lags or leads the component of another series at the same frequency. Similarly, the sample cross-amplitude spectrum shows how large the amplitudes of the corresponding components at a certain frequency are in two rows. Note that is a non-negative even function and is an odd frequency function.
Sample cospectrum and quadrature spectrum. Since the function from (8.3.8) is complex-valued, it can be written as a product of amplitude and phase functions, as in (8.3.7). Expression (8.3.8) can also be written in another form, separating the real and imaginary parts:
Note that - an even, - an odd function of frequency due to the fact that - an even, - an odd function. To illustrate, consider the 2D cosine wave example above.
