Pulse modulation (PM) is widely used in radar, in the transmission of telemetry information and in other cases. The signal emitted by the RPDU, modulated by a sequence of rectangular pulses, is shown in fig. 23.1. The spectrum of the radio signal with MI is wide, so it is used in the microwave remote control.
Rice. 23.1. Radiated IM signal
With MI, the signal is determined by the following parameters: t - pulse duration; T - pulse repetition period; q \u003d (T–t) / t - duty cycle; f 0 - carrier frequency; P and - signal power in a pulse; P cf =P and (t/T) - average signal power; Df c p - the width of the spectrum of the emitted signal; type of pulse modulation. Let's reveal the content last parameter. The pulses modulating the carrier frequency f 0 can in turn be modulated themselves. In this case, there are: pulse amplitude modulation (AIM), pulse width modulation (PWM), pulse time modulation (PWM), pulse code modulation (CMM), intrapulse modulation - frequency or phase. The spectrum of the signal in MI is determined in two stages. At the first stage, the spectrum of the periodic sequence of pulses modulating the carrier is determined; at the second stage - the spectrum of the carrier modulated by impulses. With a periodic sequence of rectangular pulses (Fig. 23.1, a), the spectrum can be obtained by expanding the function into a Fourier series. As a result, we obtain for the amplitudes of the components in this spectrum following through the intervals W=2p/T or F=1/T:
, (23.l)
where E is the amplitude of the pulse (Fig. 23.1, a); k is a positive integer.
a:= 0.1 N:= 20 AM:= 1
An example of calculating the line spectrum at AM=E=1, a=t/T=0.1, N=20 is shown in fig. 23.3. From (23.1) and the considered example it follows that at w=2pk/t or f=k/t the amplitude A k =0.
Rice. 23.2 Example of line spectrum calculation for MI
The spectrum of a periodic sequence of radio pulses (Fig. 23.1, b) is similar to the spectrum in fig. 23.2, but symmetrical and offset relative to the origin by the carrier frequency f 0 . An example of the central part of such a spectrum is shown in Fig. 23.3. Theoretically, the width of the spectrum of the considered signal is infinite. However, most of its energy is concentrated in the band Df cn \u003d 6 / t (according to Fig. 23.3, the main and two side "lobes" of the spectrum are taken into account).
Rice. 23.3. An example of the central part of the spectrum of a periodic
sequences of radio pulses
Converting an analog signal to a discrete one is called sampling. The result is a sequence periodic impulses. The simplest kind modulation of this sequence - amplitude-pulse. There are amplitude-pulse modulation of the first (AIM-1) and second kind (AIM-2).
In this term paper it is necessary to carry out AIM of the first kind. In this case, the amplitude of each carrier pulse is determined by the law of change of the modulating signal, i.e.
The following notation is used in the formula:
U0 - amplitude of unmodulated rectangular pulses;
mAIM - pulse modulation depth (AIM coefficient);
Normalized modulating signal;
Sequence of unmodulated pulses, repetition period T0;
The moment of appearance of the k-th impulse is relative to:
where is the start time of the first pulse.
Let us determine the spectrum of the AIM-1 signal if the modulating signal has the form, where is the amplitude of the harmonic signal.
In this case, the expression takes the form:
Since the function is periodic, it can be expanded into a Fourier series. As a result of decomposition, it will take the form:
DC component;
Harmonic amplitude, V;
Circular frequency of the main (first) harmonic of rectangular pulses (sampling frequency), rad/s;
The initial phase of the harmonic.
Substitute the expression into equality and transform:
Thus, the following components are observed in the spectrum of the AIM-1 signal:
DC component;
Carrier;
and - lower and upper side stripes respectively.
Now, based on the obtained formulas, we will perform the calculation for the given numbers of harmonics (1st, 2nd, 3rd, 15th, 30th). Let us give examples of a complete calculation for the zero and first harmonics.
1) DC component:
2) The amplitude of the side spectrum of the constant component:
3) Carrier, lower and upper frequencies:
4) The amplitude of the first harmonic at the carrier frequency:
Side spectrum amplitudes:
- 5) Sideband Frequencies:
- 6.1) left side lane
- 6.1.1) lower frequency:
- 6.1.2) upper frequency:
- 6.2) right side band.
- 6.2.1) lower frequency:
- 6.2.2) upper frequency:
Similarly, the calculation is carried out for the remaining harmonics. For clarity, we summarize the results in table 1. This table contains:
- ? harmonic numbers (marked with a letter in the table);
- ? their respective carrier and side frequencies;
- ? signal amplitudes at specified frequencies (i.e. all carriers and sidebands).
Table 1 - Results of calculating the spectrum of the modulated AIM signal
|
Meaning |
Amplitude value, V |
Value of component frequency, rad/s |
|||||
Based on the data obtained, we construct spectral characteristic. In order to obtain a clear and understandable image on this characteristic, we will break the abscissa axis in two places in compliance with the dimensions. The graph shows that each harmonic has a carrier at the frequency that accounts for most of the energy (large amplitude) and two sidebands. Their lower amplitudes are much smaller, and their upper ones are accepted zero. The values of all amplitudes gradually decrease with increasing harmonic number; so, for the first harmonic, the value of the carrier amplitude is 0.0835 V, and for the thirtieth - 0.06937 V.
The abscissa shows the frequency in radians per second with a scale. Breaks are made on the axis for a more visual representation of the diagram. Maximum value along this axis. The values of the amplitudes of harmonics in volts with a scale are located along the ordinate axis.
With pulse modulation (Fig. 6.1), various periodic pulse sequences are used as a carrier wave (more precisely, a subcarrier), one of the parameters of which contains information about the transmitted message. For discrete signals the modulation process is commonly called the manipulation of pulse parameters.
Pulse modulation. Theoretical basis construction of all methods of pulse modulation is the Kotelnikov theorem, according to which a continuous primary signal e(t) with limited spectrum width F B can be transmitted by its samples (a sequence of short pulses) following at intervals (in radio engineering when presenting pulsed, discrete and digital signals period notation is often used T through D t) T=D t =1/(2 F B). Sufficiently large time intervals between pulses are used to transmit operating pulses from other sources, i.e. to implement multichannel transmission with time division channels. Let us assume that the subcarrier oscillation in the information transmission system with pulse modulation is a periodic sequence of rectangular pulses with amplitude U n, duration t and and repetition period T(fig.6.1, A). For clarity and simplification of mathematical calculations, we choose as a modulating signal harmonic oscillation e(t) = at whom initial phase q 0 \u003d 90 o (Fig. 6.1, b).
Pulse modulation, depending on the choice of the variable parameter of the modulated pulse sequence, is usually divided into the following types:
Amplitude-pulse (AIM), when, according to the law of the transmitted message, the amplitude of the pulses of the original sequence changes (Fig. 6.1, V);
Pulse-width (PWM), when the duration (width) of the pulses of the original sequence changes according to the law of the transmitted message (Fig. 6.1, G);
Phase-pulse (PIM), or time-pulse (VIM), if, according to the law of the transmitted message, the time position of the pulses changes (Fig. 6.1, d);
Frequency-pulse modulation (PFM), when the frequency of the subcarrier pulses changes according to the law of the transmitted message (Fig. 6.1, e);
Pulse code modulation (PCM) is a type of discrete (digital) modulation ( digital manipulation), at which the transmitted analog primary signal is converted into digital code- a sequence of pulses (1 - "ones") and pauses (0 - "zeroes"), having the same duration, is most widely used in modern radio electronics and communication systems. This type of pulse modulation is shown in Fig. 6.1, and.
Amplitude-pulse modulation. As an example that allows us to evaluate the parameters and characteristics of pulse-modulated oscillations, consider the AIM - signal and determine its spectrum when the pulse sequence is modulated by a harmonic oscillation e(t) = E 0 cosW t.
From an analytical point of view, the procedure for obtaining an AIM signal And AIM(t) can be conveniently viewed as a direct multiplication of a continuous transmitted signal And(t) to the auxiliary sequence at(t) rectangular video pulses of unit amplitude.
Rice. 6.1. Pulse modulation:
A- periodic sequence of initial impulses; b– modulating signal; V- AIM; G- PWM; d- FIM; e- CHIM; and– PCM
Let us imagine a periodic sequence of rectangular unmodulated video pulses And(t) having an amplitude U H , duration t and and repetition period T, trigonometric Fourier series. We give the formula for the carrier vibration
And n ( t) = U n cosw 0 t (6.1)
and a generalized function And(t) describing a sequence of rectangular pulses. Then AIM-signal can be written as:
u AIM ( t) = u(t)y(t). (6.2)
u AIM ( t) = (1 + M cos W t)u(t), (6.3)
In this ratio, the parameter M =DU/U m is the coefficient (depth) of pulse modulation. Substituting value And(t) in (6.3), after simple transformations, we write the expression for the AIM signal:
u AIM ( t)= (6.4)
Fig.6.2. Signal spectrum with amplitude-pulse modulation
From relation (6.4) it follows that with single-tone amplitude-pulse modulation of a sequence of rectangular video pulses, the spectrum of the AIM - signal contains a constant component A 0, a harmonic A 0 M of the frequency W of the modulating oscillation and higher harmonic components A n of the pulse repetition rate of the carrier nw 1, about each of which lateral components with frequencies nw 1 + W and nw 1 - W are symmetrically arranged in pairs (Fig. 6.2).
The main types of AIM - signals. AIM signals are divided into two main types: a signal of the first kind - AIM -1 (see Fig. 6.3, b) and a signal of the second kind - AIM -II (Fig. 6.3, V)
The instantaneous value of the amplitude of the pulses of the signal AIM -1 depends on instant value modulating oscillation e(t) (Fig. 6.3, A), and the amplitude of the AIM-II signal pulses is determined only by the value of the modulating oscillation at clock points (Fig. 6.3, b). Clock moments can coincide with the beginning of the impulse, any point of its middle or end. Therefore, with AIM-II, the carrier sequence is characterized by one more parameter - the position of the pulses relative to the clock points.
The difference between the AIM-1 and AIM-II signals turns out to be significant if the duration of the pulses t and is comparable with the period of their repetition of the AIM methods for message transmission, it is necessary to know the frequency band of the signals used.
Fig.6.3. Formation of AIM signals: A– impulse carrier; b- AIM-I; V- AIM-II
AIM-1 signals with the simplest, single-tone harmonic modulating signal, defined by formula (6.3), are rarely used in practice in communication systems. Let us estimate the spectrum of a pulsed radio signal of the AIM-1 type for a real modulating oscillation.
Literature: 1, 2; 6[ 46-61].
1. What is the process of pulse modulation?
2. What types of pulse modulation do you know?
3. How is pulse-amplitude modulation carried out?
4. How is frequency-pulse modulation carried out?
5. How are AIM-I, AIM-II signals formed?
6. How is pulse-phase modulation carried out?
7. What features does relative phase modulation have?
An idea of the spectral composition of pulse-modulated oscillations can be obtained by examining the spectrum during AIM.
The spectrum of the modulating oscillation is represented by one component at a frequency (Fig. 6.2, a). The spectrum of the carrier oscillation is determined by a periodic sequence of pulses (Fig. 6.2, b).
The amplitude-frequency spectrum of the AIM signal is shown in fig. 6.2. Note that the spectrum contains a constant component, a component at the frequency of the modulating signal, and components at frequencies , , while near each component at frequencies , , there are side frequencies spaced by the frequency of the modulating signal .
The presence in the spectrum of a component with the frequency of the modulating signal allows you to select it using a low-pass filter. If the sequence of video pulses is modulated not by a simple harmonic oscillation, but by a tone frequency signal ( speech signal) with a band , then in the spectrum of the AIM signal, instead of frequencies, there will be spectral components in the band (Fig. 6.3). Due to the relatively low noise immunity, AIM is usually used not independently, but as an intermediate procedure in the formation of signals.
The amplitude-frequency spectrum of the ODIM signal is shown in fig. 6.2, g. The composition of the spectrum is similar to the considered case of AIM, but has a more complex structure. However, the values of the amplitudes of the higher spectral components rapidly decrease, and a low-pass filter can also be used during demodulation. In this case, it is possible to limit the pulses in amplitude; this makes the system more robust.
The amplitude-frequency spectrum of the PIM signal is shown in fig. 6.2, d. In its structure, it is close to the DIM spectrum, however, the spectral component at the frequency of the modulating signal is 50 or more times less than with DIM and AIM. This is explained by the fact that the information is embedded in the position of the pulses, and their shifts during modulation are small. Therefore, the average value of the baseband frequency of the received PPM sequence is also small. In this case, it is not advisable to use the LPF. For PIM demodulation, the signals are first converted to AIM or DIM, and then a standard low-pass filter is applied.
Analog impulse types modulation
Used in time division systems.
are used as a modulating signal. analog signals, and as a carrier - periodic sequences of pulses.
Pulse modulation means double modulation:
1. Primary modulation (carrier - pulse sequence
2. Secondary modulation (modulating signal - signal obtained as a result of primary modulation, carrier - harmonic oscillation)
Primary modulation
During primary modulation, according to the law of the modulating signal, one of the parameters of the pulse sequence changes:
Pulse Amplitude - Pulse Amplitude Modulation (AIM)
Pulse Width - Pulse Width Modulation (PWM)
· Temporal position of impulses – pulse-phase modulation.
Pulse-amplitude modulation
Graphs of the modulating, carrier and AIM signal are shown on slide 2 of the presentation. There are two types of AIM signal: AIM-I and AIM-II. In AIM-I, the upper surface of the pulses exactly repeats the shape of the modulating signal. With AIM-II, the pulses have a rectangular shape, and their amplitude is equal to the value of the modulating signal in this moment time.
If the modulating signal changes slowly, and the pulse duration is short, then AIM-I and AIM-II practically do not differ from each other.
The expression for the AIM-I signal can be represented as follows:
The pulse sequence can be described by the expression (slide 3):
where m is the modulation index.
Most often, a sequence of rectangular pulses is chosen as a carrier, which can be represented by a Fourier series (see topic spectral analysis signals):
 |
The spectrum of a single-tone AIM signal (slides 5-7).
The demodulation of the AIM signal is carried out using a low-pass filter.
PULSE WIDTH MODULATION
With PWM, according to the law of the modulating signal, the duration of the carrier pulses changes. There are two types of PWM (slide 8):
1. PWM-I - one-way modulation (the duration is changed only by shifting the pulse cutoff)
2. PWM-II - two-way modulation (duration is changed by shifting the cutoff and the front of the pulse)
The PWM signal can be described by the following expression:
The spectrum of the PWM signal with single-tone modulation, see slides 9.10.
The spectrum of the PWM signal has a more complex structure. He contains:
1. DC component
2. Harmonics that are multiples of the carrier frequency
3. Spectrum of the modulating signal
4. Harmonics with frequencies k W n ± n W With
Under certain conditions, the part of the spectrum occupied by the useful signal can be clogged with frequencies W n - n W With, which can lead to distortion of the modulating signal.
Demodulation of the PWM signal is carried out using a low-pass filter.
PULSE PHASE MODULATION
With PIM, according to the law of the modulating signal, the time position of the pulses changes.
Very often, to facilitate demodulation and synchronization, the PIM signal is presented in the form of reference (Sch) and measuring pulses (I).
O - motionless on the time axis
And - move along the time axis depending on the value of the signal.
The time interval (Dt) between O and I is the carrier of information (slide 11).
PWM and PWM signals are closely related: the fixed edge of the PWM pulse coincides with the moment of appearance of O, and the cutoff of the PWM pulse coincides with the moment of appearance of I.
PIM signal spectrum
Analytic expression spectrum of the PIM signal is very difficult. The spectrum includes the following components:
1. DC component
2. Spectrum of the modulating signal
3. Components with frequencies kWh
4. Components with frequencies kWн ± nWс
An approximate expression for the amplitude of a harmonic with a frequency, equal to the frequency modulating signal for single-tone modulation is as follows:
where W c is the frequency of the modulating signal
Dt is the deviation of the time position of the measuring pulse.
It can be seen from this expression that the amplitude of the useful component in the PIM signal spectrum is very small and is a function of the modulating frequency, i.e. distorted. Therefore, demodulation of PIM signals with a low-pass filter is not possible. During demodulation, the PIM signal is first converted to PWM or AIM, and then the useful component is extracted using a low-pass filter.
Secondary modulation
To ensure high noise immunity in radio engineering systems, AIM-FM and PIM-AM modulation are most widely used.
AIM-WCH
When using this type of modulation transmitted message is first converted into a sequence of amplitude modulated pulses (AIM modulation). The received AIM signal is modulated using a high-frequency gramonic oscillation in frequency (FM modulation) (slide 14). The receiver first demodulates the FM signal and then demodulates the AIM signal.
The spectrum of the AIM-FM signal has a very complex structure and its width is theoretically infinite. The effective width of the spectrum is determined by the expression.
