EXTRACTION OF THE ENVELOPE AND PHASE OF A BROADBAND SIGNAL

N. F. KUCHER, V. M. RAZIN, V. S. SHMITOV

(Presented by the Department of Computer Engineering)

According to [1], signals carrying an information load in communication and control systems can be represented in the form

\[ U(t)=A(t)\cdot \cos \varphi(t), \tag{1} \]

where

\(A(t)\) is the envelope of the signal amplitude, defined in accordance with [1] as

\[ A(t)=\sqrt{U^2(t)+\hat{U}^2(t)}, \tag{2} \]

where

\[ \hat{U}(t)=A(t)\sin \varphi(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{U(\tau)}{t-\tau}\,d\tau \tag{3} \]

and \(\varphi(t)\) is the envelope of the signal phase, equal to

\[ \varphi(t)=\operatorname{arctg}\frac{\hat{U}(t)}{U(t)}. \tag{4} \]

Signals of the form (1) may be narrowband or broadband; signals for which the condition

\[ \frac{f_0}{\Delta F}\gg 1 \]

is satisfied are considered narrowband

(\(f_0\) is the central frequency of the signal spectrum, \(\Delta F\) is the width of the signal spectrum).

All other signals are called broadband.

As is evident from expression (1), a signal is completely characterized by the functions \(A(t)\) and \(\varphi(t)\), i.e., in order to obtain complete information about a signal, it is sufficient to determine these functions. Determining \(A(t)\) and \(\varphi(t)\) according to expressions (2) and (4) leads to ambiguity owing to the presence of the square-root extraction operation, and also in connection with the periodicity of the function \(\operatorname{arctg} x\).

A signal written in the form (1) is a process with amplitude and phase modulation; therefore the envelopes \(A(t)\) and \(\varphi(t)\)

could be extracted by amplitude and phase detection, the detection operations being readily implemented by hardware methods. This method is applicable only to narrowband processes, since the spectra of the envelopes \(A(t)\) and \(\varphi(t)\) of a broadband process overlap.

Consequently, in order to extract the envelope and phase of a broadband signal by amplitude and phase detection, it is necessary first to transform the broadband signal into a narrowband one.

The transformation is carried out by shifting the signal spectrum into the high-frequency region, so that the narrowband condition is satisfied. The operation of shifting the spectrum can be implemented by means of single-sideband modulation.

Consider Fig. 1. The initial signal \(U_1(t)=A(t)\cdot\cos\varphi(t)\), with a spectrum of width \(\Delta\omega\), is broadband in the low-frequency region (Fig. 1, a).

The process of single-sideband modulation proceeds as follows: signal (1) is multiplied by an oscillation of constant frequency \(\omega_0\)

\[ \begin{aligned} U'_2(t) &= U_1(t)\cdot\cos\omega_0 t =\\ &= A(t)\cdot\cos\varphi(t)\cdot\cos\omega_0 t =\\ &= \frac{A(t)}{2}\cos[\omega_0 t+\varphi(t)] + \frac{A(t)}{2}\cos[\omega_0 t-\varphi(t)], \end{aligned} \tag{5} \]

as a result of which we obtain signal (5), which is a balanced-modulated oscillation with a suppressed carrier and two symmetric side components:

\[ \frac{A(t)}{2}\cos[\omega_0 t+\varphi(t)] \]
— the upper sideband, and

\[ \frac{A(t)}{2}\cos[\omega_0 t-\varphi(t)] \]
— the lower sideband (Fig. 1, b).

If one now suppresses one of the side components, for example the upper one, we obtain a single-sideband signal of the form

\[ U_s(t)=\frac{A(t)}{2}\cos[\omega_0 t-\varphi(t)] \tag{6} \]

with the spectrum shown in Fig. 1, c.

When the condition

\[ \frac{\Delta\omega}{\omega_0}\ll 1 \]

is satisfied, the obtained single-sideband signal \(U_s(t)\) is narrowband, and its envelopes coincide, up to a constant factor, with the envelopes of the original broadband signal \(U_1(t)\). In this connection, the functions \(A(t)\) and \(\varphi(t)\) can be extracted by amplitude and phase detection of the narrowband signal.

Fig. 1. Illustration of the process of spectrum shifting by means of single-sideband modulation

It should be noted that phase detection can be replaced by frequency detection, since according to [1] the instantaneous frequency of the signal \(\omega(t)\) is the derivative of the instantaneous phase \(\varphi(t)\), i.e.,

\[ \omega(t)=\frac{d\varphi(t)}{dt}. \tag{7} \]

The functional block diagram of the instrument for extracting the envelope and phase is shown in Fig. 2. Extraction of \(\dot A(t)\) and \(\varphi(t)\) is performed as follows: the input broadband signal \(U_1(t)\) and the high-frequency signal of constant frequency from the carrier generator are multiplied by a balanced modulator, at whose output a band-pass filter is connected, suppressing the upper sideband of frequencies. The resulting single-sideband signal of the form (6) is fed to an amplitude detector, where the envelope \(\dot A(t)\) is extracted, and to a clipper, which hard-limits signal (6), converting it into an oscillation with constant amplitude and variable instantaneous frequency \(\omega(t)=\dfrac{d\varphi(t)}{dt}\), extracted by the frequency detector connected at the output of the clipper.

Fig. 2. Functional block diagram of the instrument for extracting the envelope and phase

Figs. 3–8 show circuit diagrams of the individual functional units of the instrument.

Fig. 3. Balanced modulator

The balanced modulator is assembled according to a ring circuit using carefully selected D2B diodes. The carrier-frequency generator of 100 kHz is a quartz-controlled multivibrator with a passive resonant filter connected at the output.

An eight-crystal quartz band-pass filter with a passband width of 3.7 kHz is taken from the B-12 equipment set.

The radio-frequency clipper is a five-stage amplifier with an emitter follower at the output; moreover, four stages...

\((T_2—T_5)\) are loaded by limiting circuits, each of which consists of a limiting resistor \((R8, R13, R18, R23)\) and a pair of antiparallel-connected D310 diodes \((D_1—D_8)\).

Fig. 4. Carrier oscillator

Fig. 5. Band-pass filter

Fig. 6. Amplitude detector

The amplitude and frequency detectors are made according to a conventional circuit and have no essential circuit-specific features.

Fig. 7. Clipper

The instrument is characterized by the following technical specifications:
input-signal amplitude \(U_{\text{in}} = 3 \div 200\ \text{mV}\);

Fig. 8. Frequency detector

input-signal spectrum \(100—4000\ \text{Hz}\);
carrier frequency \(f_0 = 100\ \text{kHz}\);
maximum amplitude of the output signals \(0.5\ \text{V}\).
The instrument can be used for the analysis of random and regular processes, including speech signals.

REFERENCES

A. A. Zinov’ev, L. I. Filippov. Methods of Analytical Representation of Radio Signals. “Vysshaya shkola,” 1966.