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Signal envelope. Physical envelope, total phase and instantaneous frequency of a narrowband signal How to obtain the envelope of a signal

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We represent the complex envelope (2.124) in exponential form Where Uu(t) - real positive function time, called physical envelope

(often - envelope);

It is very important that the concept of the physical envelope of a narrowband signal coincides with the concept of the modulated waveform envelope. Where Physical envelope

and phase u (f) are related to the in-phase and quadrature amplitudes of the narrowband signal by the following relations: From relations (2.127) another, generalized form follows mathematical model

narrowband signal, which is used in modulation theory:

According to relation (2.128), a narrowband signal is a complex oscillation resulting from simultaneous modulation of the carrier harmonic signal both in amplitude and phase angle.

Example 2.10 A narrowband signal is given, which has the form of a single-tone LM oscillation:and(C)= U t ( 1 + McosQ/)cos(co 0 / + i/4). Let's define the complex envelope Uu(t), in-phase A and (?) and quadrature B u(t)

amplitude of this signal.

Solution

Let us choose the value c 0 as the reference frequency of the narrowband signal. Then, according to formula (2.126), we obtain the following expression for the complex envelope of a narrowband signal:

Since cos(rc/4) = sin(K/4) = У2/2, then according to formulas (2.127) we find

By analogy with signals with angle modulation, we introduce the concept of instantaneous (full) phase of a narrowband signal Let's define instantaneous frequency

as a derivative of the total phase of the signal:

Basic properties of the physical envelope of a narrowband signal. Where Using relations (2.127), we express the physical envelope

through the in-phase and quadrature amplitudes of an arbitrary narrow-band signal:

Comparing formulas (2.124) and (2.130), it is easy to see that the physical envelope is the modulus of the complex envelope of a narrowband signal. Let us evaluate the influence of the reference frequency from 0 on both envelopes of the narrowband signal. In the general case, the complex envelope of a narrowband signal is determined ambiguously. If instead of the reference frequency с 0 included in formula (2.125), we take a certain frequency C0j = со () + Дсо, then the original signal u(t)

takes the form Then the new value of the complex envelope

However, the physical envelope of the narrowband signal will remain unchanged when the frequency changes, since |e_yLo) "| = 1.

The second property of the physical envelope is at any time for a narrowband signal Let us evaluate the influence of the reference frequency from 0 on both envelopes of the narrowband signal. In the general case, the complex envelope of a narrowband signal is determined ambiguously. If instead of the reference frequency с 0 included in formula (2.125), we take a certain frequency C0j = со () + Дсо, then the original signal Uu(t). The validity of this statement follows from relation (2.128). The equal sign here corresponds to the moments in time when the factor cos|co 0? + f u (?)] = 1. We can assume that the physical envelope actually “envelopes” the amplitudes of the narrowband signal and is its instantaneous amplitude. The value of the concept of envelope is due to the fact that in communication systems amplitude detectors (demodulators) are widely used, capable of reproducing the envelope of a narrow-band signal with high accuracy.

Basic properties of the instantaneous frequency of a narrowband signal. If the complex envelope of a narrowband signal is represented by a vector that rotates on the complex plane with some constant angular velocity Q, i.e. analytically the signal is described by the function U u (t) = = U u (t)e ±jnt , then, according to formula (2.129), the instantaneous frequency of this oscillation is constant in time and therefore cd m = с 0 ± Q.

It can be shown that, in the general case, the instantaneous frequency of a narrowband signal varies with time according to the law

Relationship between the spectra of a narrowband signal and its complex envelope. Let 5(co) be the spectral density of the narrowband signal u(t)y complex envelope Where which, in turn, has a spectral density Y u ( to). Using relation (2.125), we determine the relationship between the spectral densities of the physical signal and its complex envelope by writing the direct Fourier transform:

We represent the complex envelope (2.124) in exponential form U*(t) - complex conjugate envelope; U m *(co) - complex conjugate spectral density of the complex envelope of a narrowband signal Uu(t).

From formula (2.131) it follows that the spectral density of a narrow-band signal 5(co) can be found by transferring the spectrum of the complex envelope V m (co) from the vicinity of co = 0 to the vicinity of the reference frequencies co = ±co (). In this case, the amplitudes of all spectral components of the signal are reduced by half. Note that to determine the signal spectrum in the region of negative frequencies, the complex conjugation operation is used.

Formula (2.131) allows us to use the known spectral density of a narrowband signal to find the spectrum of its complex envelope, which, in turn, fully determines its physical envelope and instantaneous frequency.

Example 2.11

The narrowband signal is a radio pulse of exponential shape, analytically written as u(t) = U m e“"since/. Let us define the complex envelope 1 + McosQ/)cos(co 0 / + i/4). Let's define the complex envelope the spectral density of a given signal S(co) and the spectral density V/co) of its complex envelope.

amplitude of this signal.

Let the reference frequency be 0. Since sin co/ = cos(co/ - l/2), then initial phase u(t) =-l/2. Using relation (2.126) and Euler’s formula, we obtain the following expression for the complex signal envelope:

Using the direct Fourier transform, we find the spectral density of the complex envelope:

We similarly calculate the spectral density of a narrowband signal.

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Spectral density of the analytical signal.

Let us examine the spectral density of the analytical signal, i.e., the function associated with the Fourier transform:

Based on formula (5.38), it can be argued that this function is nonzero only in the region of positive frequencies:

If is the spectral density of the conjugate signal, then due to the linearity of the Fourier transform

Therefore, equality (5.42) will be satisfied only in the case when the spectral densities of the original and conjugate signals are related to each other as follows:

Abstractly, one can imagine this way of obtaining a conjugate signal: the original oscillation is fed to the input of some system, which rotates the phases of all spectral components by an angle of -90° in the region of positive frequencies and by an angle of 90° in the region of negative frequencies, without changing their amplitude. A system with similar properties, is called a quadrature filter.

Hilbert transform.

Formula (5.44) shows that the spectral density of the conjugate signal is the product of the spectrum of the original signal and the function -. Therefore, the conjugate signal is a convolution of two functions: , which is the inverse Fourier transform of the function .

For ease of calculation, let's represent this function as a limit:

Thus, the conjugate signal is related to the original signal by the relation

You can do it differently, expressing the signal through which it is supposed to be known. To do this, it is enough to note that the following relationship between spectral densities follows from (5.44):

Therefore, the corresponding formula will differ from (5.45) only by sign:

Formulas (5.45) and (5.46) are known in mathematics as direct and inverse Hilbert transforms.

Their symbolic notation is as follows:

Since the function, called the kernel of these transformations, has a discontinuity at integrals (5.45) and (5.46) should be understood in the sense of the principal value. For example:

Some properties of Hilbert transforms.

The simplest property of these integral transformations is their linearity:

for any constants, which can be verified directly.

The kernel of the Hilbert transform is an odd function of the argument relative to the point a, which means that the signal conjugate to the constant is identically equal to zero:

An important property of the Hilbert transform is the following: if at any t the original signal s(t) reaches an extremum (maximum or minimum), then in the vicinity of this point the conjugate signal passes through zero. To verify this, you need to combine the graphs of s(t) and kernels in one drawing. Let the value of t be close to the one at which the function is extremal. Because the signal is here even function, and the kernel is odd, compensation will be observed for the areas of the figures limited by the horizontal axis and the curve that describes the integrand function of the Hilbert transform. Figuratively speaking, if the original signal changes over time “like a cosine,” then the signal associated with it will change “like a sine.”

Note that the Hilbert transforms are non-local in nature: bringing the conjugate signal into the vicinity of any point depends on the properties of the original signal along the entire time axis, although the greatest contribution, of course, comes from a fairly close neighborhood of the point in question.

Hilbert transforms for harmonic signals.

Let us calculate the signals associated with harmonic oscillations and The results can be obtained directly from formula (5.45). However, it is easier to do it this way. Let some arbitrary signal be given by its Fourier representation:

Based on relation (5.44), we find a similar representation of the conjugate signal:

Considering formulas (5.48) and (5.49) together, we find the following Hilbert transformation laws:

Hilbert transform for narrowband signal

Let the function be known - the spectral density of the complex envelope of a narrow-band signal s(t) with a reference frequency . According to formula (5.36), the spectrum of this signal

The first term on the right side corresponds to the frequency range of the second - Then, based on formula (5.44), the spectrum of the conjugate signal

from which it can be seen that the spectral density of the complex envelope of the conjugate signal

So, the conjugate signal in this case is also narrowband. If the complex envelope of the original signal

then, in accordance with equality (5.53), the complex envelope of the conjugate signal

differs from the complex bending of the initial vibration only by the presence of a constant phase shift of 90° towards the retardation.

It follows that the narrowband signal

corresponds to the Hilbert conjugate signal

Calculate envelope, total phase and instantaneous frequency.

Within the framework of the Hilbert transform method, the envelope of an arbitrary signal is defined as the modulus of the corresponding analytical signal:

The feasibility of such a definition can be verified using the example of a narrowband signal. Using formulas (5.54) and (5.55), we find that the envelope of such a signal

In § 5.3 this formula was obtained from other considerations.

By definition, the total phase of any signal is equal to the argument of the analytical signal

(5-57)

Finally, the instantaneous frequency of the signal is the derivative of the total phase with respect to time:

Let's consider examples illustrating the calculation of the indicated characteristics of narrowband signals.

Example 5.7. Given a simple harmonic oscillation

In this case, the conjugate signal is the envelope of the original signal

naturally, does not depend on time and is equal to its amplitude.

Total phase and, finally, instantaneous frequency This example shows that determining the envelope, total phase and instantaneous frequency through the Hilbert transform leads to results consistent with the usual ideas about the properties of harmonic oscillations.

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Ministry of Education of the Russian Federation

NOVOSIBIRSK STATE UNIVERSITY

Faculty of Mechanics and Mathematics.

Department of Programming.

ABSTRACT

Signal envelope.

group 7126

Scientific supervisor Kulikov A.I. __________

Novosibirsk 2009

Content:

Introduction.
Signal processing.
Finding the signal envelope.
Applying an envelope.
Conclusion.
List of sources used.

1. Introduction.

The number of means of transmitting information is constantly increasing. One of the ways to effectively use the radio frequency resource is to compress the spectrum of transmitted signals, which occupy a significant portion of the signals.

Despite the fact that the problem of companding (compression - restoration of the spectrum of speech signals when processing them based on a mathematical model of modulation theory) of the spectrum of speech signals (RS) today is quite successfully solved by means of statistical theory, the search for solutions to this problem on the basis of alternative theoretical concepts is not Not only has it not lost its relevance, but it has acquired even greater urgency with the development of telecommunications technologies, which is explained by the limited capabilities of known methods with increasing demand.

Development of new effective ways RS spectrum companding is relevant, first of all, for radio communication systems, including specialized mobile radio communication systems. This is also relevant for systems for recording and storing large amounts of speech information.

Also one of most important tasks radio monitoring systems is

determining the presence of one or more signals in

analyzed frequency band. In this case, various temporary

characteristics of the signal envelope.

2. Signal processing.

The basis of signal research is spectral analysis. The concept of spectral analysis is quite broad. It is applicable to the consideration of any functions in the form of a generalized Fourier series. Signal analysis typically uses a Fourier transform or series to move the analysis into the frequency domain. The signal is considered as an infinite or finite collection of harmonic components.

Spectral analysis of non-periodic signals is based on the use of the Fourier transform. Direct and inverse Fourier transforms establish a one-to-one correspondence between the signal (time function describing the signal s(t)) and its spectral density:

, . (2.1)

The function is generally complex

(2.2)

where Re, Im are the real and imaginary parts of the complex quantity;

Modulus and argument of a complex quantity.

. (2.3)

The modulus of the spectral density of the signal describes the distribution of the amplitudes of the harmonic components over frequency, called the amplitude spectrum. The argument gives the phase distribution over frequency, called the phase spectrum of the signal.

Formation of the signal envelope in time is the most effective way to isolate the modulating component in cases where the spectral composition of the modulating and carrier components is different and does not intersect in frequency domain, i.e. The frequency domain of the carrier is much higher than the frequency domain of the modulating component.

Envelope amenities:

storing information about the signal shape and its main peaks in the envelope;
the ability to reduce the amount of information when moving to envelopes due to local averaging;
using envelopes as templates.

Therefore, the use of signal envelope has found wide application in various fields of activity.

At the first stages of the development of vibration diagnostics, spectral analysis of the vibration envelope was used to determine the frequencies and amplitudes of harmonic components that have similar frequencies, which do not allow separating these components in the spectrum of the vibration signal due to the limited resolution of the analyzers.

With the advent of digital spectral analyzers with high frequency resolution, diagnosticians began to abandon the analysis of the envelope spectra of those multiplicative vibration components in which both components are strictly periodic. In practice, this type of analysis is sometimes used in the diagnosis of rolling bearings of pumps and other flow-creating machines, in order to detect the modulation of the strongest vibration components at the harmonics of the impeller rotation speed by lower modulating frequencies, for example, the separator rotation frequency. The reason is that in the low-frequency vibration of machines of this type there are significant random components that make it difficult to detect weak side components in the spectrum of vibration at the rotor rotation frequency.

Also today the problem of RS spectrum compression is very acute. The necessity of continuing the development of the modulation theory of sound signals, which studies the properties of natural acoustic signals, is substantiated. The necessity of compressing the spectrum of speech signals to increase the efficiency of using the frequency resource of speech transmission channels is substantiated. The development and current state of solving the problem of companding the RS spectrum for the purpose of broadcasting them over communication channels is shown. The dependence of speech quality on the degree of compression of the RS spectrum by the most popular modern methods is given.

Compression of the RS spectrum is possible by reducing their statistical and psychoacoustic redundancies. In modern radiotelephony systems, in order to compress the spectrum of speech signals, hybrid vocoders have found the most widespread use, reducing both psychoacoustic and statistical redundancy. The rather low quality of the received speech with a relatively low degree of compression of its spectrum using modern methods justifies the need to find new ways to effectively solve this problem on the basis of alternative theoretical concepts.

3. Finding the signal envelope.

When mathematically analyzing the signal envelope, it is very often convenient to use an equivalent complex representation of signals instead of real signals in order to simplify the mathematical apparatus of data conversion.

In the general case, an arbitrary dynamic signal s(t), given on a certain section of the time axis (both finite and infinite) has a complex two-way spectral density S(ω). With a separate inverse Fourier transform of the real and imaginary parts of the spectrum S(ω), the signal s(t) is divided into even and odd components, which are two-sided with respect to t = 0, and the summation of which completely restores the original signal. In Fig. Figure 2 shows an example of a signal (A), its complex spectrum (B) and obtaining the even and odd parts of the signal from the real and imaginary part of the spectrum (C).

Rice. 3.1. Signal, signal spectral density, even and odd components.

You can also perform the inverse Fourier transform in another form - separately for positive and negative spectrum frequencies:

s(t) = S(ω) exp(jωt) dω + S(ω) exp(jωt)dω (3.1)

Information in the complex signal spectrum is redundant. Due to complex conjugacy full information about the signal s(t) contains both the left (negative frequencies) and the right (positive frequencies) part of the spectrum S(ω). The analytical signal representing the real signal s(t) is the second integral of expression (3.1), normalized to π, i.e. inverse Fourier transform of the signal spectrum s(t) only at positive frequencies:

z s (t) = (1/π) S(ω) exp(jωt). (3.2)

The duality of the properties of the Fourier transform determines that the analytical signal z s (t), obtained from a one-way spectral function, is always complex and can be represented in the form:

z s (t) = Re z(t) + j·Im z(t). (3.2")

A similar transformation of the first integral of expression (3.1) gives the signal z s *(t), complex conjugate to the signal z(t):

z s *(t) = Re z(t) - j·Im z(t),

which is clearly visible in Fig. 3.2 when reconstructing signals from one-sided parts of the spectrum shown in Fig. 2-B.

Rice. 3.2. Signals z(t) and z*(t).

From Figure 3.2 one can see that when adding the functions z s (t) and z s * (t), the imaginary parts of the functions cancel each other out, and the real parts, taking into account normalization only to π, and not to 2π, as in (3.1), in the sum give the complete original signal s(t):

/2 = Re z(t) = =

= (1/2π) S(ω) cos ωt dt = s(t).

It follows that the real part of the analytical signal z s (t) is equal to the signal s (t) itself.

To identify the nature of the imaginary part of the signal z s (t), we translate all terms of the function (3.2") into the spectral region with separate representation by positive and negative frequencies (indices – and +) of the real and imaginary parts of the spectrum:

Z s (ω) = A - (ω) + A + (ω) + jB - (ω) + jB + (ω) + j,

where the indices A" and B" denote the transformation functions Im(z(t)). In this expression, the functions on the left side of the spectrum (at negative frequencies) must mutually compensate each other according to the definition of the analytical signal (3.2), i.e.:

B" - (ω) = A - (ω), A" - (ω) = -B - (ω).

From here, taking into account the parity of the real A" - (ω) and the oddity of the imaginary B" - (ω) spectrum functions, the equalities also follow:

B" + (ω) = - A + (ω), A" + (ω) = B + (ω).

But these four equalities are nothing more than the Hilbert transform in the frequency domain of the spectrum of the function Re z(t)Û A(ω)+jB(ω) into the spectrum of the function A"(ω)+jB"(ω)Û Im z(t) by multiplying by the signature function -j× sgn(ω). Consequently, the imaginary part of the analytical signal z s (t) is analytically conjugate with its real part Re z(t) = s(t) through the Hilbert transform. This part of the analytical signal is called quadrature complement signal s(t):

Im z(t) = = TH = s(t) * hb(t), (3.3)

hb(t) = 1/(πt),

z s (t) = s(t) + j × . (3.4)

where the index denotes the signal, analytically conjugate with signal s(t), hb(t) is the Hilbert operator.

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