Decomposition of signals by Walsh functions. Frequency spectrum
Basic functions
Mathematical representation
The signal spectrum can be written through the Fourier transform (it is possible without the coefficient 1 / 2 π (\displaystyle 1/(\sqrt (2\pi )))) in the form:
S (ω) = ∫ − ∞ + ∞ s (t) e − i ω t d t (\displaystyle S(\omega)=\int \limits _(-\infty )^(+\infty )s(t)e^ (-i\omega t)dt), Where ω (\displaystyle \omega )- angular frequency equal 2 π f (\displaystyle 2\pi f).
The signal spectrum is a complex quantity and is represented as: S (ω) = A (ω) e − i ϕ (ω) (\displaystyle S(\omega)=A(\omega)e^(-i\phi (\omega))), Where A (ω) (\displaystyle A(\omega))- amplitude spectrum of the signal, ϕ (ω) (\displaystyle \phi (\omega))- phase spectrum of the signal.
If under signal s (t) (\displaystyle s(t)) understand
From (2.48) we obtain
(2.49)
Taking into account that the Walsh functions are equal to ±1, we write expression (2.49) in the form
(2.50)
where a n (k) = 0 or 1, determines the sign of the Walsh function on the interval 
Examples of Walsh spectra.
1. Walsh spectrum of a rectangular pulse s(t) = 1, 0 ≤ t ≤ t (Fig. 2.9)
From (2.50) we find
The Walsh spectrum of a rectangular pulse depends on the relationship between m and T. For τ/T = 2 v where v is a positive integer, taking into account the values of the Walsh functions we obtain
The expansion of a rectangular pulse in terms of Walsh functions has the form
The spectrum consists of 2 V components with equal amplitudes equal to 1/2 V. The spectrum contains a finite number of components. At t/T≠ 2 V, the structure of the spectrum will change.
2. Walsh spectrum of a triangular pulse (Fig. 2.10) When describing a triangular pulse
it is convenient to go to dimensionless time x = t/T
In accordance with (2.50) we find:
Walsh spectra with Harmuth and Paley numbering are shown in Fig. 2.10, b and c.
3. Walsh spectrum of a sinusoidal pulse (Fig. 2.11)
For a sinusoidal pulse
passing to dimensionless time x = t/T, we write 
From (2.50) in the Harmuth system we find (Fig. 2.11):
The Walsh spectra of the signal under consideration with Harmuth and Paley numbering are shown in Fig. 2.11.6 and c.
2.7A. Properties of Walsh spectra
When analyzing signals using Walsh functions, it is useful to take into account the properties of signal decomposition in the Walsh basis - Walsh spectra.
1. The spectrum of the sum of signals is equal to the sum of the spectra of each signal.
The spectrum of the signal in the system of Walsh functions is determined by the expansion coefficients (2.47). For the sum of signals, the expansion coefficients are determined by the expression
(2.52)
where a pk are the expansion coefficients of the signal s k (t).
2. Multiplying the signal by the Walsh function with number n changes the numbers of expansion coefficients with k according to the law of binary shift modulo two
3. Walsh spectrum of the product of signals s 1 (t) and s 2 (t). defined on the interval . Such functions describe periodic signals with limited power.
For an even function s(t), as follows from (3.2),
(3.3)
for an odd function s(t):
(3.4)
Typically, when analyzing signals, the expansion s(t) is used in the form
(3.5)
A periodic signal is represented as a sum of harmonic components with amplitudes A n and initial phases.
The set of amplitudes (D,) determines the amplitude spectrum, and the set of initial phases (φ n) determines the phase spectrum of the signal (Fig. 3.1, a). As follows from (3.5), the spectra of periodic signals are discrete or line, the frequency sampling interval is equal to the signal frequency ω 1 = 2π/ T.
The trigonometric Fourier series can be written in complex form
(3.7)
(3.8)
The transition from (3.1) to (3.7) is obvious taking into account Euler’s formula
(3.9)
Coefficients with n are generally complex quantities
When using the complex form of the Fourier series, the signal is determined by the set of complex amplitudes (with n). Modules of complex amplitudes |с n | describe the amplitude spectrum, arguments φ n - the phase spectrum of the signal (Fig. 3.1,6).
Representing (3.8) in the form
(3.11)
As follows from the written expressions, the amplitude spectrum has even symmetry, and the phase spectrum has odd symmetry.
(3.13)
From a comparison of expressions (3.2) and (3.11) it follows
As an example, consider a periodic sequence of rectangular pulses (Fig. 3.2,a). When expanding a periodic sequence of rectangular pulses into a trigonometric Fourier series from (3.2), we obtain amplitude and phase spectra in the form (Fig. 3.2,b):
When using the complex form of the Fourier series 
from (3.8) it follows:
The amplitude and phase spectra of the signal are equal
The limiting form of the Fourier series is the Fourier integral. A periodic signal at T → ∞ becomes non-periodic. Substituting (3.8) into (3.7), we write
(3.16)
Harmonic signal analysis
Transforming (3.16), as T→∞ (in this case ω 1 → dω and pω 1 = ω), we obtain
(3.17)
The Fourier integral is written in square brackets; it describes the spectral density of the signal
Expression (3.17) will take the form
The written relations represent the direct and inverse Fourier transforms. They are used in harmonic analysis of non-periodic signals.
3.2. Harmonic analysis of non-periodic signals
Direct and inverse Fourier transforms establish a one-to-one correspondence between the signal (the time function describing the signal s(t)) and its spectral density S(ω):
(3.18)
We denote the Fourier correspondence:
(3.19)
The condition for the existence of the Fourier transform is the absolute integrability of the function s(t)
(3.20)
In practical applications, the condition of integrability of the square of this function is more convenient
(3.21)
For real signals, condition (3.21) is equivalent to condition (3.20), but has a more obvious physical meaning: condition (3.21) means limited signal energy. Thus, we can consider it possible to apply the Fourier transform to signals with limited energy. These are non-periodic (pulse) signals. For periodic signals, harmonic decomposition
nic components are produced using a Fourier series.
The function S(ω) is generally complex
where Re, lm are the real and imaginary parts of the complex quantity; |s(w)|, f(oo) - module and argument of a complex value:
Signal spectral density modulus |S(ω)| describes the distribution of amplitudes of harmonic components by frequency, called the amplitude spectrum. The argument φ(ω) gives the phase distribution over frequency, called the phase spectrum of the signal. The amplitude spectrum is an even function, and the phase spectrum is an odd function of frequency
Taking into account Euler’s formula (3.9), we write the expression for S(ω) in the form
(3.24)
If s(t) is an even function, then from (3.24) we obtain
(3.25)
The function S(ω), as follows from (3.25), is a real function. The phase spectrum is defined as
(3.26)
For an odd function s(t) from (3.24) we obtain
(3.27)
The function S(ω) is purely imaginary, the phase spectrum
(3.28)
Any signal can be represented as the sum of even s h (t) and odd s H (t) components
(3.29)
The possibility of such a representation becomes clear taking into account the following equalities:
From (3.24) and (3.29) we obtain
(3.30)
Therefore, for the real and imaginary parts of the signal spectral density we can write:
Thus, the real part of the spectral density represents the Fourier transform of the even component, the imaginary part - of the odd component of the signal. The real part of the complex spectral density of the signal is even, and the imaginary part is an odd function of frequency.
Signal spectral density at ω = 0
(3.31)
equal to the area under the curve s(t).
As examples, we obtain the spectra of some signals.
1. Rectangular pulse (Fig. 3.3, a)
where τ and is the pulse duration.
Signal spectral density
Graphs of the amplitude and phase spectra of the signal are shown in Fig. 3.3,b,c.
2. Signal described by the function
The spectral density of the signal is determined by the expression 
Integrating by parts n-1 times, we get
Signal (Fig. 3.4a)
has spectral density
The graphs of the amplitude and phase spectra are shown in Fig. 3.4, b, c.
Signal (Fig. 3.5,a)
has spectral density
Graphs of amplitude and phase spectra - Fig. 3.5, b, c.
The number of examples increases in the table. 3.1.
Comparison of (3.18) and (3.8) shows that the spectral density of a single pulse at τ< Taking into account this relationship, the determination of the spectrum of a periodic signal in a number of cases can be simplified using the Fourier transform (3.18). The coefficients of the Fourier series are found as where S(ω) is the spectral density of one pulse. Thus, when determining the amplitude and phase spectra of periodic signals, it is useful to keep in mind the following equalities: The coefficient 1/T can be considered as the frequency interval between adjacent spectrum components, and spectral density as the ratio of the amplitude of the signal component to the frequency interval to which the amplitude corresponds. Taking this into account, the term “spectral density” becomes more understandable. Continuous amplitude and phase spectra of a single pulse are envelopes of discrete amplitude and phase spectra of a periodic sequence of such pulses. Using relations (3.33), the results given in table. 3.1 can be used to determine the spectra of periodic pulse trains. The following examples illustrate this approach. 1. Periodic sequence of rectangular pulses (Table 3.1, item 1), Fig. 3.2. The written expression repeats the result of example step 3.1. 2. Periodic sequence of meander pulses (Table 3.1, item 2), Fig. 3.6, fig. 3.2. 3. Periodic sequence of exponential pulses (Table 3.1, paragraph 8), Fig. 3.7. Table 3.1 Signals and their spectra 3.3. Frequency spectra of signals presented in the form of a generalized Fourier series When representing a signal as a generalized Fourier series, it is useful to have the Fourier transform of the basis functions. This will allow us to move from the spectrum in the basis of various orthogonal systems to the frequency spectrum. Below are examples of frequency spectra of some types of signals described by the basis functions of orthogonal systems. 1.Legendre's signals. The Fourier transform of the Legendre polynomial (Section 2) has the form n= 1,2, ... - Legendre polynomial; Using (3.34), from the signal represented as a series with odds Expression (3.35) describes the spectral density of the signal s(f) in the form of a series. Graphs of spectrum components with numbers 1 - 3 are shown in Fig. 3.8. 2. Laguerre signals. The Fourier transform of the Laguerre function has the form n= 1,2,... are Laguerre functions. Using (3.36), from the signal represented as a series of expansions in Laguerre polynomials (Section 2) with odds you can go to the spectral density of the signal 3. Hermite signals. The Fourier transform of the Hermite function has the form From (3.38) it follows that the Hermite functions have the property of transformability, i.e. the functions and their Fourier transforms are equal (up to constant coefficients). Using (3.38), from the signal represented as a series of expansions in Hermite polynomials with odds you can go to the spectral density of the signal 4. Walsh signals. The frequency spectra of Walsh signals (signals described by Walsh functions) are determined by the following Fourier transform: where wal(n,x) is the Walsh function. Since the Walsh functions have N regions of constant values, where x k is the value of x on the k-th interval. From (3.41) we obtain Where Since the Walsh functions take values ±1, we can write (3.42) in the form where a n (k) = 0 or 1 determines the sign of the function wal(n,x k). In Fig. Figure 3.9 shows graphs of the amplitude spectra of the first six Walsh signals. 3.4. Spectra of signals described by non-integrable functions The Fourier transform exists only for signals with finite energy (for which condition (3.21) is satisfied). The class of signals analyzed using the Fourier transform can be expanded by a purely formal technique based on the introduction of the concept of spectral density for the impulse function. Let's look at some of these signals. 1. Pulse function. The impulse function (or δ - function) is defined as From the definition of the impulse function follows its filtering property We define the spectral density of the impulse function as The amplitude spectrum is equal to unity, the phase spectrum φ(ω) = ωt 0 (Fig. 3.10). The inverse Fourier transform gives By analogy with (3.47), for the frequency domain we write Using the obtained expressions, we determine the spectral densities of some types of signals described by functions for which there is no Fourier transform. 2. Constant signal s(t) = s 0 . Taking (3.48) into account, we obtain (Fig. 3.11) 3. Harmonic signal. The spectral density of the signal will be obtained taking into account (3.48) in the form At φ = 0 (Fig. 3.12) For signal by analogy with (3.52) we find 4. Unit step function. We will consider the unit step function σ(t) as the limiting form of exponential momentum Let us represent the exponential momentum as the sum of even and odd components (3.29) 1. Spectrum of a sinusoid (Fig. 14.14, a) in the basis of Walsh functions. In this case, it is advisable to equate the decomposition interval to the value of T. Moving on to dimensionless time, we write the oscillation in the form Let's limit ourselves to 16 functions, and first choose the Walsh ordering. Since the given function is odd with respect to the point , all coefficients for even Walsh functions in the series (14.27), i.e., for are equal to zero. Those of the remaining eight functions that coincide with the Rademacher functions and have periodicity within the interval lead to a zero coefficient due to parity in the indicated intervals. So, only four coefficients out of 16 are not equal to zero: A (1), A (5), A (9) and A (13). Let us determine these coefficients using formula (14.28). The integrand functions, which are products of signals (see Fig. 14.14, a) and the corresponding function, are presented in Fig. 14.14, b - d. Piecewise integration of these products gives The spectrum of the signal under consideration in the basis of Walsh functions (ordered by Walsh) is presented in Fig. 14.15, a. Rice. 14.14. Gating a sinusoid segment using Walsh functions Rice. 14.15. Spectra of a sinusoid in the basis of Walsh functions ordered by Walsh (a), Paley (b) and Hadamard (c). Base size When ordered by Paley and Hadamard, the spectrum of the same signal takes the form shown in Fig. 14.15, b and c. These spectra are obtained from the spectrum in Fig. 14.15, but by rearranging the coefficients in accordance with the table (see Fig. 14.13), showing the relationship between the ways of ordering the Walsh functions (for ). To reduce distortions when reconstructing oscillations using a limited number of Walsh functions, preference should be given to ordering, which ensures a monotonic decrease in the spectrum. In other words, the best ordering is one in which each subsequent spectral component is no greater (in absolute value) than the previous one, i.e. In this sense, the best ordering when representing a sinusoid segment, as follows from Fig. 14.15, is the Paley ordering, and the worst is Hadamard. Restoration of the original signal (see Fig. 14.14, a) with sixteen Walsh functions is presented in Fig. 14.16 (twelve spectral coefficients vanish), This construction, of course, does not depend on the method of ordering the functions. Obviously, for a more satisfactory approximation of a sinusoidal oscillation in the Walsh basis, a significant increase in the number of spectral components is required. Outside the interval (0,1), series (14.27), as noted in § 14.4, describes a periodic continuation, in this example a harmonic function. 2. Spectrum of harmonic vibration (Fig. 14.17) in the basis of Walsh functions. As in the previous example, one cycle of harmonic oscillation with period is considered. Moving on to dimensionless time, we write the vibration in the form The Walsh spectrum of a function is defined in Example 1. The definition of the spectrum of a function on the interval is completely similar) 
(3.32)
(3.34)
- Bessel function.
(3.35)
(3.36)
(3.37)
(3.38)
n= 1,2,... are Hermite functions.
(3.39)
(3.40)
(3.43)
(3.44)
(3.45)
(3.46)
(3.48)
(3.49)
(3.53)
(3.55)
