Frequency spectrum of the Walsh function. Frequency spectrum

1. The spectrum of a sinusoid (Fig. 14.14, a) in the basis of the Walsh functions.

In this case, it is advisable to equate the interval of decomposition with the value of T.

Passing to dimensionless time, we write the oscillation in the form We restrict ourselves to 16 functions, and first we choose the Walsh ordering. Since the given function is odd with respect to the point, all the 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 a periodicity within the interval lead to a zero coefficient due to the parity in the indicated intervals.

So, only four of the 16 coefficients are not equal to zero: A (1), A (5), A (9) and A (13). Let us determine these coefficients by the formula (14.28). The integrands that are the products of signals (see Fig. 14.14, a) and the corresponding function are shown in Fig. 14.14, b - e. Piecewise integration of these products gives

The spectrum of the signal under consideration in the basis of Walsh functions (ordered according to Walsh) is shown in Fig. 14.15, a.

Rice. 14.14. Gating a segment of a sinusoid with Walsh functions

Rice. 14.15. Spectra of a sinusoid in the basis of Walsh functions ordered according to (a) Walsh, (b) Paley, and (c) Hadamard. Basis size

When ordered according to Paley and Hadamard, the spectrum of the same signal takes the form shown in Fig. 14.15, b and c. These spectra were 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 restoring an oscillation with a limited number of Walsh functions, preference should be given to ordering, which ensures a monotonic decay of the spectrum. In other words, the best ordering is such that each next spectral component is not greater (in absolute value) than the previous one, i.e. In this sense, the best ordering when representing a segment of a sinusoid, as follows from Fig. 14.15, is Paley's ordering, and the worst is Hadamard's.

The restoration of the original signal (see Fig. 14.14, a) by sixteen Walsh functions is shown in Fig. 14.16 (twelve spectral coefficients vanish). This construction, of course, does not depend on the way the functions are ordered. Obviously, for a more satisfactory approximation of the sinusoidal oscillation in the Walsh basis, a significant increase in the number of spectral components is required.

Outside the interval (0,1), the series (14.27), as noted in § 14. 4, describes a periodic continuation, in this example a harmonic function.

2. The spectrum of harmonic oscillations (Fig. 14.17) in the basis of the Walsh functions. As in the previous example, one harmonic cycle with a period is considered. Passing to dimensionless time, we write the oscillation in the form

The Walsh spectrum of the function is defined in example 1. The definition of the spectrum of a function on the interval)