Sine-cosine Function Research Articles

Frequency analysis based on general periodic functions

Following the sine–cosine function, the sawtooth wave, square wave, triangular wave, trapezoidal wave, and so on become new easily generated periodic functions in modern electronics. Similar to Fourier’s idea, a natural question is whether a signal can be considered as a superposition of easily generated functions with different frequencies. Therefore it is necessary to generalize Fourier analysis based on sine–cosine functions into frequency analysis based on general periodic functions. In this paper, we introduce the frequency series and frequency transformation based on general periodic functions. We discuss when a frequency system is a complete system or an unconditional basis in L2[−π,π], and when a frequency transformation can be carried out in L2(−∞,+∞). For practical convenience almost all easily generated functions in electronics are considered carefully as examples. As a new and practical generalization of classical Fourier analysis, these results will become a theoretical foundation for the technique of easily generated function analysis in signal processing.

Triangular function analysis

This paper concerns triangular function analysis including triangular function series and triangular function transformation, which is very similar to Fourier analysis based on sine and cosine functions. Besides sine-cosine functions, triangular functions are frequently-used and easily-generated periodic functions in electronics as well, so it is an urgent practical problem to study the basic properties of triangular functions and the fundamental theory of triangular function analysis. We show that triangular functions and sine-cosine functions not only have the similar graphs, but also possess similar analysis properties. Any continuous periodic function may be approximated uniformly by linear combinations of triangular functions as well as trigonometric functions, and every function f( x) ϵ L 2[− π, π] has a triangular function series as well as a Fourier series. Since the triangular functions are nonorthogonal in L 2[− π, π], the orthonormalization is discussed so that a function f( x) ϵ L 2[− π, π] can be approximated best by a superposition of given finite triangular functions. Finally, we introduce the theory of the triangular function transformation in L 2(−∞, ∞), which has a close relation with Fourier transformation. These results form the theoretical foundation of the technique of triangular function analysis in modern electronics.

Square wave analysis

This paper concerns square wave analysis, which is in no small part a generalization of the Fourier analysis based on sine and cosine functions. Following sine and cosine functions square waves become another frequently used and easily generated waveform in electronics, so it is an urgent practical problem to study the basic properties of square waves and the fundamental theory of square wave analysis. Like the sine-cosine function system, the square wave system, i.e., the function system of square waves with different frequencies (or periods), is linearly independent and complete in the real Hilbert spaces L2(−π,π), so square waves can approximate an arbitrary quadratically integrable function with a vanishing mean-square error. Since the square wave system is nonorthogonal, its biorthogonal functions are presented so that a function with a certain condition can be expanded as a square wave series easily. Meanwhile the orthogonalization of the square wave system is discussed so that a quadratically integrable function can be approximated best by a superposition of finite square waves. These results form the theory basis of the square wave analysis technique in modern electronics.

Quantification of directional and orientational selectivities of visual neurons to moving stimuli

Directional and orientational components usually coexist and are mixed in the cell's overall responses when moving optical stimuli are used to study the response characteristics of visual neurons. While these two properties were quantified with all the previous methods for data analysis, their effects could not be efficiently separated from each other, and thus the analyses were imperfect. In this paper, theoretical evidence and examples are provided to show the defects of the old methods. In order to separate the two components completely, we propose to apply optimal regression analysis with the sine-cosine function series as the fundamental variables. Based on this separation, we defined the orientational selectivity as variation of response strength with orientation and performed integration and averaging to quantify the two properties [cf. Eqs. (5) and (6)]. The present method has the advantages of completeness and accuracy, and can detect some details which would have been missed by other methods. An explanation of the intrinsic implications of the method and our comprehension of directional and orientational selectivities and preferred direction and orientation are also given.

Graphic recording and averaging the wind direction

A continuous record of the wind direction, independent of the number of wind vane rotations, is obtained by coupling to the vane a linear continuous potentiometer supplied at two opposite taps. This gives an output which linearly increases with the azimuth α between 0 and π and symmetrically decreases when α veers between π and 2π radians. A rotary switch coupled to the potentiometer acts as a control device for the tracing and colouring upon a point recorder. When the azimuth increases between 0 and π, α is represented by a black graph shifting to the right, until the upper chart limit is reached when α=π. If the veering goes up to 2π, the printer inverts its direction moving to the left and traces a red line until the lower chart limit. When α=2π, the same condition reoccurs as for α=0. Thus the elongation represents the azimuth and the colour indicates whether an Eastward or Westward direction is computed. The instantaneous wind direction is, therefore, represented with no chart paintings or gaps. The results obtained by this method are highly accurate because of a better utilization of the strip chart width which is employed for a range of π instead of 2π. If this device is the final stage of a syncromotor system, by filtering the voltages at the delta windings, even the mean wind direction obtained is represented with continuity. In measuring the mean wind direction, an alternative method especially designed for light winds is presented. This is accomplished with a potentiometer whose output can be directly integrated. This is obtained by coupling in the previously mentioned potentiometer two wipers outphased by π/2. The two transfer functions are identical in shape but outphased by π/2. One of them at least is linear in each part of the wind rose and can be directly integrated without difficulty. Two outphased outputs are needed in order to maintain a one-to-one correspondence with α (as well as the sine-cosine function) and to take an accurate mean value when the wind fluctuations cause one of the wipers to oscillate about a tape. The largest error is estimated to be of the order of 1 degree.

Sine-cosine squared-magnitude function for maximally flat digital bandpass filters with adjustable skirt selectivities

A sine-cosine squared-magnitude function for a digital bandpass filter is proposed. It offers a maximally flat passband magnitude response and more flexibility on the number of exchangeable zeros at ωT = 0 and ωT = π rad.

Asynchronous Filters and Mobile Radio Communication Based on Walsh Functions

It is well known that the usual frequency filters based on sine-cosine functions do not require synchronization. Furthermore, sinusoidal electromagnetic waves of different frequency can be separated in mobile communication without any synchronization between the various transmitters and receivers, It is widely believed that this is a result of certain shift invariant features of sinusoidal functions that do not exist for other functions, The purpose of this paper is to show by means of simple examples that there are infinitely many systems of functions on which one may base asynchronous filters and mobile radio commnuication. The difference in the usefulness of these systems of functions depends only on the state of technology, no system is inherently better than others.

Conversion of sequency-limited signals into frequency-limited signals and vice versa (Corresp.)

Sequency-limited signals, consisting of a superposition of Walsh functions, and frequency-limited signals, consisting of a superposition of sine-cosine functions, can readily be converted from one form to the other without wasting either frequency or sequency bandwidth.

Applications of Walsh functions in communications

Communication theory was founded on the system of sine-cosine functions. A more general theory has become known more recently; it replaces the sine-cosine functions by other systems of orthogonal functions, and the concept of frequency by that of sequency. Of these systems, the Walsh functions are of great practical interest since they lead to equipment that is easily implemented by semiconductor technology. Filters, multiplexing equipment, and a voice analyzer/synthesizer have been built successfully for Walsh functions. Some interesting applications of electromagnetic Walsh waves have been found theoretically.