## 1 Introduction  
Optical encoders are usually the most reliable devices for precise displacement measurement and motion control. Due to their high accuracy, resolution, and repeatability, as well as their ability to work under different environmental conditions and their relatively low price, they are used in a variety of applications. Some examples are manually controlled machine tools and computer-numerical-control (CNC) machines [1; 2; 3; 4], robotics [5; 6; 7; 8; 9; 10], servosystems [11; 12; 13; 14], monitoring and fault diagnosis systems [15; 16; 17; 18], tracking systems [19; 20], linear and rotary positioning stages [21; 22; 23; 24], and other precision-positioning applications [25]. The overall accuracy and precision of the linear encoder mainly depend on the quality of the measuring scale graduation and nonlinear subdivisional error (SDE) in one signal period [26]. Standard photoelectric linear encoders typically have a measuring scale with a period of 20 or 40 \(\upmu\)m [27; 28]. In order to increase resolution, two analog electric signals generated by an encoder are converted to digital and interpolated [29; 30]. The smallest increment that can be detected may then reach the nanometer level. High accuracy and resolution are very important for applications that require precise positioning and good repeatability. In some applications, a linear encoder is used not only for position estimation, but also to provide feedback for speed control. A higher resolution ensures more stable and accurate position and speed loop. The SDE (also called the interpolation error) is a cyclic error that repeats with each period of encoder grating. It depends on the quality of a main measuring scale and of the generated electric signals. This error appears during the interpolation process because the encoder's distorted analog electric signals prevent the formation of a fine quadrature signal. In practice, the SDE does not cause any issues until its magnitude reaches the size of a measuring step. In other words, the magnitude of the SDE is a limit for the highest resolution. The minimal measurable incremental step has no point if the position error is greater. It is particularly important for applications requiring precise positioning and repeatability. In machines of which the feed axes or rotation tables are direct-driven, these nonlinear interpolation errors can cause not only positioning inaccuracies, but also result in loud noise, additional heat generation, and speed ripple [31]. The quality of the electrical signals highly depends on various aspects, like the optical-scanning principle, the design of a reading head and its ability to handle various deformations [32], mechanical vibrations [33; 34; 35], or temperature variations [36; 37]. In order to improve signal quality and to make encoders more robust, manufacturers use advanced optical-scanning methods such as the single-field or interferential scanning principle [38; 39], or they implement special-configuration multiple-track analyzer grating to eliminate higher-order harmonic signals [40]. Unfortunately, all such improvements require more complex encoder configurations and expensive optical parts. Another way to minimize SDE is electronic-signal correction. Nonideal signal parameters, such as relative amplitude, DC offset, phase shift, and nonsinusoidal shape, must be corrected before the interpolation process. Many studies dealt with this problem by using digital filtering and look-up tables, or by creating different algorithms for offline and dynamic compensation [41; 42; 43; 44; 45]. Because SDE is a short-range error that repeats each period, its compensation is a hard task. It is very important to know the actual limits of the used displacement-measurement encoder so that the application can work smoothly and be properly tuned. SDE is one of the parameters that should be investigated first, especially if the electrical signals of the encoder are not corrected or compensated before an interpolation process.