Explain how the resolution of a shaft encoder could be improved by pulse interpolation. Specifically, consider the arrangement shown in Figure P6.47. When the masking windows are completely covered by the opaque regions of the moving disk, no light is received by the photosensor. The peak level of light is received when the windows of the moving disk coincide with the windows of the masking disk. The variation of the light intensity from the minimum level to the peak level is approximately linear (generating a triangular pulse), but more accurately sinusoidal, and may be given by

where θ denotes the angular position of the encoder window with respect to the masking window, as shown, Δθ is the window pitch angle. Note that, in the sense of rectangular pulses, the pulse corresponds to the motion in the interval Δθ/4 ≤ θ ≤ 3Δθ/4. By using this sinusoidal approximation for a pulse, as given above, show that one can improve the resolution of an encoder indefinitely simply by measuring the shape of each pulse at clock cycle intervals using a high-frequency clock signal.

FIGURE P6.47