Interpolation Technology in DSO (I)

   Interpolation is an important function in today's digital oscilloscopes. The primary purpose of a DSO is to analyze and view analog waveforms. To do this, the DSO samples the waveform at some finite sampling rate and generates a voltage vector with respect to time. Since this vector represents a set of sample points (rather than the actual smooth analog waveform), it is often necessary to modify the acquired waveform by generating predicted sample points between the actual acquired points. The process of generating sample points between the actual waveform sample points is called interpolation. In normal operation, interpolation results in a waveform with a higher sampling rate that is a closer approximation of the analog waveform being analyzed.

   This article will discuss the two most popular interpolation methods and explain their basic workings. It will then explain how to ensure good interpolation results and how to determine when to use interpolation. This article uses some simple experiments to compare the interpolation performance of three high-end oscilloscopes. Finally, this article compares the differences in interpolation operations.

Linear interpolation

The simplest form of interpolation is linear interpolation. Linear interpolation assumes that a straight line connects each waveform sample point, which is a very simple and natural method that provides limited results.

Note: Definition from Merriam-Webster dictionary 2003 Linear interpolation can be viewed as the convolution of an oversampled waveform using a triangular window. The triangular window is constructed by generating a triangle with a height of 1.0 and a width of twice the sampling period. As the window slides to the right, the interpolated point is obtained by multiplying the sum of the window values ​​by the actual sample value at the intersection of the window and the sample. The interpolated sample is placed at the time of the window vertex. The window width defines its memory, that is, the time when the actual sample affects the interpolated sample. Since the window width is twice the sampling period, only the samples that fall within the interpolated sample time affect the interpolated value. This convolution can also be achieved by using a digital filter with an oversampling arrangement. The arrangement shown in the figure is a 5-point oversampler. In this arrangement, 5 new samples are generated for each new sample. The filter output depends only on the new input and the last input. The filter coefficients are generated by sampling the window. By examining these figures, it is easy to see the pattern of establishing the filter coefficients. SinX/X Interpolation A popular and more complex form of interpolation is called SinX/X (also called Sync or simply SinX interpolation). SinX interpolation gets its name from the well-known shape of the window function used for convolution. Unlike the narrow, pointed triangle of linear interpolation, the SinX interpolation window is theoretically a damped sine wave that never ends. [Graphic content:] point 0: point 0 point 1: point 1 point 2: point 2 point 3: point 3 point 4: point 4 This window shape is derived from an important assumption, namely that the Nyquist criterion is followed in the sampling of the original waveform. In other words, it assumes that all frequency components in the sampled analog waveform are below half the sampling rate of the waveform, which is a reasonable assumption. When this assumption is made and the inverse Fourier transform of this hypothetical spectrum is taken, this well-known function is obtained. Generally speaking, this assumption is the best assumption that can be made, but as we will see, it is not always the correct assumption. As a result, SinX interpolation is actually the most efficient interpolation method. This can be seen by examining the significance of the Nyquist criterion. Nyquist claimed that a continuous analog signal can be fully determined from the sampled points when all frequency components of the signal are below half the sampling rate. SinX interpolation is a technique for obtaining a continuous analog signal. SinX interpolation is subject to certain mathematical and practical techniques that make this method impossible to be perfect. First, the Sync function is infinite and must be truncated at a point where the truncation error is acceptably low. This is because a truly bandwidth-limited signal must have an infinite length, meaning that all sample points must be known all the time. It turns out that the influence of points farther and farther from the interpolation point drops off rapidly, and truncation provides highly acceptable results. Another disadvantage is that in the system being sampled, noise and spurious signals generated by the DSO structure (such as channel multiplexing) can mix into the signal, resulting in noise and distortion that exceeds the Nyquist limit. The resulting errors can be kept at an acceptably low level.