You must always avoid the temptation to “connect the dots” (to naively interpolate between samples) when working with discrete signals.1 Let’s recall the example from the previous section, and see how connecting the dots can get us into trouble.
Imagine someone asks you what the altitude of the plane was at 65 minutes into the flight. How should you respond? You don’t actually have a sample for the altitude at 65 minutes, but you do have measurements for the altitude at 60 minutes and 70 minutes. You might feel tempted to draw a line between these two samples, perform some simple linear interpolation, and infer that the altitude was about 31,000 feet. This sort of temptation is completely natural, but really unhealthy when working with discrete signals. The most appropriate response is simply to say, “I don’t know”. Anything else would be a fib.
Given our measurements and context, we cannot confidently report an altitude for 65 minutes into the flight. Our discrete signal tells us nothing about the altitude of the plane at 65 minutes into the flight. Think about it like this: our discrete signal can actually represent many potential flight histories, most of which have different altitudes at 65 minutes into the flight. In actuality, there are an infinite number of possible flight histories which intersect with the samples of our discrete signal but assume decidedly different altitudes at the 65 minute mark. Click the Play button to see four examples. I hope you’ll notice that connecting the dots in a naive way can be extremely misleading.
Figure 1. Altitude of a Plane as it Traveled from Paris to Berlin
Samples connected dubiously with a grey line
We would call the red, blue, green, and orange curves aliases of one another since they are indistinguishable from one another when sampled with a period of 10 minutes per sample. In other words, the red, green, blue, and orange signals all look exactly the same after being sampled every ten minutes.