DIGITAL SIGNAL PROCESSING (DSP) 193
So here’s a question. How often must we sample the parked cars to feel comfortable about driving by them at this speed? Remember, we are driving past one car per second. Let’s assume we close our eyes and only open them briefly at a fixed sampling rate. How often do we have to open them to feel comfortable? Well, to confess, I tried this stupid experiment. It’s a little bit like a doctor injecting himself with germs to test out his new vaccine. I did it safely though. Here’s my report. Keeping my eyes closed was intensely uncomfortable, and I didn’t try it very long, which was certainly to be expected. Opening my eyes once a second was uncomfortable. I could only see each car once as I passed it. Opening my eyes twice a second was more comfortable in that I felt I could control the car properly. In this experiment, I experienced the Sampling Theorem firsthand in a conscious manner. To observe the cars properly, I had to sample the cars twice a second in a situation where the cars were going by once per second. Critics of this experiment might say, “That’s great, but what if a fast-moving car came darting out of a side street? Wouldn’t that cause an accident?” The answer is yes. Sampling might not work properly if an unexpected car appeared on the street. If we got lucky, we would notice the fast car when our eyes were open and we might be able to avoid it. We would probably not be able to tell how fast it was going though. Worst case, we would never even see the fast car; it would both appear and hit us while our eyes remained closed. The key here is an antialias filter, which, in our example, would be a speed limit sign. Town planners automatically protect the quiet side streets (those with rows of parked cars) by surrounding the neighborhood with speed limit signs. The fast-moving vehicles are therefore filtered out of the situation. If fast-moving cars were the norm in the neighborhood, we would be on guard and sample the road ahead much more frequently. We react instinctively as we apply the Sampling Theorem in this way. Let’s summarize the driving experiment in DSP terms. Cars are driven at all different speeds; these are our input signals. To protect our sampling system, we put in an antialiasing filter (speed limit signs) so we do not have to deal with cars moving faster than one car length a second. Driving past parked cars at one car per second, we sample the cars visually two times a second. Per the Sampling Theorem, this gives us enough information to process the data and to drive carefully. Let’s try another experiment. We will use pure sine waves as input signals to the DSP system and will sample at a fixed rate every 0.3 seconds. This works out to a sampling rate of 3.33 Hz or roughly 20 radians per second. We will vary the frequency of the analog input signals from 3 to 15 radians per second. With a fixed sampling rate of 20 radians per second, the Sampling Theorem predicts we will do a good job of sampling sine wave input signals with frequencies as high as 10 radians per second. By looking at sine waves from 3 to 15 radians per second, we should see a breakdown in the sampling