I can't think of a better way for asking this question so I will start with an example. Suppose that I have an input signal with a max frequency of 50Hz (sampled at 100Hz). Now the signals of interest lie in the range 0-5Hz, so I can add a low-pass filter with a cut-off of 5Hz, and use the resulting signal for further processing. My understanding is that now I can downsample the filtered signal by a factor of 10 and hence reduce processing load. Am I right? If yes, why is downsampling not ALWAYS performed after filtering because it seems to me as the obvious way to go? And if I am wrong in my assumption, where am I mistaken?
Answer
You are correct that if your signal is bandlimited to <5 Hz, then you can perfectly represent it with a 10Hz sampling rate. This is the well-known sampling theorem
But ... there may be practical considerations for why one would not be able and/or inclined to use critically sampled data.
One reason is the difficulty of making a signal critically sampled. Any operation you perform to change the rate of the signal is going to have some filter with a non-zero transition bandwidth. In your example, this limits the unaliased frequency content to 5-ftrans This transition bandwidth can be made very narrow with long impulse response filters but this has costs both in terms of processing and in transients (ringing) at signal start and end.
Another reason is the efficacy of algorithms that work on the resulting signal. If you need to work with a blackbox component that can only choose the nearest sample, then you'll be better off feeding it oversampled data.
Most (all?) non-linear operations will behave differently with critically sampled vs oversampled data. One example is squaring a signal, a well known method of BPSK carrier recovery. Without a 2x oversampled condition, the multiplication of the time domain signal with itself causes wraparound garbage aliasing when the frequency domain convolves with itself.
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