QA measurements

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Questions & Answers for the exam Measurement exam – Tarabini Sampling Describe what is the aliasing problem and how can we avoid it. Aliasing is a problem related to the choice of the wrong fs, if the fs is not high enough the signal cannot be properly reconstructed starting from its samples: - In time it can be explained by Shannon’s theorem: fs > 2 fmax - In frequency it can be explained through convolution: sampling in time is equal to multiplying by a train of unitary pulses at distance dT, so in frequency it is equivalent of a convolution of the signal’s spectrum with a train of pulses at distance 1/dT=fs. If fs is not high enough (>2 fmax) the convolution of the spectrum will overlap Describe what is the leakage problem and how can we avoid it. The leakage is an error of the spectrum due to the wrong choice of the observation time T. - In time it can be explained by the circular convolution, because the signal has to be made periodic, but if the length of the observation is not equal to the period of the signal (or an integer multiple) the circular convolution of the signal will reconstruct a signal different from the original one, that will show different frequency component - In frequency it can be explained by convolution, because observing the signal for a period T means multiply it by a rectangular window of length Tw, which in frequency corresponds to convoluting the signal with the spectrum of the window. If the period of the signal T is not an integer multiple of the frequency resolution (=1/Tw), then the zero crossing of the window’s spectrum won’t correspond with the spectral lines and there will be a distortion of the spectrum à to avoid leakage problem we can use different types of windows like Hanning window Short time leakage/long time leakage…