I wanted to share the above image to show why aliasing occurs if you sample below the Nyquist frequency. In each of the Fourier spectrum graphs in the image, the horizontal-axis represents frequency. Lower frequencies are close to the origin and higher frequencies are far from the origin. On the vertical axis, we have the magnitude of the the frequency present in the signal. The top graph is the the Fourier spectrum of a contrived continuous signal. The middle graph is the spectrum you'll get if you sample that signal at a frequency below the Nyquist frequency. The bottom is the spectrum you'll get if you sample it at a frequency above the Nyquist frequency. You'll notice that the spectrums of the sampled versions of the signals look like a bunch of copies of the spectrum of the original signal. The reasons for that will make sense if you ever choose to take EE120, but I think they are a little too much to explain for the purposes of this comment. The main thing to realize is that the highest frequencies present in the original signal are W and -W. All other higher frequencies are zero. That means that the Nyquist frequency should be 2W or higher. You'll notice that if the sampling period Ts > 1/2W, then the sampling frequency (equivalent to 1/Ts), will be below the Nyquist frequency. You'll see that the copies of the of the original Fourier spectrum in this case overlap with each other and distort the shape of the original spectrum. This is aliasing! In the bottom image, we are sampling at a rate above the Nyquist frequency. This causes the Fourier spectrum of the sampled signal to preserve the original shape of the original signal. We can reconstruct the original signal from the Nyquist sampled signal by truncating all the higher frequencies using an ideal low-pass filter! This is not the case with the aliased sampled signal because of how the shape of the spectrum gets distorted.
muuncakez
really great summary of the math! much appreciated and the visual really helped cement the mathematical aspect.
sueyoungshim
@aeave Great explanation of aliasing and the Nyquist frequency! Your breakdown of the Fourier spectrum graphs really helps understanding how sampling below the Nyquist frequency leads to aliasing, distorting the original signal's spectrum. The comparison between sampling below and above the Nyquist frequency explains the importance of meeting or exceeding the Nyquist criterion to avoid aliasing.
https://i.stack.imgur.com/KiTeP.png
I wanted to share the above image to show why aliasing occurs if you sample below the Nyquist frequency. In each of the Fourier spectrum graphs in the image, the horizontal-axis represents frequency. Lower frequencies are close to the origin and higher frequencies are far from the origin. On the vertical axis, we have the magnitude of the the frequency present in the signal. The top graph is the the Fourier spectrum of a contrived continuous signal. The middle graph is the spectrum you'll get if you sample that signal at a frequency below the Nyquist frequency. The bottom is the spectrum you'll get if you sample it at a frequency above the Nyquist frequency. You'll notice that the spectrums of the sampled versions of the signals look like a bunch of copies of the spectrum of the original signal. The reasons for that will make sense if you ever choose to take EE120, but I think they are a little too much to explain for the purposes of this comment. The main thing to realize is that the highest frequencies present in the original signal are W and -W. All other higher frequencies are zero. That means that the Nyquist frequency should be 2W or higher. You'll notice that if the sampling period Ts > 1/2W, then the sampling frequency (equivalent to 1/Ts), will be below the Nyquist frequency. You'll see that the copies of the of the original Fourier spectrum in this case overlap with each other and distort the shape of the original spectrum. This is aliasing! In the bottom image, we are sampling at a rate above the Nyquist frequency. This causes the Fourier spectrum of the sampled signal to preserve the original shape of the original signal. We can reconstruct the original signal from the Nyquist sampled signal by truncating all the higher frequencies using an ideal low-pass filter! This is not the case with the aliased sampled signal because of how the shape of the spectrum gets distorted.
really great summary of the math! much appreciated and the visual really helped cement the mathematical aspect.
@aeave Great explanation of aliasing and the Nyquist frequency! Your breakdown of the Fourier spectrum graphs really helps understanding how sampling below the Nyquist frequency leads to aliasing, distorting the original signal's spectrum. The comparison between sampling below and above the Nyquist frequency explains the importance of meeting or exceeding the Nyquist criterion to avoid aliasing.