Remember how we do sinc sampling. So we take a sinc function, we scale it by t s, we shift it by n times t s, and we take the inner product with x of t to get the sample x n. This can be written as the convolution between sinc. Ts, so we denote the scaling of the sinc function by Ts at location nTs. In a block diagram, we take x(t), we go through a low pass filter, an ideal low pass filter of bandwidth omega N. And we take samples every Ts seconds to derive a sequence x[n]. Raw sampling is when we don't care about first taking the inner product with the sinc function. So we just take x(t), and every Ts seconds, we take a sample, x(n). So you must remember the wagon wheel effect. We had seen simulated movie how a wheel could go backwards if we were not careful about having a high sampling rate. The complex exponential is a femular character. Now we look at the continuous time complex exponential xt is e to the j. Capital omega naught times 't'. It's always periodic. It has a period of 2 pi over capital omega. Not all angular speeds are allowed. The full transform of this complex exponential is a delta that is seated at omega is equal to omega naught. It is obviously bandlimited to omega naught. If we look at this continuous time complex exponential on the unit circle, then it is a phaser that runs around the unit circle. If we take samples of this continuous time complex exponential so x[n] is e to the j omega naught ts times n. Then the raw samples are snapshots at regular intervals of these rotating point. So resulting digital frequency is small omega naught which is equal to omega naught times ts okay. So please note the difference between small omega and capital omega. When Ts is more than pi over capital omega 0 or small omega 0 is smaller than pi, then the phaser will advance in small steps as we can see here on this figure. When Ts is between pi omega 0 and 2 pi omega 0 that is small omega 0 is between pi and 2 pi then the phaser advances in big steps. However this looks as if it were going in a negative direction as you can see here it steps in small steps and it looks like it's going in the opposite direction than what it is actually going. Finally, when TS is bigger than 2 pi over omega not, or small omega not is bigger than 2 pi, then again we think it goes in the positive direction, but it does actually go full circle. Plus the little step. So we see that this large frequency looks actually like a small frequency. Okay, we see x[1], x[2], x[3], x[4], and so on, as if it were going in small steps. And so, this large frequency actually mimics a small frequency. And that's the phenomenon we call aliasing. Let's look what happens when we reconstruct or interpolate based on the samples we have just seen. Let's look at this in a block diagram. So we have x(t). We sample with a sampling period Ts. Then we interpolate again, with a spacing of Ts and we get x hat of t. What is x hat of t with respect to x of t? No let's look at the output x hat of t and what frequency will be reproduced. So the first case is easy, that's when the sampling period Ts is smaller than Pi over omega naught. Why, because we are meeting the sampling theorem and so the same frequency that went in should actually also come out. So the digital frequency in this case, the digital frequency small omega naught, is between zero and pi, and the output is indeed what went in, e to the j omega naught t. The second case is when ts is in the intermediate range so between pi over omega not and 2 pi over omega not. The digital frequency now is beyond pi but smaller than 2 pi so we see this in this formula here. And then the output will have a different frequency in the input, namely frequency omega one, which is equal to omega not, minus two pi over T S. Okay? So there is already aliasing happening, because you have this shift in frequency here that transform the original omega not into the output. Omega 1. Finally, the rep case, the third one here, when Ts is bigger than 2 pi over omega naught then the digital frequency would be beyond 2 pi. Okay, now you know the frequencies are between 0 and 2 pi so that doesn't really have a meaning except that it shows that this frequency will be folded back through everything. And it results in an output here e to the j omega 2t where omega 2 is simply omega naught. Now take module 2 pi over ts. Okay and this is definitely a different frequency then what went in.
