 A warm welcome to the third lecture on the subject of digital signal processing and its applications. Let us take a couple of minutes to review what we did in the previous lecture. In the previous lecture, we had initiated our discussion on sampling. We had tried to answer the question, when can we retain information even after sampling? To expand a little bit, when can we consider values of a signal only at sampling instance and yet not create any confusion or not lose information? We had not completed our answer to that question. We have only got thus far and we shall now summarize that. So let us put down the theme of our lecture today, reconstruction from sampling or from samples to be more appropriate, aliasing and discrete systems. This is the theme of our lecture today. Well, what we had seen the last time is that, if you have an original sinusoid of frequency 0.1 kHz and if you have a sampling rate of 1 kHz, then you lined up with a confusion of sine waves and the confusion is as follows. On the kHz scale, the samples could have come either from a 0.1 kHz frequency or from a 1.1 or a 0.9 kHz frequency and of course, if the amplitude of this is a0 and the phase phi0, the amplitude of this is a0, phase phi0, the amplitude of this a0, phase minus phi0, the amplitude of this at 2.1 is a0, phase phi0 and we can keep doing this, amplitude a0, phase minus phi0 and we can continue. So at every multiple of 2 kHz, we have these confusing sine waves if you want to call them that which contribute to the same samples. Now I want to say a little more about this observation. You see, what we had shown in the previous lecture was that all these sine waves generated the same samples at those instance. However, we need to show something more. We need to show that if I indeed add up these sine waves as I have shown here with amplitudes a0 and phase either phi0 or minus phi0 depending on whether it is after the sampling frequency or before, if I add up all these sine waves, I generate what I really call a train of samples and what do you mean by a train of samples. In fact, this is a challenge which I put before you. It is a practice that I follow in courses that I teach to challenge your imagination right from the beginning to put before you exercises which will make you think and work certain things out on your own so that you get a much better grasp of the ideas that are being taught. And we are beginning with the first challenge, take these sine waves and add them one by one. What I mean by that is take the sum a0 cos omega nut nt plus phi0 or in fact let us put sample you know at nt of course you will get infinity there is nothing much to be seen but let us instead add up the continuous so you know we should say not these sinusoid at the samples but sinusoids the continuous sinusoids so 2 pi by t as we would have it minus omega0 times t minus phi0 plus a0 cos 2 pi by t plus omega0 times t plus phi0 plus now what will come next let me do it for a few terms for you so what I mean is you see you have taken the original sine wave as a continuous sine wave here you have taken the next one the next one is the sampling frequency in radians per second 2 pi by t minus omega0 and then the phase would be minus phi0 sampling frequency in radians plus omega0 plus phi0 now the next term would be 2 times the sampling frequency in radians minus phi0 and the next term after that would be 2 pi by t times 2 plus phi0 let me write down 2 more terms to make the concept clear let me write down completely you can keep on writing more and more terms anyway we got the basic idea now the challenge before you is to prove that as you add more and more terms you have a constructive interference at the point of sampling and a destructive interference at all points other than that of sample in other words what I want you to prove in the challenge and let me write that down prove that this sum is constructive at the points of sampling that means t equal to nt and destructive at others so what is going to happen as you add more and more terms essentially this sum is going to become localized near the points of sample it is going to rise higher and higher in fact it is very easy to see that at the point of sampling the sum is divergent it would go towards infinity we knew right in the beginning from the previous lecture that the samples are the same when they come from all of these sine waves so at the points of sampling of course constructive interference is obvious because all the samples are the same all the values all the sine waves take the same values the key thing is to prove that at other points the sum is destructive that means as you add more and more and more and more terms they all compete and cancel one another out so I leave this as a challenge to you and I give you a hint you know you will have to use trigonometric identities you are summing up sine waves of the same amplitude of different frequencies use trigonometric identities and show and try and see what they are converging to what that sum converges to if you like you could decompose a sine wave into a sum of 2 oppositely rotating complex numbers as you often do so that is also acceptable we will be dwelling on that idea in more detail later they are called phasers anyway for the moment I am putting this before you as a challenge why I am saying this is because it tells you what happens when you sample when you sample what you are doing is essentially to create small pulses at the instance of sampling of value or strength equal to the sample value of the waveform at that point at all other points you essentially have a 0 signal so sampling means to replace a continuous signal by a train of pulses.