I have just started with Fourier and Signal Processing . Here are few problems I am not able to understand. Thanks !
Deriving Fourier Transform Pairs is a royal pain in the ass, so you try to minimize how often you have to do it. Thus, you go through the work of deriving the transform pair for a very simply function (the "unit" version of the function, for instance the "unit impulse" or "unit step" or "unit pulse" or "unit triangle") and then figure out how scaling and shifting the unit version in the time-domain changes the Fourier transform without having to rederive a new transform pair. Then you use scaled and shifted versions of the unit functions to construct the time-domain signal you are working with. When you are done, you have your arbitrarily complex time-domain signal, which would be a real nightmare to analytically take the Fourier transform of, represented as the sum of a whole bunch of simpler signals, each of which you already know the transform of. As for convergence, that concept is critical to the whole thing. You are representing a signal as the sum of an infinite number of terms and relying on the Fourier transform of that sum to be finite. That can only happen if, as you add more and more terms, the sum converges to a finite result. As a crude example, imagine that I have a pin at one end of a meter stick and then, each second, I move the pin one half of the distance from whereever it is to the other end. The total distance traveled after N steps is the sum of the distance travelled in each of those steps and, in the limit that N goes to infinity, the sum has an infinite number of terms. However, the sum itself is finite and approaches one meter. So it converges. But if I change the rules just a little bit and say that in the first step I travel 1/1 a meter and in the second step 1/2 of a meter and in the third step 1/3 of a meter and so on (in the Nth step I travel 1/N meter), then this series does not converge and, in fact, given any desired distance, I can tell you a value of N that will put you past that distance.
Thank You Bahn. The needle example is a very practical one. Helps a lot. For the shifting. I am able make out from your explanations the idea behind the shifting of signals. But I am still not able to tackle the problem. Should I leave it for now and come back to it later after I have some concrete basics about signals ?
No, I'd struggle through it now. The key is to keep in mind that you can scale the triangles that you are putting together by a negative scale factor as well as positive. So, in the first one, note that you can use one triangle to get the sides just fine, but now there is a peak where it should be flat. But the difference between the peak and the flat is simply a small triangle! So subtract off a second, small triangle that is just big enough and shifted just enough to make that peak go away. The same thing is true for the others. In general, use a triangle to capture as much of the behavior as you can and then look at the difference between what you have and what you want. Then add or subtract another triangle to eliminate as much of the difference as you can and keep repeating the process. Note that there is no single correct answer because you can construct each of them multiple ways. Heck, even the unit triangle can be composed of multiple scaled and shifted unit triangles! Having said this, you are usually expected to find a solution that uses the minimum number of superimposed triangles, but even this is not always the "simplest" or "best" solution.
I get it now. The picture much clearer. Thanks for the explanation .The whole idea behind practicing such questions is that we are able to formulate the functions of the signals by looking at them. Am I wright here or is there something else ?
Not only visually, but just getting a feel for how to combine variations on a basic signal to form much more complex signals and, later, how to combine variations on a handful of basic signals to form even more complex signals and, conversely, to tear them apart.