The Eternal Thing

Shifting, reversal and scaling are the common operations we perform on an analog (or) digital signal. In this post, we will see how can we visualize shifted version of a time-reversed signal. To put it simply, here is what we are gonna do: Given a signal x(t) , how to plot x(-t+1) Note: If you are already good at this, maybe this post is not for you. You can safely stop reading now. But if you still want to proceed, you are always welcome!! Before discussing how to plot our signal, let's have a small talk about - Time Shifting: For a given x(t) , we can say that x(t+1) is a time-shifted signal. In general, we talk about the shifting of the waveform relative to the time axis, and we use the terms "left shift" and "right shift" to denote two types of movements possible for the waveform. However, we have to note that the terms indicate only half the story. There is an underlying assumption for these to work, which is the dir...

  The divisibility rule of 3 is simple: For a given number, add the individual digits. If the sum is divisible by 3, then the overall number is divisible by 3. Otherwise, it's not. But why the sum of the digits? How can we prove that this is robust for any given number? If you wanna think it out yourself, take a pause. Otherwise, just continue reading below... The Proof: Consider a generalized decimal number, with the total number of digits as n+1, and the coefficients being a 0 , a 1 , a 2 , and so on up to a n . So, our number can be expressed as: a 0 10 0 + a 1 10 1 + a 2 10 2 + ... + a n 10 n . Now, the powers of 10 can be written in terms of 9s, right? 10 1 = 10 = 9 + 1 10 2 = 100 = 99 + 1 10 3 = 1000 = 999 + 1 and similarly, 10 n = 100..0 = 99..9 + 1. Then, our number becomes: a 0 + a 1 (9 + 1) + a 2 (99 + 1) + .... + a n (99..9 + 1) = [a 0 + a 1 + a 2 + ... + a n ] + [9*a...