How to visualise shifting of a time-reversed signal?

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 direction of the time axis (or whatever the quantity we have in the x-axis) i.e., these "left" and "right" notations work only when the axis is running from left to right!

So, if we have to define a broader definition, we can say:

"signal is shifted towards positive t-axis" instead of "signal is shifted towards right", and

"signal is shifted towards negative t-axis" instead of "signal is shifted towards left".

Okay, that's enough intro; let's go and get our waveform.

Plotting x(-t+1):

First, consider an arbitrary signal x(t) such as shown below:

 

If we time-reverse our signal, we get x(-t), and it looks like this:

Now, just make a simple substitution: let's say η = -t.

Then our x(-t) becomes x(η)!

So, it looks like this:

Looks good right? 

NOO..!!

The direction of the axis was not changed! When we say that η = -t, means the positive t-axis becomes the negative η axis, and vice versa...

So, our x(η) should look like this:

Now, now, now, we are almost done.

So, what do we want? x(-t+1)

What is our signal in terms of η? x(η+1)

But what is x(η+1)? A left-shifted signal of x(η) right?

Yeah, absolutely. But what does left shifting actually means? Moving towards the negative direction of the axis, i.e., shifting towards the -η axis..!

Ta-da! There we are, and our target signal, x(-t+1) (or) x(η+1), looks like this:

That's how we can think about shifting a reversed signal.

Of course, this seems a bit lengthy procedure, but at least this can help in the initial stages till we develop an intuition for it.

Yup, that's it. You can leave the page now.

Or...

Hey, thank you for reading! Means a lot. Feel free to share your views (or) point out any mistakes in the comments, and help us all get the 'Eternal Thing'!