Why Is Time Compression Represented by x(2t) in Signal Processing?

[physio]

I have just started a course on signals and systems and am finding the subject confusing.
This question pertains to the transformations of the independent variable which is time in this case. I don't know why it is transformation of the "independent variable" as the time axis is the same as it originally was. The time axis is not "transformed". For example in time reversal of a signal, the signal is simply flipped about the origin and the time axis is unaltered (i.e. 't' stays the same but doesn't become '-t') yet the book says it is a transformation of the independent variable. Am I missing something?

After a reasonable amount of thought can anyone intuitively explain why time compression has a>1 i.e for a signal x(t) why and how x(2t) is the time compressed signal?? I think I will be able to answer the first question if I know why x(2t) is the compressed version of x(t).

 

[uart]

I have just started a course on signals and systems and am finding the subject confusing.
This question pertains to the transformations of the independent variable which is time in this case. I don't know why it is transformation of the "independent variable" as the time axis is the same as it originally was. The time axis is not "transformed". For example in time reversal of a signal, the signal is simply flipped about the origin and the time axis is unaltered (i.e. 't' stays the same but doesn't become '-t') yet the book says it is a transformation of the independent variable. Am I missing something?

Hi physio. Think of it in two steps.
Step 1. Just re-label your time axis as "-t" instead of "t" in the right hand direction. Essentially this is all that the transformation actually involves, however convention requires that the "t" axis to run the other way (so that's why we need step 2).

Step 2. Flip both the "t" axis and the graph together. This step doesn't really change the graph, it merely puts it into the form people expect to see.

After a reasonable amount of thought can anyone intuitively explain why time compression has a>1 i.e for a signal x(t) why and how x(2t) is the time compressed signal?? I think I will be able to answer the first question if I know why x(2t) is the compressed version of x(t).

The best way to see this is to pick any particular point on your graph, say $x_1 = x(t_1)$ and notice that this same x point on the transformed graph occurs at a transform "t" value of $t^* = t_1/2$, (as $x_1 = x(2(t_1/2))$.

 

[physio]

Thanks a lot for your reply uart! I understood the time reversal operation with your explanation but I yet have problems understanding the time scaling operation. I understood that the transformed variable t*=t₁/2 i.e the transformed time variable is a scaled version (in this case t₁ /2) of the original time variable. Hence shouldn't the transformed graph should be x(t*)?? Do we have to consider the graphs together for calculating the new function i.e. x(2t) or x(t/2) i.e. with reference with the original signal? Thanks in advance..!