We talked in class about how, in a cascaded system, the "time coordinates" change and that you have to keep track of them as they propagate down the system through other transforms if you are trying to find the final output of the systems. This explanation seems somewhat confusing, so I tried looking at it in a different manor. It seems like we are trying to find the equation for the output by starting at the input, like a signal would, and writing all the transforms or "things" that happen to the signal using different variables, then we go back and substitute so it all works out. I did the reverse and started with the output and "built up" the total effect (output equation) from what happened "most recently" (that is to say the present is at the output and the input is the past), so you don't have to worry about keeping track of different substitution variables (Mimi used squares and squiggles) or the "time coordinates".
A very simple method is as such:
1) Start from the output, take the last transform and put it in parenthesis. It's in a nice package now, it's done, don't touch it.
2) Take that "package" and drop it right into the next "most recent" (one to the left) transform (substitute it for t). Put that in parenthesis, it's your new package.
3) Keep going until you run out of transforms.
4) If so inclined, simplify.
Example:
Sys 1: y1(t)= x(2t) Sys 2: y2(t)= x(t-3)
Input -> Sys 1 -> Sys 2 -> Output
Start from the output, take the "most recent" transform (Sys 2) and put it in parenthesis, so: (t-3)
Next, take the next most recent transform (Sys 1), and drop your (t-3) in it (substitute your "package" of (t-3) for t): 2(t-3)
Simplify: 2t-6
Done! Don't forget it is a transform of a function, not a function itself so you need to state so, as such: z(t) = x(2t-6)
So, put very simply, start at the output and substitute, in iterations, towards the input. That's all you really need to know, I just thought if I was verbose and used analogies I might score bonus points. Keep in mind the all the transforms deal with the independent variable, in this case time. Also, in the spirit of the kiwi, I could be completely wrong about everything. :) So seriously, someone check my work.
From michael.a.mitchell.2 Sun Sep 2 12:20:52 -0400 2007 From: michael.a.mitchell.2 Date: Sun, 02 Sep 2007 12:20:52 -0400 Subject: Determining the effects of Transforms of the Independent Variable (Time) in the form x(at + b) Message-ID: <20070902122052-0400@https://engineering.purdue.edu>
If you are trying to find the effect of a transform in the form of x(at + b), you should:
1) Delay or advance x(t) by the value of b. (Advance if b>0, delay if b<0) 2) Then scale/reverse time by the value of a. (Compress if |a|>1, Stretch if |a|< 1, Reverse time if a<0)
