We reverse a signal x(t) by flipping it over the verticalaxis to form a new signal y(t) = x(t).
Example: Find
y(t) = x(t)
where x(t) is:
The signal y(t) = x(at) is a timescaled version of x(t).
If a > 1, we are SPEEDING UP x(t) by a factor of a.
If a < 1, we are SLOWING DOWN x(t) by a factor of a.
Example: Given x(t), find y(t) = x(2t). This SPEEDS UP x(t)
What happens to the period T?
The period of x(t) is 2 and the period of y(t) is 1, because
Example: Given x(t), find z(t) = x(t/2). This SLOWS DOWN x(t)
Example: Given y(t), find w(t) = y(3t) and v(t) = y(t/3).
y(t) = x(t  t_{0})
Here, the original signal x(t) is shifted by an amount t_{0 }.
Rule: set t  t_{0}=0 and move the origin of x(t) to t_{0}.
Given x(t) = u(t+2)  u(t2), find x(tt_{0}) and x(t+t_{0}).
Example: Determine
x(t) + x(2t)
where x(t) = u(t+1) u(t2)
Step 1: which of these functions is x(t)?
A, B, C or D
Step 2: Find x(2t)
Method I to find x(2t): Reverse in time, then delay.
Let y(t) = x(t)
And y(t2) = x((t2)) = x(2t)
Method II to find x(2t): Advance, then reverse in time.
Let w(t) = x(t +2)
As the last step, add up x(t)and x(2  t)
Combinations of Scale and Shift
Find x(2t+1) where x(t) is:
Method 1: Shift then scale: x(at+b):(i) v(t)=x(t+b); (ii) y(t) =v(at)= x(at+b).
v(t) 
= 
x(t+1) 

y(t) 
= 
v(2t) 

Method 2: Scale then shift: x(at + b) = x(a (t + b/a)) :
( i ) w(t) = x(at) ; 
( ii ) y(t) = w(t + b/a) = x(at+b) 
w(t) 
= 
x(2t) 

y(t) 
= 
w(t+1/2)) = x(2(t+1/2 )) = x(2t+1) 

Example: Given x_{1}(t), find x_{1}(t), 2x_{1}(t), and .5x_{1}(t)
Example: Given x_{2}(t), find 1  x_{2}(t). This is a case of adding together two signals.
Addition of two signals: x_{1}(t) + x_{2}(t)
To add two signals together, we simply add the values of the signals pointbypoint.
Multiplication of two signals:x_{2}(t)u(t)
To multiply two signals together, we simply multiply the values of the signals pointbypoint.