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Signal Operations

Module by: Richard Baraniuk. E-mail the author

Summary: This module will look at two signal operations, time shifting and time scaling. Signal operations are operations on the time variable of the signal. These operations are very common components to real-world systems and, as such, should be understood thuroughly when learning about signals and systems.

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This module will look at two signal operations, time shifting and time scaling. Signal operations are operations on the time variable of the signal. These operations are very common components to real-world systems and, as such, should be understood thoroughly when learning about signals and systems.

Time Shifting

Time shifting is, as the name suggests, the shifting of a signal in time. This is done by adding or subtracting the amount of the shift to the time variable in the function. Subtracting a fixed amount from the time variable will shift the signal to the right (delay) that amount, while adding to the time variable will shift the signal to the left (advance).

Figure 1: f⁢t−T f t T moves (delays) f f to the right by T T.
Figure 1 (sigops1.png)

Time Scaling

Time scaling compresses and dilates a signal by multiplying the time variable by some amount. If that amount is greater than one, the signal becomes narrower and the operation is called compression, while if the amount is less than one, the signal becomes wider and is called dilation. It often takes people quite a while to get comfortable with these operations, as people's intuition is often for the multiplication by an amount greater than one to dilate and less than one to compress.

Figure 2: f⁢a⁢t f a t compresses f f by a a.
Figure 2 (sigops2.png)

Example 1

Actually plotting shifted and scaled signals can be quite counter-intuitive. This example will show a fool-proof way to practice this until your proper intuition is developed.

Given f⁢t f t , plot f⁢−(a⁢t) f a t .

Figure 3
(a) Begin with f⁢t f t (b) Then replace t t with a⁢t a t to get f⁢a⁢t f a t (c) Finally, replace t t with t−ba t b a to get f⁢a⁢(t−ba)=f⁢a⁢t−b f a t b a f a t b
Figure 3(a) (sigops4.png)Figure 3(b) (sigops5.png)Figure 3(c) (sigops6.png)

Time Reversal

A natural question to consider when learning about time scaling is: What happens when the time variable is multiplied by a negative number? The answer to this is time reversal. This operation is the reversal of the time axis, or flipping the signal over the y-axis.

Figure 4: Reverse the time axis
Figure 4 (sigops3.png)

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