Time Scaling of a Signal

Time scaling refers to compressing or expanding a signal along the time axis, without changing its amplitude. Mathematically, this is achieved by replacing the time variable with a scaled version of itself.

Mathematical Definition

For a continuous-time signal \( x(t) \), time scaling is defined as:

\[ y(t) = x(a t) \]

where \( a \neq 0 \) is the time-scaling factor.

Interpretation of the Scaling Factor

1. Time Compression (Speed-Up)

If \( |a| > 1 \), the signal is compressed in time and occurs faster.

\[ y(t) = x(2t) \]

Events that originally occurred at time \( t_0 \) now occur at \( t_0 / 2 \).

2. Time Expansion (Slow-Down)

If \( 0 < |a| < 1 \), the signal is expanded in time and occurs more slowly.

\[ y(t) = x(0.5t) \]

Events are stretched and take twice as long.

3. Time Reversal

If \( a < 0 \), the signal is reversed in time.

\[ y(t) = x(-t) \]

Perspectives

Rescaling the time axis

Fixing the time axis (Used for engineering purposes)

Normal signal

Time compression (Speed-Up)

Time expansion (Slow-Down)

Time reversal

Normal signal

Time compression (Speed-Up)

Time expansion (Slow-Down)

Time reversal

Effect on Frequency

Time scaling directly affects the frequency content of a signal. If \( x(t) \leftrightarrow X(f) \), then:

\[ x(at) \leftrightarrow \frac{1}{|a|} X\!\left(\frac{f}{a}\right) \]

• Time compression increases frequency
• Time expansion decreases frequency
• The factor \( \frac{1}{|a|} \) preserves signal energy

Example

Consider the signal:

\[ x(t) = \sin(2\pi t) \]

Applying time scaling with \( a = 2 \):

\[ y(t) = \sin(4\pi t) \]

• The signal happens twice as fast
• The frequency is doubled
• The amplitude remains unchanged

Summary

• Time scaling modifies the time axis, not amplitude
• \( |a| > 1 \): compression (speed-up)
• \( 0 < |a| < 1 \): expansion (slow-down)
• \( a < 0 \): time reversal
• Frequency scales inversely with time