Time Scaling

Introduction

Time scaling stretches or compresses a signal along the time axis while preserving its shape. This operation directly demonstrates one of the most fundamental principles in signal processing: the time-bandwidth tradeoff. When you speed up a recording, it plays faster (compressed in time) but sounds higher-pitched (expanded in frequency). When you slow it down, the duration increases but the pitch drops. You cannot independently control both duration and frequency content — nature enforces a strict relationship between them.

This tradeoff is not merely a curiosity; it has profound engineering implications. Radar systems exploit pulse compression to achieve fine range resolution using long transmitted pulses. Communication systems use spread-spectrum techniques where the time-bandwidth product determines the processing gain. Understanding time scaling mathematically equips you to reason about all these applications.

Mathematical Definition

Given a signal $x(t)$, the time-scaled version is:

$$y(t) = x(at)$$

where $a$ is a real, non-zero constant called the scaling factor:

• $|a| > 1$: Time compression — the signal is squeezed toward $t = 0$, playing out faster
• $0 < |a| < 1$: Time expansion — the signal is stretched away from $t = 0$, playing out slower
• $a < 0$: Includes time reversal in addition to scaling

Duration Effect

If $x(t)$ occupies the interval $[t_1, t_2]$, then $x(at)$ occupies $[t_1/a, t_2/a]$:

• For $a = 2$: a signal on $[0, 4]$ becomes $[0, 2]$ — half the duration
• For $a = 0.5$: a signal on $[0, 4]$ becomes $[0, 8]$ — double the duration
• For $a = -1$: a signal on $[0, 4]$ becomes $[-4, 0]$ — time reversal

Effect on the Fourier Transform

The scaling property of the Fourier Transform states:

$$\mathcal{F}\{x(at)\} = \frac{1}{|a|} X\left(\frac{j\omega}{a}\right)$$

This reveals the time-bandwidth tradeoff mathematically:

Time compression ($|a| > 1$):

• Duration decreases by factor $|a|$
• Bandwidth increases by factor $|a|$ (spectrum stretches)
• Amplitude decreases by factor $1/|a|$

Time expansion ($|a| < 1$):

• Duration increases by factor $1/|a|$
• Bandwidth decreases by factor $|a|$ (spectrum compresses)
• Amplitude increases by factor $1/|a|$

The Time-Bandwidth Product

The product of signal duration and bandwidth remains constant:

$$\Delta t \cdot \Delta f = \text{constant}$$

This is a fundamental limit that cannot be violated. A Gaussian pulse achieves the minimum possible time-bandwidth product of $1/(4\pi)$, making it the most "efficient" signal in the time-frequency sense.

Graphical Procedure for Sketching $x(at)$

Step 1: Identify all critical time values in $x(t)$: start point, end point, peaks, zero crossings.

Step 2: Divide each time value by $a$ to get the new positions.

Step 3: Plot the signal using the new time positions. Amplitudes remain unchanged.

Example

Given $x(t)$ defined as a triangle from $t = 0$ to $t = 6$ with peak at $t = 3$:

For $x(2t)$: Peak at $t = 3/2 = 1.5$, extends from $0$ to $3$. Same height, half the width.

For $x(t/2)$: Peak at $t = 3/(1/2) = 6$, extends from $0$ to $12$. Same height, double the width.

Combined Operations: $y(t) = x(at - b)$

When time scaling is combined with shifting, the correct interpretation requires factoring:

$$y(t) = x(at - b) = x\left(a\left(t - \frac{b}{a}\right)\right)$$

This means: first scale by $a$, then shift right by $b/a$.

Procedure for Combined Operations

Method 1 (Recommended): Factor the expression to separate scaling and shifting.

• $x(2t - 6) = x(2(t-3))$: scale by 2 (compress to half width), then shift right by 3.

Method 2: Apply operations sequentially.

• Start with $x(t)$
• Replace $t$ with $at$: get $x(at)$ (scale)
• Replace $t$ with $t - b/a$: get $x(a(t - b/a)) = x(at - b)$ (shift)

Common Mistake

Students often write $x(2t - 6)$ and shift by 6 instead of 3. Always factor out the coefficient first: the shift is $b/a = 6/2 = 3$, not 6.

Energy Relationship

If $x(t)$ has energy $E_x$:

$$E_y = \int_{-\infty}^{\infty}|x(at)|^2 dt$$

Using the substitution $\tau = at$, so $dt = d\tau/|a|$:

$$E_y = \frac{1}{|a|}\int_{-\infty}^{\infty}|x(\tau)|^2 d\tau = \frac{E_x}{|a|}$$

Compression ($|a| > 1$): Energy decreases (signal exists for shorter duration) Expansion ($|a| < 1$): Energy increases (signal exists for longer duration)

Note: The peak amplitude is unchanged by time scaling, but the total energy changes because the signal's temporal extent changes.

Effect on the Laplace Transform

$$\mathcal{L}\{x(at)\} = \frac{1}{a}X\left(\frac{s}{a}\right), \quad a > 0$$

The ROC scales accordingly: if the original ROC was $\text{Re}(s) > \sigma_0$, the new ROC is $\text{Re}(s) > a\sigma_0$.

Worked Example

Problem: A rectangular pulse $x(t) = 1$ for $|t| < 1$ (duration 2, bandwidth $\approx 1/2$ Hz using first-null bandwidth) is time-scaled by $a = 4$.

Solution: $y(t) = x(4t) = 1$ for $|4t| < 1$, i.e., $|t| < 0.25$

• Original: duration = 2, first-null bandwidth = $1/\pi \approx 0.318$ Hz
• Scaled: duration = 0.5, first-null bandwidth = $4/\pi \approx 1.273$ Hz

Time-bandwidth product check: $2 \times 0.318 = 0.636$ and $0.5 \times 1.273 = 0.636$ ✓ (preserved)

Applications

Audio playback speed: Playing at 2× applies $x(2t)$ — duration halves and all frequencies double (pitch goes up one octave). Modern "time-stretch" algorithms modify speed without changing pitch using phase vocoders.

Radar pulse compression: A long chirp signal (high energy, wide duration) is transmitted. Upon reception, matched filtering "compresses" it into a narrow pulse, achieving fine range resolution. The compression ratio equals the time-bandwidth product.

Doppler effect: Motion causes effective time scaling of received signals. A source moving toward you compresses the received waveform ($a > 1$), increasing frequencies.

Video slow motion: High-speed cameras capture at fast rates and play back at normal rates, effectively expanding time by the ratio of capture-to-playback frame rates.

Key Takeaways

• $x(at)$: $|a| > 1$ compresses in time (shorter, wider bandwidth); $|a| < 1$ expands (longer, narrower bandwidth)
• Fourier Transform scaling: $\mathcal{F}\{x(at)\} = \frac{1}{|a|}X(j\omega/a)$
• Time-bandwidth product is invariant — compression in one domain means expansion in the other
• For combined operations $x(at-b)$, factor as $x(a(t-b/a))$ to identify the true shift
• Energy changes: $E_y = E_x/|a|$
• The time-bandwidth tradeoff is fundamental and cannot be circumvented