Time Scaling Property FT

The time scaling property of the Fourier transform explains how compressing or stretching a signal in time reshapes its spectrum. This is the mathematical foundation behind bandwidth calculations in communication systems and pulse compression in radar.

Core Concept

The time scaling property says that squeezing a signal along the time axis spreads its spectrum wider in frequency, and stretching a signal in time compresses its spectrum. This is the uncertainty principle of signal processing: you cannot make a signal both short in time and narrow in frequency simultaneously.

When you replace t with at where |a| > 1, the signal repeats its variations faster. Faster variations require higher frequency components, so the spectrum broadens. The amplitude must also drop — total energy is shared across more frequencies — which introduces the 1/|a| scaling factor.

In radar, a transmitted pulse is compressed in the receiver by a matched filter. This improves range resolution without increasing peak power. The time-bandwidth product of the pulse remains constant, governed exactly by this scaling property.

Key Formula

If x(t) has Fourier transform X(f), then x(at) has transform (1/|a|)·X(f/a). Here a is any nonzero real constant: a > 1 means time compression, 0 < a < 1 means time expansion, and a = -1 gives time reversal x(-t) ↔ X(-f). The absolute value |a| in the denominator prevents negative amplitude for negative a. Using ω convention: x(at) ↔ (1/|a|)X(ω/a).

Problem: x(t) = rect(t) — a pulse of width 1 centred at origin.
  X(f) = sinc(f)  [standard pair]

Find the transform of y(t) = x(3t) and z(t) = x(t/2).

Case 1 — y(t) = x(3t), so a = 3:
  Y(f) = (1/|3|) X(f/3)
       = (1/3) sinc(f/3)
  Interpretation: pulse is 3x narrower in time, spectrum is 3x wider and 1/3 taller.

Case 2 — z(t) = x(t/2), so a = 1/2:
  Z(f) = (1/|1/2|) X(f/(1/2))
       = 2 X(2f)
       = 2 sinc(2f)
  Interpretation: pulse is 2x wider in time, spectrum is 2x narrower and twice as tall.

Check time-bandwidth product:
  y(t): duration = 1/3, first null bandwidth = 3  → product = 1
  z(t): duration = 2,   first null bandwidth = 1/2 → product = 1
  Product is invariant — confirmed.

Exam Tip: The most common error is writing x(at) ↔ X(af) instead of X(f/a). Remember: the frequency variable is divided by a, not multiplied. For time reversal a = -1, so (1/|a|) = 1 and X(f/-1) = X(-f). In GATE problems, you may be given Y(f) = X(2f) and asked for y(t) — that means a = 1/2, so y(t) = 2x(2t) after inverse scaling.

Properties Summary

• Basic rule: x(at) ↔ (1/|a|)X(f/a) for any real nonzero constant a.
• Compression (|a|>1): signal shorter in time, spectrum broader in frequency.
• Expansion (|a|<1): signal longer in time, spectrum narrower in frequency.
• Time reversal (a=-1): x(-t) ↔ X(-f); for real even signals X(-f)=X(f).
• Time-bandwidth product: duration × bandwidth is approximately constant regardless of a.
• Combined shift and scale: x(at-b) ↔ (1/|a|)e^(-j2πfb/a)X(f/a).

Quick Revision

• Scaling factor a appears in denominator of f, not in numerator.
• Amplitude scales by 1/|a| to preserve total signal energy.
• Time compression ↔ spectral expansion; time expansion ↔ spectral compression.
• Time reversal is the special case a = -1.
• The time-bandwidth product is an invariant under scaling.
• Combining shift and scale: apply scale first, then shift for x(at-b).
• Exam trap: writing Y(f) = X(af) instead of X(f/a) — dividing f by a, not multiplying, is the correct scaling direction.