Multiplication Property FT

The multiplication property of the Fourier transform describes what happens in the frequency domain when two signals are multiplied in time. It is the mathematical basis for amplitude modulation, windowing in spectral analysis, and mixer circuits in radio receivers.

Core Concept

When you multiply two signals in the time domain, you do not simply multiply their spectra. Instead, the spectrum of the product is the convolution of the individual spectra. This is the exact dual of the convolution theorem, which says convolution in time produces multiplication in frequency.

In amplitude modulation, a message signal x(t) is multiplied by a sinusoidal carrier cos(2πf₀t). The carrier has a spectrum consisting of two impulses at ±f₀. Convolving X(f) with those impulses simply shifts copies of X(f) to the carrier frequency and its negative — this is the physical origin of upper and lower sidebands.

Windowing in spectral analysis is another application. A finite observation of a signal is equivalent to multiplying by a rectangular window. The resulting spectrum is X(f) convolved with the sinc-shaped window spectrum, which introduces spectral leakage. Understanding this explains why different window functions reduce leakage.

Key Formula

If y(t) = x(t)·g(t), then Y(f) = X(f)*G(f) = ∫ X(λ)G(f-λ)dλ. Here λ is a dummy frequency variable in hertz. For the common case g(t) = cos(2πf₀t): G(f) = (1/2)[δ(f-f₀)+δ(f+f₀)], so Y(f) = (1/2)[X(f-f₀)+X(f+f₀)]. Convolution with an impulse δ(f-f₀) shifts the spectrum to f₀.

Problem: x(t) = sinc(2Wt) with X(f) = (1/2W)rect(f/2W), bandwidth W.
  Multiply by carrier: y(t) = x(t)·cos(2πf₀t), f₀ >> W.

Step 1 — Carrier spectrum:
  cos(2πf₀t) ↔ (1/2)[δ(f-f₀) + δ(f+f₀)]

Step 2 — Apply multiplication property:
  Y(f) = X(f) * (1/2)[δ(f-f₀) + δ(f+f₀)]

Step 3 — Convolve X(f) with each impulse:
  Convolution with δ(f-f₀) shifts X(f) to centre at f₀
  Convolution with δ(f+f₀) shifts X(f) to centre at -f₀

Step 4 — Result:
  Y(f) = (1/2)X(f-f₀) + (1/2)X(f+f₀)
       = (1/4W)[rect((f-f₀)/2W) + rect((f+f₀)/2W)]

Final Answer:
  Spectrum consists of two rect pulses centred at ±f₀, each of width 2W and height 1/4W.
  Total bandwidth occupied = 2W (upper sideband) + 2W (lower sideband) = 4W.

Exam Tip: The multiplication property is often called the modulation theorem. For multiplication by cos(2πf₀t), the factor is 1/2 in front of each shifted copy — students frequently omit it and get the amplitude wrong. For multiplication by a general periodic signal, decompose the signal into its Fourier series first, then shift X(f) to each harmonic frequency and scale by the corresponding coefficient.

Properties Summary

• Multiplication property: x(t)·g(t) ↔ X(f)*G(f) — dual of convolution theorem.
• Modulation by cosine: x(t)cos(2πf₀t) ↔ (1/2)[X(f-f₀)+X(f+f₀)].
• Modulation by complex exponential: x(t)e^(j2πf₀t) ↔ X(f-f₀) — single shift, no (1/2).
• Windowing effect: finite duration observation = multiplication by window → spectral leakage.
• Bandwidth: multiplying by a bandlimited signal of bandwidth B adds ±B to original bandwidth.
• Symmetry: convolution in frequency is commutative, just as in time.

Quick Revision

• Time multiplication → frequency convolution (dual of convolution theorem).
• Multiplying by cos(2πf₀t) shifts spectrum to ±f₀ with factor 1/2 on each.
• Multiplying by e^(j2πf₀t) shifts spectrum to +f₀ only, no amplitude change.
• AM signal bandwidth is twice the message bandwidth.
• Windowing a signal convolves its spectrum with the window's transform.
• Demodulation uses another multiplication to shift the spectrum back to baseband.
• Exam trap: forgetting the 1/2 factor when multiplying by cosine — e^(j2πf₀t) has no such factor, but cos does.