Signal Multiplication

Introduction

Signal multiplication — the pointwise product of two signals — is one of the most powerful operations in signal processing because it creates new frequency components that did not exist in either original signal. This frequency-generating property is what makes communication systems possible: it allows us to shift a low-frequency voice signal to a high-frequency carrier for radio transmission. The mathematical foundation behind this is the multiplication-convolution duality, which states that multiplication in the time domain corresponds to convolution in the frequency domain.

Unlike signal addition (which preserves existing frequencies), multiplication fundamentally alters the spectral content of signals. This makes it simultaneously useful (for modulation and frequency conversion) and dangerous (as a source of unwanted distortion and intermodulation products).

Mathematical Definition

For continuous-time signals: $$y(t) = x_1(t) \cdot x_2(t)$$

For discrete-time signals: $$y[n] = x_1[n] \cdot x_2[n]$$

At each time instant, the output is simply the arithmetic product of the two input values at that same instant. This is a sample-by-sample operation — no memory of past or future values is involved.

The Multiplication-Convolution Duality

The Fourier Transform of a product of two signals is:

$$\mathcal{F}\{x_1(t) \cdot x_2(t)\} = \frac{1}{2\pi} X_1(j\omega) * X_2(j\omega)$$

This states that multiplication in the time domain becomes convolution in the frequency domain (scaled by $1/2\pi$). Conversely, convolution in time corresponds to multiplication in frequency.

The consequence is profound: if $x_1(t)$ occupies bandwidth $B_1$ and $x_2(t)$ occupies bandwidth $B_2$, then their product $y(t)$ can have bandwidth up to $B_1 + B_2$. New spectral components appear at sum and difference frequencies.

Amplitude Modulation: The Primary Application

The most important application of signal multiplication is amplitude modulation (AM). When a baseband message signal $x(t)$ is multiplied by a high-frequency carrier:

$$y(t) = x(t) \cdot \cos(\omega_c t)$$

The Fourier Transform of the cosine carrier is: $$\mathcal{F}\{\cos(\omega_c t)\} = \pi[\delta(\omega - \omega_c) + \delta(\omega + \omega_c)]$$

Using the multiplication property: $$Y(j\omega) = \frac{1}{2\pi} X(j\omega) * \pi[\delta(\omega - \omega_c) + \delta(\omega + \omega_c)]$$

$$Y(j\omega) = \frac{1}{2}[X(j(\omega - \omega_c)) + X(j(\omega + \omega_c))]$$

The baseband spectrum is shifted to appear centered at $+\omega_c$ and $-\omega_c$. This is the mathematical foundation of AM radio: a voice signal (20 Hz to 4 kHz) is moved to a carrier frequency (say, 1 MHz) for efficient antenna radiation and channel separation.

Demodulation

To recover the original signal, multiply the received AM signal by the same carrier again:

$$z(t) = y(t) \cdot \cos(\omega_c t) = x(t)\cos^2(\omega_c t) = \frac{x(t)}{2} + \frac{x(t)\cos(2\omega_c t)}{2}$$

A low-pass filter removes the double-frequency term, leaving $x(t)/2$ — the original message (scaled by 1/2).

Windowing: Controlled Truncation

In spectral analysis, signals must be truncated to finite duration before computing the DFT. This truncation is mathematically equivalent to multiplying by a window function:

$$y(t) = x(t) \cdot w(t)$$

In the frequency domain: $$Y(j\omega) = \frac{1}{2\pi} X(j\omega) * W(j\omega)$$

The window's spectrum convolves with and smears the signal's spectrum. This smearing effect is called spectral leakage. Different window shapes (rectangular, Hamming, Hanning, Blackman) offer different trade-offs between main lobe width (frequency resolution) and side lobe level (leakage suppression).

Example: Rectangular Window

The rectangular window $w(t) = 1$ for $|t| < T/2$ and $0$ elsewhere has a sinc-shaped spectrum: $$W(j\omega) = T \cdot \text{sinc}(\omega T / 2\pi)$$

The sinc function's side lobes cause spectral leakage, which is why more sophisticated windows with lower side lobes are preferred in practice despite their wider main lobes.

Gating and Switching

Multiplying a signal by a unit step function "turns on" the signal:

$$y(t) = x(t) \cdot u(t) = \begin{cases} x(t), & t \geq 0 \\ 0, & t < 0 \end{cases}$$

This gating operation is fundamental in communication systems where signals are transmitted in bursts, and in measurement systems where data acquisition begins at a trigger event.

Sampling as Multiplication

Ideal sampling can be viewed as multiplication by an impulse train:

$$x_s(t) = x(t) \cdot \sum_{n=-\infty}^{\infty}\delta(t - nT_s)$$

In the frequency domain, this produces the periodically repeated spectrum that is central to the Sampling Theorem. The frequency convolution creates spectral replicas at multiples of the sampling frequency $\omega_s = 2\pi/T_s$.

Worked Example

Problem: Two signals $x_1(t) = \cos(100\pi t)$ and $x_2(t) = \cos(600\pi t)$ are multiplied. Find the frequency content of the output.

Solution: Using the trigonometric identity: $$\cos A \cdot \cos B = \frac{1}{2}[\cos(A-B) + \cos(A+B)]$$

$$y(t) = \cos(100\pi t) \cdot \cos(600\pi t) = \frac{1}{2}\cos(500\pi t) + \frac{1}{2}\cos(700\pi t)$$

The input frequencies were 50 Hz and 300 Hz. The output contains 250 Hz and 350 Hz — entirely new frequencies created by the multiplication. This is the principle behind frequency mixers in radio receivers.

Practical Applications

• AM/FM Broadcasting: Baseband audio multiplied by carrier for radio transmission
• RF Mixers: Multiply received signal by local oscillator for frequency downconversion
• Lock-in Amplifiers: Multiply a noisy signal by a reference to extract buried periodic components
• Sampling Systems: Multiplication by impulse train creates discrete samples
• Chopper Amplifiers: Multiply DC signal by square wave to convert to AC for amplification

Key Takeaways

• Multiplication in time equals convolution in frequency (scaled by $1/2\pi$)
• Signal multiplication creates new frequencies at sum and difference of input frequencies
• Amplitude modulation: $x(t)\cos(\omega_c t)$ shifts the spectrum to $\pm\omega_c$
• Windowing before spectral analysis causes spectral leakage due to frequency convolution
• Product signal bandwidth can be up to the sum of individual bandwidths
• Multiplication is the foundation of modulation, demodulation, mixing, and sampling