4.3 Convolution and Multiplication Properties

Convolution Property of Fourier Transform

Mathematical Formulation and Implications

The convolution property connects time-domain convolution to frequency-domain multiplication. It's one of the most practically useful properties in signal processing because it turns a difficult integral operation into simple multiplication.

The formal statement: if $f(t)$ and $g(t)$ have Fourier transforms $F(\omega)$ and $G(\omega)$, then:

$\mathcal{F}\{f(t) * g(t)\} = F(\omega) \cdot G(\omega)$

Convolution in time becomes multiplication in frequency.

This directly applies to linear time-invariant (LTI) systems. For any LTI system with impulse response $h(t)$ and frequency response $H(\omega)$, the output $y(t)$ for an input $x(t)$ satisfies:

$Y(\omega) = X(\omega) \cdot H(\omega)$

So instead of evaluating the convolution integral in time, you can multiply two functions in frequency and inverse-transform the result.

Applications and Advantages

Filter design and analysis. To build a low-pass, high-pass, or band-pass filter, you specify the desired frequency response $H(\omega)$ and find the corresponding impulse response $h(t)$. The output for any input is then just the product $X(\omega) \cdot H(\omega)$ in the frequency domain.

Cascaded systems. When you chain multiple LTI systems together, the overall frequency response is the product of the individual frequency responses:

$H_{\text{total}}(\omega) = H_1(\omega) \cdot H_2(\omega) \cdot H_3(\omega) \cdots$

This makes modular design straightforward. You can analyze each stage independently and multiply to get the combined behavior.

Fast convolution via FFT. Direct time-domain convolution of two length-$N$ signals requires on the order of $N^2$ operations. The convolution property enables a faster approach:

1. Compute the FFT of both signals: $F(\omega)$ and $G(\omega)$
2. Multiply the transforms point-by-point: $F(\omega) \cdot G(\omega)$
3. Compute the inverse FFT of the product to get the convolution result

This reduces computational cost to roughly $N \log N$ operations, which is a major speedup for long signals.

Multiplication Property for Product Signals

Mathematical Formulation and Implications

The multiplication property is the dual of the convolution property. It states that multiplying two signals in time corresponds to convolving their spectra in frequency, with a scaling factor.

Formally, if $f(t)$ and $g(t)$ have Fourier transforms $F(\omega)$ and $G(\omega)$:

$\mathcal{F}\{f(t) \cdot g(t)\} = \frac{1}{2\pi} \left[ F(\omega) * G(\omega) \right]$

Multiplication in time becomes convolution in frequency (scaled by $\frac{1}{2\pi}$).

The $\frac{1}{2\pi}$ factor comes from the Fourier transform convention using $\omega$ (angular frequency). If you use the convention with ordinary frequency $f$ (Hz), the factor disappears.

Amplitude modulation (AM) is the classic application. When you multiply a message signal $m(t)$ by a sinusoidal carrier $\cos(\omega_c t)$, the multiplication property tells you the resulting spectrum consists of shifted copies of $M(\omega)$ centered at $\pm \omega_c$. That's exactly how AM transmission works: the message spectrum gets relocated to the carrier frequency for transmission.

Applications and Signal Analysis

Deriving transforms of common pulses. You can express pulse shapes as products of simpler functions. A rectangular pulse, for instance, is a constant multiplied by a rect window. The multiplication property lets you find its Fourier transform from the transforms of those simpler components.

Windowing effects. Multiplying a signal by a window function (rectangular, Hamming, Hanning, etc.) in time means convolving the signal's spectrum with the window's spectrum in frequency. This is why windowing causes:

• Spectral leakage: the window's spectrum has sidelobes that spread energy across frequencies
• Main lobe widening: the convolution broadens spectral peaks, reducing frequency resolution

Different windows trade off main lobe width against sidelobe level, and the multiplication property is how you analyze those tradeoffs quantitatively.

Sampling. Sampling a continuous-time signal can be modeled as multiplying it by a periodic impulse train. The multiplication property then shows that the sampled signal's spectrum is a periodic repetition of the original spectrum. When those repetitions overlap, you get aliasing, which is the motivation behind the Nyquist sampling theorem.

Signal Analysis in Frequency Domain

Combining Convolution and Multiplication Properties

Together, these two properties form a complete duality between time and frequency domains:

This duality means you can always choose whichever domain makes the math easier. Filtering problems are simpler in frequency (just multiply). Modulation analysis also uses frequency-domain convolution to see how spectra shift and combine.

For communication systems, the workflow often looks like this: modulation (multiplication in time) creates the transmitted spectrum, and the receiver applies a filter (convolution in time, multiplication in frequency) to recover the original signal.

Advanced Signal Processing Techniques

Truncation and spectral effects. Truncating a signal in time is equivalent to multiplying it by a rectangular window. By the multiplication property, the truncated signal's spectrum is the original spectrum convolved with a sinc function. This convolution causes spectral broadening and Gibbs-phenomenon ringing near discontinuities.

Multi-stage filter design. Cascading filters with frequency responses $H_1(\omega), H_2(\omega), \ldots$ gives an overall response that's their product (convolution property). This lets you decompose a complex filter specification into simpler stages that are easier to design and implement individually.

Complex modulation schemes. Techniques like quadrature amplitude modulation (QAM) and orthogonal frequency-division multiplexing (OFDM) involve both multiplication (modulation onto carriers) and convolution (filtering at transmitter and receiver). Analyzing these systems requires applying both properties together:

• The multiplication property explains how data symbols modulate subcarriers and shape the transmitted spectrum
• The convolution property governs how channel filtering and receiver processing affect the signal

The core takeaway: convolution and multiplication are mirror operations across the time-frequency divide. Recognizing which property applies in a given situation lets you move the problem to whichever domain makes it tractable.