CTFT convolution property

Statement

If $x(t) \xleftrightarrow{F} X(\omega)$ and $h(t) \xleftrightarrow{F} H(\omega)$, then

$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau)\,d\tau \;\xleftrightarrow{F}\; Y(\omega) = X(\omega)H(\omega)$$

Convolution in the time domain becomes multiplication in the frequency domain.

Proof

By the definition $X(\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t}dt$:

$$Y(\omega) = \int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty} x(\tau)h(t-\tau)d\tau\right]e^{-j\omega t}dt$$

Change the order of integration:

$$Y(\omega) = \int_{-\infty}^{\infty} x(\tau)\left[\int_{-\infty}^{\infty} h(t-\tau)e^{-j\omega t}dt\right]d\tau$$

In the inner integral put $\sigma = t - \tau$ ($dt = d\sigma$, limits unchanged):

$$\int_{-\infty}^{\infty} h(\sigma)e^{-j\omega(\sigma + \tau)}d\sigma = e^{-j\omega\tau}H(\omega)$$

This is the time-shift property. Substituting back:

$$Y(\omega) = H(\omega)\int_{-\infty}^{\infty} x(\tau)e^{-j\omega\tau}d\tau = H(\omega)X(\omega)$$ $$\boxed{x(t)*h(t) \xleftrightarrow{F} X(\omega)H(\omega)}$$

Significance

• LTI analysis: the output of an LTI system with impulse response $h(t)$ is $Y(\omega) = H(\omega)X(\omega)$. $H(\omega)$ is the frequency response: each frequency component of the input is scaled by $|H(\omega)|$ and phase-shifted by $\angle H(\omega)$.
• Filtering: an ideal low-pass filter has $H(\omega) = 1$ for $|\omega| < \omega_c$ and 0 elsewhere. It removes the high-frequency parts of $X(\omega)$ simply by multiplication.
• Cascades: systems in series give $H(\omega) = H_1(\omega)H_2(\omega)$.
• Computation: a convolution integral can be replaced by: transform, multiply, inverse transform.

Example

$x(t) = e^{-at}u(t)$ is applied to $h(t) = e^{-bt}u(t)$, with $a \ne b$ and both positive:

$$Y(\omega) = \frac{1}{(a+j\omega)(b+j\omega)} = \frac{1}{b-a}\left[\frac{1}{a+j\omega} - \frac{1}{b+j\omega}\right]$$

So $y(t) = \frac{1}{b-a}\left(e^{-at} - e^{-bt}\right)u(t)$, found without evaluating the convolution integral.

Dual (multiplication/modulation) property: $x(t)p(t) \xleftrightarrow{F} \frac{1}{2\pi}X(\omega) * P(\omega)$.