8.5: Continuous Time Convolution and the CTFT
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This module discusses convolution of continuous signals in the time and frequency domains.
Continuous Time Fourier Transform
The CTFT transforms a infinite-length continuous signal in the time domain into an infinite-length continuous signal in the frequency domain.
CTFT
\[\mathcal{F}(\Omega)=\int_{-\infty}^{\infty} f(t) e^{-(j \Omega t)} d t \nonumber \]
Inverse CTFT
\[f(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} \mathcal{F}(\Omega) e^{j \Omega t} d \Omega \nonumber \]
Convolution Integral
The convolution integral expresses the output of an LTI system based on an input signal, \(x(t)\), and the system's impulse response, \(h(t)\). The convolution integral is expressed as
\[ y(t)=\int_{-\infty}^{\infty} x(\tau) h(t-\tau) d \tau \nonumber \]
Convolution is such an important tool that it is represented by the symbol *, and can be written as
\[y(t)=x(t) * h(t) \nonumber \]
Convolution is commutative. For more information on the characteristics of the convolution integral, read about the Properties of Convolution (Section 3.4).
Demonstration
Convolution Theorem
Let \(f\) and \(g\) be two functions with convolution \(f*g\). Let \(F\) be the Fourier transform operator. Then
\[F(f * g)=F(f) \cdot F(g) \nonumber \]
\[F(f \cdot g)=\frac{1}{2 \pi} F(f) * F(g) \nonumber \]
By applying the inverse Fourier transform \(F^{−1}\), we can write:
\[f * g=F^{-1}(F(f) \cdot F(g)) \nonumber \]
Conclusion
The Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain (e.g., time domain) corresponds to point-wise multiplication in the other domain (e.g., frequency domain).