Thanks to the

convolution theorem, we have two alternate ways to
perform

cyclic convolution in practice:

- Direct calculation in the time domain using (8.13)
- Frequency-domain convolution:
- Fourier Transform both signals
- Perform term by term multiplication of the transformed signals
- Inverse transform the result to get back to the time domain

For short convolutions (less than a hundred samples or so), method 1
is usually faster. However, for longer convolutions, method 2 is
ultimately faster. This is because the computational
complexity of direct

cyclic convolution of two

-point signals is

, while that of FFT convolution is

. More
precisely, direct cyclic convolution requires

multiplies and

additions, while the exact FFT numbers depend on the
particular FFT algorithm used [

80,

66,

224,

277].
Some specific cases are compared in §

8.1.4 below.

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