Radians versus Cycles

Our usual frequency variable is $\omega$ in radians per second. However, certain Fourier theorems are undeniably simpler and more elegant when the frequency variable is chosen to be $f$ in cycles per second. The two are of course related by

$$\omega = 2\pi f.$$ (B.1)

As an example, $e^{j\omega t}$ is more compact than $e^{j2\pi f t}$ . On the other hand, it is nice to get rid of all normalization constants in the Fourier transform and its inverse:

$$X(f) = \ensuremath{\int_{-\infty}^{\infty}}x(t)e^{-j2\pi f t} dt$$ (B.2)

$$x(t) = \ensuremath{\int_{-\infty}^{\infty}}X(f)e^{j2\pi f t} df$$ (B.3)

The ``editorial policy'' for this book is this: Generally, $\omega$ is preferred, but $f$ is used when considerable simplification results.