In the frequency domain, which variable denotes angular frequency?

• A. s
• B. ω
• C. jω
• D. f

Answer: (C)

The moment-to-moment concept here is that the frequency-domain description of a signal uses complex exponentials e^{jωt}, so the parameter that carries frequency information is tied to the imaginary axis. The angular frequency ω tells you how rapidly the sinusoid cycles per radian of time, and when we write the Fourier/Laplace transform in the form F(jω) we’re indicating that we’re evaluating the transform along the imaginary axis where the complex frequency is s = jω. The j factor encodes the 90-degree phase relationship inherent in complex exponentials, while ω is the actual rate of rotation (the angular frequency). That combination, jω, is the standard way the frequency content is parameterized in these transforms, which is why it’s the conventional variable used to denote angular frequency in the frequency-domain notation. The other symbols serve different roles: s represents the full complex frequency in the Laplace domain, f would denote ordinary frequency in cycles per second, and ω alone is the angular frequency concept but not the usual transform argument by itself in this notation.