# Parseval's Theorm and Effective Bandwidth

This question says that RMS bandwidth (effective bandwidth) is defined based on the carrier frequency of a signal. This makes intuitive sense to me that the carrier frequency shouldn't determine the value of the RMS bandwidth, just the pulse shape. However looking at Parseval's Theorem it is not immediately clear to me that upconversion of a signal to bandpass plays out in this way. For example let $x(t)$ be a low pass signal with energy from -B to B in the frequency domain and let

$$s(t) = x(t)\cos(2\pi f_c t).$$

Then from Parseval's Theorem we have $$\int_{0}^{\infty}\bigg(\frac{\partial s(t)}{\partial t}\bigg)^2 dt = \int_{-\infty}^{\infty} f^2|S(f)|^2 df$$ $$= \int_{-(f_c + B)}^{-(f_c - B)} f^2 |X(f)|^2 df + \int_{f_c - B}^{f_c + B} f^2 |X(f)|^2 df$$

which leads me to believe that the energy relies upon the carrier frequency. What am I missing here? Please forgive any scaling factors, I'm concerned with the general behavior not if there is a constant multiplier. Thanks!

• I'm unfamiliar with that form of Parseval's theorem... do you have a reference? – MBaz Mar 7 '18 at 23:34
• I'm not sure I understand your question. In the accepted answer to the question you linked to, I explained that the definition of RMS bandwidth depends on the center frequency. So you can't use the same formula for defining the RMS bandwidth of $s(t)$ (band pass signal) and $x(t)$ (low pass signal) as you did. – Matt L. Mar 8 '18 at 8:12
• @MattL. I mean that shifting the carrier frequency of a pulse does not change the effective bandwidth as your answer states. However in the example I posted above this seems to be untrue and that is where my confusion lies. thanks for taking the time to look at this! – EE_13 Mar 8 '18 at 19:09
• @MBaz if you use the derivative property of the FT as well as the partial derivative of the signal then you can derive this from Parseval's theorem in two steps. See page 339 of Whalen's "Detection of Signals in Noise" for a direct reference. – EE_13 Mar 8 '18 at 19:10
• @MattL. a reference for where you got the definition of effective bandwidth for a bandpass signal would be a huge help to me – EE_13 Mar 8 '18 at 19:17