frequency spectrum - What is the bandwidth of a (real) sinusoidal tone, and pulse?

I would like to know how to go about calculating the bandwidth of:

1. 
   

   A constant (real) sinusoidal tone

   
   


2. 
   

   A (real) sinusoidal pulse.

   
   

The question is as simple as that, but I am having a hard time with the concept of what exactly the bandwidth of a constant tone should be to begin with, and from there, what the bandwidth of a pulse should be.

• In the frequency domain, a constant real tone of frequency $f$ exists as two delta functions, located at $f$ and $-f$, but how does one go about calculating its bandwidth?


• Furthermore, in regards to the pulse, this is rectangular function in time, and thus a sinc in frequency domain, so would not its bandwidth simply be $\frac{1}{T}$, where $T$ is the duration of the pulse?

Answer

The spectrum of a continuous tone is, as you said, of the form $\delta(f-f_0) + \delta(f+f_0)$: 2 impulses at frequencies $f_0$ and $-f_0$.

As a lowpass signal, this is said to have bandwidth $f_0$ (the one-sided spectrum has components up to $f_0$).

As a bandpass signal, it has zero bandwidth (there's nothing around the carrier frequency $f_0$).

If you multiply the sine wave by a pulse, this makes it time-limited, and therefore frequency-unlimited. Infinite bandwidth in theory.

In practice, you must define some criteria for estimating your bandwidth. Examples are:

• 3 dB drop (of the sinc function around $f_0$)


• 10 dB drop


• drop below the noise level