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I'm studying for my finals , and I'm struggling with finding bandwidths.

I have this energy spectral density : Ψ(f)=rect(f) , and I need to find the bandwith, 3 dB band and the band with 90 % power.

About the 90 % case, do I need to find the total energy and then ∫W0Ψ(f)=0.9×TotalEnergy?

If I had a power signal and wanted to find the 3 dB bandwidth, would I need to find the average power and then solve ∫W0Sx(f)=0.5×TotalPower ?

Thanks in advance.

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    $\begingroup$ all these terms are well-googleable, and a drawing of the spectral density would instantly answer your questions – what really is your question? $\endgroup$ – Marcus Müller Dec 28 '19 at 22:26
  • $\begingroup$ You do realize that this is 3 entirely separate questions? Just take them one by one. You know the bandwidth; it was given in your knowns. 90% is 90% of that. 3 DB is where it crosses 3 DB; which in your case is at the band limit. Of course, this presumes that you have a coherent signal; if the signal is real noise then the calculations differ. Basically noise adds as the square root of bandwidth. $\endgroup$ – rrogers Dec 31 '19 at 22:37
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The concept of the bandwidth has several definitions.

For simple filters, one definition of bandwidth is the -3 dB frequency (for a lowpass filter) $f_c$ where the filter gain at $f_c$ is $H(f_c) = H(0)/\sqrt(2)$; i.e. $f_c$ is the half-power or cut-off frequency. This definition could be applied for a bandpass filters too but not for highpass filters. Note again that this definition of bandwidth requires that the filter frequency response be a continuous waveform, otherwise the the frequency will be undefined.

For signals, a 3dB bandwidth concept will not work and instead something like a 90 % energy (or power) definition is invoked. In this case the frequency spectrum of the signal is used to find the minimum frequency $f_b$ at which 90% of the total enegy of the (baseband, real) signal is contained.

In your case you have a signal with a rectangular frequency spectrum, which defies a 3 dB bandwidth, but enables the computation of a 90 % bandwidth definition.

However note that all these definitions make sense under typical waveform assumptions, but for your case neither -3dB nor 90% enegy definitions adequately define the signal's bandwidth.

You signal is bandlimited in the sense that the ractangular function is exactly zero out of its lobe and that frequency shall be taken as the bandwidth of this signal.

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