The rise time of a high speed signal is 1ns. What is the band-width of the signal?

The rise time and band-width are related by the formula Tr = 0.35/Band-Width

So, for 1ns rise time, Band-Width = 0.35/(1*10^-9) = 350MHz

The above formula is derived from the fact that the signal input to particular load can be assumed as a circuit with some load impedance and capacitance there by forming a R-C circuit Low Pass Filter as shown below:

For the RC circuit, the response is Vout = Vin(1-e^(t/RC)), where RC is the time constant of the circuit.

The band-width of this RC circuit is, BW = 1/(2*pi*RC)

The rise time is defined as the time taken for the signal to reach 90% of the maximum voltage from 10% of the value.

So, rise time is T90% - T10%.

T90%  means , Vout = 0.9*Vin

So, 0.9*Vin = Vin(1-e^(T90%/RC))   => T90% = RC*ln0.1

T10%  means , Vout = 0.1*Vin

So, 0.1*Vin = Vin(1-e^(T10%/RC))   => T10% = RC*ln0.9

Rise Time, T90%-T10% = RC(ln0.9-ln0.1)   => 2.2RC   => RC = Rise-Time/2.2

So, BW = 1/(2*pi*Rise-Time/22)  => Rise-Time = 0.35/BW

The approximation here is that single circuit is assumed as single order Low Pass Filter. The same formula goes for multiple order filters as well.