bandwidth between the signal and the channel I have a signal with a bandwidth $B=4$ kHz and the bandwitdh is between $6$ kHz and $10$ kHz. I want to transmit this signal in a channel with bandwidth $B'=4$ kHz but for the channel the bandwidth is between $1$ kHz and $5$ kHz.

This is my question:
Can we pass a signal in this channel even if the bandwidth support is not the same?
If I understand correctly you would like to pass a bandpass signal of 4Khz bandwidth in the frequency range 6 to 10 Khz, but you would like to transmit this directly over a channel which has good channel gain in 1 to 5Khz. Clearly, the signal when passed over a channel with this frequency response will suffer complete degradation. 

Hence, we cannot transmit this signal directly over this channel. You could first downcovert the signal by 5 Khz and then transmit.