Find the band pass signal for a given low pass range and cutoff frequency?

Question

We know the fact that a bandpass signal is defined for real values. But for a typical scenario where we have a signal in the lowpass range:say : 𝑥𝐿(𝑡)=𝑗𝑋0 sinc(Bt) and the transformation of the signal is done at cut off frequency say 5B, how can we represent the band pass signal for such a case scenario? Can anyone suggest any representation formula?

I don't understand how the real part can be handled in the above signal representation because bandpass is always defined for real and the given signal has only imaginary component. Please suggest your opinion.

Any clue or suggestions will be of real help.

matlab signals signal-processing fft continuo

Answer 1

First to clarify, a bandpass signal need not be real, but this would be the case when the positive and negative frequencies are complex conjugate symmetric. We can and do have bandpass signals where the positive and negative frequencies are independent of each other, but this would require two “wires” in implementation to track the real and imaginary components.

I believe the OP may be referring to the relationship between real bandpass signals and the complex baseband equivalent. In this case we can translate the complex baseband signal to the real bandpass carrier frequency. This can be done in the frequency domain by shifting the baseband spectrum at $f=0$ to the passband carrier at $f=f_c$ and replicating at the negative frequency carrier as the complex conjugate.

In the time domain the above operation is done by simply multiplying the complex baseband signal by $e^{j2 \pi f_c t}$ and then taking the real part of this product.