Transfer function of a frequency shifting system

There is a system which shifts frequencies of input by $$-F_c$$ such that:

$$Y(S) = X(S).H(S)$$

But $$X(S)$$ has value zero from $$0$$ to $$F_c$$.

I am confused on how the product of $$X(S)$$ and $$H(S)$$ becomes a positive value in $$Y(S)$$ in that frequency range, for any $$H(S)$$?

How transfer function of the system $$H(S)$$ will look like in the frequency domain?

While the answers that point out that a system needs to be LTI to have a transfer function is correct, there isn't a lack of trying.

Tymerski, Richard. "Application of the time-varying transfer function for exact small-signal analysis." IEEE Transactions on Power Electronics 9.2 (1994): 196-205.

Kamen, Edward W., Pramod P. Khargonekar, and K. R. Poolla. "A transfer-function approach to linear time-varying discrete-time systems." SIAM journal on control and optimization 23.4 (1985): 550-565.

these are only a few Google hits. There is more.

The system you're looking for cannot be described by a transfer function because it is a time-varying system. Only linear time-invariant (LTI) system can be fully characterized by a transfer function. However, there is no LTI system that can shift frequencies. The output of a (stable) LTI system can only have frequency components that are already present in the input signal.

Frequency shifts are usually achieved by modulation, i.e., by multiplying the input signal with a sinusoid or with a complex exponential. Such a system is linear but not time-invariant.

The frequency shifting system is not LTI, and therefore neither its impulse response nor its frequency response exists. A continuous-time frequency shifting system can be described by the following I/O relationship:

$$y(t) = \mathcal{T} \{x(t) \} = e^{j\omega_0 t } x(t)$$ which is clearly not time-invariant and hence not LTI.

• The impulse response is defined as the system response to an impulse stimulation. That very much exists for this system. In fact, it exists for all continuous-time linear systems. – Jazzmaniac Feb 13 at 14:57