# What is the effect of carrier frequency offset (CFO) on the zeros of the z-transform?

Suppose I have a discrete-time signal vector, for example, x(n)=[1,a1,a2,…,aN]. The signal is then transmitted by using the single carrier pulses, constituting a single-carrier communication over a multipath channel. Hence, at the receiver side, the received signal experiences also the zeros of the z-transform of the channel impulse response. However, due to a frequency offset caused by the wireless multipath channel, the received signal is affected. Upon reception, the signal exhibits a CFO effect.

1- Could you please help me how to interpret this carrier frequency offset effect on the zeros of the z-transform of the received signal?

2- How a carrier frequency offset (CFO) causes a movement of all zeros of the received signal’s z−transform, from their original positions on the z-domain?

Assuming it is the former case (a shift before the channel) and if we clarify a positive carrier offset to mean that the signal has been shifted higher in frequency, then if we hold the signal as a reference, the zeros of the channel shift in frequency in the opposite direction, relative to the signal. So the carrier offset is just a rotation of the Z-plane. Given a frequency offset $$\omega_\Delta$$ in normalized angular frequency radians/sample, the rotation will be $$-\omega_\Delta$$ radians.
For example consider if we had a channel zero directly at the frequency of the carrier. On the z-plane this would be located at $$z=1$$, which is DC as the baseband equivalent of the passband signal. If we had a carrier offset of +1 KHz for the signal, meaning the carrier was shifted 1 KHz higher, and if the received signal was sampled at 100 KHz, then the zero will now be at -1KHz relative to the original carrier frequency, or $$-\pi/50$$ radians/sample. (The graphic above shows a significantly larger rotation than this.)