You may have a look at Estimate Sine Frequency under White Noise.

The CRLB for a real harmonic signal is:

$$ \operatorname{var} \left( \hat{f} \right) \geq \frac{12}{ {\left( 2 \pi \right)}^{2} \cdot SNR \cdot \left( {N}^{3} - N \right) } {f}_{s}^{2} $$

For your case, a complex harmonic signal it is given by:

$$ \operatorname{var} \left( \hat{f} \right) \geq \frac{6}{ {\left( 2 \pi \right)}^{2} \cdot SNR \cdot \left( {N}^{3} - N \right) } {f}_{s}^{2} $$

In both the SNR is defined as:

$$ SNR = \frac{ {A}^{2} }{ {\sigma}_{n}^{2} } $$

Where $A$ is the amplitude of the harmonic signal and ${\sigma}_{n}^{2}$ is the AWGN noise variance.

You may have a look at Estimate Sine Frequency under White Noise.

The CRLB for a real harmonic signal is:

$$ \operatorname{var} \left( \hat{f} \right) \geq \frac{12}{ {\left( 2 \pi \right)}^{2} \cdot SNR \cdot \left( {N}^{3} - N \right) } {f}_{s}^{2} $$

For your case, a complex harmonic signal it is given by:

$$ \operatorname{var} \left( \hat{f} \right) \geq \frac{6}{ {\left( 2 \pi \right)}^{2} \cdot SNR \cdot \left( {N}^{3} - N \right) } {f}_{s}^{2} $$

In both the SNR is defined as:

$$ SNR = \frac{ {A}^{2} }{ {\sigma}_{n}^{2} } $$

Where $A$ is the amplitude of the harmonic signal and ${\sigma}_{n}^{2}$ is the AWGN noise variance.

You may have a look at the code to create such analysis in my StackExchange GitHub Repository (Look at the SignalProcessing\Q76644 folder). You may look at Q76644.m for the real harmonic case and Q76644C.m for the complex case.