Does down-mixing a signal before taking the fourier transform aid in signal detection? Let us assume that we have a narrowband signal $s[n]$, at frequency $f_1$, and let us further assume that this $f_1$ is just below $\frac{f_s}{2}$, where $f_s$ is the sampling rate. 

Furthermore, let us also assume that we have a fixed number of time-domain samples of this signal, $N$, and that our FFT length is also $N$. 

The objective is to detect this signal $s[n]$'s presence in the frequency domain. (Further assume that we know to expect this signal in or around that frequency $f_1$).

My question is simple: With all things being equal, which is better from a detection perspective:


Simply do the FFT of this signal and look at the peak at that frequency bin


Or


BPF a band of where we expect the signals' frequency $f_1$ to be, (eg, BPF this band $f_1 - \epsilon \text{ to} f_1 + \epsilon$). Then, mix this band down such that $f_1 + \epsilon$ is guaranteed to be just above DC, and LPF that entire band. Now, take the FFT, and detect energy. 


Would the second method be better from a detection point of view because we have more samples per cycle, whereas the first one has just over 2 samples per cycle?

Thanks,

TLDR: Is it better from a detection standpoint to FFT a signal or down-mix it and then FFT it (with appropriate filtering)?
If you are doing are doing this in both the analog and digital domain, then there may be significant differences due to trade-offs between the noise sources in the analog stage(s) in the filters and the mixer oscillator frequency, and the corresponding sampling jitter and quantization noise in converting to digital.

Even strictly in the digital domain, the bandpass and lowpass filter responses may have different effects on how potential stop-band noise might affect your detection accuracy, compared to how windowing (rectangular or otherwise) affects the continuous frequency response of each FFT bin.