BJT and signal source
It can be done that way.
That is : the following circuit will work:



simulate this circuit &ndash; Schematic created using CircuitLab
(assuming there is a low impedance DC path through the AC input source V1).
However notice that R1 and R2 form a potential divider, so only 90% of the input voltage makes it as far as Q1 base, reducing the overall gain.
More seriously, R1 is a Johnson noise source in series with V1.
If you connect it to ground instead, and connect V1 to Q1 base by capacitor as normally done, the AC impedance to GND is only the source resistance of V1 which may be only a few ohms - in parallel with R1 and R2. And the Johnson noise resistance is this combined - low - resistance of a few ohms, making for a much less noisy amplifier.
(R3 is also a noise source : however a capacitor across it normally takes care of that, again by decreasing the AC resistance, while also increasing gain. R4 is also a noise source but after the input has been amplified, so it doesn't matter).