Kinda sorta! But not really!
What comes out of your wall is AC. Bias voltage needs DC. This much you know!
The diode is what converts it to DC-- it's what's called a "half-wave rectifier." But the way it does that is that it just chops off the voltage every time it swings negative-- in other words, only the positive bits remain after passing through the diode. Like this:
So what you have left is not smooth battery-like DC, but rather pulsing
DC. What's worse is that since it's half-wave rectified, it pulses only 50 times a second in Europe, or 60 times a second in the US. In other words, there's a big "gap" in between each pulse.
Predictably, if you didn't do anything to this, it would hum
at 50 (or 60) Hz. Note that the waveform you have left is "sine" on one side, and "square" on the other (where the rectifier lopped it off). This means that in addition to the basic 50 or 60 Hz tone, you'll have lots of upper harmonics present, too (square waves have tons and tons of harmonics--- fourier's theorem
states that any periodic waveform can be mathematically represented as the sum of several sine waves).
Your filter cap's job is to charge on the pulses, and discharge in the "gaps." This serves to "even out" the DC so that it's more like battery DC. This is why they're sometimes called "smoothing capacitors..." they take the "lumpy" input signal and even it out so that it's more like a straight line.
If you installed an extra small-value film bypass cap, it would be able to discharge and charge very quickly-- and ideally be 'fast' enough to deal with the upper harmonics imparted by the square half of the waveform.
blah blah blah!