Circuit Idea/How to Make the Simplest Transistor Amplifier Bipolar?

What do we mean by "bipolar" here? The word has lots of uses, and even is part of the "official" name of the most common type of transistor, Bipolar Junction Transistors (BJTs). In our present context think of waves of sound coming to your ear... pressure will increase compared with normal during part of each cycle, and decrease during the other half. The simplest of transistor circuits will have a single power supply, perhaps positive (usually for NPN transistors) so it can make the output signal go positive to varying degrees, but how could it make it go negative when needed for the rest of the signal waveform? Looking at circuits for common emitter and common collector amplifier configurations (the jargon isn't all that important - let's just say: both of the most frequently used simple circuits!) it should be clear the circuit simply cannot drag the output voltage down any lower than the negative side of the battery, which is connected to "ground" (the "zero" volts reference point for signals). If we used a PNP transistor and a negative power supply, we would still only be able to handle half of the signal, this time the negative half.

One idea is to have two transistors and two power supplies (one positive, one negative), with each half of the cycle being handled by, in effect, a different amplifier. This is called push pull and is common in many power amplifiers to drive loudspeakers, for instance. But it is easier said than done! How do you connect the outputs of both amplifiers so one has control on the positive half of the wave but not the other, without the two "fighting" over what the output should be or passing wrong polarity currents through the other transistor that might damage it? There are solutions, but they are not entirely simple. Let's come back to them later, and think about easier solutions.

What about "redefining" zero on the signal to be half way between the power supply rails? It is something of what was going on with push-pull anyway, but we could do it with one transistor. We add some voltage (or better talked of in terms of current for BJT transistors) to have the transistor half on with no signal, so positive signal turns it more on, negative parts of the wave tends to...