That's exactly right, although if you take the Fourier transform of that modulated signal you'll see that it uses infinite bandwidth due to the square-wave modu

That's exactly right, although if you take the Fourier transform of that modulated signal you'll see that it uses infinite bandwidth due to the square-wave modulation steps at each half second. That can be sent through a narrow bandwidth channel, which will filter out the higher harmonics. But then, as the Fourier transform suggests, you'll see at the receiver a different signal, with rounded edges instead of steps in the modulated amplitude. Those half second intervals set what's called the symbol rate to 2sym/s, and the Nyquist frequency for them is 1Hz. If your channel bandwidth (the filter) is > 1Hz, the information will still be there in the modulated, filtered signal to recover, but you'll have an interesting time doing it if the bandwidth is close to 1Hz. If your channel bandwidth is < 1Hz (the filter strictly removes all frequencies above 1Hz), you won't be able to recover all the transmitted bits. I recommend building a demodulator and varying the channel bandwidth to get an intuition for how it gets trickier to demodulate close to 1Hz. Note that making the strict bandwidth filter is also quite an interesting challenge, especially close to 1Hz; this is called a brick wall filter. Doing the Fourier transform on the entire signal before and after filtering is a good way to visualise the bandwidth taken by a modulation scheme and the effects of a channel, and to see if it's correctly removed everything above the threshold frequency. Modifying those Fourier values in the frequency domain then using an inverse Fourier transform to construct a filtered version of the entire signal is a good way to construct an ideal filter for testing the demodulator.