Can't understand Frequency and Bandwidth [duplicate]

Question

I just don't get it. How bandwidth is the factor that the data rate and the capacity are based on? Let's say we're using Amplitude Modulation. The main factor here is the ability to change the amplitude! The difference between the maximum and minimum frequency can be 0. Who cares? How can Niquist come and say that Capacity=2*B*n? Just tell me how bandwidth states the speed and the capacity, please. Thank you! Oh, Please use understandable English :)

Answer 1

I think what you are asking is "Since the carrier frequency of a AM signal is always the same, and we're only changing its amplitude, not frequency, how come the bandwidth isn't 0 when modulation is applied? No matter what we do, everything should be at the single carrier frequency.". That's a good question, and the answer is unintuitive until you learn about Fourier analisys and to think in frequency space.

It turns out that there there is non-zero frequency content in changing a sine wave in any way, including changing its amplitude. The detailed proof of this goes into mathematics that you probably aren't ready for yet, since if it had been covered in class you'd already understand the answer to your question. Instead, I'll try to give you a conceptual justification.

Let's say you have a nice fixed steady 1 kHz sine signal of ±1 V. Suppose you add another 1 kHz sine of ±1/2 V to that. What do you get? Think about it a little. The answer is not as simple as a ±1.5 V signal. It depends on where the second signal is within its cycle relative to the first, otherwise known as its phase relationship. If the two are in phase, meaning the peaks line up in time, then you do get the ±1.5 V signal. However, if the low points of the second signal happen to line up with the high points of the first signal, you get ±1/2 V instead. Depending on the phase relationship, you can actually get any amplitude from ±0.5 V to ±1.5 V.

Now imagine that the second signal is just a little different in frequency, let's say 1001 Hz instead of the 1000 Hz of the original signal. Sometimes the second signal will be in phase, and you get ±1.5 V, and sometimes it will be out of phase and you get ±0.5 V. Since the second frequency is off by 1 Hz from the first, the result go thru a full 1.5 - 0.5 - 1.5 amplitude cycle every 1 second. Put another way, we are sortof changing the amplitude of the first signal over time. (The reason I say sortof is because we're also messing with its phase a little, but that's a detail I want to ignore at this point). If you were to look at the resulting signal on a scope, or even just listen to it, it would look like or sound like someone adjusting the volume control up and down every second. In effect, we've created a AM modulated signal.

It turns out (you'll have to trust me on the math for now), that true AM modulation is actually adding a signal at 999 Hz and 1001 Hz. That avoids messing with the phase of the 1 kHz signal that we did inadvertantly before. The result is a 1 kHz signal that just goes up and down in amplitude with no phase shift or other artifacts. That is exactly AM modulation. The result is the same as you would get by passing the signal thru a volume control and having someone rotate the knob back and forth at 1 Hz.

The point is that what appears to be a single frequency going up and down in amplitude can be broken into three steady sine waves, one at the carrier frequency and one each at the carrier frequency plus and minus the modulation frequency. And yes, it's possible with the right math to start with the AM modulated signal and derive the three steady sine waves from it. Put another way, it doesn't matter how you actually produced the AM modulated signal, it always has the same frequency content as if it were the three steady sines added together.

Note that the faster you change the amplitude, the further the frequency distance from the carrier the added signals are. The faster the amplitude is changed, the more bandwidth is required. With 1 Hz modulation, we needed 2 Hz bandwidth (999 to 1001 Hz). In general, with true AM modulation, you need a bandwidth twice of the modulation signal. If you have a 1 MHz radio carrier that can carry voice and music up to 8 kHz, for example, then the total signal will be spread out over a frequency range from 992,000 Hz to 1,008,000 Hz. And yes, the transmitter is actually spewing frequencies within that range into the air. If you tuned a very narrow receiver to just 1,000,990 to 1,001,100 Hz, for example, you'd get some signal whenever the radio station was sending 990-1100 Hz tones.

I know this is a lot to take in and the whole thing sounds rather unintuitive, but go look up something called Fourier analisys for more information. In electronics and most other engineering disciplines, it is very usefull to be able to think of signals not just in the time domain, but also in the frequency domain.

Answer 2

I think you're assuming that in AM the bandwidth of the signal is zero.

It is not.

Standard AM produces sidebands at the modulating frequency as follows (Wikipedia), where fc is the carrier frequency and fm is the bandwidth of the modulating frequency: