1. Mathematical principles

The mathematical relationship of amplitude modulation is simple and intuitive: it involves multiplying the carrier signal by the baseband signal. The frequency of the carrier itself does not change, but its amplitude varies according to the baseband signal. There is a subtle detail here, which involves shifting the baseband signal.

If we have a baseband waveform that varies between -1 and +1, the mathematical relationship can be expressed as follows:

xAM is the waveform of amplitude modulation, composed of the carrier wave xC and the baseband signal xBB. Assuming the carrier wave is an infinite, constant-amplitude, fixed-frequency sine wave with an amplitude of 1, we can replace xC with sin(ωCt).

This modulation relationship encounters an issue: the inability to control the "intensity" of modulation. Specifically, the relationship between baseband variation and carrier amplitude change is fixed. For instance, we cannot design a system where a small change in the baseband value results in a significant change in carrier amplitude. To overcome this limitation, we introduce the modulation index

m.

Now, by varying m, we can control the intensity of the baseband signal's influence on the carrier amplitude. However, it's important to note that  m multiplies with the original baseband signal, not the shifted baseband signal. Therefore, if xBB ranges from -1 to +1, any m value greater than 1 will cause  (1 + m×xBB) to extend into the negative part of the y-axis — which is exactly what we initially sought to avoid by upward shifting. So, remember, if using the modulation index, the signal must be offset based on the maximum amplitude of m×xBB , not xBB itself.

2. Time domain

Here is the final time-domain waveform of the AM signal (red represents the baseband signal, and blue represents the AM waveform).

Now, let's examine the effect of the modulation index. Here is a similar chart, but this time I have offset the baseband signal by 3 units instead of 1 (the original range still remains from -1 to +1).

Now we will introduce the modulation index. The following chart illustrates the scenario when  m = 3 .

Now, the carrier amplitude is more sensitive to variations in the baseband signal. Due to the DC bias I chose based on the modulation index, the shifted baseband signal does not extend into the negative part of the y-axis.

3. Frequency domain

This is our expected unmodulated carrier: a single peak at 10 MHz. Now, let's examine the spectrum of the signal produced by amplitude-modulating the carrier with a sinusoidal wave of constant frequency 1 MHz.

From the figure below, you can observe the typical characteristics of an amplitude-modulated waveform: the baseband signal has shifted the carrier's frequency. You can also think of this as adding the baseband frequencies to the carrier signal, which is essentially what we do with amplitude modulation—keeping the carrier frequency constant while varying its amplitude introduces new frequency components corresponding to the spectral characteristics of the baseband signal.

If we examine the spectrum after modulation more closely, we can see two new peaks located respectively 1 MHz above and below the carrier frequency which is the baseband frequency:

4. Negative frequency

In summary, amplitude modulation shifts the baseband spectrum to a frequency band centered around the carrier frequency.

The reason for having two peaks at different positions is as follows: Typically, we only display positive frequencies, but the negative part of the x-axis corresponds to negative frequencies. These negative frequencies are often overlooked when dealing with the original spectrum but are crucial when analyzing the shifted spectrum.

The following chart should clarify this situation.

The baseband spectrum and the carrier spectrum are symmetric about the y-axis. For the baseband signal, this results in the spectrum extending continuously from the positive to the negative parts of the x-axis. For the carrier, we observe two peaks: one at +ωC and another at –ωC. The AM spectrum is symmetric again: the shifted baseband spectrum appears in both the positive and negative parts of the x-axis.

Amplitude modulation causes the bandwidth to double. We measure bandwidth using only positive frequencies, hence the baseband bandwidth is BWBB as seen in the figure below. However, after shifting the entire spectrum both positive and negative frequencies, all original frequencies become positive. Therefore, the bandwidth after modulation is 2BWBB.

Conclusion

Amplitude modulation corresponds to multiplying the carrier wave by the shifted baseband signal.

The modulation index controls how sensitive the carrier amplitude is to changes in the baseband signal more or less.

In the frequency domain, amplitude modulation shifts the baseband spectrum to a frequency band centered around the carrier frequency.

Since the baseband spectrum is symmetric about the y-axis, this frequency shift causes the bandwidth to double.