Frequency spectrum |S(f)|
Theory & Key Formulas
$$s_{AM}(t) = A_c[1+m\cos(2\pi f_m t)]\cos(2\pi f_c t)$$
AM変調：$A_c$ 搬送波振幅、$m$ 変調度（0〜1）、$f_c$ 搬送波周波数 [Hz]
$$s_{FM}(t) = A_c\cos\left(2\pi f_c t + \beta\sin(2\pi f_m t)\right)$$
FM変調：$\beta = \Delta f/f_m$ 変調指数、$\Delta f$ 最大周波数偏移 [Hz]
$$B_{FM} \approx 2(\Delta f + f_m) = 2f_m(\beta+1)$$
カーソンの法則によるFM帯域幅 [Hz]
What is Signal Modulation?
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What exactly is "modulation"? Why can't we just send the original sound signal directly over the air?
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Great question! Basically, the original audio signal (like your voice) is a low-frequency signal. In practice, efficient radio transmission requires much higher frequencies. Modulation is the process of "piggybacking" that information onto a high-frequency carrier wave. Try moving the "Carrier Frequency" slider in the simulator above—you'll see the fast, wiggly wave. That's the carrier we're modifying.
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Wait, really? So AM and FM are just different ways of modifying that carrier wave? What's the actual difference I should look for in the simulator?
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Exactly! In AM, the amplitude (the height) of the carrier wave goes up and down to match the message. In FM, the carrier's frequency (how close together the wiggles are) changes slightly. A common case is FM radio having better sound quality. In the simulator, switch between "AM" and "FM" modes and watch how the shape of the red modulated wave changes while the blue message stays the same.
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I see the "Modulation Index" slider. What does that do? Is it like a volume knob for the effect?
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You're on the right track! The modulation index controls how strongly the message affects the carrier. For instance, in AM, a higher index means the amplitude varies more dramatically. If you set it to zero, the modulation disappears and you just get the pure carrier wave. Try sliding it up and down while watching the waveform and the spectrum display—you'll see the sidebands grow and shrink.
Physical Model & Key Equations
The core idea is a carrier wave, $c(t)$, which is a pure sinusoid. The message signal, $m(t)$, modifies a specific property of this carrier. The general form of the modulated signal, $s(t)$, depends on the type.
$$ c(t) = A_c \cos(2\pi f_c t) $$
Here, $A_c$ is the carrier amplitude, $f_c$ is the carrier frequency (adjustable in the simulator), and $t$ is time. This is the wave you start with before any modulation is applied.
For Amplitude Modulation (AM), the message signal directly multiplies the carrier's amplitude. The equation for a standard AM (DSB-LC) wave is:
$$ s_{AM}(t) = A_c \big[ 1 + k_a m(t) \big] \cos(2\pi f_c t) $$
Here, $k_a$ is the amplitude sensitivity, and the term $k_a m(t)$ creates the amplitude variations you see. The "1 +" ensures the amplitude never goes negative, which simplifies receiver design. The modulation index $\mu = k_a \cdot \max|m(t)|$ controls the depth of these variations.
For Frequency Modulation (FM), the message signal changes the instantaneous frequency of the carrier. The equation is:
$$ s_{FM}(t) = A_c \cos\Big(2\pi f_c t + 2\pi k_f \int_0^t m(\tau) d\tau \Big) $$
Here, $k_f$ is the frequency sensitivity. The integral means the message changes the phase of the carrier, and the derivative of phase is frequency. The modulation index in FM is $\beta = (k_f \cdot \max|m(t)|) / f_m$, where $f_m$ is the message frequency. It determines the bandwidth of the FM signal.
Real-World Applications
AM Radio Broadcasting (535–1705 kHz): This is the classic application of Amplitude Modulation. Its advantage is simple receiver circuitry, which is why it was the first commercially successful broadcast technology. However, it's susceptible to static from lightning and electrical interference, which also affect amplitude.
FM Radio Broadcasting (88–108 MHz): FM provides high-fidelity audio because noise typically affects a signal's amplitude, not its frequency. This superior noise immunity is why music stations use FM. The trade-off is a more complex receiver and a wider bandwidth requirement per station.
Aviation Voice Communication (VHF Band): Aircraft communicate with towers using AM, not FM. This might seem counterintuitive, but in crowded airspace, the weaker signal from a distant aircraft on the same frequency is still audible as background static in an AM system, alerting others to its presence. In FM, a weaker signal might be completely rejected by the receiver ("capture effect"), which is a safety hazard.
Phase Modulation in Digital Communications (Wi-Fi, Bluetooth): While our simulator shows analog PM, its digital cousin (Phase Shift Keying, PSK) is everywhere. By shifting the phase of the carrier to specific, discrete values, it can represent digital bits (0s and 1s). This is a fundamental technology in modern wireless data transmission due to its spectral efficiency and robustness.
Common Misconceptions and Points to Note
First, there is a misconception that "a larger modulation index is always better." While increasing the "maximum frequency deviation" in FM improves sound quality, it also widens the occupied frequency bandwidth. For example, if the highest frequency of an audio signal is 5 kHz and the modulation index β is set to 5, Carson's rule gives a bandwidth of approximately 2*(25 kHz+5 kHz)=60 kHz. This is an extremely "generous" usage within limited radio resources and carries a high risk of interfering with other communications. In practice, the basic principle is to choose the minimum modulation index that meets the required quality.
Next, you need to be cautious about the behavior when using a square wave as the "message signal" in a simulator. Abrupt changes like those in a square wave theoretically contain infinite high-frequency components. Applying FM modulation to this state theoretically generates infinite sidebands, causing the bandwidth to diverge. While a simulator displays this within a finite bandwidth, in real circuits, such steep changes cause distortion and unwanted radiation. In practical systems, it is essential to process the message signal with a low-pass filter (pre-emphasis) to suppress high-frequency components.
Finally, there is the issue of incorrect "modulation depth" settings in AM. In an "overmodulation" state where the modulation depth m_a exceeds 1, the waveform's peaks and troughs are clipped, and unwanted components spread in the spectrum. Demodulating in this state not only causes severe distortion in the audio but also leads to interference (crosstalk) in adjacent frequency bands. Radio station operations strictly monitor this. If you set m_a to, say, 1.2 in a simulator and observe the changes in the waveform and spectrum, the impact becomes immediately clear.
Worked Example
FM broadcast scenario: fc=88.5 MHz carrier, fm=5 kHz audio tone, modulation index β=5, Ac=10 V. Resulting FM bandwidth = 2(5+1)×5 = 60 kHz, matching Carson's rule. Spectrum displays sidebands at 88.495, 88.500, 88.505, 88.510 MHz. Phase modulation with same parameters but Δφ=2.5 radians produces narrower spectrum concentration near carrier due to lower effective bandwidth of 2fm(1+Δφ/π).