Why don’t the spectrum lines look infinitely sharp?

A **finite time window** convolves the true spectrum with a **sinc-like** kernel, **spreading** energy across bins. This simulator does not apply windowing beyond the rectangular capture, so **leakage** is expected.

What happens if f_c is set close to Nyquist?

The discrete model **folds** energy around **f_s/2**. Raise **f_c** too high and **aliases** appear in the plotted band—use it as a **Nyquist demo**.

Is Carson’s rule exact?

No—it is a **practical rule of thumb** for sinusoidal FM. Real audio FM uses much richer messages and bandwidth regulations; treat the number as **pedagogical**.

Why separate sliders for AM depth and FM β?

They play **analogous** roles (how strongly the message affects the waveform) but enter **different equations**; keeping both avoids confusion when switching modes.

AM / FM Modulation

Amplitude modulation (AM) impresses a message onto a high-frequency carrier by varying the envelope of the RF waveform: in the canonical teaching form s(t) = [1 + m cos(ω_m t)] cos(ω_c t), small m keeps sidebands symmetric about f_c at f_c ± f_m (ideal sinusoidal message). Frequency modulation (FM) holds amplitude nearly fixed and encodes information in instantaneous frequency, modeled here as s(t) = cos(ω_c t + β sin(ω_m t)) with modulation index β; the spectrum spreads into Bessel sidebands, and Carson’s rule 2(β+1)f_m is a common bandwidth estimate. This page uses a short fixed window of samples and a 512-point DFT magnitude with log-style vertical scaling for readability—spectral leakage and finite window effects are visible and are part of the teaching point when comparing to textbook line spectra. Nyquist for the internal sample rate limits how high you can set f_c before aliasing appears in the plot.

Who it's for: Introductory signals & systems, communications, or physics labs linking time-domain waveforms to frequency-domain intuition.

Key terms

Amplitude modulation
Frequency modulation
Carrier
Modulating signal
Sideband
Modulation index
DFT / FFT
Carson bandwidth rule
Nyquist frequency

How it works

Amplitude modulation (AM) scales the carrier by a slowly varying envelope (1 + m cos ω_m t). Frequency modulation (FM) keeps amplitude roughly constant but drives instantaneous frequency through phase β sin ω_m t. The plots use a fixed time window and a DFT magnitude snapshot—useful for seeing carrier and sidebands, not for broadcast-standard masking or leakage-free spectra.

Key equations

AM: s(t) = [1 + m cos(ω_m t)] cos(ω_c t)

FM: s(t) = cos(ω_c t + β sin(ω_m t))

Frequently asked questions

Why don’t the spectrum lines look infinitely sharp?: A finite time window convolves the true spectrum with a sinc-like kernel, spreading energy across bins. This simulator does not apply windowing beyond the rectangular capture, so leakage is expected.
What happens if f_c is set close to Nyquist?: The discrete model folds energy around f_s/2. Raise f_c too high and aliases appear in the plotted band—use it as a Nyquist demo.
Is Carson’s rule exact?: No—it is a practical rule of thumb for sinusoidal FM. Real audio FM uses much richer messages and bandwidth regulations; treat the number as pedagogical.
Why separate sliders for AM depth and FM β?: They play analogous roles (how strongly the message affects the waveform) but enter different equations; keeping both avoids confusion when switching modes.

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