Modulation Workbench — AM, FM, PM in the Frequency Domain

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Three modulation types, one frequency-domain view. The continuous-time spectrum (carrier + sidebands) is shown as impulses, and the sampled-signal spectrum is the same impulses replicated at every n·f_samp. Watch what happens when you lower f_samp below 2·(f_c + BW): the replicas start to overlap and the signal aliases. Drag f_c, f_m, β, and f_samp to see this in real time.

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AM Parameters

Modulation type:

A_c (carrier): 2.0

f_c (carrier): 20.0 kHz

Message Signal m(t) = Σ A_i·cos(2πf_i·t)

#	f_i (Hz)	A_i

Sampling

f_samp (Hz): 60.0 kHz

f_Nyq = f_samp/2: 30.0 kHz

Replicas shown:

Lower f_samp until the replicas overlap → aliasing. To avoid aliasing for AM, choose f_samp > 2·(f_c + f_m).

① Time-domain signal x(t)

Reference view of the modulated waveform before we go to the frequency domain. The AM envelope is visible (DSB-FC, SSB) or absent (DSB-SC).

② Frequency-domain spectrum |X(f)| with sampled-signal replicas at n·f_samp

Blue = carrier, green = upper sideband, orange = lower sideband, gray = replicas at n·f_samp. Dashed lines at ±f_Nyq mark the Nyquist boundary. When the replicas overlap (red highlight), the signal aliases.

carrier USB (f_c+f_m) LSB (f_c−f_m) replica at n·f_samp aliasing

Modulation

Type

DSB-FC

μ (mod. index)

—

f_c

—

f_m

—

Bandwidth + Power

BW (occupied)

—

P_carrier

—

P_sidebands

—

P_total

—

η (efficiency)

—

η = P_sidebands/P_total. DSB-SC and SSB are 100% efficient (no carrier waste).

Sampling

f_samp

—

f_Nyq = f_samp/2

—

Signal at DC fold:

—

Aliasing?

—

FM Parameters

A_c: 1.0

f_c: 20.0 kHz

f_m: 2.00 kHz

β (mod. index): 2.00

Δf = β·f_m (peak dev.): 4.0 kHz

Max sidebands:

β = Δf/f_m. β < 1 is NBFM; β >> 1 is WBFM.

Sampling

f_samp: 100.0 kHz

f_Nyq = f_samp/2: 50.0 kHz

Replicas shown:

To avoid aliasing, f_samp > 2·(f_c + β·f_m + f_m).

① Time-domain signal x(t)

x(t) = A_c·cos(2πf_ct + β·sin 2πf_mt). The instantaneous frequency oscillates between f_c−Δf and f_c+Δf.

② Frequency-domain spectrum |X(f)| (Bessel J_n(β) sidebands) with sampled-signal replicas

The Bessel expansion: x(t) = A_c·Σ J_n(β)·cos(2π(f_c+n·f_m)·t). Sideband amplitudes follow J_n(β); only the first ~⌈β+1⌉ are significant. Replicas at n·f_samp shown in gray.

J₀(β) (carrier) J_n(β) (positive sideband n) J_n(β) (negative sideband −n) replica at n·f_samp

Modulation

β (mod. index)

—

Δf = β·f_m

—

f_c

—

f_m

—

Bandwidth + Power

P_total (constant)

—

Significant sidebands

—

BW (Carson's rule)

—

BW (99% power)

—

Carson's rule: BW = 2(β+1)·f_m (98% power criterion).

Sampling

f_samp

—

f_Nyq

—

Aliasing?

—

Modulation Workbench — AM, FM, PM in the Frequency Domain

AM Parameters

Message Signal m(t) = Σ Ai·cos(2πfi·t)

Sampling

FM Parameters

Sampling

PM Parameters

Sampling

Message Signal m(t) = Σ A_i·cos(2πf_i·t)