Pulse Modulation: Analog Communication Formula
Sheet for GATE
Pulse Amplitude Modulation (PAM)
• PAM Signal: Samples of the message signal m(t) at regular intervals.
∑
∞
s(t) = m(nTs )p(t − nTs )
n=−∞
1
where p(t) is the pulse shape, Ts is the sampling period, fs = Ts
is the sampling
frequency (Hz).
• Nyquist Sampling Theorem:
fs ≥ 2fm
where fm is the maximum frequency of the message signal.
• Bandwidth:
BW ≥ fm
(Depends on pulse shape; ideal rectangular pulses require infinite bandwidth).
• Reconstruction: Low-pass filter with cutoff frequency fm to recover m(t).
Pulse Width Modulation (PWM)
• PWM Signal: Pulse width varies proportionally to m(t).
∑
∞
s(t) = p(t − nTs , wn ), wn ∝ m(nTs )
n=−∞
where wn is the pulse width at the n-th sample.
• Bandwidth: Depends on pulse shape and modulation depth, typically:
BW ≈ fs
• Demodulation: Low-pass filtering after converting pulse widths to amplitudes.
Pulse Position Modulation (PPM)
• PPM Signal: Pulse position shifts proportional to m(t).
∑
∞
s(t) = p(t − nTs − km(nTs ))
n=−∞
where k is the position sensitivity.
• Bandwidth: Similar to PWM, typically:
BW ≈ fs
• Demodulation: Convert position shifts to amplitude, then low-pass filter.
1
Pulse Code Modulation (PCM)
• PCM Process: Sampling, quantization, and encoding.
• Quantization Levels:
L = 2b
where b is the number of bits per sample.
• Quantization Step Size:
Amax − Amin
∆=
L
where Amax , Amin are the maximum and minimum signal amplitudes.
• Quantization Noise Power:
∆2
σq2 =
12
• Bit Rate:
Rb = bfs , fs ≥ 2fm
where Rb is in bits per second.
• Bandwidth:
Rb bfs
BW ≥ =
2 2
• Signal-to-Quantization Noise Ratio (SQNR):
Psignal Psignal
SQNR = = 3L2
σq 2 (Amax − Amin )2
SQNR (dB) ≈ 6.02b + 1.76 (for uniform signal)
Noise in Pulse Modulation
• PAM SNR: Similar to AM, depends on channel noise.
Psignal
SNRoutput ∝
N0 f m
where N0 is noise PSD (W Hz−1 ).
• PCM SNR: Dominated by quantization noise for high channel SNR.
SNRtotal ≈ SQNR (if channel noise is negligible)
Key Notes
• PCM is digital, while PAM, PWM, and PPM are analog pulse modulation techniques.
• Sampling frequency fs ≥ 2fm is critical for signal reconstruction.
• PCM’s SQNR improves by 6 dB per additional bit.
• Use consistent units: frequency in Hz, power in W, bit rate in bit s−1 .
