Sampling and Pulse Modulation

Introduction

Communication systems often transmit a continuous-time message through digital or pulse-based equipment. Sampling creates a discrete-time representation of an analogue signal, while pulse modulation uses pulses whose parameters are varied according to that message. These operations form the conceptual bridge between analogue information sources and digital communication, pulse-code modulation (PCM), time-division multiplexing, digital telephony, instrumentation, and modern software-defined radios.

Sampling is not merely a periodic measurement operation. It is a spectral transformation that can either preserve all information in a band-limited signal or irreversibly create aliasing. Pulse modulation is likewise not a single technique: pulse amplitude, width, position, or code may carry the message. A sound engineering design must therefore distinguish ideal theory from practical limitations such as aperture effects, clock jitter, quantisation noise, bandwidth, synchronization, and regulatory constraints.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Explain impulse, natural, and flat-top sampling in time and frequency domains.
2. Derive and apply the sampling theorem and determine Nyquist rates.
3. Analyse aliasing, reconstruction, aperture effects, and anti-alias filtering.
4. Compare PAM, PWM, PPM, PCM, DPCM, and delta modulation.
5. Calculate quantisation step size, bit rate, signal-to-quantisation-noise ratio, and transmission bandwidth.
6. Design a practical sampling and pulse-modulation chain and evaluate its limitations.
7. Interpret engineering trade-offs involving bandwidth, complexity, power, latency, and robustness.

1. Signal Sampling as a Spectral Operation

Let an analogue message be x(t), with Fourier transform X(f). Ideal uniform sampling at interval T_s is represented by multiplying x(t) by an impulse train:

Here T_s is the sampling interval, f_s = 1/T_s is the sampling frequency in hertz, and δ(t) is the Dirac impulse. The sampled signal is therefore

The Fourier transform of the impulse train is also an impulse train:

Because multiplication in time corresponds to convolution in frequency,

Thus, sampling replicates the message spectrum around integer multiples of f_s. Perfect reconstruction is possible when these replicas do not overlap. If x(t) is strictly band-limited to |f| ≤ B, where B is its highest frequency in hertz, the sufficient condition is

This is the sampling theorem. The quantity 2B is the Nyquist rate, while f_s/2 is the Nyquist frequency of a sampler operating at f_s. The strict inequality is preferred in engineering practice because equality leaves no transition band for a realizable filter and is sensitive to phase and timing errors.

Figure: Sampling produces translated replicas of the baseband spectrum at integer multiples of the sampling frequency.

A reconstruction low-pass filter selects the central replica. For ideal impulse sampling, its gain must compensate the factor 1/T_s. In practice, the reconstruction filter is not ideal, so a guard band between the message bandwidth and the first spectral replica is valuable.

Band-pass sampling

The condition f_s > 2B is sufficient for a low-pass signal whose spectrum extends from zero to B; it is not always necessary for a band-pass signal. If a signal occupies a band of width W centred around f_c, intentional undersampling may be possible provided translated bands do not overlap. This is called band-pass, or harmonic, sampling. The sampling frequency must be selected from allowable frequency intervals determined by the band edges; the carrier frequency and occupied bandwidth, rather than only the highest frequency, govern the design. Band-pass sampling is useful in software-defined receivers but requires careful analysis of spectral inversion, clock phase noise, and interference.

2. Aliasing and Reconstruction

Aliasing occurs when spectral replicas overlap. Once two distinct analogue frequencies produce the same samples, no linear reconstruction filter can determine which original frequency was present. For a sinusoid x(t) = A cos(2πf_0t + φ), samples are

Frequencies separated by integer multiples of f_s generate the same sampled sequence. The observable alias frequency may be expressed as the frequency folded into the interval [0, f_s/2]. For example, a 7 kHz tone sampled at 10 kHz appears as a 3 kHz tone because 7 kHz = 10 kHz − 3 kHz.

The anti-aliasing filter is an analogue low-pass filter placed before the sampler. It attenuates components above the intended band limit. It cannot restore information already lost, and its transition band must be included in system design. A practical specification identifies passband edge, stopband edge, allowable passband ripple, required stopband attenuation, and sampling frequency.

The reconstruction filter is placed after a digital-to-analogue converter or ideal sampler. It removes spectral images and smooths the staircase or pulse waveform. Its response must account for the zero-order hold, whose frequency response is approximately

where sinc(u) = sin(πu)/(πu), and the exponential represents a linear phase delay under the stated Fourier-transform convention. The magnitude therefore has a sin(x)/x droop, which may require equalisation in wideband systems.

Worked Example 1: Selecting a sampling frequency

An audio signal is specified as band-limited to 15 kHz. The theoretical Nyquist rate is 30 ksample/s. A practical design using f_s = 40 ksample/s provides a 5 kHz transition interval between 15 kHz and the first image beginning at 25 kHz. Sampling exactly at 30 ksample/s would satisfy the ideal inequality only marginally and would leave no practical filter transition band. If an anti-alias filter has a passband to 15 kHz and must reach its stopband by 20 kHz, 40 ksample/s is a more feasible choice than 32 ksample/s because the Nyquist frequency is 20 kHz.

3. Forms of Sampling

Ideal impulse sampling

In ideal sampling, the signal is multiplied by zero-width impulses. It is a mathematical model that clearly demonstrates spectral replication, but physical circuits cannot generate impulses of zero duration.

Natural sampling

In natural sampling, a train of finite-width pulses is used as the switching waveform. During each pulse, the output follows the instantaneous message waveform. The pulse top is therefore not flat. Natural sampling preserves more within-pulse waveform detail but is less convenient for some hold-and-convert circuits.

Flat-top sampling

A sample-and-hold circuit takes a value at the beginning, middle, or another defined instant and holds it for the pulse duration. The output has flat-topped pulses. This is the dominant practical model for analogue-to-digital conversion. The hold aperture produces amplitude droop, especially at high input frequencies, and introduces a sin(x)/x response.

Important practical parameters include sampling-clock accuracy, aperture time, acquisition time, hold settling, switch resistance, leakage, kickback, and clock jitter. For a sinusoidal input of frequency f_in, timing jitter with rms value σ_t produces an approximate jitter-limited signal-to-noise ratio

This approximation assumes small, uncorrelated timing errors and a sinusoidal input. It shows why high-frequency converters require especially clean clocks.

4. Pulse Modulation: Concept and Classification

A pulse train has several controllable attributes. Pulse amplitude modulation (PAM) varies amplitude; pulse-width modulation (PWM), also called pulse-duration modulation, varies duration; pulse-position modulation (PPM) varies time location; and pulse-code modulation (PCM) represents quantised samples with binary words.

Pulse amplitude modulation

For PAM, the amplitude of each pulse is proportional to a sample of x(t). In ideal natural PAM, the pulse amplitude follows the message during its duration; in flat-top PAM, it is held constant. PAM is often an intermediate signal rather than the final transmission format. It is used in time-division multiplexing, optical intensity systems, and multilevel wired links.

Pulse-width modulation

In PWM, pulse amplitude and repetition period are generally fixed while pulse width τ varies with the message. If the pulse period is T_p and 0 ≤ τ ≤ T_p, the duty cycle is D = τ/T_p. A suitable modulator must impose a monotonic mapping between message amplitude and τ and must prevent overmodulation. PWM is widely used in motor drives, switching power converters, class-D amplifiers, and control systems. Its average value over a switching period is related to duty cycle, but the exact relationship depends on the load and filtering network.

Pulse-position modulation

PPM uses a fixed-amplitude, usually fixed-width pulse whose position within a frame is varied. It can be generated by converting PWM width into a delay. PPM is less affected by amplitude disturbances than PAM, but timing noise directly affects the recovered message. A synchronisation reference or framing mechanism is essential.

5. Pulse-Code Modulation

PCM consists of sampling, quantisation, encoding, transmission or storage, decoding, and reconstruction. The encoder maps each sample to one of L discrete levels. For a uniform N-bit quantiser,

If the quantiser input range is from −V_max to +V_max, the total range is V_R = 2V_max and the nominal step size is

The quantisation error e_q is the difference between the input sample and its reconstructed level. Under the common assumptions of a rounding quantiser, an error uniformly distributed over [−∆/2, +∆/2], and an input sufficiently busy to exercise many levels,

where P_q is the mean-square quantisation-error voltage for a normalized one-ohm reference or, more generally, the mean-square value in the same units as x². The rms error is ∆/√12. The uniform-error model is not valid for very small signals, overload, correlated signals, or poorly dithered quantisers.

For a full-scale sinusoid that spans the quantiser range without overload, the ideal quantisation signal-to-noise ratio is approximately

This result assumes an ideal uniform quantiser and measures the sinusoidal signal power against quantisation-noise power over the relevant bandwidth. Real converters additionally exhibit thermal noise, reference noise, distortion, nonlinearity, clock jitter, and missing codes.

The PCM bit rate is

for one channel, excluding framing, line coding, forward-error correction, and protocol overhead. For M multiplexed channels with overhead factor η, a simple engineering estimate is R_total = ηMNf_s, where η ≥ 1 represents overhead.

Worked Example 2: PCM design

A signal occupies 3.4 kHz and is sampled at 8 ksample/s using 8 bits per sample. The bit rate is

R_b = 8 × 8,000 = 64,000 bit/s = 64 kbit/s.

The number of levels is 2⁸ = 256. If the quantiser input range is −1 V to +1 V, then ∆ = 2/256 = 7.8125 mV. The ideal full-scale sinusoidal SQNR is 6.02(8) + 1.76 ≈ 49.92 dB. The 8 ksample/s rate exceeds the 6.8 ksample/s Nyquist rate, leaving a 1.2 kHz transition interval for the anti-alias filter. These calculations exclude framing and channel-coding overhead.

6. Differential and Adaptive Pulse-Code Methods

Speech and many sensor signals are correlated from sample to sample. DPCM predicts the current sample from previous reconstructed samples and quantises the prediction error:

where x[n] is the current sample and x̂[n] is the predicted value. The receiver applies the same predictor and adds the decoded error. Because the error usually has smaller variance than x[n], fewer bits may be required for a selected quality level. Prediction mismatch and channel errors can degrade subsequent samples until the predictor resynchronises.

Adaptive DPCM changes the quantiser or predictor parameters according to signal statistics. Companding is another important approach. A compressor gives relatively finer resolution to low-amplitude signals and coarser resolution to large signals, after which an expander restores the amplitude scale. The μ-law and A-law families are widely associated with digital telephony, although their exact parameters are defined by standards rather than by the general PCM theory presented here.

7. Delta Modulation

Delta modulation transmits one bit per sample indicating whether the reconstructed signal should increase or decrease by a fixed step δ. A basic discrete model is

where s[n] is +1 or −1. The method is simple and can use a high sampling rate with one-bit quantisation. Its two characteristic errors are slope overload and granular noise.

If the input slope is too large for the maximum trackable slope, the staircase cannot follow it. For a sinusoidal input x(t) = A sin(2πf_m t), the maximum slope is 2πf_mA. A rough no-slope-overload condition is

where δ/T_s is the maximum staircase slope. The inequality is a design approximation and assumes a simple fixed-step modulator. Increasing δ reduces slope overload but increases granular noise when the input is small or slowly varying. Adaptive delta modulation changes δ to balance these errors.

8. Transmission, Detection, and System Design

Pulse signals may be transmitted in baseband or used to modulate a carrier. Baseband pulse transmission requires a channel pulse shape that controls intersymbol interference (ISI). The Nyquist zero-ISI criterion can be satisfied by suitable raised-cosine or root-raised-cosine filtering, with roll-off factor α between 0 and 1. For binary signalling, the ideal Nyquist bandwidth depends on pulse shaping and signalling convention; therefore bandwidth claims must specify whether one-sided or null-to-null bandwidth is intended.

Regeneration is a major advantage of PCM. A repeater can decide whether a received symbol is a zero or one and transmit a clean replacement, preventing gradual accumulation of amplitude noise. However, incorrect decisions, timing slips, synchronization failures, and framing errors remain significant. Forward-error correction can improve reliability at the cost of redundancy, latency, and processing complexity.

A practical sampling and PCM chain normally includes:

source → anti-alias filter → sample-and-hold → ADC quantiser/encoder → framing and line coding → channel → clock and carrier recovery → decoder/DAC → reconstruction filter → destination.

Design decisions should be made jointly. Increasing f_s relaxes the analogue anti-alias filter but increases data rate, storage, processing, and channel bandwidth. Increasing N reduces quantisation noise but increases rate and may reveal analogue noise that was previously masked. Oversampling and noise shaping can move quantisation noise out of the signal band, where digital filtering can remove it; this is central to sigma-delta converters.

9. Illustrative Case: Digitising an Industrial Vibration Sensor

Background

Consider a condition-monitoring unit measuring machine vibration. The useful band extends from 10 Hz to 18 kHz, while strong interference occurs near 30 kHz. The system must support spectral diagnostics and transmit measurements over a packet network.

Problem

A direct 36 ksample/s design would meet the theoretical Nyquist rate for 18 kHz, but the analogue filter would have almost no practical transition band before 18 kHz. The 30 kHz interference could alias into the diagnostic band. In addition, the vibration amplitude varies widely, so a fixed uniform quantiser may waste codes during quiet operation.

Analysis and approach

The engineering team selects 48 ksample/s, giving a Nyquist frequency of 24 kHz. An analogue low-pass anti-alias filter passes the diagnostic band with controlled ripple and strongly attenuates the 30 kHz interferer. A 16-bit ADC is chosen to provide adequate dynamic range, with gain staging and overload monitoring. Digital decimation is performed only after filtering; decimating before filtering would cause digital aliasing. For low-amplitude signals, programmable gain or a suitable floating-point processing path improves effective resolution.

The samples are packetised with timestamps and sequence numbers. The receiver detects missing packets, applies a defined concealment policy, and records quality indicators rather than silently treating missing data as valid measurements. Calibration data, sensor sensitivity, ADC reference accuracy, and filter characteristics are stored with the measurement metadata.

Outcome

The 48 ksample/s architecture provides a realizable filter transition region and prevents the known interference from entering the useful band. The 16-bit quantiser supports diagnostic analysis, while gain control protects against occasional high vibration. Timestamping permits phase-consistent spectral analysis and exposes clock drift.

Learning points

The case demonstrates that the theoretical Nyquist rate is a minimum condition, not a complete design. Anti-alias filtering, dynamic range, clock quality, metadata, and post-acquisition processing are equally important. It also illustrates why decimation must follow adequate filtering and why measurement integrity includes provenance and calibration information.

10. Limitations, Governance, and Contemporary Developments

Sampling theory assumes a well-defined signal band, stable timing, and a known sampling process. Real systems face nonstationary spectra, electromagnetic interference, converter nonlinearity, clock drift, and uncertain sensor behaviour. A safety-critical measurement system should document requirements, calibration intervals, uncertainty budgets, filter verification, firmware versions, and failure modes.

In regulated or safety-related applications, sampled data must not be presented with false precision. Missing samples, saturation, clipping, and invalid timestamps should be flagged. Security is also relevant: an attacker may inject signals above the nominal band that alias into the monitored band, or manipulate clock and timestamp information. Anti-alias filters, input protection, authenticated data, anomaly detection, and independent safety limits are complementary controls.

Contemporary converters increasingly use oversampling, noise shaping, time-interleaved ADCs, digital calibration, and machine-learning-assisted correction. Time interleaving increases aggregate sampling rate but can produce mismatch spurs from gain, offset, and timing differences between channels. Machine learning may estimate nonlinear correction or detect anomalous sampling behaviour, but it does not remove the need for a verified sampling theorem, bounded operating conditions, and interpretable failure responses. In communication receivers, direct RF sampling and band-pass sampling reduce analogue mixing stages, but they intensify requirements on clock jitter, front-end linearity, and interference management.

Best-Practice Design Checklist

• Specify the occupied signal band from measurements and system requirements, not assumptions alone.
• Choose f_s with a realistic transition band; do not design at exactly 2B unless the ideal model is genuinely applicable.
• Place and verify an anti-alias filter before every sampling operation.
• Select ADC range and gain so normal signals use the available codes without frequent overload.
• Distinguish quantisation noise from analogue noise, distortion, jitter, and interference.
• Use synchronized clocks and quantify aperture-jitter effects at the highest input frequency.
• Filter before digital decimation and reconstruct with an appropriate output filter.
• Include framing, timestamps, calibration data, error detection, and valid-data indicators.
• State bandwidth conventions and account for coding, framing, and error-correction overhead.
• Validate the complete chain with sinusoidal, multitone, transient, overload, and interference tests.

Chapter Summary

Sampling converts a continuous-time signal into a sequence of values. Ideal sampling replicates the spectrum at integer multiples of f_s, and a band-limited signal can be reconstructed when f_s > 2B and spectral replicas do not overlap. Aliasing is irreversible, so anti-alias filtering must precede the sampler. Practical sample-and-hold circuits introduce aperture effects and zero-order-hold droop, while clock jitter becomes increasingly important at high frequencies.

Pulse modulation varies amplitude, width, position, or code. PAM is a useful intermediate representation; PWM and PPM provide timing-based alternatives; PCM combines sampling, quantisation, and binary encoding to enable regeneration, multiplexing, coding, and digital processing. Quantiser resolution, bit rate, SQNR, filter transition bands, synchronization, and channel bandwidth must be evaluated together. DPCM, adaptive methods, companding, delta modulation, oversampling, and noise shaping exploit signal structure or move quantisation noise to improve performance. A reliable professional design also addresses calibration, metadata, cybersecurity, regulation, and failure handling.

Key Terms

Sampling: Measuring a continuous-time signal at discrete time instants.

Sampling frequency: Number of samples acquired per second, f_s.

Nyquist rate: Twice the highest frequency of a strictly band-limited low-pass signal.

Nyquist frequency: Half the sampling frequency of a sampler.

Aliasing: Irrecoverable frequency ambiguity caused by overlapping spectral replicas.

Anti-alias filter: Analogue filter that limits input bandwidth before sampling.

Reconstruction filter: Filter that removes spectral images and recovers a continuous waveform.

Impulse sampling: Mathematical sampling using a train of Dirac impulses.

Natural sampling: Sampling in which the pulse top follows the input during the pulse interval.

Flat-top sampling: Sampling in which each measured value is held constant for a finite duration.

PAM: Pulse amplitude modulation; information is carried by pulse amplitude.

PWM: Pulse-width modulation; information is carried by pulse duration or duty cycle.

PPM: Pulse-position modulation; information is carried by pulse timing location.

PCM: Pulse-code modulation; quantised samples are represented by binary words.

Quantisation: Mapping continuous-valued samples to a finite set of levels.

Quantisation error: Difference between an input sample and its quantised representation.

DPCM: Differential PCM using a prediction error rather than the absolute sample.

Delta modulation: One-bit differential method that transmits the direction of a fixed step.

Aperture jitter: Uncertainty in the instant at which a sample is taken.

Oversampling: Sampling substantially above the minimum rate to ease filtering or enable noise shaping.

Review Questions

Short Answer Questions

1. Define the Nyquist rate and distinguish it from the Nyquist frequency.
2. Why can aliasing not be removed by a reconstruction filter?
3. State two differences between natural and flat-top sampling.
4. What information-bearing parameter is varied in PWM and PPM?
5. List the principal operations in a PCM system.

Long Answer Questions

1. Derive the spectrum of an ideally sampled signal using the impulse-train model and explain the reconstruction condition.
2. Explain the operation, assumptions, and limitations of a uniform PCM quantiser.
3. Compare PAM, PWM, PPM, PCM, DPCM, and delta modulation in terms of bandwidth, noise immunity, complexity, and synchronization.
4. Discuss slope overload and granular noise in delta modulation and explain how adaptive step size addresses them.
5. Develop a practical design procedure for sampling a wideband sensor, including filtering, clock selection, ADC range, data rate, and validation.

Analytical / Application Questions

1. A 12 kHz low-pass signal is sampled at 20 ksample/s. Determine whether ideal reconstruction is possible and identify the minimum theoretical sampling frequency.
2. An 8-bit uniform PCM quantiser spans −2 V to +2 V. Calculate the number of levels and quantisation step size, and state the assumptions needed before estimating SQNR.
3. Design an illustrative digitisation chain for a sensor with a 25 kHz occupied bandwidth and a strong interferer at 45 kHz. Justify the sampling frequency, anti-alias transition band, ADC resolution, and any subsequent decimation.

Multiple Choice Questions with Answers

Question 1

A low-pass signal is strictly band-limited to 12 kHz. Which sampling frequency satisfies the ideal sampling theorem?

A. 12 ksample/s

B. 20 ksample/s

C. 24 ksample/s exactly in every practical system

D. 30 ksample/s

Correct Answer: D. 30 ksample/s

Question 2

What is the principal function of an anti-aliasing filter?

A. To increase ADC bit depth

B. To limit analogue input bandwidth before sampling

C. To encode samples into binary words

D. To remove quantisation noise after decoding

Correct Answer: B. To limit analogue input bandwidth before sampling

Question 3

A 7 kHz sinusoid is sampled at 10 ksample/s. Its baseband alias is

A. 1 kHz

B. 3 kHz

C. 7 kHz

D. 17 kHz

Correct Answer: B. 3 kHz

Question 4

For an ideal N-bit uniform PCM quantiser, the number of quantisation levels is

A. N²

B. 2N

C. 2^N

D. N/2

Correct Answer: C. 2^N

Question 5

An 8-bit PCM system samples one channel at 8 ksample/s. Ignoring overhead, its bit rate is

A. 8 kbit/s

B. 16 kbit/s

C. 64 kbit/s

D. 256 kbit/s

Correct Answer: C. 64 kbit/s

Question 6

In PWM, the primary information-bearing quantity is the

A. pulse width or duty cycle

B. binary word length

C. carrier phase only

D. quantiser step number only

Correct Answer: A. pulse width or duty cycle

Question 7

Under the ideal full-scale sinusoidal quantisation model, the approximate SQNR of an 8-bit quantiser is

A. 8 dB

B. 24.1 dB

C. 49.9 dB

D. 96 dB

Correct Answer: C. 49.9 dB

Question 8

Which error occurs when a delta-modulator staircase cannot follow a rapidly changing input?

A. Granular noise

B. Slope overload

C. Image rejection

D. Framing gain

Correct Answer: B. Slope overload

Question 9

DPCM can reduce the required bit rate primarily because it encodes

A. the optical carrier

B. the prediction error between correlated samples

C. only the sampling clock

D. the anti-alias filter response

Correct Answer: B. the prediction error between correlated samples

Question 10

For a sinusoidal input, increasing rms sampling-clock jitter generally causes the jitter-limited SNR to

A. improve at higher input frequency

B. remain independent of input frequency

C. decrease as input frequency increases

D. become equal to the number of quantisation levels

Correct Answer: C. decrease as input frequency increases