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Pulse width

Pulse width, also referred to as pulse duration or pulse length, is the time interval over which a pulse in an electrical or electromagnetic signal remains above a specified amplitude threshold, commonly measured from the 50% points on the rising (leading) and falling (trailing) edges of the pulse.^[1] This measurement captures the effective active period of the pulse, distinguishing it from related terms like rise time or fall time, and is typically expressed in units such as microseconds (μs) or nanoseconds (ns).^[2] In radar systems, pulse width plays a pivotal role in determining key performance characteristics, including the minimum detectable range—approximately half the spatial equivalent of the pulse length—and the range resolution, where narrower pulses enable finer discrimination between closely spaced targets.^[3] For instance, typical pulse widths in weather radars range from 0.5 to 2 μs, balancing energy for long-range detection against resolution needs.^[3] Shorter pulses improve resolution but reduce signal energy, potentially limiting maximum detection range unless compensated by higher peak power or pulse compression techniques. Beyond radar, pulse width is fundamental to digital signal processing and communications, where it influences data encoding, timing jitter, and bandwidth requirements in pulsed transmission schemes.^[4] In power electronics, it forms the basis of pulse-width modulation (PWM), a technique that varies the width of pulses in a fixed-frequency waveform to control average power delivery, enabling efficient regulation of motor speeds, LED brightness, and voltage in inverters without dissipative losses.^[5] The duty cycle, defined as the ratio of pulse width to the total period, directly quantifies this control, with values from 0% to 100% corresponding to off to full-on states.^[6] PWM's adoption spans applications like DC motor drives and switch-mode power supplies due to its simplicity and energy efficiency.^[5]

Fundamentals

Definition

Pulse width, also known as pulse duration, refers to the temporal extent of a single pulse within a signal, whether periodic or aperiodic, defined as the interval between the leading and trailing edges where the instantaneous amplitude reaches a specified fraction of the peak value, most commonly at the half-maximum (50%) level.^[7] This measurement captures the effective "on" time of the pulse, distinguishing it from the overall signal envelope or transient behaviors.^[8] In practical terms, it quantifies how long the pulse maintains its elevated amplitude relative to the baseline, providing a key metric for signal analysis in electronics and physics.^[9] Unlike related parameters such as rise time—the duration for the signal to transition from 10% to 90% of its peak amplitude—or fall time, which measures the symmetric descent from 90% to 10%, pulse width specifically emphasizes the sustained portion of the pulse, excluding the sloped edges.^[8] This focus on the core duration avoids conflating the pulse's body with its initiation or termination dynamics, ensuring precise characterization of energy delivery or information content.^[7] The concept gained prominence in radar systems developed during the 1930s and 1940s for range determination.^[10] Formal standardization emerged with IEEE efforts, culminating in definitions within IEEE Std 194-1951 and subsequent revisions like IEEE Std 181-2003, which refined measurement protocols for consistency across applications.^[11]^[7] For instance, in an ideal rectangular pulse of amplitude A and duration \tau, the pulse width \tau denotes the flat interval where the signal exceeds the zero baseline, approximating the full extent in the absence of edge distortions.^[12] This parameter also informs duty cycle, the ratio of pulse width to the total period in repetitive signals.^[9]

Key Parameters

Pulse width is typically measured in units of time, such as seconds (s), nanoseconds (ns), microseconds (μs), or picoseconds (ps), reflecting the duration of the pulse at a specified amplitude level, often the full width at half maximum (FWHM) in signal processing contexts. In periodic signals, it is frequently normalized as a fraction of the signal period when expressed through the duty cycle.^[13] A key parameter derived from pulse width is the duty cycle D, defined as the ratio of the pulse width \tau to the signal period T, expressed as a percentage:

D = \left( \frac{\tau}{T} \right) \times 100\%.

This metric quantifies the fraction of the period during which the signal is active, typically above a threshold level.^[6] In applications like pulse width modulation (PWM) for electronics, the duty cycle directly influences power efficiency by allowing precise control of average power delivery to loads such as motors or LEDs; for instance, a 50% duty cycle delivers half the maximum power, minimizing energy loss through switching rather than resistive dissipation.^[6] For pulse trains consisting of multiple successive pulses, the pulse repetition interval (PRI) represents the time from the start of one pulse to the start of the next, while the pulse repetition frequency (PRF) is the reciprocal:

PRF = \frac{1}{PRI}.

These parameters govern the spacing and rate of pulses within the train, enabling the accommodation of several pulse widths per cycle in systems like radar, where PRI determines the maximum unambiguous detection range as c \times PRI / 2 (with c the speed of light).^[3]^[14] The relationship between peak and average power in pulsed signals is given by

P_{avg} = D \times P_{peak} = \left( \frac{\tau}{T} \right) \times P_{peak},

where P_{peak} is the instantaneous power during the pulse and D is the duty cycle (or duty factor).^[3] This formula highlights how short pulse widths relative to the period reduce average power while preserving high peak power for applications requiring intense but intermittent energy bursts, such as in radar transmitters.^[14]

Signal Characteristics

Pulse Shapes

Pulse shapes refer to the specific morphologies of transient signals where the duration, or width, is a defining characteristic. These shapes determine how the signal's amplitude varies over time within the pulse duration, influencing its propagation, detection, and processing in various systems. Common pulse shapes include rectangular, Gaussian, triangular, and sawtooth forms, each with distinct profiles that affect the effective width measurement and signal integrity.^[15] Rectangular pulses, also known as square waves in periodic contexts, feature a constant amplitude during the pulse width τ, rising abruptly from zero to a peak value A and falling similarly at the end. This idealized shape assumes instantaneous transitions, making it the fundamental unit in digital transmission schemes where the pulse represents binary data bits. In such signals, the width τ directly corresponds to the bit duration, with the duty cycle briefly noting the ratio of τ to the full period for repetitive pulses.^[15]^[15] Gaussian pulses exhibit a smooth, bell-shaped profile symmetric around the peak, commonly observed in laser emissions and certain natural transient phenomena due to their minimal spectral occupancy. The pulse width is typically defined as the full width at half maximum (FWHM), denoted as τ_FWHM, which measures the duration where the amplitude exceeds half its peak value. For a Gaussian pulse, the intensity follows the form I(t) = I_0 \exp\left(-4 \ln 2 \left(\frac{t}{\tau_{FWHM}}\right)^2\right), ensuring the FWHM precisely captures the effective temporal extent. This shape arises from the Gaussian beam profile in optics, where the pulse energy E relates to peak power P_peak by P_peak \approx 0.94 E / \tau_{FWHM}.^[16]^[17]^[16] Triangular pulses consist of linear ramps, with the width encompassing the full duration of the rise and fall, often symmetric for balanced profiles. The shape can be defined piecewise: for a symmetric triangular pulse of amplitude A and base width τ, it rises with slope 2A/τ from t = 0 to τ/2, reaching A, then falls with slope -2A/τ to zero at t = τ. This linear variation provides a simple model for signals in testing and synthesis, where the slope directly influences the rate of amplitude change across the width. Sawtooth pulses, in contrast, feature an asymmetric profile with a gradual linear rise over most of the width τ and an abrupt drop, defined by a constant slope m = A/τ during the ramp phase, followed by an instantaneous reset. Such shapes are useful for generating linear sweeps, with the width τ spanning the entire ramp duration.^[18]^[19]^[19] Non-ideal pulse shapes introduce variations like jitter and distortion, which alter the effective width beyond the nominal τ. Timing jitter manifests as random or deterministic shifts in pulse edges, effectively broadening the perceived width and degrading timing precision, particularly in high-speed systems where even femtosecond variations accumulate. Distortion, such as from nonlinear propagation or amplification, warps the shape—e.g., compressing or stretching ramps in triangular pulses—leading to discrepancies in width measurement and increased error in signal recovery. In intensity-modulated signals, pulse shape distortion correlates directly with induced timing jitter, amplifying width variability. These effects necessitate compensation techniques to maintain accurate pulse width characterization.^[20]^[21]

Frequency Domain Representation

The frequency domain representation of a pulse reveals how its temporal characteristics, particularly width, influence spectral properties. For a rectangular pulse of duration τ and unit amplitude, the Fourier transform yields the sinc function:

S(f) = \tau \cdot \mathrm{sinc}(\pi f \tau),

where \mathrm{sinc}(x) = \sin(x)/x. This spectrum features a central lobe with first nulls at f = \pm 1/\tau, resulting in a null-to-null bandwidth of approximately $2/\tau. Consequently, the bandwidth \Delta f is inversely proportional to the pulse width \tau, such that narrower pulses occupy wider frequency ranges, a direct consequence of the duality in Fourier analysis.^[22]^[23] This inverse relationship is formalized by the uncertainty principle in signal processing, originally articulated by Gabor, which bounds the time-bandwidth product as \Delta t \cdot \Delta f \geq 1/(4\pi), with \Delta t and \Delta f defined as the root-mean-square durations in time and frequency domains, respectively. The principle implies that achieving high temporal resolution (small \Delta t) inherently requires a broad spectrum (large \Delta f), limiting the simultaneous precision in both domains for any pulse-like signal. This trade-off governs the design of signals where localization in time must balance with frequency selectivity.^[24] In discrete Fourier transform (DFT) analysis, the finite pulse width exacerbates spectral leakage, as the implicit rectangular windowing of the signal convolves its true spectrum with the sinc function's side lobes in the frequency domain. These side lobes cause energy to "leak" into neighboring frequency bins, reducing resolution and introducing spurious peaks, particularly when the pulse does not align perfectly with the analysis window. Windowing techniques can mitigate this, but the underlying effect stems from the non-periodic nature of finite-duration pulses.^[25] An illustrative application occurs in radar systems, where a short pulse width produces a wide spectral bandwidth, enabling high range resolution; the minimum resolvable range \Delta R \approx c / (2 \Delta f), with c the speed of light, allows distinction of closely spaced targets through the pulse's broad frequency content.^[26]

Applications

Electronics and Control Systems

In electronics and control systems, pulse-width modulation (PWM) is a technique used to control the average power delivered to a load by varying the width of pulses in a periodic signal while keeping the amplitude and frequency constant. This method allows precise regulation of voltage or current in digital circuits, enabling efficient power management without the need for linear dissipation. The average output voltage V_{out} of a PWM signal is determined by the duty cycle D, which is the ratio of the pulse width \tau to the period T, according to the equation V_{out} = D \times V_{max}, where V_{max} is the maximum supply voltage.^[27] PWM finds widespread application in motor speed control, where varying the duty cycle adjusts the average voltage applied to the motor windings, thereby regulating speed and torque without mechanical components. For instance, in DC motor drives, PWM signals from microcontrollers or dedicated ICs enable smooth speed variation, improving energy efficiency in industrial automation and electric vehicles. Similarly, PWM is employed for LED dimming, where rapid on-off switching at frequencies above 100 Hz controls perceived brightness by modulating the average current through the LED, preserving color consistency and extending device lifespan compared to resistive methods.^[28]^[29] The development of PWM in electronics traces back to early telecommunications applications before World War II, but its significant evolution for power electronics occurred in the 1960s, with F. G. Turnbull at General Electric introducing selective harmonic elimination PWM in 1964 to control harmonics in inverters. By the 1970s, further advancements, such as Martti Harmoinen's work at Strömberg (now ABB), integrated PWM into variable frequency drives for precise AC motor control, marking a pivotal shift toward efficient power conversion. In practical circuits like buck converters, PWM generates the switching signal for the high-side switch, where the duty cycle directly sets the output voltage ratio V_{out}/V_{in} = D, achieving efficiencies exceeding 90% at moderate loads due to minimized conduction losses.^[30]^[31]^[32] A key advantage of PWM over analog control methods is the reduction in switching losses, as power devices operate in full saturation (low voltage drop when on) or complete cutoff (no current when off), avoiding the dissipative linear region that plagues analog regulators and can limit efficiency to below 70% in high-power scenarios. This results in lower heat generation, smaller cooling requirements, and higher overall system reliability in applications like switch-mode power supplies.^[33]^[34]

Communications and Sensing

In digital communications, pulse width coding can encode binary data by varying the duration of transmitted pulses, where the width of each pulse distinguishes between bit values such as 0 and 1. This approach is used in certain serial protocols and remote control systems to transmit information over noisy channels.^[35] In radar systems, pulse width τ directly governs range resolution, defined as \delta r = \frac{c \tau}{2}, where c is the speed of light, allowing the system to distinguish between closely spaced targets along the line of sight. Typical pulse widths of 0.1 to 10 μs are employed, providing resolutions from approximately 15 meters to 1.5 kilometers while delivering adequate energy for long-range detection in applications like air traffic control and weather monitoring.^[36] Sonar and ultrasound imaging follow analogous principles for range and axial resolution, respectively, with \delta r = \frac{v \tau}{2}, where v is the speed of sound in the propagation medium such as water or tissue.^[37]^[38]^[39] Shorter pulse widths improve resolution for precise target separation or tissue differentiation but diminish total signal energy, creating a trade-off between high accuracy in shallow or near-field sensing and sufficient penetration for deeper imaging in medical diagnostics or underwater navigation. A pivotal historical advancement occurred at the MIT Radiation Laboratory during the 1940s, where engineers developed short-pulse radar technologies, such as the SCR-584 system, to enable accurate ranging and fire control for Allied forces in World War II.^[40] These innovations, producing pulses in the microsecond range, laid the groundwork for modern sensing systems by demonstrating the feasibility of high-resolution detection under combat conditions.^[41]

Optics and Physics

In optics, pulse width refers to the temporal duration of electromagnetic pulses, particularly those generated by lasers, where ultrashort pulses on the femtosecond scale (10^{-15} s) enable high temporal resolution in various phenomena.^[42] These pulses are typically characterized using autocorrelation techniques, which involve splitting the pulse, delaying one part relative to the other, and measuring the resulting second-harmonic generation signal to infer the intensity envelope; for instance, autocorrelation has been employed to measure durations as short as 85 fs from Ti:sapphire lasers.^[43] Such femtosecond pulses find critical applications in time-resolved spectroscopy, allowing observation of ultrafast molecular dynamics and chemical reactions that occur on picosecond to femtosecond timescales.^[44] A key technique for generating and amplifying these ultrashort pulses without excessive broadening is chirped pulse amplification (CPA), which stretches the pulse temporally using a grating pair to introduce a linear chirp, amplifies the lower-peak-power pulse in a gain medium, and then compresses it back to its original duration via a second grating pair, thereby avoiding damage to the amplifier while achieving petawatt-level peak powers.^[45] Developed by Donna Strickland and Gérard Mourou in 1985, CPA revolutionized high-intensity laser systems and earned them the 2018 Nobel Prize in Physics for enabling applications from precision micromachining to attosecond science.^[45] In practice, CPA systems often assume an initial Gaussian pulse shape for optimal compression efficiency, though real pulses may deviate slightly.^[42] In particle physics, pulse width describes the longitudinal extent of particle bunches in accelerators, treated as temporal durations via the speed of light; for example, in the Large Hadron Collider (LHC), proton bunches have a root-mean-square length of approximately 8.4 cm at top energy, corresponding to a pulse width of about 0.28 ns.^[46] This bunch duration is crucial for maintaining collision luminosity and minimizing beam-beam interactions, with the LHC operating bunches at relativistic speeds near 0.99999999c, where the lab-frame length reflects Lorentz contraction of the proper bunch size.^[46] Relativistic effects impose fundamental limits on observed pulse widths, particularly in high-speed reference frames, where length contraction shortens the apparent spatial extent of a moving pulse or bunch in the direction of motion, thereby reducing the measured temporal width by the Lorentz factor γ = 1 / √(1 - v²/c²).^[47] For instance, in relativistic astrophysical sources like gamma-ray burst jets, Doppler boosting and time contraction can shorten observed pulse durations by factors of 10 or more compared to the source frame, enhancing variability timescales.^[48] This effect is also relevant in accelerator physics, where boosting to the bunch's rest frame would elongate the observed width due to the inverse contraction.^[47]

Measurement Techniques

Time-Domain Methods

Time-domain methods for measuring pulse width involve direct observation and quantification of the temporal extent of a pulse in the time domain, typically using hardware instruments that capture or digitize the signal's waveform over time. These approaches are particularly suited for applications requiring high temporal fidelity, such as electronics testing and ultrafast physics experiments, where the pulse duration is determined by identifying the interval between specific amplitude thresholds on the captured signal. Unlike frequency-domain techniques, time-domain methods provide straightforward, real-time visualization and precise edge detection without relying on spectral transformations.^[49] Oscilloscopes are among the most common instruments for pulse width measurement, employing triggering mechanisms to stabilize the display and cursors or automated functions to quantify the duration. Triggering synchronizes the acquisition to the pulse's leading edge or a specific level, ensuring consistent capture of repetitive signals, while measurements are typically taken at the 50% amplitude points of the leading and trailing edges to define the full width at half maximum (FWHM). Digital oscilloscopes achieve resolutions down to the picosecond range through high sample rates and advanced analog-to-digital converters; for instance, the Keysight Infiniium UXR-Series supports minimum pulse width triggering as fine as 40 ps (software). This enables accurate assessment of pulses in high-speed digital systems, where timing jitter and edge uncertainty must be minimized.^[50]^[49] Time-to-digital converters (TDCs) offer specialized, high-precision alternatives for pulse width determination, particularly in applications demanding sub-picosecond accuracy beyond standard oscilloscope capabilities. Implemented as application-specific integrated circuits (ASICs), TDCs digitize time intervals by converting analog pulse edges into digital counts using techniques like delay-line interpolation or vernier methods, directly yielding the width as the difference between start and stop times. These devices routinely resolve widths to less than 10 ps, with examples achieving 1.92 ps resolution and root-mean-square precision below 3.3 ps, making them essential for high-precision timing in particle physics detectors and laser ranging systems. TDCs excel in low-power, compact setups where continuous waveform storage is unnecessary, focusing instead on event-based interval measurements. For ultrafast pulses in the femtosecond regime, streak cameras provide a unique optical-electronic approach to capture temporal profiles with sub-picosecond resolution. In a streak camera, the input pulse illuminates a photocathode to generate electrons, which are then accelerated and swept across a phosphor screen by a high-voltage deflection ramp, effectively converting time into spatial position for direct imaging of the intensity versus time. This sweeping beam technique resolves widths down to 300 fs in advanced systems, enabling characterization of sub-picosecond laser pulses in nonlinear optics and plasma diagnostics. Hamamatsu's C11200 series, for example, supports time responses down to 800 fs for fluorescence lifetime and ultrafast spectrophotometry applications.^[51]^[52] Calibration of time-domain pulse width measurements ensures traceability and accuracy, commonly achieved by applying known reference pulses generated by calibrated function or arbitrary waveform generators. These instruments produce pulses with precisely controlled widths—typically adjustable from nanoseconds to microseconds via direct digital synthesis—allowing verification of the measurement system's response against traceable standards. For instance, connecting a Tektronix AFG to an oscilloscope enables comparison of the displayed width against the generator's specified value, correcting for systematic errors like trigger delay or bandwidth limitations. Such procedures align with NIST guidelines for waveform characterization, maintaining measurement uncertainty below 1% for reference pulses.^[49]^[53]

Frequency-Domain Methods

Frequency-domain methods for measuring pulse width involve analyzing the spectral characteristics of the signal to infer temporal properties indirectly, which is particularly advantageous when direct time-domain access to the waveform is challenging, such as in remote sensing or high-frequency optical systems. These techniques leverage the Fourier transform relationship between time and frequency domains, where the bandwidth of the spectrum provides an estimate of the pulse duration. By examining the power spectral density or phase information, the pulse width can be derived without capturing the full temporal profile, though accuracy depends on the pulse shape and linearity assumptions. One common approach uses a spectrum analyzer to measure the 3 dB bandwidth, denoted as Δf, which is the frequency range where the power spectrum drops to half its maximum value. For Gaussian-shaped pulses, the full width at half maximum (FWHM) pulse duration τ can be estimated using the relation τ ≈ 0.44 / Δf, derived from the time-bandwidth product inherent to Gaussian profiles. This method is widely applied in RF and microwave engineering for characterizing pulsed signals, as the narrower the spectral bandwidth, the longer the corresponding pulse width. Spectrum analyzers facilitate this by resolving the envelope of the frequency spectrum, enabling quick assessments in applications like radar pulse analysis. Autocorrelation interferometry provides another indirect route by performing intensity correlation on the signal to obtain the coherence time, which is inversely related to the spectral bandwidth via the Fourier transform. In this technique, the pulse is split into two paths with a variable delay, recombined in a nonlinear medium to generate a second-harmonic signal, and the autocorrelation trace's width yields the coherence time τ_c ≈ 1 / Δf. For transform-limited pulses, this coherence time closely approximates the pulse width, allowing estimation without resolving the carrier oscillations. This method is prevalent in ultrafast optics for femtosecond pulse characterization, where direct detection is limited by detector bandwidth. Hilbert transform methods extract the signal envelope from the frequency-domain representation to infer pulse width, particularly by analyzing the phase spectrum. The Hilbert transform constructs the analytic signal by applying a 90-degree phase shift to the positive frequency components of the spectrum, suppressing negative frequencies to isolate the envelope as the magnitude of this analytic signal. The resulting envelope's temporal width then directly indicates the pulse duration, useful for modulated or complex pulses where phase information aids in distinguishing the carrier from the modulating envelope. This approach is implemented in digital signal processing tools for precise envelope detection in communications and biomedical signals. These frequency-domain techniques assume stationary signals with well-defined Fourier transforms and are most accurate for Fourier-limited pulses, where the time-bandwidth product matches the minimum value for the given shape (e.g., 0.44 for Gaussians). Deviations arise in non-Fourier-limited cases, such as chirped pulses, leading to overestimated widths due to additional spectral broadening from phase variations.

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[53]
[PDF] Full pulse characterization
Abstract. We present a method for calibrating the voltage that a pulse generator produces at a load at every point in the measured waveform epoch.Missing: width | Show results with:width