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A Dirac comb is a periodic train of Dirac delta distributions. With Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$ and period $D>0$,The identity holds as a tempered distribution. Truncating the real-space train by a finite window convolves the reciprocal train with the transform of that window, broadening its peaks. Convolving the train with one slit's transmission instead multiplies the reciprocal peaks by the slit's field transform, the basic construction of a diffraction grating pattern.

Let $D={\sum }_{k\in \mathbb{Z}}{\delta }_{2\pi k}$ be the Dirac comb. Pair the given identity $D=(2\pi {)}^{-1}{\sum }_{n\in \mathbb{Z}}{e}^{in\cdot }$ with the Schwartz function $y\mapsto \varphi (x-y)$. On the left this gives ${\sum }_k\varphi (x-2\pi k)=P\varphi (x)$. On the right,Therefore the Fourier series isThe Fourier bounds for Schwartz functions imply $|\hat{\varphi }(n)|\le C_N(1+|n|{)}^{-N}$ for every $N$. For each nonnegative integer $j$, choose $N>j+1$ to obtain ${\sum }_n|n{|}^j|\hat{\varphi }(n)|<\infty$. Thus the Fourier series and every differentiated Fourier series converge absolutely and uniformly on all of $\mathbb{R}$. In particular convergence holds in the topology of smooth $2\pi$ periodic functions, defined by uniform convergence of every derivative on one period, and also in tempered distributions.

Uniform signal sampling at rate $f_s$ multiplies a continuous signal by a time-domain Dirac comb, giving frequency-domain replicas spaced by $f_s$:If a baseband signal is restricted to $|f|\le B$ and $f_s>2B$, these replicas can be separated by an ideal reconstruction filter. Frequencies differing by integer multiples of $f_s$ have the same sampled phase progression; for real signals they fold into the interval from zero to $f_s/2$. This ambiguity is aliasing.

For example, $0.8f_s$ and $0.2f_s$ cosines produce identical sample values. Apply an analogue Anti-aliasing filter before digitization and choose enough sample-rate margin for its finite transition band. A digital filter applied afterward cannot distinguish components already folded onto one another. Out-of-band noise also aliases unless filtered. The sample rate sets temporal/spectral coverage; quantization resolution sets amplitude precision and is a separate limitation. Bandpass Signal sampling has additional conditions beyond the simple baseband rule.^[ref]

Use Fourier transform convention $\hat{f}(q)=\int f(x)e^{-iqx}dx$. The convolution theorem givesA finite array of very narrow slits at $x=jD$, $j=0,\dots ,N-1$, has transmission proportional to ${\sum }_j\delta (x-jD)$. It can be constructed by multiplying an infinite Dirac comb by a finite window. The infinite comb transforms to a reciprocal comb; the finite window convolves each reciprocal peak with a narrow sinc-shaped amplitude. Thus increasing the illuminated length narrows each Fraunhofer diffraction peak.

The exact finite sum makes the argument quantitative:Its normalized intensity is $[\sin (NqD/2)/(N\sin (qD/2)){]}^2$. Principal maxima occur at $qD=2\pi m$; their first zeros are displaced by $|\Delta (qD)|=2\pi /N$. Replacing the ideal narrow slit by a finite slit function convolves the array in real space, multiplying this field by the single-slit Fourier transform. Field amplitudes are transformed and combined before their squared modulus is taken.

A pulse train is a sequence of pulses. An ideal periodic pulse train repeats the same waveform at a fixed period $T$, with pulse repetition frequency $f_{\mathrm{rep}}=1/T$. A finite burst, varying pulse heights or pulse timing jitter can break exact periodicity without removing the useful description as a pulse train. The repetition interval describes spacing; it is distinct from one pulse's duration.

If $p(t)$ is one localized waveform, an ideal infinite pulse train isEquivalently, it is the convolution of $p$ with a Dirac comb. Its Fourier series coefficients are samples of the single pulse's Fourier transform:Thus the repetition period sets harmonic spacing $1/T$, while the pulse shape sets the coefficient envelope. This statement concerns the repeated waveform itself; an optical electric field additionally has carrier phase information that a measured optical-power trace does not contain.

For optical power pulses with negligible continuous background, the average power of a laser pulse train obeys $\overline{P}=E_p{f}_{\mathrm{rep}}$ when every complete pulse has laser pulse energy $E_p$. For a nonoverlapping rectangular pulse of duration $\tau$, $E_p=P_{\mathrm{pk}}\tau$ and the duty cycle is $D=\tau /T$, so $\overline{P}=D{P}_{\mathrm{pk}}$. For other shapes, a width such as the full width at half maximum needs a pulse energy shape factor before it determines peak power.^[ref]

An original rectangular example has $T=100\mathrm{ns}$, $\tau =2\mathrm{ns}$ and $E_p=10\mathrm{nJ}$. The pulse repetition frequency is $10\mathrm{MHz}$, the duty cycle is $0.02$, the peak power is $5\mathrm{W}$ and the average power is $0.10\mathrm{W}$. A slow power meter can report that average while failing to resolve either the $2\mathrm{ns}$ width or individual pulse-to-pulse energy fluctuation.

A fast photodetector and oscilloscope instead display successive electrical responses. Their bandwidth and impulse response limit the optical timing information available. Counting the spacing tests the pulse repetition frequency; integrating calibrated individual responses tests laser pulse energy. Pulse energy measurement with a photodiode explains the charge-integration step. Separating those measurements avoids inferring the pulse width from an average-power reading alone.