Sampling formulas

In $\omega$: the transform is ${\omega }_s{\sum }_k\delta (\omega -k{\omega }_s)$, with ${\omega }_s=2\pi /T_s$.

Replicas of $X(f)$ at every multiple of $f_s$, each scaled by $1/T_s$. In $\omega$: $X_s(\omega )=\frac{1}{T_s}{\sum }_{k}X(\omega -k{\omega }_s)$ — the scale factor is still $1/T_s$, not ${\omega }_s$.

Put $f=0$ in the spectrum formula above. The first equality holds generally; the second needs $X(k{f}_s)=0$ for every $k\ne 0$, which a signal band-limited to $f_m<f_s$ guarantees — the rate does not have to reach the Nyquist rate for this. It turns an infinite sum of sinc-type samples into a single transform value.

For a signal band-limited to $f_m$ Hz. If $X(f)$ has impulses exactly at $\pm f_m$ (the highest component is a sinusoid), the inequality must be strict: $f_s>2f_m$.

Nyquist rate = twice the HIGHEST frequency present (equal to twice the bandwidth only for a lowpass signal). Nyquist interval = the largest allowed sampling interval. In rad/s: ${\omega }_N=2{\omega }_m$.

$\mathrm{rect}(f/2B)=1$ for $|f|<B$; $\mathrm{tri}(f/2B)=1-|f|/(2B)$ for $|f|\le 2B$. So $\sin (2\pi Bt)/(\pi t)$ reaches $B$ Hz and its square reaches $2B$ Hz. In $\omega$: $\sin (Wt)/(\pi t)$ is band-limited to $W$ rad/s $=W/2\pi$ Hz, so its Nyquist rate is $W/\pi$ samples/s.

The frequency an ideal reconstruction filter of cutoff $f_s/2$ outputs for a tone at $f_0$ sampled at $f_s$. If $f_0<f_s/2$, $k=0$ and $f_a=f_0$ — no aliasing.

The samples depend only on $A\cos \theta$: a sine ($\theta =-\pi /2$) gives all-zero samples, and different $(A,\theta )$ pairs give identical samples. A sinusoid therefore needs $f_s>2f_0$ strictly. In the spectrum, the impulses at $\pm f_0$ fall on each other's replicas at $\pm f_s/2$.

Passband gain $T_s$ restores the amplitude. The range exists only when $f_s>2f_m$; at $f_s=2f_m$ it shrinks to the single value $f_c=f_s/2$. The endpoints are also acceptable when $X(f)$ has no impulses at $\pm f_m$.

Exact for $x(t)$ band-limited to $f_m<f_s/2$, using $f_c=f_s/2$. For any other allowed cutoff: $x(t)=2f_c{T}_s{\sum }_{n}x(n{T}_s)\mathrm{sinc}(2f_c(t-n{T}_s))$.

$x(t)$ times a unit-height rectangular pulse train (width $\tau$, period $T_s$, pulses centred on $n{T}_s$). $c_k=(\tau /T_s)\mathrm{sinc}(k\tau /T_s)$ are the pulse train's Fourier-series coefficients — one constant per replica. The baseband copy is $(\tau /T_s)X(f)$, undistorted; an ideal LPF with gain $T_s/\tau$ recovers $x(t)$.

Impulse samples convolved with the hold pulse $\mathrm{rect}((t-\tau /2)/\tau )$ (width $\tau$, starting at each sampling instant), whose transform is $\tau \mathrm{sinc}(f\tau )e^{-j\pi f\tau }$. The factor varies with $f$, so the baseband copy itself is distorted — the aperture effect — and delayed by $\tau /2$.

Attenuation relative to DC, worst at the top of the band $f=f_m$. The equaliser, placed with the reconstruction filter, undoes the magnitude distortion; the delay $\tau /2$ is simply left in. The effect is small when $\tau \ll T_s$: with $\tau /T_s=0.1$ the loss even at $f_s/2$ is under 0.04 dB.

Flat-top with $\tau =T_s$: each sample is held until the next, giving a staircase. It droops across the band (needs a $1/\mathrm{sinc}$ equaliser) and passes attenuated images (needs a smoothing lowpass filter).

$\lfloor \cdot \rfloor$ is the greatest-integer function. If $f_H$ is an integer multiple of $B$, $f_{s,\min }=2B$. In general $2B\le f_{s,\min }<4B$.

For $k=1$ the upper limit is $\infty$: ordinary sampling at $f_s\ge 2f_H$. Rates in the gaps between these windows alias, so a rate above the minimum is not automatically safe.

One-to-one only on the baseband, $|\Omega |\le \pi \leftrightarrow |f|\le f_s/2$. $\cos (2\pi f_{0}t)$ sampled gives $\cos ({\Omega }_{0}n)$ with ${\Omega }_0=2\pi f_0/f_s$; reduce ${\Omega }_0$ modulo $2\pi$ into $(-\pi ,\pi ]$ to read off the alias.

A continuous-time sinusoid is always periodic; its samples need not be — $\cos (n)$, from $f_0/f_s=1/2\pi$, never repeats. Examples: 1 kHz sampled at 3 kHz gives $p/q=1/3$, $N=3$; 700 Hz sampled at 1 kHz gives $7/10$, $N=10$. A sum of such sequences with periods $N_1$ and $N_2$ repeats every $\mathrm{lcm}(N_1,N_2)$ samples.