Nyquist rate

The Nyquist rate of a bandlimited signal with maximum frequency $f_m$ is twice that maximum:

$f_{\text{Nyq}}=2f_m.$

This is the critical rate above which sampling preserves the signal completely. Sample above the Nyquist rate → reconstruction is possible. Sample below → aliasing destroys information.

The “Nyquist frequency” sometimes refers to half the sampling rate ($f_s/2$), the highest frequency that can be unambiguously represented at sampling rate $f_s$. The two terms get used interchangeably, but the distinction matters: $2f_m$ is the rate you need for a signal of max frequency $f_m$; $f_s/2$ is the upper limit of what a sampler at rate $f_s$ can resolve.

Finding it for given signals

The recipe is always the same:

1. Find the Fourier transform $X(f)$ of the signal.
2. Determine the highest frequency $f_m$ in $X(f)$ (the largest $|f|$ for which $X(f)\ne 0$).
3. The Nyquist rate is $2f_m$.

Examples

$x(t)=25\cos (500\pi t)$: cosine at $f_0=500\pi /(2\pi )=250\text{Hz}$. Spectrum: two impulses at $\pm 250\text{Hz}$. So $f_m=250\text{Hz}$, Nyquist rate $=500\text{Hz}$.

$x(t)=15\mathrm{rect}(t/2)$: rectangle of width 2 in time. Spectrum: $30\mathrm{sinc}(2f)$ — a sinc that never strictly reaches zero. So formally the signal isn’t bandlimited and the Nyquist rate is infinite. In practice, the sinc decays past its first zero at $f=1/2$, so you make a judgment call about how much high-frequency content you can afford to lose, then anti-alias filter and sample.

Pattern: any time-limited signal (zero outside a finite interval) is not bandlimited, so technically has infinite Nyquist rate. And reverse: bandlimited signals extend over all time. This is the time-bandwidth tradeoff again.

$x(t)=10\mathrm{sinc}(5t)$: spectrum is $(10/5)\mathrm{rect}(f/5)=2\mathrm{rect}(f/5)$ — a rectangle of width 5 in frequency. So $f_m=5/2=2.5\text{Hz}$, Nyquist rate $=5\text{Hz}$.

$x(t)=2\mathrm{sinc}(5000t)\sin (500,000\pi t)$: spectrum is the convolution of the sinc’s spectrum (rectangle from $-2500$ to $+2500\text{Hz}$) with the sin’s spectrum (impulses at $\pm 250,000\text{Hz}$). Each impulse drags a copy of the rectangle with it, giving two rectangles centered at $\pm 250,000\text{Hz}$. The rectangle near $+250,000$ spans $247,500$ to $252,500\text{Hz}$. So $f_m=252,500\text{Hz}$, and Nyquist rate $=505,000\text{Hz}$.

Note: this last example is a sinc carrying narrowband information ($\pm 2.5\text{kHz}$ bandwidth) modulated onto a 250 kHz carrier. To sample it faithfully, you need $2\times 252,500\text{Hz}$, determined by the carrier, not the bandwidth. That’s why broadcast radio needs such high sampling rates even when the message is narrowband.

Boundary subtlety

The sampling theorem requires strictly $f_s>2f_m$, not $\ge$. Sampling exactly at the Nyquist rate doesn’t always work. For pure tones, sampling at exactly $2f_m$ can hit the zero crossings of $\cos$, missing the signal entirely. Always sample strictly above.

Practical undersampling

In practice, signal engineers often choose $f_s$ a few times larger than the Nyquist rate for safety margin. CD-quality audio uses 44.1 kHz to capture frequencies up to ~22 kHz (a small margin above the human hearing limit of ~20 kHz). Professional audio uses 48, 96, or 192 kHz for additional headroom in mixing and processing.