# Sampling at the Nyquist Rate

August 1, 2019

In Example 1, the time waveform x(t) was sampled at 𝖥s = 4 × 𝖿h. The resulting DTFT spectral images were fairly spread out in the DTFT frequency domain. By examining these results, it is apparent that the minimum sample rate is 𝖥s = 2 × 𝖿h. Anything less would result in the overlap of spectral images.

The 𝖥s = 2  × 𝖿h rate is called the Nyquist rate. Example 5 (below) is similar to Example 4 but the sample rate 𝖥s is reduced to the Nyquist rate.

### Example 5

#### Frequency Content of a Waveform Sampled at the Nyquist Rate

Given the time waveform illustrated in Figure 2.1:

(1) The Fourier transform 𝖷̃(𝜔) is the triangle function illustrated in Figure 2.3 with the highest frequency component of 𝖿h = 1/2𝜋 Hz.

The waveform x(t) sampled at 𝖥s ≜ 2 × 𝖿h = 4 × 1/2𝜋 = 1/𝜋 (giving a sample period of 𝜏 = 1/𝖥s = 𝜋 ) and is illustrated in Figure 2.5. Figure 2.5. Sample sequence.

Using this sample rate and Theorem 1 results in a DTFT 𝖸̃(𝜔) of y(n) expressed as:

(2) (3) ### In conclusion:

• 𝖷̃(𝜔) repeats every 2𝜋 (this is true of all DTFTs)
• 𝖷̃(𝜔) reaches 0 at the place where (1/𝜋) 𝜔 = 1 or 𝜔 = 𝜋
• The height of each triangle is or about 0.798 (see Figures 2.6 and 2.7) Figure 2.6. Fourier transform. Figure 2.7. DTFT.