# Sampling at the Nyquist Rate

August 1, 2019

In Example 1, the time waveform x(t) was sampled at π₯s = 4 Γ πΏh. The resulting DTFT spectral images were spread out in the DTFT frequency domain.

By examining these results, it is apparent that the minimum sample rate is π₯s = 2 Γ πΏh. Anything less would result in the overlap of spectral images.

The π₯s = 2Β  Γ πΏh rate is called the Nyquist rate. Example 5 is similar to Example 4, but the sample rate π₯s is reduced to the Nyquist rate.

### Frequency Content of a Waveform Sampled at Nyquist Rate (Ex. 5)

Given the time waveform illustrated in Figure 2.1:

(1)

The Fourier transform π·Μ(π) is the triangle function illustrated in Figure 2.3 with the highest frequency component of πΏh = 1/2π Hz.

The waveform x(t) sampled at π₯s β 2 Γ πΏh = 4 Γ 1/2π = 1/π gives a sample period of π = 1/π₯s = π and is illustrated in Figure 2.5.

Figure 2.5. Sample sequence.

Using this sample rate and Theorem 1, a DTFT πΈΜ(π) of y(n) is expressed as:

(2)

(3)

### Conclusion

• π·Μ(π) repeats every 2π (this is true of all DTFTs)
• π·Μ(π) reaches 0 when (1/π) π = 1 or π = 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.