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The Nyquist Frequency

Module by: Don Johnson

Summary: The Nyquist Frequency corresponds to the highest frequency at which a signal can contain energy and still be compatible with the Sampling Theorem.

The frequency 12 T s 1 2 T s , known today as the Nyquist frequency and the Shannon sampling frequency, corresponds to the highest frequency at which a signal can contain energy and remain compatible with the Sampling Theorem. High-quality sampling systems ensure that no aliasing occurs by unceremoniously lowpass filtering the signal (cutoff frequency being slightly lower than the Nyquist frequency) before sampling. Such systems therefore vary the anti-aliasing filter's cutoff frequency as the sampling rate varies. Because such quality features cost money, many sound cards do not have anti-aliasing filters or, for that matter, post-sampling filters. They sample at high frequencies, 44.1 kHz for example, and hope the signal contains no frequencies above the Nyquist frequency (22.05 kHz in our example). If no aliasing occurs, we can use digital signal processing to reduce the sampling rate without incurring further aliasing. If, however, the signal contains frequencies beyond the sound card's Nyquist frequency, the resulting aliasing can be impossible to remove.

Exercise 1

To gain a better appreciation of aliasing, sketch the spectrum of a sampled square wave. For simplicity consider only the spectral repetitions centered at -1 T s -1 T s , 00, 1 T s 1 T s . Let the sampling interval T s T s be 1; consider two values for the square wave's period: 3.5 and 4. Note in particular where the spectral lines go as the period decreases; some will move to the left and some to the right. What property characterizes the ones going the same direction?

Solution 1

Figure 1
Figure 1 (spectrum16.png)

The square wave's spectrum is shown by the bolder set of lines centered about the origin. The dashed lines correspond to the frequencies about which the spectral repetitions (due to sampling with T s =1 T s 1 ) occur. As the square wave's period decreases, the negative frequency lines move to the left and the positive frequency ones to the right.

If we satisfy the Sampling Theorem's conditions, the signal will change only slightly during each pulse. As we narrow the pulse, making ΔΔ smaller and smaller, the nonzero values of the signal st p T s t s t p T s t will simply be sn T s s n T s , the signal's samples. If indeed the Nyquist frequency equals the signal's highest frequency, at least two samples will occur within the period of the signal's highest frequency sinusoid. In these ways, the sampling signal captures the sampled signal's temporal variations in a way that leaves all the original signal's structure intact.

Exercise 2

What is the simplest bandlimited signal? Using this signal, convince yourself that less than two samples/period will not suffice to specify it. If the sampling rate 1 T s 1 T s is not high enough, what signal would your resulting undersampled signal become?

Solution 2

The simplest bandlimited signal is the sine wave. At the Nyquist frequency, exactly two samples/period would occur. Reducing the sampling rate would result in fewer samples/period, and these samples would appear to have arisen from a lower frequency sinusoid.

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