THE SAMPLING THEOREM

8.1 THE SAMPLING THEOREM

We now show that a real signal whose spectrum is bandlimited to B Hz [X ( ω)= 0 for |ω| > 2πB] can be reconstructed exactly (without any error) from its samples taken uniformly at a rate f s > 2B samples per second. In other words, the minimum sampling frequency is f s = 2B Hz. [ † ]

Figure 8.1: Sampled signal and its Fourier spectrum. To prove the sampling theorem, consider a signal x (t) ( Fig. 8.1a) whose spectrum is band-limited to B Hz ( Fig. 8.1b) . [ ‡ ] For

convenience, spectra are shown as functions of ω as well as of f (hertz). Sampling x (t) at a rate of f s Hz (f s samples per second) can

be accomplished by multiplying x (t) by an impulse train δ T (t)( Fig. 8.1c) , consisting of unit impulses repeating periodically every T seconds, where T= 1/f s . The schematic of a sampler is shown in Fig. 8.1d . The resulting sampled signal x (t) is shown in Fig. 8.1e . The sampled signal consists of impulses spaced every T seconds (the sampling interval). The n th impulse, located at t= nT, has a strength x (nT), the value of x (t) at t= nT.

Because the impulse train δ T (t) is a periodic signal of period T, it can be expressed as a trigonometric Fourier series like that already obtained in Example 6.7 [ Eq. (6.39) ]

Therefore

To find X ( ω), the Fourier transform of x (t), we take the Fourier transform of the right-hand side of Eq. (8.3) , term by term. The To find X ( ω), the Fourier transform of x (t), we take the Fourier transform of the right-hand side of Eq. (8.3) , term by term. The

2 ω s )+X( ω + 2ω s ), which represents the spectrum X ( ω) shifted to 2ω s and −2ω s , and so on to infinity. This result means that the spectrum X ( ω) consists of X (ω) repeating periodically with period ω s =2 π/T rad/s, or f s = 1/T Hz, as depicted in Fig. 8.1f . There is also a constant multiplier 1/T in Eq. (8.3) . Therefore

If we are to reconstruct x (t) from x (t), we should be able to recover X ( ω) from X (ω). This recovery is possible if there is no overlap between successive cycles of X ( ω). Figure 8.1f indicates that this requires

Also, the sampling interval T= 1/f s . Therefore

Thus, as long as the sampling frequency f s is greater than twice the signal bandwidth B (in hertz), X ( ω) consists of nonoverlapping repetitions of X ( ω). Figure 8.1f shows that the gap between the two adjacent spectral repetitions is f s − 2B Hz, and x (t) can be recovered from its samples x (t) by passing the sampled signal x (t) through an ideal lowpass filter having a bandwidth of any value

between B and f s − B Hz. The minimum sampling rate f s = 2B required to recover x (t) from its samples x (t) is called the Nyquist rate for x (t), and the corresponding sampling interval T= 1/2B is called the Nyquist interval for x (t). Samples of a signal taken at its Nyquist

rate are the Nyquist samples of that signal. We are saying that the Nyquist rate 2B Hz is the minimum sampling rate required to preserve the information of x (t). This contradicts

Eq. (8.5) , where we showed that to preserve the information of x (t), the sampling rate f s needs to be greater than 2B Hz. Strictly speaking, Eq. (8.5) is the correct statement. However, if the spectrum X ( ω) contains no impulse or its derivatives at the highest frequency B Hz, then the minimum sampling rate 2B Hz is adequate. In practice, it is rare to observe X ( ω) with an impulse or its

derivatives at the highest frequency. If the contrary situation were to occur, we should use Eq. (8.5) . [ † ] The sampling theorem proved here uses samples taken at uniform intervals. This condition is not necessary. Samples can be taken

arbitrarily at any instants as long as the sampling instants are recorded and there are, on average, 2B samples per second. [ 2 ] The essence of the sampling theorem was known to mathematicians for a long time in the form of the interpolation formula [see later, Eq. (8.11) ]. The origin of the sampling theorem was attributed by H. S. Black to Cauchy in 1841. The essential idea of the sampling theorem was rediscovered in the 1920s by Carson, Nyquist, and Hartley.

EXAMPLE 8.1

In this example, we examine the effects of sampling a signal at the Nyquist rate, below the Nyquist rate (undersampling), and above the Nyquist rate (oversampling). Consider a signal x (t)= sin 2 (5 πt) ( Fig. 8.2a) whose spectrum is X ( ω)= 0.2 Δ (ω/20π) ( Fig. 8.2b) . The bandwidth of this signal is 5 Hz (10 π rad/s). Consequently, the Nyquist rate is 10 Hz; that is, we must sample the signal at a rate no

less than 10 samples/s. The Nyquist interval is T= 1/2B= 0.1 second.

Figure 8.2: Effects of undersampling and oversampling. Recall that the sampled signal spectrum consists of (1/T) X ( ω) = (0.2/T) Δ (ω/20π) repeating periodically with a period equal to the

sampling frequency f s Hz. We present this information in Table 8.1 for three sampling rates: f s = 5 Hz (undersampling), 10 Hz (Nyquist rate), and 20 Hz (oversampling).

Table 8.1

Open table as spreadsheet

Sampling Frequency f s (Hz)

Sampling Interval T (second)

Comments

5 0.2 Undersampling

10 0.1 Nyquist rate

20 0.05 Oversampling

In the first case (undersampling), the sampling rate is 5 Hz (5 samples/s), and the spectrum (1/T) X ( ω) repeats every 5 Hz (10π rad/s). The successive spectra overlap, as depicted in Fig. 8.2d , and the spectrum X ( ω) is not recoverable from X (ω); that is, x (t)

cannot be reconstructed from its samples x (t) in Fig. 8.2c . In the second case, we use the Nyquist sampling rate of 10 Hz ( Fig. 8.2e) . The spectrum X ( ω) consists of back-to-back, nonoverlapping repetitions of (1/T) X (ω) repeating every 10 Hz. Hence, X (ω) can be recovered from X ( ω) using an ideal lowpass filter of bandwidth 5 Hz ( Fig. 8.2f) . Finally, in the last case of oversampling (sampling rate

20 Hz), the spectrum X ( ω) consists of nonoverlapping repetitions of (1/T) X (ω) (repeating every 20 Hz) with empty bands between successive cycles ( Fig. 8.2h) . Hence, X ( ω) can be recovered from X (ω) by using an ideal lowpass filter or even a practical lowpass filter (shown dashed in Fig. 8.2h) . [ † ]