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The Sampling Theorem

Module by: Don Johnson. E-mail the author

Summary: Converting between a signal and numbers.

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Digital transmission of information and digital signal processing all require signals to first be "acquired" by a computer. One of the most amazing and useful results in electrical engineering is that signals can be converted from a function of time into a sequence of numbers without error: We can convert the numbers back into the signal with (theoretically) no error. Harold Nyquist, a Bell Laboratories engineer, derived this result, known as the Sampling Theorem, first in the 1920s. It found no real application back then. Claude Shannon, also at Bell Laboratories, revived the result once computers were made public after World War II.

We begin with the periodic pulse signal p T s t p T s t , this time centered about the origin so that its Fourier series coefficients are real (the signal is even).

p T s t= k = c k ej2πkt T s p T s t k c k e j 2 π k t T s
c k =sinπkΔ T s πk c k π k Δ T s π k
To sample the signal st s t , we multiply it by the periodic pulse signal. The resulting signal xt=st p T s t x t s t p T s t , as shown in Figure 1, has nonzero values only during the time intervals n T s Δ2 n T s +Δ2 n T s Δ 2 n T s Δ 2 , n=101 n 1 0 1 .

Figure 1: The waveform of an example signal is shown in the top plot and its sampled version in the bottom.
Sampled Signal
Sampled Signal (sig17.png)

If the properties of st s t and the periodic pulse signal are chosen properly, we can recover st s t from xt x t by filtering.

To understand how signal values between the samples can be "filled"; in, we need to calculate the sampled signal's spectrum. Using the Fourier series representation of the periodic sampling signal,

xt= k = c k ej2πkt T s st x t k c k e j 2 π k t T s s t
Considering each term in the sum separately, we need to know the spectrum of the product of the complex exponential and the signal. Evaluating this transform directly is quite easy.
stej2πkt T s e(j2πft)d t =ste(j2π(fk T s )t)d t =S(fk T s ) t s t e j 2 π k t T s e j 2 π f t t s t e j 2 π f k T s t S f k T s
Thus, the spectrum of the sampled signal consists of weighted (by the coefficients c k c k ) and delayed versions of the signal's spectrum (Figure 2).
Xf= k = c k S(fk T s ) X f k c k S f k T s
In general, the terms in this sum overlap each other in the frequency domain, rendering recovery of the original signal impossible. This unpleasant phenomenon is known as aliasing.

Figure 2: The spectrum of some bandlimited (to W W  Hz) signal is shown in the top plot. If the sampling interval T s T s is chosen too large relative to the bandwidth W W , aliasing will occur. In the bottom plot, the sampling interval is chosen sufficiently small to avoid aliasing. Note that if the signal were not bandlimited, the component spectra would always overlap.
aliasing (spectrum9.png)

If, however, we satisfy two conditions:

  • The signal st s t is bandlimited—has power in a restricted frequency range—to W W  Hz, and
  • the sampling interval T s T s is small enough so that the individual components in the sum do not overlap— T s <1/2W T s 12 W ,
aliasing will not occur. In this delightful case, we can recover the original signal by lowpass filtering xt x t with a filter having a cutoff frequency equal to W W  Hz. These two conditions ensure the ability to recover a bandlimited signal from its sampled version: We thus have the Sampling Theorem.

Exercise 1

The Sampling Theorem (as stated) does not mention the pulse width Δ Δ . What is the effect of this parameter on our ability to recover a signal from its samples (assuming the Sampling Theorem's two conditions are met)?


The only effect of pulse duration is to unequally weight the spectral repetitions. Because we are only concerned with the repetition centered about the origin, the pulse duration has no significant effect on recovering a signal from its samples.

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