The next question we need to answer is what should be the sampling rate, ƒ_s which is the inverse of the sampling period T in Figure 3. The rate at which a signal should be sampled in order to make a complete and unambiguous recovery of the signal from the samples was first proposed by Nyquist. Called the Nyquist rate, it depends on the highest frequency present in the signal and is given by an expression:

ƒ_s >=2 ƒ_s

If a signal is sampled at a frequency meeting the above criteria, we can recover the signal without any loss in quality. Hence, unlike quantization, sampling is a lossless process. Now we can see why a low-pass filter is required before the sampler in Figure 2. The maximum frequency in analog signals is usually higher than what we can appreciate. For example, in telephone applications frequencies up to 3400 Hz are found quite adequate although speech signal contains frequency components higher than 3400 Hz. The frequencies above 3.4 kHz can be removed from speech without significantly affecting the quality. The low-pass filter does exactly this job. We sample the speech, bandlimited to 3.4 kHz, at 8 kHz satisfying the Nyquist rate requirement given in Equation 2.

What really is the sampling process? Mathematically, sampling is a process of multiplying the analog signal x(t), with a sampling signal s(t), which is a train of impulses as shown in Figure 4. The impulse train is a sequence of delta function, which has a value of unity at instants ‘t’ and zero elsewhere. The impulse train is an infinite sum of individual delta functions, each delayed by nT seconds. Thus, the sampled signal is expressed as:

Since the impulse has unit magnitude only at nT instants, the product of x(t) and the impulse train has non-zero value only at nT instants and is zero at all other instants. At the instants nT the product has the value of x(nT).

What happens if the sampling frequency chosen does not satisfy Equation 2? We find the answer to this question in the frequency domain. Figure 5 shows the sampling process in frequency domain. The amplitude spectrum X(Tω) of signal x(t), which is obtained through the Fourier transform of x(t), is shown in Figure 5 (a). Figure 5 (b) shows the amplitude spectrum of the sampled signal, which is obtained by convolving the Fourier transforms X(jω) of x(t) and S(jω) of s(t) as given in Equation 4.

Clearly, the spectrum of the sampled signal stretches from minus infinity to plus infinity. Figure 5 shows only the positive spectrum. Observing the spectrum in Figure 5 (b) we see that the signal spectrum appears on either side of the sampling frequency ƒ_s, and all its harmonics, 2ƒ_s, 3 ƒ_s, etc. It is just like the amplitude modulation of the sampling frequency and its harmonics by the analog signal.

The spectral window from 0 to 0.5ƒ_s Hz is important because whatever happens in this region is simply repeated all over the spectrum. In fact, as will be apparent soon, we can only see what appears in this window. Our digital world is virtually limited to this band of frequencies. It is easy to see that if the signal to be sampled has a spectrum exceeding 0.5ƒ_s then the different bands will overlap causing a distortion, called aliasing as shown in Figure 5 (c). This is due to the fact that all the frequencies in the signal, which are above the spectral window of 0-0.5ƒ_s are reflected back in to the window. A low-pass filter is invariably placed before the sampler to ensure that the signal applied to the sampler does not have any frequency components above 0.5ƒ_s. This bandlimiting filter must have sharp cut-off characteristics near 0.5ƒ_s, which is difficult to realize in practice. This is the reason why we choose the sampling rate more than twice the highest frequency in the signal, ignoring the equal sign in Equation 2. A sampling frequency much higher than the Nyquist rate is used in Figure 5 (d). As a result, we have a large gap in the sampled signal spectrum. Although this facilitates the design of bandlimiting filter, which can now have a slow fall in the transition characteristics, we pay a penalty in the higher speed required for the quantizer and encoder. Higher sampling rate also means lower sampling period. The sampling period is critical in the signal processing applications because it dictates the speed of the processing devices in a given application. (See “Real-time processing or off-line processing?”)

We said that we can only see what is within the spectral window. Just what we mean by this is made clear by taking some examples such as a digital filter. Let's say we wish to have a low-pass filter with passband up to 3 kHz. An analog low-pass filter would have a response shown in Figure 6 (a) with a passband up to 3 kHz and stop band going up to infinity. If we use a digital filter for the same purpose first we have to digitize our signal. Assume that we sample the signal at 10 kHz creating a spectral window of 5 kHz. Our digital filter would then have a response with a passband up to 3 kHz and stop band up to 5 kHz. What happens beyond 5 kHz? The response from 0 kHz to 5 kHz is mirrored from 5 kHz to 10 kHz, which is our sampling frequency ƒ_s. Beyond 10 kHz the entire picture form 0 kHz to10 kHz is repeated infinitely as shown in Figure 6 (b). Similarly, if we design a high-pass analog filter with passband from 3 kHz the response would be as shown in Figure 6 (c). A digital high-pass filter would have a response shown in Figure 6 (d), with the passband from 3 kHz to 5 kHz. Again we are not concerned about what happens to frequencies beyond 5 kHz because when we decided to use a sampling frequency of 10 kHz we knew that our horizon is limited only up to 5 kHz. The high-pass response, in fact, repeats infinitely.

Interestingly, in the digital filter specification the passband frequency is defined as a fraction of the sampling frequency, 0.3 in our case. So, if the sampling frequency is changed the passband frequency is automatically scaled. For example, we can use the same design of the low-pass filter for passband of 3 MHz if we use the sampling frequency of 10 MHz.

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