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Sampling theory represents one of the fundamentals in Digital Signal Processing and is treated as a separate chapter by almost every textbook on the market. Despite this large coverage, most of the literature emphasizes the basic low-pass sampling concept based on the Nyquist theorem. The general assumption is that the signal has frequencies components from dc to some upper value fa. However, for signals which do not extend to dc, the minimum required sampling rate is a function of the bandwidth of the signal as well as its position in the frequency spectrum. This technique is often called undersampling, harmonic sampling, band-pass sampling or Super-Nyquist sampling. In this paper we are going to examine both aspects with more, in depth analysis of the band-pass sampling technique and its applications. In addition, an equation will be given that can be used to easily calculate the minimum sampling rate of a digitally sampled signal.
Introduction: Sampling Theory
The concept of discrete time and amplitude sampling of an analog signal is obvious. The continuous analog data must be sampled at discrete intervals that are carefully chosen to insure an accurate representation of the original analog signal. It is clear that the more samples taken (faster sampling rates), the more accurate the digital representation. On the other hand, if fewer samples are taken (lower sampling rates), a point is reached, where critical information about the signal is actually lost or distorted. Then, a phenomenon called aliasing and alias overlapping will occur. Aliases could cause problems but as we’ll explain in paragraph 3, they are used to our advantage in undersampling applications.
Here is the question: Given any analog periodic signal, how should we select the sampling period T or, equivalently, the sampling rate fs?
In 1928, Henry Nyquist published an important paper titled "Certain Topics in Telegraph Transmission Theory." In it, he postulated a theorem which proposed that a sample of twice the highest signal frequency rate, captures the signal’s frequency content in a manner that enables it to be reconstructed without fear of aliasing. Another way of saying this is the highest frequency which can be accurately represented is less than one-half of the sampling rate. The theorem is based upon the assumption that, transmission is in a noiseless channel - we know that some noise is inevitable, but this does not stop us from applying the theorem. This is a very important concept, and is widely known and used in the field of electrical engineering, as well as computer science and engineering.
Nyquist's theorem for a noiseless channel is expressed as a mathematical model:
max data rate = C = 2fmaxlog2(L) bits/sec
- where 2fmax is 2 times the highest frequency contained in the noiseless channel, and
- L is the number of discrete levels (e.g., binary = two levels, 0 and 1)
For a practical example of the application of his theorem, we can examine the technologies used for digital voice transmission. Voice data for telephony purposes is limited to frequencies less than 4,000 Hz (cycles per second). Because of the requirement for guardbands, the actual frequency that can be utilized over voice grade lines is limited to 3,000 Hz. This limitation has nothing to do with Nyquist's Theorem, but instead, is based upon the assumption made by Bell Labs that this frequency rate was good enough to make the voice intelligible (agreed it's not CD quality). According to Nyquist, it would take greater then 8,000 samples per second (2 * 4,000) to capture a 4,000 Hz signal without aliasing. This does not, however, dictate how precise each sample is. For the purposes of digital representation, let’s assume a range of 256 values for each sample or 8 bits per sample (28). Therefore, if we take a 4K Hz voice grade signal sampled at 8K Hz and quantize it into 256 levels, it requires 64,000 bits per second to transmit this digitally encoded voice signal in real time (8 bits * 8,000 samples per second = 64Kbps).
Sampling Low-Pass Signals
Consider sampling a continuous signal whose spectrum is shown in Figure 1(a). The spectrum is symmetrical about zero Hz, and the spectral amplitude is zero above +fmax Hz and below –fmax Hz, i.e., the signal is band-limited. From now on in order to simplify the theory we are going to consider only the positive spectrum from 0 (zero) to a fmax. Given that the signal is sampled at a rate of fs samples per second, we can see the spectral replication effects that occur in Figure 1(b). Both the original spectrum as well as an infinite number of replications (aliases) whose period of replication is fs Hz can be visualized.
Figure 1 Spectral replications
(a) original continuous signal spectrum;
(b) spectral replications of the sampled signal when fs/2>fmax
The replicated spectra are not just anomalies of the mathematics; they exist, and enable many of the digital signal processing techniques that we will discuss later.
It is natural to ask: “Why should we care about spectral replications outside the frequency band within 0 to fs/2?”. The problem is that if we perform a frequency translation operation or induce a change in sampling rate through decimation or interpolation, the spectral replications will shift up or down right in the middle of the bandwidth of interest 0-fs/2 and could cause aliasing. As already stated in the introduction, if we keep the sampling frequency greater than twice the bandwidth of the signal (fs ≥ 2fmax), then we avoid the spectral replications that overlap the original baseband spectrum, centered about 0Hz.
From the same analysis, it is obvious that any signal energy located above +fmax Hz in the original continuous spectrum will always end up in the band of interest after sampling, regardless of the sample rate.
This is the reason in practice we need continuous (analog) low-pass filters also called anti aliasing filters. We illustrate this in Figure 2(a) by showing a continuous signal of bandwidth fmax in presence of noise energy.
Figure 2 Spectral replications with noise
(a) original continuous signal plus noise;
(b) discrete spectrum with noise altering the signal of interest.
Sampling this composite continuous signal at a rate greater than 2fmax, prevents replications of the signal of interest from overlapping each other, but all of the noise energy still ends up in the range between zero and +fs/2 of our discrete spectrum, shown in Figure 2(b). As I said, this problem is solved in practice by using an analog low-pass anti-aliasing filter, prior to A/D conversion, to attenuate any unwanted signal energy above +fmaxHz. Some of the Analogue to Digital converters today contain built in anti aliasing filters. In certain situations these anti aliasing filters are programmable.
Sampling Band-Pass Signals
As stated before, a continuous-time signal with highest frequency fmax can be uniquely represented by samples taken at the minimum rate (Nyquist rate) of 2fmax samples per second. However, if the signal is a band-pass signal with frequencies components in the band f1 ≤ f ≤ f2 as shown in Figure 3, a blind application of the sampling theorem would have us sampling the signal at a rate of 2f2 samples per second. If that were the case and f2 was an extremely high frequency, the sampling would be more difficult to perform.
Figure 3 Band-pass signal with frequency components in the range f1 ≤ f ≤ f2
where fc = (f1 + f2)/2 (the carrier frequency).
To understand the whole theory behind sampling band-pass signals, let’s start analyzing a practical example. Imagine a 14KHz sine waveform (ωc -carrier frequency) AM modulated by a 1KHz cosine waveform (ωm –modulation frequency).
According to the Amplitude Modulation equation, the carrier is suppressed and we end up with two different frequencies: 14KHz-1=13KHz and 14KHz+1=15KHz:
Figure 4(a) represents the continuous spectrum of the resulted signal while 4(b) represents the discrete spectrum of the same signal, sampled at 32KHz. To simplify the problem only the positive frequencies of the spectrum are represented.
Note that in the continuous spectrum, only 13KHz and 15KHz are present. However, sampling at 32 KHz we end up with two additional aliases, one at 17KHz and the other one at 19KHz, which are symmetrical about half the sampling frequency, 16KHz.
Figure 4 AM Modulated signal (fs=32KHz)
(a) Continuous spectrum
(b) Discrete spectrum at 32KHz sampling frequency
The question is: “Can we recover the original signal?” The answer is yes, and for this we simply have to pass the digitized signal, through a low-pass digital filter (15.5KHz) and eliminate the aliases as in Figure 4(b). Keep in mind that at this point all of the filtering is accomplished in the digital domain, using standard filtering algorithms.
Let’s lower the sampling frequency to 16KHz and analyze the results. As we can see in Figure 5, the continuous spectrum remains the same, but in the discrete spectrum there will be additional aliases at 1KHz, 3KHz, 29KHz, 31KHz and so on.
Figure 5 AM Modulated signal (fs=16KHz)
(a) Continuous spectrum
(b) Discrete spectrum at 16KHz sampling frequency
If we want to recover the original signal, we again pass the data through a digital band-pass filter and get rid of the unwanted aliases.
Given our previous example, is the Nyquist criteria wrong? We should have sampled at more than twice the highest frequency in the signal (2x15KHz = 30KHz) but instead, we sampled at 16KHz and we have been able to recover the original signal characteristics. However we have to know where the signal of interest is located in the spectrum and apply a specific band pass filter to get rid of all the aliasing introduced by process of digitizing the analog signal.
The reason for this is that band-pass sampling, performs digitization and frequency translation in a single process, often called sampling translation.
The process of digital sampling and frequency translation are intimately bound together in the world of digital signal processing, where every sampling operation, inherently, results in spectral replications. The inquisitive reader may ask, “Can we sample at some still lower rate and avoid overlapping aliasing?” The answer is again yes, but to find out how, we have to do a more in depth analysis of the band-pass sampling technique.
All this work will pay off considering the practical aspects of band-pass sampling that will be presented later on.
Let’s assume then, that we have a continuous input band-pass signal of bandwidth B. Its carrier frequency is fcHz i.e., the band-pass signal is centered at fc and it’s sampled spectrum is that shown in Figure 6 (The description herein of bandpass sampling (undersampling) closely follows the original derivation and description presented in "Understanding Digital Signal Processing", by Richard Lyons, 1997, Prentice Hall Publishing). To simplify the analysis we’ll consider again only the positive spectrum. For a fixed signal in the spectrum, by varying the sampling frequency fs, the aliasing components of the discrete spectrum will shift left and right as shown in Figure 6(a) and 6(b). With an arbitrary number of replications, say n, in the range of 0 to fc-B/2, we see that:
 or 
Let’s illustrate this situation for n=3 in Figure 6(a). Of course, n can be any positive integer as long as fs is never less than 2B.
Figure 6 Band-pass sampling frequency limits
(a) variable sample rate
(b) maximum sample rate without aliasing
(c) minimum sampling rate without aliasing
If the sample rate increases, the original spectra (bold) do not shift, but all the replications will shift up to a point, where the lowest frequency will touch 0Hz and the highest one will touch the signal of interest. Increasing beyond that point will result in replications overlap as in Figure 6(b).
Thus, from Eq. (2-1), for an arbitrary n, there is a frequency that the sample rate must not exceed, or
 (2-1)
If we reduce the sampling rate, the spacing between replications will decrease as in Figure 6(c). Again, the original spectrum do not shift. At some new sample rate f2s where f2s<f1s, the replications will coincide with each other. Then, we know that:
 or 
Should f2s decrease even further down in frequency, the replicas will start overlapping.
For n+1 there is a frequency that the sample rate must always exceed, or
 (2-2)
We now can combine Eqs. (2-1) and (2-2) to say that sampling frequency fs may be anywhere between f2s and f1s, to avoid alias overlapping, or
 (2-3)
where as I already said n is an arbitrary, positive integer ensuring that fs ≥ 2B.
An interesting way of illustrating the nature of Eq. (2-3) is to plot the minimum sampling rate, (2fc+B)/2(m+1), for various values of n as a function of fmax/B. (*)
Calculating the minimum sampling rate
From this, a simple way to calculate the minimum sampling rate in order to avoid overlapped aliases is illustrated by the following equation:
 (2-4)
Equation 2-4 consist of two components. The first is widely recognized as the basic criteria used in low-pass sampling and is simply twice the signal bandwidth. This guarantees that spectral replications around the sampling rate will not fall inside the bandwidth of interest.
The second term is required for signals that are no longer low-pass but band-pass limited because fm is greater than the signal bandwidth, B. In this case, it is possible for the sampling rate to be below the bandwidth of interest.
For this reason, the second term guarantees that spectral replications around the sampling rate and all integer multiples of that sampling rate will not fall within the bandwidth of interest, causing overlapping to occur. By dividing this equation by the signal bandwidth and plotting the ratio of the minimum sampling rate (fs) to bandwidth against the ratio of the maximum frequency (fm) to bandwidth, you get the plot in Figure 7. In the limit as the ratio fm/B becomes large, the second term goes to zero and you are again left with the most commonly understood criterion of 2B as the minimum sampling rate.
Figure 7 Minimum band-pass sampling rate from Eq. (2-4)
So surprisingly, we start out with the criterion of 2B as the minimum acceptable sampling frequency and then ultimately, we end up there again as the maximum signal frequency increases and the total signal bandwidth becomes relatively much smaller.
Note that we have to make sure that the selected bandwidth, used for the formula, includes all frequencies components and filter transition bands as in Figure 8.
Figure 8 Bandwidth of a given signal has to overlap all signal components
Now, coming back to the AM modulated signal given at the beginning of this paragraph, let’s calculate the minimum sampling frequency using Eq. (2-5). As we recommended in Figure 7, the bandwidth and of course maximum frequency need to be increased to allow for filtering: fmax=15.5KHz and B=2+1=3KHz.
At this sampling frequency, the spectrum of the digitized signal is shown in Figure 9.
Again we can recover the information from the signal of interest, by passing this data through the same band-pass digital filter. The advantage this time is that the signal was sampled at a much lower frequency, therefore the cost of the Analogue to Digital converter may be a lot less than if we would have used a high speed A/D. However, it is important to remember that the input analog bandwidth of the sample and hold must be capable of passing the actual signal bandwidth.
Figure 9 AM Modulated signal - review
(a) Continuous spectrum
(b) Discrete spectrum at 6.18KHz sampling rate
Practical Applications of Undersampling
A popular application of undersampling is in digital receivers. A simplifies block diagram of a traditional digital receiver using baseband sampling is shown in Figure 10. The mixer in the RF section of the receiver mixes the signal from the antenna with the RF frequency of the local oscillator. The desired information is contained in relatively small bandwidth of frequencies “Delta f” (∆f). In actual receivers, ∆f may be as high as a few Megahertz. The local oscillator frequency is chosen such that the ∆f band is centered about the IF frequency at the band pass filter output. Popular IF frequencies are generally between 10 and 100MHz. The detector then, translates the ∆f frequency band down to baseband where it is filtered and processed by a baseband ADC. Actual receivers can have several stages of RF and IF processing but the simple diagram serves to illustrate the concept.
Figure 10 Simplified Digital Receiver using Baseband Sampling
In a receiver that uses IF-to-digital techniques (often called undersampling, harmonic, band-pass or IF sampling), the IF signal is applied directly to a wide bandwidth ADC as shown in Figure 11. The ADC sampling rate is chosen to be at least 2∆f. The process of sampling the IF frequency at the proper rate causes one of the aliased components of ∆f to appear in the dc to fs/2 Nyquist bandwidth of the ADC output.
DSP techniques can now be used to process the digital baseband signal.
This approach eliminates the detector and its associated noise and distortion. There is also more flexibility in the DSP, because the ADC sampling rate can be shifted to tune the exact position of the ∆f signal within the baseband. The obvious problem with this approach is that the ADC must now be able to accurately digitize signals, which are well outside the dc to fs/2 Nyquist bandwidth which most ADCs were designed to handle.
Figure 11 Simplified Digital Receiver using IF Sampling
Special techniques are available however, which can extend the dynamic range of ADCs to include IF frequencies.
Another large opportunity for band-pass sampling is in digital cellular radio base stations. For systems, which have RF frequencies at 900MHz, 70 MHz is a popular first-IF frequency. For systems using an RF frequency of 1.8GHz, first-IF frequencies between 200 and 240 MHz are often used. In this kind of application, one ADC digitizes multiple channels in the receive path. Individual channel selection and filtering is done in the digital domain. Narrowband channel characteristics such as bandwidth, passband ripple, and adjacent channel rejection can be controlled with changes to digital parameters (i.e. filter coefficients). Such flexibility is not possible when narrowband analog filters are in the receive path.
Conclusions
The major focus of this paper was sampling and reconstruction of signals. In particular, we treated the sampling of continuous periodic signals, and the subsequent operation of A/D conversions. These are necessary operations in the digital signal processing of analog signals, either on a general-purpose computer or on a custom-designed Digital Signal Processor.
The sampling theorem was introduced by Nyquist in 1928 and later popularized by Shannon in 1949. Since then, things have changed dramatically in the world of Digital Signal Processing. The main focus here was how to sample signals of a finite bandwidth, that are located on the spectrum at high frequencies. There are some things to remember when considering the undersampling technique discussed in this paper.
First, using the criteria of sampling at a frequency that is twice the signal bandwidth instead of twice the highest signal frequency, assumes that the input analog signal is indeed limited to that bandwidth. Any components that fall outside the assumed bandwidth will potentially create overlapping aliasing in the sampled spectrum if not attenuated appropriately.
Second, when using Eq. 2-4 to calculate the minimum sampling rate, any increase to that sampling rate must be an integer multiple of the minimum sampling rate until you reach a frequency that is twice the highest frequency component in the input signal. At that point, you are no longer undersampling and the common Nyquist criteria holds true.
Third, it is important to remember, that even though undersampling implies a lower sampling rate, the input bandwidth of the analog waveform must still be accommodated. Many manufacturers are now coming out with A/D’s that have input analog bandwidths that are much greater than half the available sampling rate exactly for this reason.
References
[1] Lyons, G. R. “Understanding Digital Signal Processing”, Prentice Hall NJ, 1997
[2] Webb, R.C. “IF Signal Sampling Improves Receiver Detection Accuracy”, Microwave & RF, March 1989
[3] Proakis, G. J, Manolakis G. D. “Digital Signal Processing – Principles, Algorithms, and Applications”, Prentice Hall, NJ, 1996
[4] Groshong, R. Ruscak, S. “Undersampling Techniques Simplify Digital Radio”, Electronic Design – May 23, 1991.
[5] Ifeachor, C. E. Jervis, W. B. “Digital Signal Processing – A Practical Approach” Addison-Wesley, 1993
[6] Smith, W. S. “The Scientist and Engineer’s Guide to Digital Signal Processing” San Diego, 1999.
[7] Steiglitz, K “A Digital Signal Processing Promer” Addison-Wesley, 1995.
[8] Arrow Electronics, Signal Processing Seminar NY, 2002
Acknowledgments
The authors would like to express their profound gratitude to Richard Lyons, the author of “Understanding Digital Signal Processing” (1997, Prentice Hall Publishing). His book is an excellent presentation of many of the concepts covered in this paper and, as such, was relied on heavily in the writing of this article.
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