Chris Angove, Independent Professional Engineer

Chris Angove is a highly experienced and MSc qualified chartered electronics engineer specialising in electrical and electronics engineering. He manages and owns Faraday Consultancy Limited (FCL).

Digital Receivers and Transmitters

This document is looking at digital receivers and transmitters, or digital transceivers, as they are often described when integrated into one unit. This document is frequently updated with new information, diagrams and references after I have studied and prepared them.

Contents

How much of a digital receiver is actually analog?

What determines the necessary data channel bandwidth?

What parameters define the performance of a digital receiver?

How does the Shannon-Hartley capacity theorem relate to digital receivers?

How do we convert especially high frequencies to more manageable frequencies?

What are some more popular types of digital modulation?

Digital baseband waveforms for pulse amplitude modulation

How do SNR and Eb/N0 influence a digital receiver performance?

Where is the current ADC/DAC technology for digital receivers?

Are not negative frequencies and times unnecessarily complicated?

What are IQ mixers and how are they used?

What is bandwidth compression using multilevel codes?

How do we transmit digitally modulated frequency spectra?

Can we use similar designs for transmissions over (copper) cable or even optical fiber?

What are baseband and passband modulated frequency spectra?

Where does the clock come from?

What do you mean by 'Nyquist zones', oversampling and undersampling?

An undersampling example

How much of a digital receiver is actually analog?

First of all, what do we actually mean by a digital receiver (DRx)? Here, we will assume that the input to the DRx comes from a digitally modulated source occupying one or more sections of the radio frequency (RF) spectrum. Signals from the DRx input connector up to the associated ADC we will assume are analog with digital signals from the ADC(s) to the DRx output being digital. The DRx input would typically be fed from one or more antennas. Waves like these which propagate through the atmosphere and use antennas may be described using the adjective 'wireless'. Alternatively, a typical 'cable' input to a DRx receiver would meet one or more of the DOCSIS specifications (5). DOCSIS was developed for upgrading the former cable TV (CATV) receive only infrastructure to carry bi-directional (broadband) services for customers, mostly for internet access. DOCSIS cables do not carry digital basebands but bandwidth modulated spectra using similar techniques to those used for wireless but with different frequency ranges and powers.

By definition, the output of an ADC is digital, so there we may use suitable signal processing hardware and a toolbox of digital signaling processing (DSP) applications in the form of software (SW) or firmware (FW). Either may be programmed into the flash memory of the internal processor and used as required. Many of these have become collectively known as software defined radio (SDR). SDR is a relatively new technology and provides routes to quickly re-configure or upgrade the DRx hardware for new frequencies, types of modulation and other parameters, provided that the hardware can accommodate the changes. Previously these would have always required hardware changes, upgrades or replacement. In fact, virtually everything which was previously done in hardware can now be done, in theory at least, with SW. The constraints are usually cost, development time, physical size/mass and reliability.

Let us examine some of the possible ADC front end configurations for a DRx. Figure 1-1 shows a simple first Nyquist zone (analog baseband) sampling configuration. Can we use something like this for all the necessary frequencies?

Figure 1-1 A simple direct (first Nyquist zone) analog baseband sampling ADC configuration might be one of our aspirations but this is not generally a good solution with the present technology except for low frequencies.

The short answer is 'not yet for probably a few years' (9). Note that here we are discussing baseband sampling, either at the Nyquist sampling frequency itself or at a somewhat higher frequency with some level of over-sampling. Alternatives to this are bandpass sampling or under-sampling which are discussed with Nyquist zones in Section 17. The LPF is ideal and assumed to have a perfect 'brick wall' transmission response to reject any analog frequency components greater than one half of the ADC sampling frequency, fs/2, also known as the Nyquist frequency. This is necessary to prevent aliasing.

For example, if we wanted to receive frequencies from an antenna up to 1 GHz, which covers several digital cellular 2G, 3G and 4G services, the sampling frequency (fs) would need to be at least 2 GHz. That is quite a tall order with the present technology and an expensive ADC option. Also, if we used this, the ADC would be converting a range of lower frequencies from at or near DC which may not be required. The only real advantage of this design is that the circuit is simple and there is no frequency conversion or demodulation prior to the ADC.

Using a low noise amplifier (LNA) like that shown in Figure 1-1 at the front end of a receiver, especially for wireless signals, is useful provided the LNA is specified correctly. But, to make it effective over the full operating frequency range of 1 GHz, would be challenging. Such a huge range will be difficult to achieve using this architecture without many problems: harmonics, intermodulation, cross modulation and excessive AWGN (related to noise figure) and quantisation noise.

A more popular architecture is shown in Figure 1-2 for receiving quadrature amplitude modulation (QAM), probably the most common form of digital modulation.

Figure 1-2 A single conversion (homodyne) front end of a typical digital receiver for QAM with the current technology. Each ADC converts a quadrature component of the demodulated analog waveform.

This is known as a homodyne or zero IF front end because, for QAM, the local oscillator (LO) frequency fLO is normally tuned to the frequency at the center of the service provider's frequency spectrum. It is designed for a relatively narrow range of frequencies received at the antenna, as defined by the transmission passband of the bandpass filter, f1 to f2. For digital cellular services, this might cover a range of the allocated frequency bands, for example 900 MHz to 1 GHz. There may be several services available in this frequency range but probably only one that is required. Although the homodyne architecture has a higher part count for the analog portion than the first Nyquist zone architecture, it has several useful properties:

What about the less useful properties?

What determines the necessary data channel bandwidth?

Several factors need to be taken into account to determine the bandwidth necessary to carry a data channel. Some of these are listed below.

What parameters define the performance of a digital receiver?

The digital receiver (DRx) comprises both analog and digital parts, so there are both analog and digital properties to define. Some of these are described in the following paragraphs.

Frequency ranges and synthesizer resolution

Modulation properties, coding and information rates

Input power handling and AGC capability

Simultaneous multi-service capability

Susceptibility to interference, conducted and radiated

Emissions of interference, conducted and radiated

Phase noise

SDR capability and upgradability

Overload capability: wanted signals and impulse (lightning)

Latency

Electrical interfaces

Physical interfaces

Noise figure

Power supply requirements

Physical chartacteristics

Power Interface and Temperature Requirements

How does the Shannon Hartley capacity theorem relate to digital receivers?

The Shannon-Hartley capacity theorem relates to a bandwidth limited digital communication channel in the presence of additive white Gaussian noise (AWGN) (2). AWGN was chosen because it is inherent in the hardware itself and is a function of the product of only two parameters: the absolute temperature of the hardware and the noise bandwidth of the channel. The Shannon-Hartley capacity of the channel is the maximum theoretical data rate that such a channel will support, given its bandwidth and signal power. The formula for this is given by (4-1).

C=B log 2 1+ S N
(4-1) The Shannon-Hartley capacity theorem

where, for the channel:

With the Shannon-Hartley channel capacity theorem there are no restrictions on data rate, parity, forward error correction, modulation or any other feature of the channel that we might want to deploy to improve its error performance. As long as sufficiently error-free data is successfully transmitted, even if the net data throughput is one bit per week, it will still be compliant with the theorem. Now we will define some other important parameters.

The energy per bit (Eb) is the mean signal power of the digital signal divided by the actual data rate (R) (4-2).

E b = S R
(4-2) The energy per bit (Eb) is the signal power S (watts) divided by the bit rate R (/s), so the result is in watt-seconds (Ws) or joules (J).

where:

It is useful to normalise the total mean noise power in the channel against its (noise) bandwidth (4-3).

N 0 = N B
(4-3) The AWGN power density across frequency (N0) is the total mean AWGN power within the (noise) bandwidth (W) divided by the noise bandwidth B (Hz).

where:

The limit of operation of the channel is when the data rate is equal to the channel capacity, C=R. Substituting for this, also for S and N from (4-2) and (4-3) respectively, gives (4-4).

R=C R=B log 2 1+ E b R N 0 B
(4-4) The channel data rate in terms of Eb/N0

γ=R/B is a measure of the theoretical utilisation efficiency of the channel in bit/s/Hz (4-5).

γ= R B = log 2 1+ E b N 0 γ
(4-5) The utilisation factor (γ) for the Shannon-Hartley channel

A useful way of representing this information graphically is to plot the utilisation factor from (4-6).

2 γ =1+ E b N 0 γ
(4-6) This equation, derived from (4-5) may be used for plotting γ against Eb/N0.

The plot in Figure 4-1 is of gamma (γ) against 10log10(Eb/N0). Note that γ does not have any units but the variable Eb/N0 has been converted to the logarithmic (decibel) form so here it has units of dB. However, to further confuse things, the Eb/N0 dB values have been scaled linearly and the γ values logarithmically to base 2.

Figure 4-1 A plot of bandwidth utilisation (γ) against Eb/N0 in dB showing the Shannon limit.

Plots of this type derived from the Shannon Capacity theorem are some of the most important in communications theory. The y axis (γ) is the ratio of data rate to the associated bandwidth necessary to transmit that bandwidth, with units of bits per second per hertz (bit/s/Hz). This is sometimes referred to as the utilisation factor. The x axis is the ratio of energy per bit to noise power density which here, as we mentioned, is in logarithmic (dB) units. The plot represents the theoretical limit of possible communications through a channel limited only by AWGN. This is independent of the data rate or any type of error correction which may be used. If the dB value of Eb/N0 is less than the Shannon limit of -1.6 dB, communications will be impossible whatever form of error mitigation we try (2). The Shannon limit therefore serves as a goal in designing digital communications equipment, one which we know will never be equalled. However, the closeness a piece of equipment to the Shannon limit provides a measure of the current technology and the quality of its design. In the years since Shannon's work in 1948, the practical and achievable performances of digital receivers have come to within about 2 dB of the Shannon limit helped with the use of turbo codes (2).

The RF spectrum is a valuable but restricted resource shared across the World and sections of it are traded as an expensive commodity between service providers and government controlled communications regulators. Perhaps the clearest evidence of this is in the expansion of the analog and later digital cellular communications infrastructure starting in the 1980s. Therefore most service providers' first instinct is to demand whatever is necessary to achieve the largest possible channel capacity in the smallest possible bandwidth: what we referred to earlier as the utilisation factor (γ). From Figure 4-1 we can see that this would require ever increasing values of Eb/N0, which is analogous to signal to noise ratio if this was an analog communications system. How can we increase Eb/N0?. Increase Eb, reduce N0 or both? From (4-2) and (4-3) we may extend this to include signal power (S), actual data rate (R), total noise power in the noise bandwidth (N) and the noise bandwidth itself (B).

So, to increase the information capacity, this gives us a tradeoff amongst competing parameters S, B, R and N. We will address each of these considered separately:

How do we convert especially high frequencies to more manageable frequencies?

Figure 5-1 shows a schematic of a typical homodyne front end of a digital receiver. Conversion of the received frequency band, f1 to f2 to the quadrature (IQ) baseband is in one step as the LO is tuned to the center frequency of the wireless service. Components within the IQ mixer and the LO source itself will therefore require to be specified for frequencies including f2, preferably with a comfortable margin above this. In general, the part, design and production costs increase in proportion to their specified highest frequencies. However, there are usually more service bandwidths available at the higher frequencies so, until the technology progresses further, it may be cost effective to downconvert blocks of these higher frequencies down to more manageable frequencies before the next stage of conversion to the baseband frequency. The first frequency is known as an intermediate frequency (IF).

Figure 5-1 A single conversion (or homodyne) front end for a typical wireless digital receiver. To receive particularly high frequencies at the antenna, a frequency downconversion stage may be required.

What are some more popular types of digital modulation?

Here, we are not considering baseband modulation which is sometimes used to describe the creation of a digital baseband using pulse amplitude modulation (PAM), but we are looking at bandpass modulation. A baseband spectrum starts at or close to 0 Hz, zero frequency or DC and extends to a finite upper limit. The result of bandpass modulation is a finite width spectrum usually with a center frequency which is much greater than the maximum baseband frequency. The width or bandwidth of the spectrum would typically be about twice its maximum baseband frequency. For example, the baseband waveform of a digitally modulated transmission might occupy a spectrum from 0 Hz to 1 MHz but after bandwidth modulation at 500 MHz its spectrum would range from 499 MHz to 501 MHz.

A carrier (continuous wave or CW) may be modulated in order to carry information by altering one or more of its parameters: frequency, amplitude or phase. Digital modulation is achieved by performing the modulation with discrete steps applied to one or more of the parameters. The most common form of digital modulation is quadrature amplitude modulation (QAM) in which both the amplitude and the phase of the carrier are modulated. It is a 'M-ary' form of modulation meaning that it may have different orders 'M' each of the form M-QAM, where:

M= 2 k
(6-1) For M-QAM modulation, the relationship between the modulation order or symbol set (M) and the number of bits per symbol (k)

In QAM, the number of available discrete amplitude and phase values is known as the symbol set, equivalent to the order (M). Each symbol is made up of k bits so it is simple to convert from bits to symbols at the transmit end and vice versa at the receive end. The symbol sets may be represented as voltages on IQ axes by a constellation of dots, each dot representing one symbol's position in terms of amplitude and phase. The quadrature (IQ) axes also relate very conveniently to the quadrature architecture used in the homodyne front end, Figure 5-1. Some examples of these are shown in Figure 6-1.

Figure 6-1 QAM symbol set constellation diagrams for orders of 2, 4 and 16.

The QAM constellation diagrams shown are for 2-QAM (or binary phase shift keying, BPSK), 4-QAM (or quadrature phase shift keying, QPSK) and 64-QAM. For the related discussion we will assume that these constellation diagrams are to the same (voltage) scale. By convention, the horizontal axis is the in-phase (I) component and the vertical axis is the in-quadrature (Q) component. From (6-1), the number of bits per symbol (k) are 1, 2 and 4 respectively and the number of available symbols are 2, 4 and 16 respectively. Thus, the higher the QAM order is the closer each symbol will be to its next closest symbol. This means that once we allow for additive white Gaussian noise (AWGN), the higher order constellations have a greater probability of errors. The least error prone is 2-QAM, more commonly referred to as BPSK.

BPSK is the most resistant QAM order to AWGN because it comprises the smallest possible symbol set of 2.The voltage amplitudes of both symbols are equal, usually normalised to 1 V and they have a phase difference of 180°. As an example of BPSK in the time domain, Figure 6-2 shows the voltage time waveform of 6 consecutive symbols at a BPSK modulator output for a carrier (local oscillator) frequency of 10 MHz. If the probabilities of either a 1 or a 0 are both equal at 0.5, ther average frequency will be 10 MHz.

Figure 6-2 A BPSK IQ modulator voltage time output for 6 bits, equivalent to 6 symbols: 100101 with a local oscillator frequency of 10 MHz

For BPSK there is one bit per symbol and the amplitude of the modulated waveform remains constant, 1 V being chosen for this example. In this case a phase of zero represents a symbol of 1 and a phase of 180° represents a symbol of 0. The waveform shown is for 6 symbols 100101 starting at the most significant bit (MSB). In this case, 2 LO cycles are used for each symbol, so the duration of each is 200 ns.

Digital baseband waveforms for pulse amplitude modulation

The word 'baseband', or 'digital baseband' in the case of digital communications, refers to a frequency spectrum, the result of digital modulation, which covers a range from at or near zero (DC) to some defined upper limit (14). The upper limit could range from around 1 MHz for a low capacity system to many gigahertz for some high capacity telecommunications infrastructure equipment.

A common baseband modulation is pulse amplitude modulation (PAM). As its name implies, PAM is the process of representing the binary data (bit) stream to be transmitted by discrete (amplitude) voltage levels, or pulses. There are many ways that this could be achieved, but common forms of PAM are 'return to zero' (RZ) and non-return to zero (NRZ). For RZ, the bit waveform will settle for some finite time at 0 V and for NRZ it will never settle at 0 V but may almost instantaneously pass through 0 V. Both RZ and NRZ can be unipolar or bipolar forms. Unipolar means that each bit waveform has one voltage state other than 0 V and bipolar refers to it having two states other than 0 V. The zero state in the bipolar form is not a 1 or 0 detection state and the bipolar voltages are symmetric with a mean value of zero.

Some alternative RZ baseband waveforms for streams of periodic digital data are shown in Figure 7-1, Figure 7-3, Figure 7-5 and Figure 7-7. The first of these is a RZ unipolar waveform with a 1 state of +1 V and a 0 state of 0 V. This is shown in Figure 7-1 carrying a long string of alternating of 1s and 0s. The pulse or bit width is 1 µs so the data information rate or signaling rate is 1 Mbit/s.

Figure 7-1 Baseband waveform y11, pulse amplitude modulation, RZ unipolar, serial transmission of alternate 10 states (16 bits of 64 bits shown).

This waveform is equivalent to a periodic rectangular waveform with a frequency of 500 kHz and a duty ratio of 50% or, more simply, a square wave of frequency 500 kHz. It also has a peak to peak amplitude of 1 V and a long term mean value of 0.5 V.

In theory, the content of a digital stream could contain all 0s, all 1s or any mixture of 1s and 0s, with the highest frequency content for this repeating '1010' waveform. However, if long strings of the same bit are possible, this can cause issues with a significant build up of a DC component, sometimes called 'DC wander', in this case either 0 V or 1 V. Often the coding is therefore designed to deliberately introduce state transitions to avoid this from happening. The spectrum for the RZ unipolar '1010...' waveform shown in Figure 7-1 is shown in Figure 7-2. This was generated using the fast Fourier transform (FFT) function in Matlab®.

Figure 7-2 The logarithmic (dBuV) frequency spectrum from y11 in Figure 7-1 using the Matlab® FFT function.

This shows a DC or 0 Hz component with an ampllitude at 114 dBuV, which converts to a linear value of 0.5 V. This is consistent with the average value of the 50% duty cycle waveform for many cycles of the waveform shown in Figure 7-1. The remaining parts of the FFT result provide a guide to the maximum frequency content of the baseband spectrum with discrete components starting at 500 kHz with 1 MHz spacing. We will now extend the same RZ unipolar waveform to one comprising alternate groups of 4 1s followed by 4 0s and so on, the waveform shown in Figure 7-3.

Figure 7-3 Baseband waveform y12, pulse amplitude modulation, RZ unipolar, serial transmission of alternate 11110000 states (16 bits of 64 bits shown).

So, in this case the DC component is unchanged, the smallest discrete components are 250 kHz and the spacing is 500 kHz. Compared to Figure 7-1, Figure 7-3 has a period and pulse width of four times the value so the fundamental frequency is one quarter of the previous value. The FFT of this waveform is shown in Figure 7-4.

Figure 7-4 The logarithmic (dBuV) frequency spectrum from y12 in Figure 7-3 using the Matlab® FFT function. The spectrum is much narrower than that shown in Figure 7-2.

Compared to the RZ unipolar waveforms, an example of an RZ bipolar waveform is shown in Figure 7-5. The peak to peak voltage and bit width for this waveform were the same as those in the RZ unipolar waveforms, chosen for comparison purposes.

Figure 7-5 Baseband waveform y13, pulse amplitude modulation, RZ bipolar, serial transmission of alternate 1010... states (16 bits of 64 bits shown).

The waveform for each bit is a high to low transition for a 1 and a low to high transition for a 0, each state being one half of a bit width. Therefore, the average voltage for a 1 is +0.25 V and for a 0 is -0.25 V. So a long string of 1s or 0s, if either was allowed, would generate average voltages of these respective values. However, for an average data throughput, the probability of either a 1 or a 0 occurring is usually about 0.5, so the average waveform voltage should be close to 0 V. A small average voltage reduces the risk of DC voltages building across coupling capacitors, inadvertently biasing the datastream and causing detection errors. The FFT for this waveform is shown in Figure 7-6.

Figure 7-6 The logarithmic (dBuV) frequency spectrum from y13 in Figure 7-5 using the Matlab® FFT function.

Clearly there is no DC component, nor would we expect one, as the average voltage for a data stream of many bits is 0 V. Another important advantage of the RZ bipolar waveform is that it creates a significant and consistent frequency spectrum components, in this case starting at 500 kHz and increasing with 1 MHz spacing. These support synchronous communication systems in which the clock is extracted from the data.

For completeness, Figure 7-7 shows the RZ bipolar waveform but carrying alternative groups of 4 ones followed by 4 zeros.

Figure 7-7 The logarithmic (dBuV) frequency spectrum from y13 in Figure 7-3 using the Matlab® FFT function.

The FFT of this waveform is shown in Figure 7-8 with an expanded frequency scale to emphasise the frequency spacing of the discrete components.

Figure 7-8 The logarithmic (dBuV) frequency spectrum from y14 in Figure 7-4 using the Matlab® FFT function.

There is clearly no DC component as we would expect with a long term average being 0 V. For each side of the spectrum, the first discrete is at 250 kHz and the spacing is 500 kHz.

How do SNR and Eb/N0 influence a digital receiver performance?

Figure 8-1 shows a NRZ bipolar baseband waveform with the 1 and 0 states of +1 V and -1 V respectively with superimposed additive white Gaussian noise (AWGN).

Figure 8-1 A detector input waveform for NRZ bipolar with AWGN superposed, shown in red. For symmetric signaling with a threshold of 0 V, errors may be potentially detected at E1 and E2. The error frequency is exagerated for clarity.

AWGN, by definition, has a Gaussian distribution of the noise voltage amplitude over time. This will be added to the digital voltage state as shown in Figure 8-1. Therefore, there is a finite probability that the instantaneous magnitude of the noise voltage will sometimes be quite high. It might even exceed the threshold for the unintended bit and cause an error should the noisy waveform be sampled at that point.

In order to calculate the probability of errors occuring on a baseband NRZ bipolar waveform which is subject to AWGN, Figure 8-2 shows how the Gaussian probability density functions (PDFs) for each of the states may be used.

Figure 8-2 The additive property of AWGN applied to Figure 8-1 results in two overlapping Gaussian PDFs with means of -1 V and +1 V.

This is a classic application of conditional probability summarised in Bayes' Theorem. In this case we assume that the probability of receiving a 1 is equal to that for receiving a 0, or 0.5 in each case. Then the shaded area under the Gaussian PDF curve gives the probability of detecting a 1 if a 0 was sent. Both values of probability are included in Bayes' theorem to get the result.

Where is the current ADC/DAC technology for digital receivers?

Content planned and under construction.

Are not negative frequencies and times unnecessarily complicated?

Content planned and under construction.

What are IQ mixers and how are they used?

An 'IQ' mixer refers to an in-phase, in-quadrature mixer. IQ mixers are frequently used in digital receivers as demodulators and in digital transmitters as modulators. Circuit schematics for each of these are shown in Figure 11-1 and Figure 11-2 respectively.

Figure 11-1 The IQ mixer used as a demodulator. LO, the local oscillator is tuned to precisely the carrier frequency.
Figure 11-2 The IQ mixer used as a modulator. LO is the local oscillator providing the carrier. I and Q are the 'in-phase' and 'in-quadrature' analog baseband components respectively.

IQ mixers are analog components normally found before the ADC in a digital receiver or after the DAC in a digital transmitter. They work on the principle of splitting a continuous wave (CW) local oscillator into 2 quadrature (90° separated) components, I and Q which represent the real and imaginary axes respectively of an argand diagram. This may be achieved with a quadrature splitter appropriate for the LO frequency. A vector or phasor on the argand diagram may be used to describe the amplitude (length) and spatial phase (angle) of the carrier. By convention, in the argand diagram, the I and Q quadrature components are represented by the x and y axes respectively.

Each of the I and Q (quadrature) components is fed to one port of a double balanced mixer (DBM). The DBM is shown schematically in Figure 11-3.

Figure 11-3 A schematic of the double balanced mixer

This version of DBM shows the use of 3 transformers, each one providing a single ended port to interface external components to the inner balanced circuits. Ports 1 and 2 may be used for the LO and signal to be processed and port 3 provides the analog IF input for a modulator or output for a demodulator. At the higher microwave frequencies, wire wound transformers would not be appropriate but there are alternative circuit based solutions and distributed components which can provide similar functions.

The balanced circuit topology of the DBM means that the LO component applied at port 1 should theoretically be completely suppressed and none should leak out of port 2 or port 3. However, it is difficult to achieve perfect balance and a few balance imperfections will probably cause some LO leakage from these ports.

The IQ mixer IF ports (port 3 in each case) provide the I and Q baseband components from the demodulator or receive them for the modulator. The other IQ mixer component is a zero degrees combiner or splitter. This may well be a similar physical design to the quadrature combiner/splitter but of course without the π/2 phase changing component.

What is bandwidth compression using multilevel codes

Content under construction, estimated delivery July 2021

How do we transmit digitally modulated frequency spectra?

Content under construction, estimated delivery July 2021

Can we use similar designs for transmissions over cable or even optical fiber?

Content under construction, estimated delivery August 2021

What are baseband and modulated frequency spectra?

Content under construction, estimated delivery July 2021

Where does the clock come from?

Content under construction, estimated delivery August 2021

What do you mean by 'Nyquist zones', oversampling and undersampling?

Nyquist's sampling criterion requires that the sampling frequency (fs) must be at least twice the maximum frequency component that is present in the baseband spectrum (fa). To confirm this we often assume that there is an ideal or 'brickwall' low pass filter (LPF) before the analog input of the ADC as shown in Figure 17-1. This colloquial term refers to a theoretical filter which, for a LPF, has 100% transmission in-band and zero transmission out of band. The Nyquist frequency is defined as fs/2.

Figure 17-1 For baseband sampling it is assumed that the components of any frequencies above the top baseband frequency are filtered out to prevent aliasing.

In general, it is good practice to increase the sampling frequency to somewhat above twice fa which is known as over sampling. This will provide a margin to accommodate the finite LPF attenuation roll-off seen in real life to minimise aliasing and to simplify its design. Sometimes the sampling frequency may even be several times the Nyquist frequency. This may generate digital data which is not required but the data concerned may simply be ignored, by a process of decimation later in the DSP chain. The main tradeoff required for over sampling is that the ADC and DSP stages must be specified for the higher sampling frequency and necessary information rate. These requirements will add extra design and parts costs.

Until now we have been considering baseband sampling within the Nyquist criterion, fs≥2fa. A study of the sampling process itself and how this generates pulse amplitude modulation (PAM), also shows that it generates higher order Nyquist zones in the frequency domain (6) (8). These may be exploited to achieve sampling across bands of frequencies, every component of which, is actually greater than the sampling frequency itself. Not surprisingly, this process is called undersampling or bandpass sampling.

There are various sampling methods but, to a reasonable approximation, the sampling (voltage time) waveform 'amplitude modulates' the baseband (voltage time) waveform.

The theoretical sampling function in the time domain is based upon a regular train of impulse, or Dirac delta δ(t) functions, equally spaced in time by the reciprocal of the sampling frequency. These are used to take regularly spaced samples of the baseband voltage time waveform as shown in Figure 17-2 (13).

Figure 17-2 A baseband time domain waveform (A), ideal sampling waveform (B) and the resulting sampled waveform (C), which is the product of these.

In Figure 17-2, (A) shows a typical baseband voltage time waveform x(t) and (B) the impulse sampling waveform with equally spaced time impulses of interval Ts. The sampling frequency (fs) and the sampling period (Ts) have the reciprocal relationship given by (17-1)

T s = 1 f s
(17-1) The spacing of the sampling impulse functions (Ts) is the reciprocal of the sampling frequency (fs)

The sampling waveform considered in isolation, xδ(t), is the discrete sum of the impulse samples from -∞ to +∞, given by (17-2) (8):

x δ (t)= n= δ(tn T s )
(17-2) The equation that describes a regular sampling waveform in the time domain, based on the impulse function.

The infinite sampling range is mathematically necessary to comply with the conditions required for Fourier transforms which will be applied to get the voltage against frequency spectrum.

The function which describes the pulse amplitude modulation (PAM) in the time domain is the product of x(t) and xδ(t) (17-3).

x s (t)=x(t) x δ (t) = n= x(t)δ(tn T s ) = n= x(n T s )δ(tn T s )
(17-3) Pulse amplitude modulation in the time domain is the product of the baseband waveform x(t) and the sampling waveform xδ(t)

The equations shown in (17-3) also apply the sifting or sampling property of Fourier transforms. This is shown explicitly in (17-4).

x(t)δ(t t 0 )=x( t 0 )δ(t t 0 )
(17-4) The Fourier transform sifting or sampling property. The value of x(t) at each impulse location δ(t-t0) may be multiplied.

The sifting property means that the product x(t)xδ(t) may be applied at every impulse (time) location using the instantaneous value of the baseband voltage at that time. The result of this is the impulse functions in Figure 17-2 (C) which are 'shaped' or amplitude modulated by the envelope profile of the baseband waveform.

In order to arrive at the frequency spectra, we can use some standard Fourier transform (FT) properties, one of which is convolution (12). In the following equations, we use the convention that a function symbol uses lower case in the time domain and upper case in the frequency domain so, for example, X(f) is the FT of x(t). Figure 17-3 shows the spectra created by the respective time domain waveforms that were shown in Figure 17-2.

Figure 17-3 The frequency domain spectra generated by the respective time domain waveforms shown in Figure 17-2.

In Figure 17-3, (A) shows the baseband frequency spectrum and (B) shows the frequency components of the sampling waveform given by (17-5) which is another one of the FT properties (12).

X s (f)= 1 T s n= δ(fn f s )
(17-5) The FT of a sampling waveform in the time domain with impulse spacing of Ts creates a regular series of discrete frequency components in the frequency domain with spacing of 1/Ts.
X(f)*δ(fn f s )=X(fn f s )
(17-6) One of the properties of FTs is that multiplication in the time domain (17-3) is equivalent to convolution in the frequency domain (12).
X s (f)=X(f) X δ (f) =X(f)* 1 T s n= δ(fn f s ) = 1 T s n= X(fn f s )
(17-7) This is the result of convolution in the frequency domain (17-3) including the convolution of the unit impulse waveform.

(17-7) shows that the result of the PAM applied to the spectrum for the baseband waveform is to create repeated spectra in the frequency domain, with frequency spacing of nfs/2 where n is an integer. For the moment we will assume that the sampling frequency is precisely on the threshold at exactly twice the maximum baseband frequency. The resulting frequency spectrum is shown in Figure 17-4.

Figure 17-4 Assuming that the sampling waveform comprises an impulse (Dirac delta) waveform at exactly the Nyquist frequency, the baseband frequency spectrum (Z1) will be repeated in both upper and lower sideband forms by Z2, Z3, Z4, Z5 and so on.

The repeated frequency spectra start with the first Nyquist zone (Z1) between zero frequency and fs/2, the same frequency range that the baseband occupies. The higher order zones Z2, Z3 and so on are then numbered serially for increasing frequency as shown in Figure 17-4, irrespective of whether they may be upper or lower sidebands. In fact, we are not generally concerned whether the zone is upper or lower sideband because the DSP processor(s) which follow the ADC may simply be programmed for either using just a very slightly different algorithm.

To achieve baseband sampling, we know that we must filter out all but the first Nyquist zone (Z1) by using a brickwall LPF covering the Z1 frequency range as shown in Figure 17-1. The LPF would reject any signals or noise present above one half of the sampling frequency to prevent aliasing.

However, a potentially very useful property of the sampled spectra, or higher order Nyquist zones is that we can instead select a higher order (bandpass) zone, let us say for the sake of argument, Z4 in Figure 17-4, and convert this using an ADC instead of the baseband zone, Z1. Figure 17-5 shows the essential input configuration to the ADC required to achieve this.

Figure 17-5 The front end configuration required using a bandpass filter to perform undersampling or bandpass sampling in Nyquist zone Z4

If the frequency range of Z4 was 30 MHz to 40 MHz for example, fs/2=10 MHz and the sampling frequency for this brickwall filter case would be 20 MHz.

More generally, when designing undersampling architectures some manipulation of the frequencies and a few precautions are necessary:

An undersampling example

In keeping with some excellent tutorials, datasheets and other documentation available from Analog Devices, the following discussion is based on tutorial MT-002 (6).

Referring to the Nyquist frequency zone spectra in Figure 17-4, taking one of the zones of order n, covering a frequency range from fl to fu, these are related by (18-1).

f l =(n1) f s 2 f u =n f s 2
(18-1) The lower (fl) and upper (fu) bandpass sampling frequencies in terms of the Nyquist frequency (fs/2) and the Nyquist zone order (n).

By adding the two equations in (18-1), the result for n is given by (18-2).

n= f l + f u f s + 1 2
(18-2) The Nyquist zone order (n)

For example, if we require fl=33 MHz and fu=41 MHz, then (18-2) gives the result n=5.125, which is clearly not an integer. Removing the fractional part provides the integer value of 5 which is required. The 'rounded down' integer value is chosen because this will increase fs/2 to slightly above the Nyquist threshold value to avoid aliasing. Then (18-2) is re-arranged in terms of fs, and the values for fl, fu and n are substituted into (18-3) to give the result, fs=16.44 MHz.

f s = f l + f u n1/2 = 33+41 51/2 =16.44MHz
(18-3) Calculation of the sampling frequency from fl, fu and n.

The final step is to re-calculate fl and fu from (18-1) by substituting fs=16.44 MHz and n=5. The results are 32.88 MHz and 41.10  respectively. This gives margins of 120 kHz and 100 kHz on the lower and upper frequency edges of the required bandpass filter respectively.

References

  1. Pozar, David M.
  2. Microwave Engineering - Third Edition; John Wiley & Sons Inc. (2005); pp. 487 - 493. (ISBN 0-471-44878-8.

  3. Sklar, Bernard
  4. Digital Communications, Fundamentals and Applications, Second Edition; Prentice Hall PTR, Upper Saddle River, New Jersey 07458; pp. 525 - 529. (ISBN 0130847887).

  5. Pozar (op. cit.)
  6. pp. 191 - 192.

  7. Analog Devices 14-Bit, 80 MSPS/105 MSPS A/D Converter P/N AD6645
  8. Analog Devices Inc.

  9. Digital Over Cable Service Interface Specification (DOCSIS), Wikipedia
  10. Walt Kester; Analog Devices Tutorial MT-002
  11. What the Nyquist Criterion Means to Your Sampled Data System Design

    Analog Devices Inc.

  12. Sklar, Bernard (op. cit.)
  13. Why Eb/N0 is a Natural Figure of Merit; pp 118 - 119.

  14. Sklar, Bernard (op. cit.)
  15. The Sampling Theorem; pp 63 -75.

  16. Bob Stewart, Kenneth Barlee, Dale Atkinson, Louise Crockett
  17. Software Defined Radio using Matlab & Simulink and the RTL-SDR, 1st Edition (revised); Department of Electronic and Electrical Engineering; University of Strathclyde; pp 8 -12. (ISBN 978-0-992-978-723)

  18. Sklar, Bernard (op. cit.)
  19. The 'sifting' or sampling property of Fourier transforms; Section A.4.1, p 1023.

  20. Sklar, Bernard (op. cit.)
  21. The frequency convolution property of Fourier transforms; Section A.5.3, p 1023.

  22. Sklar, Bernard (op. cit.)
  23. Fourier transform properties: convolution; pp. 1025 - 1034.

  24. Sklar, Bernard (op. cit.)
  25. The unit impulse function; p 16.

  26. Sklar, Bernard (op. cit.)
  27. PCM waveform types; pp. 85 - 89.