Figure 10.1 shows a generic transmitter and receiver pair that emphasizes the modulation and corresponding demodulation. Even assuming that the transmission path is ideal (as in Figure 10.1), the signal that arrives at the receiver is a complicated analog waveform that must be downconverted and sampled before the message can be recovered. For the demodulation to be successful, the receiver must be able to figure out both the frequency and the phase of the modulating sinusoid used in the transmitter, as was shown in (5.4) and (5.5) and graphically illustrated in Figures 9.18 and 9.19. This chapter discusses a variety of strategies that can be used to estimate the phase and frequency of the carrier and to fix the gain problem (of (5.4) and Figure 9.18) and the problem of vanishing amplitudes (in (5.5) and Figure 9.19). This process of estimating the frequency and phase of the carrier is called carrier recovery.

Figure 10.1 shows two downconversion steps: one analog and one digital. In a purely analog system, no sampler or digital downconversion would be needed. The problem is that accurate analog downconversion requires highly precise analog components, which can be expensive. In a purely digital receiver, the sampler would directly digitize the received signal, and no analog downconversion would be required. The problem is that sampling this fast can be prohibitively expensive. The happy compromise is to use an inexpensive analog downconverter to translate to some lower intermediate frequency, at which it is possible to sample cheaply enough.

As foreshadowed in Section 2.8, transmission systems cannot be fully digital because the medium through which the signal propagates is analog. Hence, whether the signal begins as analog (such as voice or music) or whether it begins as digital (such as mpeg, jpeg or wav files), it will be converted into a highfrequency analog signal when it is transmitted. In a digital receiver, the received signal must be transformed into a discrete-time signal in order to allow subsequent digital processing.

This chapter begins by considering the sampling process both in the time domain and in the frequency domain. Then Section 6.3 discusses how Matlab can be used to simulate the sampling process. This is not completely obvious because analog signals cannot be represented exactly in the computer. Two simple tricks are suggested. The first expresses the analog signal in functional form and takes samples at the desired times. The second oversamples the analog signal so that it is represented at a high data rate; the “sampling” can then be done on the oversampled signal.

Sampling and quantization are important because they translate the signal from analog to digital. It is equally important to be able to translate from digital back into analog, and the celebrated Nyquist sampling theorem shows that this is possible for any bandlimited signal, assuming the sampling rate is fast enough. When the goal of this translation is to rebuild a copy of the transmitted signal, this is called reconstruction.

Any message, whether analog or digital, can be translated into a string of binary digits. In order to transmit or store these digits, they are often clustered or encoded into a more convenient representation whose elements are the symbols of an alphabet. In order to utilize bandwidth efficiently, these symbols are then translated (again!) into short analog waveforms called pulse shapes that are combined to form the actual transmitted signal.

The receiver must undo each of these translations. First, it examines the received analog waveform and decodes the symbols. Then it translates the symbols back into binary digits, from which the original message can, with luck, be reconstructed.

This chapter briefly examines each of these translations, and the tools needed to make the receiver work. One of the key ideas is correlation, which can be used as a kind of pattern-matching tool for discovering key locations within the signal stream. Section 8.3 shows how correlation can be viewed as a kind of linear filter, and hence its properties can be readily understood both in the time and in the frequency domain.

Bits to Symbols

The information that is to be transmitted by a communication system comes in many forms: a pressure wave in the air, a flow of electrons in a wire, a digitized image or sound file, the text in a book. If the information is in analog form, then it can be sampled (as in Chapter 6).

This chapter describes a software-defined-radio design project called M6, the Mix ‘n’ Match Mostly Marvelous Message Machine. The M6 transmission standard is specified so that the receiver can be designed using the building blocks of the preceding chapters. The DSP portion of the M6 can be simulated in Matlab by combining the functions and subroutines from the examples and exercises of the previous chapters.

The input to the digital portion of the M6 receiver is a sampled signal at intermediate frequency (IF) that contains several simultaneous messages, each transmitted in its own frequency band. The original message is text that has been converted into symbols drawn from a 4-PAM constellation, and the pulse shape is a square-root raised cosine. The sample frequency can be less than twice the highest frequency in the analog IF signal, but it must be sufficiently greater than the inverse of the transmitted symbol period to be twice the bandwidth of the baseband signal. The successful M6 Matlab program will demodulate, synchronize, equalize, and decode the signal, so it is a “fully operational” software defined receiver (although it is not intended to work in “real time”). The receiver must overcome multiple impairments. There may be phase noise in the transmitter oscillator. There may be an offset between the frequency of the oscillator in the transmitter and the frequency of the oscillator in the receiver. The pulse clocks in the transmitter and receiver may differ.

There is nothing new in this chapter. Really. By peeling away the outer, most accessible layers of the communication system, the previous chapters have provided all of the pieces needed to build an idealized digital communication system, and this chapter just shows how to combine the pieces into a functioning system. Then we get to play with the system a bit, asking a series of “what if” questions.

In outline, the idealized system consists of two parts, rather than three, since the channel is assumed to be noiseless and disturbance-free.

The Transmitter

• codes a message (in the form of a character string) into a sequence of symbols,

• transforms the symbol sequence into an analog signal using a pulse shape, and

• modulates the scaled pulses up to the passband.

The Digital Receiver

• samples the received signal,

• demodulates to baseband,

• filters the signal to remove unwanted portions of the spectrum,

• correlates with the pulse shape to help emphasize the “peaks” of the pulse train,

• downsamples to the symbol rate, and

• decodes the symbols back into the character string.

Each of these procedures is familiar from earlier chapters, and you may have already written Matlab code to perform them. It is time to combine the elements into a full simulation of a transmitter and receiver pair that can function successfully in an ideal setting.

Software Receiver Design helps the reader build a complete digital radio that includes each part of a typical digital communication system. Chapter by chapter, the reader creates a Matlab® realization of the various pieces of the system, exploring the key ideas along the way. In the final chapters, the reader “puts it all together” to build fully functional receivers, though as Matlab code they are not intended to operate in real time. Software Receiver Design explores telecommunication systems from a very particular point of view: the construction of a workable receiver. This viewpoint provides a sense of continuity to the study of communication systems.

The three basic tasks in the creation of a working digital radio are

1. 1. building the pieces,

2. 2. assessing the performance of the pieces,

3. 3. integrating the pieces.

In order to accomplish this in a single semester, we have had to strip away some topics that are commonly covered in an introductory course and emphasize some topics that are often covered only superficially. We have chosen not to present an encyclopedic catalog of every method that can be used to implement each function of the receiver. For example, we focus on frequency division multiplexing rather than time or code division methods, and we concentrate only on pulse amplitude modulation and quadrature amplitude modulation.

Once the received signal is sampled, the real story of the digital receiver begins.

An analog bandpass filter at the front end of the receiver removes extraneous signals (for instance, it removes television-frequency signals from a radio receiver) but some portion of the signal from other FDM users may remain. While it would be conceptually possible to remove all but the desired user at the start, accurate retunable analog filters are complicated and expensive to implement. Digital filters, on the other hand, are easy to design, inexpensive (once the appropriate DSP hardware is present), and easy to retune. The job of cleaning up out-of-band interferences left over by the analog BPF can be left to the digital portion of the receiver.

Of course, there are many other uses for digital filters in the receiver, and this chapter focuses on how to “build” digital filters. The discussion begins by considering the digital impulse response and the related notion of discrete-time convolution. Conceptually, this closely parallels the discussion of linear systems in Chapter 4. The meaning of the DFT (discrete Fourier transform) closely parallels the meaning of the Fourier transform, and several examples encourage fluency in the spectral analysis of discrete data signals. The final section on practical filtering shows how to design digital filters with (more or less) any desired frequency response by using special Matlab commands.

Discrete Time and Discrete Frequency

The study of discrete-time (digital) signals and systems parallels that of continuous-time (analog) signals and systems.

Several parts of a communication system modulate the signal and change the underlying frequency band in which the signal lies. These frequency changes must be reversible; after processing, the receiver must be able to reconstruct (a close approximation to) the transmitted signal.

The input message w(kT) in Figure 5.1 is a discrete-time sequence drawn from a finite alphabet. The ultimate output m(kT) produced by the decision device (or quantizer) is also discrete-time and is drawn from the same alphabet. If all goes well and the message is transmitted, received, and decoded successfully, then the output should be the same as the input, although there may be some delay δ between the time of transmission and the time when the output is available. Though the system is digital in terms of the message communicated and the performance assessment, the middle of the system is inherently analog from the (pulse-shaping) filter of the transmitter to the sampler at the receiver.

At the transmitter in Figure 5.1, the digital message has already been turned into an analog signal by the pulse shaping (which was discussed briefly in Section 2.10 and is considered in detail in Chapter 11). For efficient transmission, the analog version of the message must be shifted in frequency, and this process of changing frequencies is called modulation or upconversion. At the receiver, the frequency must be shifted back down, and this is called demodulation or downconversion.

When all is well in the receiver, there is no interaction between successive symbols; each symbol arrives and is decoded independently of all others. But when symbols interact, when the waveform of one symbol corrupts the value of a nearby symbol, then the received signal becomes distorted. It is difficult to decipher the message from such a received signal. This impairment is called “intersymbol interference” and was discussed in Chapter 11 in terms of non-Nyquist pulse shapes overlapping in time. This chapter considers another source of interference between symbols that is caused by multipath reflections (or frequency-selective dispersion) in the channel.

When there is no intersymbol interference (from a multipath channel, from imperfect pulse shaping, or from imperfect timing), the impulse response of the system from the source to the recovered message has a single nonzero term. The amplitude of this single “spike” depends on the transmission losses, and the delay is determined by the transmission time. When there is intersymbol interference caused by a multipath channel, this single spike is “scattered,” duplicated once for each path in the channel. The number of nonzero terms in the impulse response increases. The channel can be modeled as a finite-impulse-response, linear filter C, and the delay spread is the total time interval during which reflections with significant energy arrive. The idea of the equalizer is to build (another) filter in the receiver that counteracts the effect of the channel. In essence, the equalizer must “unscatter” the impulse response.

The fourth step describes all the practical fixes that are needed in order to create a workable radio. One by one the various pragmatic problems are studied and solutions are proposed, implemented, and tested. These include fixes for additive noise, for timing offset problems, for clock frequency mismatches and jitter, and for multipath reflections. The order in which topics are discussed is the order in which they appear in the receiver.

The envelope of a signal is a curve that smoothly encloses the signal, as shown in Figure C.1. An envelope detector is a circuit (or computer program) that outputs the envelope when the signal is applied at its input.

In early analog radios, envelope detectors were used to help recover the message from the modulated carrier, as discussed in Section 5.1. One simple design includes a diode, capacitor, and resistor arranged as in Figure C.2. The oscillating signal arrives from an antenna. When the voltage is positive, current passes through the diode, and charges the capacitor. When the voltage is negative, the diode blocks the current, and the capacitor discharges through the resistor. The time constants are chosen so that the charging of the capacitor is quick (so that the output follows the upward motion of the signal), but the discharging is relatively slow (so that the output decays slowly from its peak value). Typical output of such a circuit is shown by the jagged line in Figure C.1, a reasonable approximation to the actual envelope.

It is easy to approximate the action of an envelope detector. The essence of the method is to apply a static nonlinearity (analogous to the diode in the circuit) followed by a lowpass filter (the capacitor and resistor).

The preceding chapters of Software Receiver Design focus on a real-valued PAM transmission protocol. Though this is one of the simplest digital transmission systems, a working radio for PAM needs to include a variety of techniques that address the nonideal behaviors encountered in reality. Fixes such as the PLL, the Costas loop, timing recovery, and equalization are necessary parts of the system. This chapter describes a complex-valued generalization of PAM called quadrature amplitude modulation (QAM), which forms the basis of a “better radio.” While the change from real-valued PAM to complex-valued QAM signals is straightforward, the bulk of this chapter fleshes out the details of operation of the resulting system. The various parts of the digital radio need to be rethought (carrier recovery, timing recovery, and equalization). The theme of this chapter is that such rethinking can follow the same path as the original designs. The basic techniques for the PAM radio of earlier chapters generalize smoothly to more sophisticated designs.

This chapter begins with the observation (already noted in Section 5.3) that PAM modulations waste bandwidth. QAM, a more clever modulation strategy, is able to halve the bandwidth requirements, which represents a significant step towards practicality. This is accomplished by sending, and then unravelling, two real PAM signals simultaneously on orthogonal carriers (sine and cosine) that occupy the same frequency band.