The fundamental goal of symbol synchronization is to sample the pulse at its peak value.
This peak can be determined by estimating the derivative of the sampled signal. The first derivative is calculated as demonstrated—taking the difference of the early and late mea-surements that encompasses the estimated location of the peak. In Figure 3.63, the early-late gate derivative measurement y1 (n - k/N) for the fcth polyphase output at time n can be computed from the outputs of fc — 1 and fc + 1 filter stages [43,44].
Section 3.5 Timing Recovery in Digital Receivers Using Multirate Digital Filters 121
Figure 3.62: The Matched Filtering Process Assuming a Positive Rectangular-Shaped Pulse.
The decision statistic here is the product of the kth polyphase stage and its derivative.
This product is necessary to invert the derivative when the output is negative. Thus, the decision statistic will be independent of the desired symbol's value. Driving this product to zero by means of a feedback loop leads to maximum likelihood timing recovery. Notice that the product goes to zero if either the derivative is zero (at the peak) or the kth stage output (signal) is zero.
The process of forming the derivative can be implemented by means of a polyphase derivative matched filter with weights defined as the difference in weights of the k — 1 and k + 1 stages shown in Figure 3.63. That is, the derivative can be implemented as a polyphase filter and thus only one filter actually needs be computed to supply information to the feedback loop. Such a filter in conjunction with a polyphase-matched filter can im-plement the system of Figure 3.63. A generic DSP-based receiver that uses the polyphase filter approach for timing recovery is shown in Figure 3.64. Note that there is no feed-back between the digital and analog domains. Furthermore, this approach is well-suited for multiple channel systems, such as FDMA systems, since each channel can have its own interpolation filter. In this case, the polyphase filters can also aid in the receiver channel-ization as well as in timing recovery.
This approach to synchronization has the maximum flexibility. The analog section becomes more straightforward since the analog processing can be generic, and matched filters for specific pulse-shaping and timing considerations are handled within the more flexible digital domain. Note that the resampling process translates the signal by multiples of the data rate. This reduces the carrier offset and subsequently reduces the complexity of the digital signal processing based carrier recovery technique since the residual frequency offset is lower after resampling.
y(n)
Polyphase Filter k - 2 Stage
/c-1 Stage
k Stage
/c+1 Stage
/(n-k/M) y'(n-k/M)y(n-k/M)
" > ( + J H, * r — •
y(n-k/M)
k + 2 Stage
Figure 3.63: Forming the Decision Statistic with a Polyphase Filter Bank.
IF Stage
Mixer -sin(o/ / rf
-90° Phase Shift
c o s c o/ F t
Mixer
Analog LPF
Analog LPF
• ADC
2 FC
2 FC
* ADC
Polyphase Matched
Filter
Polyphase Matched
Filter
Carrier &
Timing Recovery Matched
Filter Detection
Demodulated Data
Figure 3.64: A Multirate Filtering Approach to Synchronization.
124 Multirate Signal Processing Chapter 3
3.6 Conclusion
The advantages of multirate signal processing are manyfold and its application is common in many systems, such as digital filtering, spectral analysis, speech coding, and analog to digital conversion. Multirate techniques are essential in applications of high-speed data acquisition and storage. They reduce the need for expensive anti-aliasing analog filters and enable processing of different types of signals with different sampling rates. The cost of high-speed components goes up exponentially with sample rate or clock rate. Multirate sig-naling allows partitioning of the high-speed processing into parallel multiple lower-speed processing tasks, thus allowing implementation with low-speed components at a reduced cost. In the design of narrowband digital FIR filters, multirate techniques are especially appealing. They facilitate the implementation of the filter at a lower sampling rate, thereby greatly reducing the filter order and in turn the computational complexity. Wideband re-ceivers take advantage of multirate signal processing for efficient channelization, i.e., par-titioning a broadband signal into smaller individual radio channels. Since the channelizer is the most computationally intensive part of the software radio, judicious design using the multirate concept can lead to a significant saving in computational power. Finally, multi-rate filtering techniques offer flexibility for symbol synchronization and downconversion for software radios. Multirate approaches to timing recovery lead to straight-forward im-plementations.
Section 3.7 Questions 1 2 5
3.7 Questions
1. What is the benefit of using the multi-stage (rather than single-stage) structures of a decimator or interpolator when large changes of sampling rate are required?
2. Suppose we are building a cellular basestation and we have a cellular signal of width 25 MHz, sampled at a rate of 50 Msamples/sec. We wish to channelize this signal into channels of width 100 kHz. We want an adjacent channel rejection of 60 dB.
This should be implemented using the polyphase filter bank approach using a Kaiser window filter.
(a) Show the prototype filter (sub-sampled filter). Show a block diagram of the designed filter and a plot of the filter response.
(b) Show a frequency response plot of the filter bank with a white noise input to prove the adjacent channel interference rejection characteristics.
3. (a) Implement in MATLAB a CIC filter with four stages of integrate and four stages of comb, to reduce the sampling rate by a factor of four. Filter a sinu-soidal signal with a frequency of 0.2 Hz (assume a normalized sample rate of 1 Hz) to illustrate the effect of aliasing.
(b) Repeat this exercise for six stages of integrate and six stages of comb.
(c) Compare the frequency response of the filter from MATLAB with theoreti-cal results for frequency response. Does the aliasing and attenuation observed match predicted results?
4. Prove that decimating a signal by a factor of D frequency shifts the signals by mul-tiples of D/Fs.
5. Prove that insertion of I — 1 zeros between samples creates replicated images at multiples of Fs/I.
6. Design a two-stage LPF and decimator that will take a signal sampled at 96 MHz and convert it to a sample rate of 1 MHz. The passband of the signal after decimation should be 450 kHz with a passband ripple of 0.01 and a stopband ripple of 0.001.
Repeat for a single-stage decimator. Determine the filter order(s) and show the fre-quency response for each. What is the memory requirement and computational rate required for each implementation?
7. Redo Question 6 using three stages of decimation.
8. An interpolator is to be created for implementing an FDM system. Each channel of the F D M system must be interpolated to a sample rate to accommodate the composite signal. Assume that each channel is 20 MHz wide and the sample rate is 44 MHz.
There are eight channels in the composite signal sampled at a rate of 8 x 44 MHz.
Assume that at 22 MHz the stopband starts and the attenuation must be 50 dB. The passband ripple is 0.5 dB. Show the overall frequency response, computation rate, and memory of a single-stage implementation and a two-stage implementation.
126 Multirate Signal Processing Chapter 3
9. In the example represented by Figure 3.29, redesign the filter assuming that dec-imation of two is done before decdec-imation by three. Compare the computational requirements for each implementation and demonstrate the frequency response for each.
10. Show the frequency response of the filter given by Equation 3.46 for iV = 1, 2, . . . 5 . Comment on the trend in the frequency response as N increases.
11. Show the frequency response of the filter given by Equation 3.47 for N = 1,2, ...5 and M = 1, 2, ..5 and comment on the trend in the frequency response as N and M increase.
12. Show mathematically the difference in distortion between a reconstructed signal that uses a ZOH and one that doesn't, and comment on the significance of this distortion.
13. Consider a multiband system where the desired system must be able to handle GSM, CDMA, and UMTS (Universal Mobile Telecommunication System), although not all at the same time. We desire a constant sample rate for the ADC, which provides exact multiples of the highest chip rate at 16x, 32x, etc. We want to use a digital downconverter with a polyphase resampler to obtain exact multiples of the GSM symbol rate, 270.833 kHz. The polyphase resampler will be bypassed for the CDMA signals where the wider bandwidths are required. What are the required decimation rates for UMTS, CDMA, and GSM? Note that the chip rate for CDMA is 1.2288 Mc/s and for UMTS (3GPP) is 3.84 Mc/s.
14. Determine the transfer function of the CIC interpolator filter structure shown in Fig-ure 3.39 and plot its frequency response.
C h a p t e r 4
D I G I T A L G E N E R A T I O N O F S I G N A L S
S r i k a t h y a y a n i S r i k a n t e s w a r a a n d J e f f r e y H . R e e d
4 . 1 Introduction
Software radios are built using reconfigurable hardware in which the parameters of the sys-tem can be adjusted in software during runtime. This adaptability places a huge demand on the hardware, which must be more versatile than that of traditional communication sys-tems to support the different functionalities. Many of the traditional analog techniques of radio design have thus far proven to be inefficient and unsuitable for software radios, which have taken recourse to digital hardware. The development of increasingly fast and efficient digital ICs and better digital design techniques have made software radios a reality. It is important to understand the components of the underlying hardware, which actually de-fines the software radio system and its capabilities. The synthesis of waveforms, especially sinusoidal signals, is an important part of a communication system. Sinusoidal signals are typically used in many of the processing steps, such as modulation, pulse-shaping, and fil-tering. In this chapter, we shall discuss the increasing role of digital techniques in signal generation and their role in a software radio system.
Analog techniques have long dominated frequency synthesis. Analog frequency tech-niques are based on bulky analog devices such as quartz crystals, inductors, capacitors, and mechanical resonators. Digital techniques began to gain prominence in communica-tion systems because of their superior accuracy and immunity to noise and because they are easy to manufacture with very large scale integration (VLSI). Direct digital synthesis (DDS) techniques generate signals directly in discrete time. Any arbitrary waveform can be generated for digital communication systems, as the amplitude, frequency, and phase can be varied to create a modulated signal.
127
1 2 8 Digital Generation of Signals Chapter 4
DDS designs date back to the early 1970s when techniques were developed for gen-erating audio signals [45]. These early designs generated a sine wave using a read-only memory (ROM) containing sampled values of a sine wave to drive a DAC and analog in-terpolation filter. Modern DDS techniques are derivatives of this basic approach, but many improvements have been incorporated, enabling DDS to be used in practical communica-tion systems.
DDS concepts gained acceptance in the early 1980s, and their practical implementa-tion for communicaimplementa-tion systems evolved in the 1990s. By the 1980s, the highest output frequency range attainable was just a few MHz, due to the logic ICs and DACs then avail-able [46]. A prototype DDS system at the Roke Manor laboratories in 1981 occupied several boards of logic clocked at 10 MHz provided an output frequency of up to 3 MHz, with all spurious responses about 40 dB down. Since then, with advances in semiconductor technology, the attainable performance has increased exponentially. Current systems need very spectrally pure signals, e.g., in IS-95, the receiver IF PLL is required to meet a phase noise of 115 dBc/Hz at 100 kHz offset. This phase noise requirement is due to the jammer specifications. DDS chips realized in gallium arsenide now provide output frequencies of several hundreds of megahertz. DDS chips are now readily available and are commonly used in communication systems.
DDS has become the preferred implementation for signal synthesis. The main ad-vantages of using digital techniques in frequency synthesis are precision, fast switching capability, and generation of arbitrary wave shapes. Unlike bulky analog systems, DDS systems are small and low-powered, making them attractive for use in mobile applications.
With fewer components per instrument, component costs are easier to control, fewer as-sembly operations are required, and the product reject rates are reduced because of fewer opportunities for manufacturing errors. These advantages are detailed in the next section.
Direct digital synthesizers allow the implementation of digital modulation techniques, after which the signals can be converted to analog signals for transmission. A M can be created by multiplying the sinusoidal output of the ROM with the modulating signal before passing it through the DAC. PM is created by changing the instantaneous phase angle, i.e., by using the modulating signal to alter the input to the ROM (phase). Frequency modulation (FM) is created by varying the instantaneous frequency, and this is accomplished by using the modulating signal to increment the phase.
In this chapter, the basic principles of DDS are explained as are the various approaches used to implement direct digital synthesizers. The applicability of DDS to various commu-nication systems is also explained. The problems associated with DDS techniques, the most important being the spectral purity of the waveform, and the techniques that have been used to minimize these problems will be discussed. The Qualcomm Direct Digital Synthesizer Chip Q2240, which can be used for various wireless applications, w i l l be used as a case study. Pseudo noise (PN) generation will be described; it is used for both direct sequence and frequency hopping spread spectrum communication systems and for data scrambling or randomization.
Section 4.2 Comparison of Direct Digital Synthesis with Analog Signal Synthesis 129