Decimation Stages with WDFs for Flexible Digital Receivers

Dieter Brckmann -January 09, 2002

Easy adaptation to several mobile communications standards is a design goal for wireless communications transceivers, where end users require low cost, low power, and small device size. We can achieve multi-standard capability by moving channel selection into the digital domain, where it can be implemented by programmable digital filters. We need a wideband A/D-converter in such architectures so that the desired channel is digitized together with the unwanted adjacent channel interferers.

Delta-sigma modulators with oversampling are especially well suited for this application because the adjacent channel interferers are filtered out by the decimation stages along with the high-pass shaped quantization noise. You don't need additional filtering for channel selection and the anti-aliasing low-pass filter requirements are relaxed due to oversampling.

For optimal performance, the decimation stages of such a receiver must meet certain requirements, which should be taken into account during the filter design process. You should especially minimize the group delay of the filters to allow the implementation of a fast AGC in the receiver. Furthermore, no significant group-delay distortion may be introduced in the pass band. To account for the very stringent cost and low-power requirements in a mobile terminal, you must optimize the decimation filters for a cost effective FPGA- or ASIC-implementation. Avoid a general-purpose DSP for multiplications and MAC-operations in favor of a low-gate-count implementation. Finally, to account for the required flexibility, the decimation filters should be adjustable with respect to the decimation ratio.

There are several filter structures available for the decimation stages of delta-sigma-modulators. This article will show that for the lower decimation stages, properly designed cascade connections of low-order lattice wave digital filters (WDFs) are especially well suited for this application—in particular, with respect to the requirements of a flexible digital receiver. Lattice WDFs consist of a parallel connection of all pass sections. These filters have several advantageous properties, such as low-coefficient sensitivity in the pass band, low round-off noise, and an absence of parasitic oscillations. Furthermore, they are minimum-phase recursive filters.

A disadvantage, however, is the high sensitivity in the stop band with respect to the coefficients. For high stop-band attenuation, you need many bits to represent the coefficient. In order to avoid this problem, the decimation stages proposed in this article are composed of a cascade connection of low-order-wave digital sub-filters. You can significantly reduce the number of data bits and the number of bits required for the coefficient representation with this method.

Decimation Filtering in a Digital Receiver
Figure 1 shows the block diagram of the receiver architecture with simple analog preprocessing and flexible digital signal processing. The structure is a mixed-signal design with broadband analog pre-processing and digital signal processing behind the A/D-converters. Whereas the analog receiver front-end is relatively simple and mainly serves for low-noise amplification and frequency translation to an intermediate frequency or baseband, the more complex signal processing is done digitally, for example, channel filtering, demodulation, and detection. The critical parts of this receiver are the broadband A/D-converters, which must be optimized with respect to cost and power consumption. Delta-sigma modulators with oversampling are especially well suited for this application.

Figure 1:  Receiver architecture with simple analog preprocessing and flexible digital signal processing

A simple blocking filter processes the RF-signal received at the antenna. The low-noise amplifier (LNA) then pre-amplifies the signal. The quadrature downconverter in the analog domain performs the downconversion to a low IF or baseband.

The digital part consists of two hardware blocks. The upper digital signal processing module, which operates at higher frequencies, is optimized with respect to chip area and power consumption, but nevertheless offers a certain degree of flexibility by using reconfigurable hardware. The module's main functions are channel selection and decimation. These functions require only a reduced flexibility even for multi-standard capability, appropriate for an ASIC/FPGA implementation.

The processor part is used for baseband signal processing and is highly flexible and easy to program. This module can be used for detection, synchronization, equalization, and speech and video processing.

Figure 2 shows the architecture of the digital part of the receive path. Delta-sigma modulators digitize the I and Q components of the broadband receive signal after quadrature downconversion in the analog domain. The channel of interest is thus centered around DC. The decimation filters then perform channel-selection, together with the suppression of the high-pass shaped quantization noise.

Figure 2:  Receive path with decimation filters

You can minimize the required hardware for the decimation filters by performing decimation in several stages, with sample rate reduction after each stage. After quantization by the delta-sigma-modulators, the signal is digitally filtered by real low-pass filters. The first filter stage performs decimation by a programmable factor M. The following three stages each reduce the sampling rate by a factor of two. You can also bypass each decimation stage so that the decimation ratio is programmable over a large range. Since the decimation stages are designed such that the lowest stage fulfills the most stringent attenuation requirements, you should bypass one or two lower stages such that Stage 2 is bypassed before Stage 3.

Because the uppermost decimation stage is running at the highest clock frequency, you should implement this stage with rather simple filter structures to optimize cost and power consumption. By using cascaded integrator-comb (CIC) filters you can obtain simple hardware structures, which you can implement using only registers and adders. You can avoid the potential drawback of this filter type, severe pass-band droop, which is dependant on the decimation ratio, by using a modified version of these filters. The so-called sharpened CIC-filter (SCIC) requires only slightly more hardware, but shows considerably improved performance in the pass band. The stop-band attenuation of the decimation filters is designed to fulfill the attenuation requirements of a fourth-order delta-sigma-modulator that requires, in the first stage, a classical fifth-order CIC filter or a comparable SCIC-filter.

Filters with a steeper transition band are required for the lower decimation stages. You can obtain a hardware-efficient filter realization with half-band filters, having a symmetrical filter characteristic with respect to Fs/4 (Fs being the sampling frequency). In this case, 50% of the filter coefficients are zero and must not be implemented. Furthermore, when properly designed, these filters can be clocked with the decimated sampling frequency.

Cascaded Low-Order Lattice Wave Digital Filters
For the lower decimation stages we propose cascade connections of low-order bi-reciprocal lattice WDFs. These filters are a special case of lattice WDFs and have a symmetrical filter characteristic with respect to Fs/4. You can express the transfer function as

which you can exploit to obtain efficient decimators for sampling rate conversion by two. The number of filter coefficients you must implement is reduced by 50% and you can clock the filter with the lower sampling frequency.

The transfer function of N cascaded low-order bi-reciprocal lattice wave digital filters is given by

where Hi0(z) and Hi1(z) are all pass filters. The respective structure is shown in Figure 3. For approximately linear phase, you should design one of the all-pass sections in each subfilter as a pure delay.

This decomposition in a cascade connection of low-order wave digital filter sections results in a number of advantages, compared to existing solutions. Bi-reciprocal WDFs are minimum-phase filters that introduce group-delay distortion. You can obtain a lattice WDF with approximately linear phase in the pass band if one of the all-pass sub-filters is a pure delay. In the pass band of the filter, the responses of the all-pass sub-filters must be approximately equal. Since one of the branches is a pure delay, the phase response of the overall filter has approximately a linear phase in the pass band. This will be taken into account by cascading third-order cells of bi-reciprocal lattice wave digital filters, as shown in Figure 3, resulting in a superior group-delay performance compared to a direct realization of a lattice WDF.

Figure 3:  Cascaded low-order wave digital filters

You can get low coefficient sensitivity by cascading low-order sections with high stop-band attenuation. Thus, you can represent the optimized coefficients with very simple values and a word length of only a few bits is required. Due to the very simple coefficient representation, no general multiplier is needed, thus minimizing implementation cost. Furthermore, the shorter word length also holds for the signal representation, resulting in reduced implementation cost for the adders and registers. In addition, the cascaded low-order sections are very modular, making it very attractive for VLSI-implementation.

For the application considered, we designed the decimation stages for a minimum stop-band attenuation of about 95dB. The strongest requirements hold for the last stage, which can be clocked with the lowest clock frequency. We designed this fourth filter stage for a normalized stop-band edge frequency of fc/fs=0.355. For the third decimation stage, you can relax this requirement; however, the discrete coefficient optimization came up with the same filter as for the fourth stage.

Figure 4 shows the architecture of the lattice wave digital filter, designed for use in the third and fourth decimation stages. The decimation stage consists of a cascade of three third-order bi-reciprocal WDF filter blocks, resulting in a total filter order of nine.

Figure 4:  Decimation stage consisting of three cascaded third-order bi-reciprocal WDF-filters

We designed the second decimation stage for a minimum stop-band attenuation of 80dB from 0.4 to 0.45 and of 95dB from 0.45 to 0.5. You can also implement the filter with the structure in Figure 4, where only two cascaded third-order cells are needed, which results in an overall filter order of six.

The optimized values for the three lower decimation stages are listed in Table 1. You can represent nearly all coefficients with only one shift-and-add operation—only one coefficient in Stage 2 requires two shift-and-adds.

Stage 2
Stage 3
Stage 4
g1 = 2-1 - 2-3
g1 = 2-1 - 2-4
g1 = 2-1 - 2-4
g2 = 2-1 - 2-3 - 2-5
g2 = 2-1 - 2-4
g2 = 2-1 - 2-4
g3 = 2-1 - 2-3
g3 = 2-1 - 2-3

Table 1: Optimized finite-precision adaptor coefficients for the cascaded lattice WDFs

By implementing the decimator by 2 with the last filter cell of each decimation stage, you can realize Stage 2 with four registers and 11 shift-and-add operations. You need a word length of only 6 bits for the coefficient representation.

You can realize Stages 3 and 4 with seven registers, 12 adders, and three shift-and-add operations for each stage. For the respective coefficients, you need a word length of only 5 bits. You can also fulfill the attenuation requirements of the fourth decimation stage with a classical ninth-order bi-reciprocal WDF, requiring four registers, 13 adders, and 12 shift-and-add operations. The word length you need for the coefficient representation is, however, 12 bits. Furthermore, you could also fulfill the attenuation requirements with an FIR-filter of order 23. Compared to classical solutions, you obtain a considerable hardware reduction using the cascaded low-order sections.

Performance Results
Figure 5 shows the frequency responses of the lower decimation stages. The second filter stage is implemented as two cascaded bi-reciprocal lattice WDFs of order three each.

Figure 5:  Frequency response of the second decimation stage (blue line) and of the third and fourth decimation decimation stages (black line)

The third and fourth filter stages consist of three cascaded third-order WDFs. Figure 6 shows the overall frequency response of the three lower decimation stages. A minimum attenuation of 96dB is obtained. When cascaded with the first decimation stage, the minimum attenuation is even higher.

Figure 6:  Overall frequency response of the three lower decimation stages implemented with cascaded low order WDFs

Figure 7 shows the excellent pass-band behavior of the SCIC-filter compared to that of a classical CIC-filter. The severe pass-band droop of the CIC-filter is avoided. The decimation factor of the SCIC-filter is programmed to a value of M=4.

Figure 7:  Pass band behavior of the SCIC-filter compared to a classical CIC-filter

The overall pass-band behavior of the lower decimation filters is shown in Figure 8. Due to the excellent pass-band performance of bi-reciprocal WDFs, the ripple is smaller than 0.05dB. The resolution required for the ADCs of the digital receiver is determined by the dynamic range requirements of the receive path. If the RF-front end is designed properly, you can obtain the required dynamic range from the smallest wanted and the largest unwanted signal, which must be processed in common. You can find the respective values in the physical layer specification of the various wireless and mobile standards as interferer and blocking requirements.

Figure 8:  Pass band behavior of the three lower decimation stages with cascaded WDFs

The proposed decimation filter architecture enables the realization of highly optimized but flexible digital receivers. It has been shown in this contribution that you can obtain very effective realizations for the lower decimation stages, by using cascaded low-order wave digital filters. You can implement this filter type with minimum hardware costs compared to other solutions such as classical wave digital filters or FIR-filters. Further advantages of the proposed filter structures are superior sensitivity properties with respect to coefficient-quantization effects, better noise performance, and less group-delay distortion compared to classical structures.

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