# Accelerating Fixed-Point Design for MB-OFDM UWB Systems

*Martin Clark, Mike Mulligan, Dave Jackson, and Darel Linebarger, The MathWorks, Inc. Jan 26, 2005 (3:24 AM)*

Ultrawideband (UWB) wireless technology is poised to replace high-speed data cables in homes and offices.^{1} UWB technology will be capable of transmitting hundreds of megabits per second over distances of several meters. Target applications include connecting digital cameras to computers and DVD players to HDTV screens. Companies expect to ship commercial UWB products by the end of 2005.

Since UWB range is fundamentally limited by low-power and high-speed transmission, there is little margin for implementation losses. However, the success of UWB in the marketplace depends on low-cost implementation. The difference of a few meters in range, or a few dollars in cost, could make or break UWB.

Fixed-point engineering is critical in managing this all-important tradeoff between UWB range and cost. First, fixed-point word lengths strongly influence hardware size and cost. The silicon area of a multiply operation, for instance, is roughly proportional to the square of word length. Second, word lengths and scaling strongly influence the signal-to-noise ratio (SNR) degradation of the communication link. Decreasing signal-to-noise ratio (SNR) degradation by 1 dB can increase coverage by up to 25%.

Unfortunately, fixed-point engineering is challenging and time-consuming, typically demanding 25 to 50 percent of the total design time. In this article, we explore fixed-point design with a focus on a multi-band UWB implementation. We will also show techniques for accelerating the fixed-point design process.

**Model Overview**

The model described in this article is based on the multi-band orthogonal frequency division multiplexing (OFDM) UWB proposal submitted to the IEEE 802.15.3a standards group in September 2003.2,3,4 Subsequent proposals have not changed the essential technology.

The proposal supports seven data rates in the range of 55 to 480 Mbit/s. The highest mandatory rate is 200 Mbit/s. OFDM signals are transmitted using a frequency-hopping (multi-band) scheme. The proposed model captures the end-to-end physical layer (PHY) for the highest mandatory data rate and for the mandatory frequency hopping mode **(Table 1)**.

**Multiband OFDM PHY: 200 Mbit/s Mode**

The multiband OFDM proposal is, in many ways, similar to the IEEE 802.11a/g wireless LAN (WLAN) physical layer (PHY) standards. Leveraging this similarity, we adapted the proposed UWB model (see Figure 1) from an existing 802.11a model.^{5} The adapted model also includes the UWB channel Matlab code programmed by Intel and used by the IEEE 802.15.3a group.^{3}

*Figure 1: Top level view of a UWB model.*

As shown in Figure 1 above, the transmitter and receiver each comprise three sections:

- Binary data processing
- Digital baseband processing
- Baseband model of the analog front-end and channel

We are interested primarily in the fixed-point design of the digital baseband section. The rest of the model is a test harness; it enables us to quickly assess the impact of the fixed-point design on end-to-end link performance.

**OFDM Transmitter**

The purpose of the OFDM transmitter proposed subsystem **(Figure 2:)** is to transform a payload of quadrature phase-shift keying (QPSK) symbols into a large frame of OFDM symbols (165 samples each) to be passed along to the transmitter's front end.

*Figure 2: Diagram of a proposed OFDM transmitter.*

The conversion block at the subsystem input converts the incoming signal into a fixed-point data type. In reality, this operation wouldn't exist. On the contrary, the QPSK modulator would translate incoming bits directly into fixed-point data. The conversion block at the subsystem output converts to double-precision floating point (for our purposes, you can think of it as a D/A converter).

Figure 2 highlights the inverse fast Fourier transform (IFFT) block and gain block because they perform fixed-point arithmetic. All other blocks are simply fixed-point "data shufflers." The highlighting throughout the UWB model helps to quickly identify which blocks will be involved in the fixed-point design process.

**OFDM Receiver**

The OFDM receiver, as shown in **Figure 3**, involves more signal processing and thus, more fixed-point arithmetic than the transmitter. The receiver requires arithmetic in four sections:

- Cyclic processing
- FFT
- Channel estimation/compensation
- Time dispreading

Cyclic processing, channel estimation, and channel compensation are necessary to mitigate the effects of multipath channel dispersion.

*Figure 3: Diagram of a proposed OFDM receiver.*

**Figure 4** shows the proposed channel estimation and compensation subsystem. This subsystem implements a simple, low-cost phase-compensation scheme. (More sophisticated schemes exploit the channel's frequency coherence, and thus improve noise averaging.^{2}) The system does not compensate for channel magnitude variations across the OFDM tone set because such schemes are computationally expensive and also unnecessary for QPSK. The proposed scheme avoids division with a complex divisor and ensures that the magnitude of the division output has a small dynamic range.

*Figure 4:Diagram showing the proposed channel estimation and compensation subsystem.*

These arithmetic considerations constitute the most important preliminary phase of fixed-point design. Before we treat the topic of engineering word lengths and scaling, we need a floating-point reference — or "golden reference" — to serve as an upper bound on link performance.

**Floating-Point Reference**

A data-type override feature in Simulink allows a designer to switch between fixed-point and floating-point for any subsystem or for the entire model. For our initial floating-point reference, we set the channel SNR to a high value (60 dB), which helps us isolate the impact of fixed-point effects on symbol distortion. **Figure 5** shows two scopes from the UWB simulation. The first scope (top diagram) shows the power spectrum of the baseband-equivalent received signal over all three sub-bands. The second scope (bottom figure) displays the signal constellation after channel phase estimation and compensation.

*Figure 5: Diagram showing two UWB simulation scopes.*

The DC null in the power spectrum is from the OFDM transmission, but the rest of the spectrum approximately follows the frequency-selective fading characteristic of the multipath channel. The dynamic range over the OFDM tone set is about 30 dB, which is also evident in the magnitude-spread of the phase-compensated signal constellation. A clean "X" indicates almost perfect phase compensation.

**Fixed-Point Design Methodology**

The all-important next phase is to set word lengths and scaling for every fixed-point arithmetic block in the system. Together, the word length and scaling constrain the dynamic range of a signal. If poorly engineered, they will introduce overflow or underflow, and degrade link performance. It makes sense, therefore, that one of the most useful things to analyze in a fixed-point design is the dynamic range of signals.

We used the following methodology for the UWB fixed-point design:

- Work through the system in order of signal processing, enabling floating-point overrides for downstream subsystems.
- For a given arithmetic subsystem or block:
- Enable floating-point override and analyze the output signal's dynamic range.
- Adjust word length and scaling to minimize overflow and underflow.
- Disable floating-point override, re-examine dynamic range, and assess the impact on link performance.

- Enable floating-point override and analyze the output signal's dynamic range.

This procedure is an iterative process, and the work flow can be tedious and time-consuming. To accelerate the process, we used Matlab to perform dynamic range analysis/visualization, as described in the following section.

**Example: Transmitter Design**

For the UWB model, we built a block that automatically outputs a signal to a histogram, which is an invaluable way to visualize dynamic range. Figure 2 above shows this block attached to the transmitter gain's output.

**Figure 6** shows the associated histogram for the floating-point reference (both in-phase and quadrature) for the block displayed in Figure 2. The base-2 log scale is useful for visualizing dynamic range in terms of number of bits, i.e., word length.

*Figure 6: Histogram of OFDM transmitter output using a floating-point reference.*

Excluding zero-valued samples (which are mapped to 2^{-15} in the plot), the signal magnitude lies in the range 2^{-13} to 2^{2} for more than 99.9 percent of the time, and so the signal can be represented adequately with 16 bits (signed). This large dynamic range — 90 dB — is typical in OFDM, and is essentially the result of passing a random signal through an IFFT (central limit theorem).

The analysis block also automatically estimates that 2^{-14} might be a reasonable scale factor to minimize overflow and underflow. Based on this analysis, we initially set the word length to 16 bits and the scaling factor to 2^{-14} for all arithmetic blocks in the transmitter. We do this by explicitly setting the fixed-point parameters of the input gateway block and selecting "same as input" for the fixed-point parameters of all other arithmetic blocks in the transmitter. We maintain floating-point overrides in the receiver subsystem to isolate and diagnose potential problems in the transmitter design.

**Figure 7** shows the resulting histogram and phase-compensated signal constellation. Notice that the constellation is somewhat distorted compared with our floating-point reference, which is shown in Figure 5 above. The histogram reveals that the highest values are saturated to a value of 2. (The dashed lines in the histogram represent the floating-point reference; the stems represent the fixed point result.) While these high-power transmitted values occur only about 1 percent of the time, this is enough to cause, with high probability, significant distortion at the output of the 128-point receiver FFT.

*Figure 7: Diagram showing the results for 2 ^{-14} scaling.*

We need to increase the scaling by one or two bits. Such an increase will cause underflow for the small values at the gain's output, but the effect should be minimal because transmitted signal values smaller than 2^{-10} will be buried in the channel noise. **Figure 8** shows the improvement achieved with 2^{-12} scaling.

*Figure 8: Diagram showing the improved results achieved with 2 ^{-12} scaling.*

Favoring the high end of a signal's range is not always the right strategy. Small signal values sometimes play a significant role, for instance, in channel estimation and compensation algorithms. The point is that setting fixed-point scaling requires some finesse, particularly when it comes to engineering for smaller word lengths. Auto-computation tools provide coarse estimates, but fine-tuning often calls for a combination of visualization and insight.

The next design steps involve analyzing output signal dynamic ranges of individual blocks in the transmitter, tuning each block's fixed-point settings, and moving through the receiver using the same techniques.

**Smaller Word Lengths**

Beginning with 16 bits throughout the system allowed us to approach design issues incrementally and then apply what we learned to smaller word lengths. For instance, when we care more about overflow than about underflow, as is often the case, the desired number of bits above the binary point tends to be similar for different word lengths.

Using the tools and methodology discussed in this article, we were able to obtain a 10-bit design working at a bit error rate of 0.1 percent, with only a 0.5 dB SNR degradation compared with the floating-point reference. Using Matlab workspace variables and selector functions, we could quickly switch between different fixed-point designs. We also wrote simple Matlab scripts to run simulations over a range of word lengths and channel conditions.

**Wrap Up**

This article has demonstrated techniques for accelerating fixed-point design in a UWB wireless communication system. This methodology can be used to explore the all-important tradeoff between chip size (or power consumption) and wireless range.

**References**

- S. Stroh, "Ultrawideband: multimedia unplugged,"
*IEEE Spectrum*, October 2003. - J. Heiskala and J. Terry, "OFDM wireless LANs: a theoretical and practical guide," Sams Publishing, 2002.
- Multi-Band OFDM Alliance, http://www.multibandofdm.org
- IEEE 802.15 Committee, http://www.ieee802.org/15/pub/TG3a.html
- http://matlabcentral.com, File Exchange: "IEEE 802.11a WLAN Model" and "UWB Fixed-Point Model."

**About the Authors****Martin Clark** is a principal engineer at The MathWorks. Martin received his B.E. (Hons.) and Ph.D. degrees in electrical engineering from the University of Canterbury, New Zealand and can be reached at martin.clark@mathworks.com.

**Mike Mulligan** is the manager of the communications development team at The MathWorks. Mike holds a Ph.D. in Electrical Engineering from Northeastern University and can be reached at mike.mulligan@mathworks.com .

**Dave Jackson** is a product manager for the video and signal processing group at The MathWorks. Dave holds a BS/BA from Boston University in International Management and a Masters degree from Boston College in Marketing & Computer Science. He can be reached at dave.jackson@mathworks.com.

**Darel Linebarger** leads a team focused on fixed-point modeling and code generation from MathWorks products in the signal processing arena. Darel holds a Ph.D. and M.S. from Rice University and a BSEE from Southern Methodist University. He can be reached at darel.linebarger@mathworks.com.

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