Apparatus and method for rectangular-to-polar conversion

A rectangular-to-polar-converter receives a complex input signal (having X0 and Y0 components) and determines an angle φ that represents the position of the complex signal in the complex plane. The rectangular-to polar-converter determines a coarse angle φ1 and a fine angle φ2, where φ=φ1+φ2. The coarse angle φ1 is obtained using a small arctangent table and a reciprocal table. These tables provide just enough precision such that the remaining fine angle φ2 is small enough to approximately equal its tangent value. Therefore the fine angle φ2 can be obtained without a look-up table, and the fine angle computations are consolidated into a few small multipliers, given a precision requirement. Applications of the rectangular-to-polar converter include symbol and carrier synchronization, including symbol synchronization for bursty transmissions of packet data systems. Other applications include any application requiring the rectangular-to-polar conversion of a complex input signal.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention is related to digital signal processing and digital communications. More specifically the present invention is related to interpolation, angle rotation, rectangular-to-polar conversion, and carrier and symbol timing recovery for digital processing and digital communications applications.

2. Related Art

Advances in technology have enabled high-quality, low-cost communications with global coverage, and provide the possibility for fast Internet access and multimedia to be added to existing services. Exemplary emerging technologies include cellular mobile radio and digital video broadcasting, both of which are described briefly as follows.

In recent years, cellular mobile radio has experienced rapid growth due to the desire for mobility while enjoying the two-way voice services it provides. GSM, IS-136 and personal digital cellular (PDC) are among the most successful second-generation personal communications (PCS) technologies in the world today, and are responsible for providing cellular and PCS services globally. As the technology advances, customers will certainly demand more from their wireless services. For example, with the explosive growth of the world wide web over the wired networks, it is desirable to provide Internet services over mobile radio networks. One effort to specify the future global wireless access system is known as IMT-2000 (Buchanan, K., et al.,IEEE Pers. Comm.4:8-13 (1997)). The goal of IMT-2000 is to provide not only traditional mobile voice communications, but also a variety of voice and data services with a wide range of applications such as multimedia capabilities, Internet access, imaging and video conferencing. It is also an aim to unify many existing diverse systems (paging, cordless, cellular, mobile satellite, etc.) into a seamless radio structure offering a wide range of services. Another principle is to integrate mobile and fixed networks in order to provide fixed network services over the wireless infrastructure. Such systems might well utilize broadband transport technologies such as asynchronous transfer mode (ATM).

For the applications of IMT-2000, a high-bit-rate service is needed. Moreover, for multimedia applications, the system should provide a multitude of services each requiring 1) a different rate, and 2) a different quality-of-service parameter. Thus, a flexible, variable-rate access with data rates approaching 2 Mb/s is proposed for IMT-2000.

The advent of digital television systems has transformed the classical TV channel into a fast and reliable data transmission medium. According to the specifications of the DVB project (Reimers, U.,IEEE Comm. Magazine36:104-110 (1998)), digital TV is no longer restricted to transmitting sound and images but instead has become a data broadcasting mechanism which is fully transparent to all contents. Digital TV broadcasting by satellite, cable and terrestrial networks is currently under intensive development. A typical system looks like this: a DVB signal is received from a satellite dish, from cable, or from an antenna (terrestrial reception). A modem built into an integrated receiver/decoder (IRD) will demodulate and decode the signal. The information received will be displayed on a digital TV or a multimedia PC. In addition to being used as a digital TV, DVB can receive data streams from companies who wish to transmit large amounts of data to many reception sites. These organizations may be banks, chains of retail stores, or information brokers who wish to offer access to selected Internet sites at high data rates. One such system is MultiMedia Mobile (M3), which has a data rate of 16 Mb/s.

For proper operation, these third generation systems require proper synchronization between the transmitter and the receiver. More specifically, the frequency and phase of the receiver local oscillator should substantially match that of the transmitter local oscillator. When there is a mismatch, then an undesirable rotation of the symbol constellation will occur at the receiver, which will seriously degrade system performance. When the carrier frequency offset is much smaller than the symbol rate, the phase and frequency mismatches can be corrected at baseband by using a phase rotator. It is also necessary to synchronize the sampling clock such that it extracts symbols at the correct times. This can be achieved digitally by performing appropriate digital resampling.

The digital resampler and the direct digital frequency synthesizer (DDS) used by the phase rotator are among the most complex components in a receiver (Cho, K., “A frequency-agile single-chip QAM modulator with beamforming diversity,” Ph.D. dissertation, University of California, Los Angeles (1999)). Their performance is significant in the overall design of a communications modem. For multimedia communications, the high-data-rate requirement would impose a demand for high computational power. However, for mobile personal communication systems, low cost, small size and long battery life are desirable. Therefore, it would be desirable to have an efficient implementation of the phase rotator, re-sampler, and DDS in order to perform fast signal processing that operates within the available resources. Furthermore, it would be desirable to have an efficient synchronization mechanism that uses a unified approach to timing and carrier phase corrections.

For Internet services it is important to provide instantaneous throughput intermittently. Packet data systems allow the multiplexing of a number of users on a single channel, providing access to users only when they need it. This way the service can be made more cost-effective. However, the user data content of such a transmission is usually very short. Therefore, it is essential to acquire the synchronization parameters rapidly from the observation of a short signal-segment.

For applications where low power and low complexity are the major requirements, such as in personal communications, it is desirable to sample the signal at the lowest possible rate, and to have a synchronizer that is as simple as possible. Therefore, it is also desirable to have an efficient synchronizer architecture that achieves these goals.

For applications utilizing Orthogonal Frequency Division Multiplexing (OFDM), sampling phase shift error produces a rotation of the Fast Fourier Transform (FFT) outputs (Polet T., and Peters, M.,IEEE Comm. Magazine37:8086 (1999)). A phase correction can be achieved at the receiver by rotating the FFT outputs. Therefore, it is also desirable to have an efficient implementation structure to perform rotations of complex numbers.

SUMMARY OF THE INVENTION

The present invention is directed at a rectangular-to-polar-converter that receives a complex input signal (having X0and Y0components) and determines an angle φ, which represents the position of the complex input signal in the complex plane. In doing so, the rectangular-to-polar converter determines a coarse angle φ1and a fine angle φ2, where φ=φ1+φ2.

The coarse angle φ1is obtained using a small arctangent table and a reciprocal table. These tables provide just enough precision such that the remaining fine angle φ2is small enough to approximately equal its tangent value. Therefore the fine angle φ2can be obtained without a look-up table, and the fine angle computations are consolidated into a few small multipliers, given a precision requirement.

More specifically, the coarse angle computation is performed by retrieving a pre-computed Z0=1/[X0] value from a reciprocal lookup table (e.g. memory device), where [X0] is a bit truncated approximation of X0. The Z0value is multiplied by the Y0component, resulting in a [Y0Z0] value. The coarse approximation angle φ1is retrieved from a second lookup table that stores pre-computed arc tan values of [Y0Z0]. Next, the input complex signal is multiplied by the [Y0Z0] value. This multiplication effectively rotates the input complex number by the coarse angle φ1back toward the X-axis of the complex plane, resulting in an intermediate complex number having an X1component and a Y1component. Next the reciprocal lookup table is re-used to determine an approximation of Z1=1/[X1]. Then the tangent of the fine angle φ2is determined based on [Z1Y1], assuming that tan φ2can be substantially approximated as [Z1Y1]. In embodiments, the Newton Raphson method is implemented to get a more accurate tan φ2result. Finally, based on the smallness of tan φ2, the trigonometric function value tan φ2is used as an approximation to φ2, hence requiring no arc tan table.

Applications of the rectangular-to-polar converter include symbol and carrier synchronization, including symbol synchronization for bursty transmissions of packet data systems. Other applications include any application requiring the rectangular-to-polar conversion of a complex input signal.

Further features and advantages of the invention, as well as the structure and operation of various embodiments of the invention, are described in detail below with reference to the accompanying drawings. The drawing in which an element first appears is typically indicated by the leftmost character(s) and/or digit(s) in the corresponding reference number.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Table of Contents1.Introduction1.1Exemplary Modulation Schemes and Synchronization Issues1.2Overview2.Interpolation Using a Trigonometric Polynomial2.1Interpolation Using an Algebraic Polynomial2.2The Trigonometric Polynomial Method2.3Performance Comparisons2.4Efficient Implementation Structures2.4.1Using a Lookup Table2.4.2Using an Angle Rotation Processor2.5Delays in the Computation2.6Simplifications of the Preliminary Structures2.6.1The Simplified Structure for N = 42.6.2The Simplified Structure for N = 82.6.3Performance Comparisons with Other Structures2.7Trigonometric Interpolator Application2.8Trigonometric Interpolator Summary3.Interpolation Filters with Arbitrary Frequency Response3.1Formulating the Trigonometric Interpolator as an InterpolationFilter3.2Analysis of the Frequency Response3.3Implementing the Modified Algorithm3.4Conditions for Zero ISI3.5Optimization Algorithm3.6Conclusion4.Design of Optimal Resamplers4.1Motivation4.2Resampler Optimizations4.3Implementations4.4Simulation Results4.5Conclusion5.A High-Speed Angle Rotation Processor5.1The angle rotation problem5.1.1Single-Stage Angle Rotation5.1.2Rotation by a Small Angle5.1.3Partitioning into Coarse and Fine Rotations5.2Simplification of the Coarse Stage5.3Reduction of Multiplier Size in the Fine Stage5.4Scaling Multiplier Simplification5.5Computational Accuracy and Wordlength5.6Comparison with the Single-Stage Mixer5.7A Modified Structure When Only One Output is Needed5.7.1Modifications to the Coarse Stage5.7.2Scaling Multiplier Simplification5.8Application of Angle Rotation Processors5.8.1Using the Angle Rotation Processor in a QuadratureDirect Digital Frequency Synthesizer/Mixer5.8.1.1A General Angle Rotator for Arbitrary InputAngles5.8.1.2Adapting the General Angle Rotator to Make aQDDFSM5.8.2How to Use the Conditionally Negating Multipliers inthe General Angle Rotator5.8.2.1Booth Multiplier5.8.2.2How to Make a Negating Booth Multiplier5.8.2.3How to Make a Conditionally Negative BoothMultiplier5.8.3Using the Angle Rotation Processor in a QuadratureDirect Digital Frequency Synthesizer5.9Conclusion6.Symbol Synchronization for Bursty Transmissions6.1Initial Parameter Estimations for Burst Modems6.2Background Information6.3Symbol Timing Estimation Assuming θ = 06.4Bias in Symbol Timing Estimation due to Truncating theSequence6.5Carrier-Independent Symbol Timing Recovery6.6Carrier Phase Computation6.7Simulation Result6.8Conclusion7.A High-Speed Processor for Rectangular-to-Polar Conversion7.1Partitioning the Angle7.2The Two-Stage Algorithm7.2.1Simplification in the Coarse Computation Stage7.2.1.1The Reciprocal Table 71067.2.1.2The Arctangent Table 71107.2.2Hardware Reduction in the Fine Computation Stage71247.3Magnitude Calculation7.4Converting Arbitrary Inputs7.5Test Result7.6Conclusion8.Exemplary Computer System9.Appendices9.1Appendix A: Proof of the Zero ISI Condition.9.2Appendix B: Impulse Response of the Simplified Interpolators9.3Appendix C: Fourier Transform of g(nTa-μ)9.4Appendix D: Interpolation on Non-Center Intervals9.5Appendix E10.Conclusion

As discussed herein, third generation and other cutting edge communications systems require proper synchronization between the transmitter and the receiver. More specifically, the frequency and phase of the receiver local oscillator should substantially match that of the transmitter local oscillator and accurate symbol timing must be achieved. The following section discuss some exemplary modulation schemes and configurations, and their related synchronization issues. These example configurations are not meant to be limiting, and are provided for example purposes only. After which, an overview of the present invention is provided.

1.1 Exemplary Modulation Schemes and Synchronization Issues

A key to the evolution of third-generation PCS is the ability to provide higher data rates via increased spectral efficiency of the access scheme. The IS-136community intends to add a 200-KHz carrier bandwidth and adopt 8PSK modulation. This allows for data rates up to 384 Kb/s.

A simplified 8PSK transmitter102and receiver104are shown in FIG.1A andFIG. 1B, respectively. The receiver104, as shown, performs baseband sampling. Alternatively, the received signal could be sampled at an IF frequency, where the down-conversion to baseband is performed digitally. However, since it does not alter the main subject in the present invention, the baseband-sampled system is used as an example.

Referring toFIG. 1B, PSK receiver104down-converts an IF input signal106to baseband by multiplication with a local oscillator signal108using mixers110. After filtering111, A/D converters112sample the down-converted signal according to a sampling clock114in preparation for logic examination. After further filtering116and equalization118, the logic decision devices120examine the sampled signal to determine a logic output for the two channels.

During down-conversion, an undesirable rotation of the symbol constellation will occur if the frequency and phase of the oscillator signal108does not match the oscillator signal of the transmitter102. This symbol rotation can seriously degrade system performance. When the carrier frequency offset is much smaller than the symbol rate, the phase and frequency mismatches can be corrected at baseband, using a phase rotator124, as shown in FIG.1D.

The sampling clock114is generated locally in the receiver104. The logic decision devices120make more accurate decisions when the sampling instant is optimal, i.e., synchronous to the incoming symbols.

If the timing information can be extracted from the signal106, it can be used to adjust the phase of the sampling clock114. This adjustment would require a voltage controlled oscillator (VCO) to drive the A/D converters112. In this scenario, the digital portion of the circuit104needs to keep in synchronization with the A/D converters112, which places strict requirements on the VCO. Moreover, changing the phase of the sampling clock114would cause jitter. High data-rate receivers are more sensitive to such jitter when used in multimedia communications.

Another solution to timing errors is to correct them entirely in the digital domain, with the equivalent of A/D sampling adjustment performed by a digital resampler122, as shown in FIG.1D. This resampler122is controlled by a timing recovery circuit (not shown) and it attempts to supply the optimal samples (i.e. synchronous) to the decision circuits120. Using the digital resampler122, the timing recovery loop is closed entirely in the digital domain. This allows the complete separation of digital components from analog components.

The digital resampler122and a direct digital frequency synthesizer (not shown) used by the phase rotator124are among the most complex components in a receiver (Cho, K., “A frequency-agile single-chip QAM modulator with beamforming diversity,” Ph.D. dissertation, University of California, Los Angeles (1999)). Their performance is significant in the overall design of the modem. For multimedia communications, the high-data-rate requirement imposes a demand for high computational power. However, for mobile personal communication systems, low cost, small size, and long battery life are desirable. Therefore, efficient implementation is the key to implementing fast signal processing within the available resources. It is also desirable to provide an efficient synchronization mechanism by using a unified approach to timing and carrier phase corrections. This can be accomplished by sharing resources between the resampler122and the phase rotator124.

As for the digital video broadcasting system (DVB) systems, the most challenging of all DVB transmissions is the one used in terrestrial channels (DVB-T) due to the presence of strong echoes which characterize the propagation medium. A common approach for DVB-T is based on Coded-OFDM (orthogonal frequency division multiplexing). The major benefit of OFDM is that the serial baseband bitstream which needs to be transmitted is distributed over many individual subcarriers. Such spreading makes the signal robust against the effects of multipath and narrowband interference. The simplified block diagram of an OFDM modem108is shown in FIG.1C.

FIG. 1Cillustrates an orthogonal frequency division multiplexing system (OFDM)126having an OFDM transmitter128and an OFDM receiver130. For the OFDM system126, synchronization errors produce a rotation of the fast Fourier Transform (FFT) outputs of the OFDM receiver130. (Pollet T., and Peeters, M.,IEEE Comm. Magazine37:80-86 (1999)). A sampling phase correction for the received signals can be achieved by rotating the FFT outputs at the receiver. For FFT rotation, it is desirable to have an efficient implementation structure to perform rotations of complex numbers.

The example applications and modulation schemes described above in this section were provided for illustrative purposes only, and are not meant to be limiting. Other applications and combinations of such applications will be apparent to persons skilled in the relevant art(s) based on the teachings contained herein. These other applications are within the scope and spirit of the present invention.

The following is an overview of the sections that follow.

Sections 2, 3 and 4, discussed herein, present a novel interpolation method for digital resampling using a trigonometric polynomial. In Section 2, after a brief review of interpolation methods, particularly those using a conventional polynomial, the trigonometric interpolation method is introduced. Efficient implementation structures for trigometric interpolation are given. The performance, hardware complexity and computational delay are compared with conventional polynomial interpolators. The trigonometric polynomial based resampler can use the same hardware as is employed in the phase rotator for carrier synchronization, thus further reducing the total complexity in the synchronization circuitry.

In Section 3, a signal processing approach is used to analyze the interpolation method devised in Section 2. It shows how an arbitrary frequency response is achieved by applying a simple modification to the original interpolation algorithm. This enables the interpolator to also perform matched filtering of the received signal.

The approaches in Section 3 can be employed to improve the interpolator performance by optimizing the frequency response of the continuous-time interpolation filter. This method is based on optimizing the performance by conceptually reconstructing the continuous-time signal from existing samples. From the point of view of designing digital resamplers, however, what we are actually interested in are new samples corresponding to the new sampling instants. In Section 4, we optimize the interpolation filter such that the error in producing a new sample corresponding to every resampling instant is minimized, hence further improving the overall interpolation accuracy.

Section 5 presents an angle rotation processor that can be used to efficiently implement the trigonometric resampler and the carrier phase rotator. This structure can also implement the resampler for an OFDM receiver, which rotates the FFT outputs. It has many other practical applications.

The discussions in the previous Sections have assumed that the sampling mismatch that is supplied to the resampler is known The problem of obtaining the synchronization parameters is studied in Section 6. For burst mode transmissions in packet data systems, we present an efficient architecture for feedforward symbol-timing and carrier-phase estimation.

Section 7 presents an efficient implementation of a key component in the feedforward synchronizer of Section 6, as well as in many other such synchronizers. This involves computing the angle from the real and imaginary components of a complex number. The discussion, however, extends to a general problem of Cartesian-to-polar conversion, which is encountered in many communication applications. An architecture that efficiently accomplishes this conversion is presented.

Section 8 presents an exemplary computer system in which the invention can be operated.

Section 9 includes various appendices.

Further discussions related to materials in Sections 2-7 are included in Dengwei Fu, “Efficient Synchronization for Multimedia Communications,” Ph.D dissertation, University of California, Los Angeles, 2000, which is incorporated-by-reference, in its entirety.

Additionally, the following articles are herein incorporated by reference:

2. Interpolation Using a Trigonometric Polynomial

As discussed in Section 1, when an analog-to-digital converter (ADC) is clocked at a fixed rate, the resampler must provide the receiver with correct samples, as if the sampling is synchronized to the incoming symbols. How can the resampler recover the synchronized samples by digital means without altering the sampling clock? Since the input analog signal to the ADC is bandlimited, as long as the sampling rate is at least twice the signal bandwidth, according to the sampling theorem, the sampled signal carries as much information as the continuous-time signal. Therefore, the value of the original continuous-time signal at an arbitrary point can be evaluated by applying an interpolation filter (e.g., sinc interpolation) to the samples. Hence, the design of the resampler has been transformed to the design of effective interpolation filters or, in other words, fractional-delay filters with variable delay.

There are numerous methods for designing fractional-delay filters. These filters have different coefficients for different delays. Thus, to implement variable-delay interpolation filters, one could compute one set of coefficients for each quantized delay value and store them in a memory. Then, in real-time, depending on the fractional delay extracted from the incoming signal, the corresponding coefficients could be loaded. However, this method is likely to result in a large coefficient memory.

To design low-cost modems, a large coefficient memory is undesirable. Gardner, et al., have shown that polynomials can be incorporated to compute the desired samples that are synchronous with the transmitted samples (Gardner, F. M.,IEEE Trans. Comm.41:502-508 (1993); Erup, L., et al.,IEEE Trans. Comm.41:998-1008 (1993)). In this case, an extensive coefficient memory is not needed. Moreover, the polynomial-based structure can be implemented efficiently with a so-called Farrow structure (Farrow, C., “A continuously variable digital delay element,” inProc. IEEE Int. Symp. Circuits Syst. (June 1988), pp. 2641-2645). This method is reviewed in Section 2.1. Although this approach achieves reasonable performance, the hardware complexity grows rapidly as the number of samples used to calculate each new sample is increased for better accuracy. In addition, given a fractional delay μ, to produce a new sample using a degree N−1 polynomial, there will be N−1 sequential multiplications that involve μ since we must compute μ raised to the (N-1)-th power times a data value. Thus, the critical data path gets longer as N increases, thereby creating a limitation on the achievable data rate.

Starting in Section 2.2, a new approach to interpolation is introduced. Instead of approximating the continuous-time signal with a conventional (i.e., algebraic) polynomial, a trigonometric polynomial is used according to the present invention. First, some background information is given. Next, the detailed implementation is discussed. We then evaluate and compare the performance and computational complexity of the algebraic polynomial interpolation to that of our method, giving numerical results. 2.1 Interpolation Using an Algebraic Polynomial

To interpolate using function values y(n) at N equally-spaced sample points, also referred to as a “base point set,” we can fit an algebraic polynomial of degree N−1 to the data, as in FIG.2. As explained in (Gardner, F. M.,IEEE Trans. Comm.41:502-508 (1993)), there should be an even number of samples in the base point set, and the interpolation should be performed only in the center interval of the base point set. That is, N is restricted to be even. In other words, given 4 samples points inFIG. 2including y(−1), y(0), y(1), and y(2), the interpolation is performed at offset μ between the points y(0) and y(1) to determine the point202on the curve p(t).

It seems that one would have to solve for the coefficients of the (N−1)-th degree polynomial from these available samples before the synchronized (i.e., interpolated) samples can be computed. However, a method devised by Farrow (Farrow, C., “A continuously variable digital delay element,” inProc. IEEE Int. Symp. Circuits Syst. (June 1988), pp. 2641-2645) can compute the synchronized sample from the available samples efficiently with use of an algorithm that is well suited for VLSI implementation. To illustrate, we consider the following example of interpolation using a cubic Lagrange polynomial. Without loss of generality, let us assume the sampling interval is TS=1. Using the Lagrange formula for N=4, the synchronized samples can be computed as
y(μ)=y(−1)C−1(μ)+y(0)C0(μ)+y(1)C1(μ)+y(2)C2(μ)  (2.1)
where Cn(μ), n=−1, 0, 1, 2, are the third degree polynomials that are shown in FIG.3.
Obviously,Cn⁡(μ)={1μ=nTs,n⁢⁢an⁢⁢integer0all⁢⁢other⁢⁢integers.(2.2)
Thus, y(μ) in (2.1), the sum of polynomials Cn(μ) weighted by the y(n) values, must be a third degree polynomial and must go through the samples y(−1), y(0), y(1) and y(2). Writing Cn(μ) asCn⁡(μ)=∑k=03⁢cnk⁢μk(2.3)
the coefficients Cnkof Cn(μ) are fixed numbers. They are independent of is. We can re-write (2.1) asy⁡(μ)=∑n=-12⁢y⁡(n)⁢∑k=03⁢cnk⁢μk=∑k=03⁢(∑n=-12⁢y⁡(n)⁢cnk)⁢μk=∑k=03⁢v⁡(k)⁢μk⁢⁢where(2.4)v⁡(k)=∑n=-12⁢y⁡(n)⁢cnk.(2.5)
To minimize the number of multipliers, we can use a nested evaluation of (2.4), as
y(μ)=((v(3)μ+v(2))μ+v(1))μ(0)  (2.6)

A Farrow structure400(for N=4) that implements equations (2.5) and (2.6) is shown in FIG.4. It consists of multiplications of data by fixed coefficients, and data multipliers, as well as addition operations.

2.2 The Trigonometric Polynomial Method

To solve the problems discussed in the section 2.1, the present invention utilizes a trigonometric polynomial to fit the asynchronous samples in FIG.2. Using WN=e−j2π/Nnotation, for tε[−N/2+1, N/2], the polynomial may be written as:y⁡(t)=1N⁢(∑k=-N/2+1N/2-1⁢ck⁢WN-kt+12⁢cN/2⁢WN-(N/2)⁢t+12⁢c-N/2⁢WN(N/2)⁢t).(2.7)

The polynomial must cross the N samples. Thus, the coefficients ckcan be determined by solving the N linear equations in N unknowns:y⁡(n)=1N⁢∑k=-N/2+1N/2⁢ck⁢WN-kn,⁢n=-N2+1,…⁢,N2(2.8)
whose solution isck=∑n=-N/2+1N/2⁢y⁡(n)⁢WNkn,⁢k=-N2+1,…⁢,N2.(2.9)

The expression in (2.9) is simply the N-point discrete Fourier transform (DFT). This suggests that, given N equally-spaced samples, we can compute the DFT of these samples as in (2.9) to obtain the coefficients of the interpolating trigonometric polynomial in (2.7). Then, for a given offset μ, the synchronized sample y(μ) can be computed using that polynomial as:y⁡(μ)=1N⁢(∑k=-N/2+1N/2-1⁢ck⁢WN-k⁢⁢μ+cN/2⁢cos⁢⁢π⁢⁢μ).(2.10)
Since ckand c−kare conjugates, this equation can be simplified asy⁡(μ)=1N⁢Re⁡(c0+2⁢∑k=1N/2-1⁢ck⁢WN-k⁢⁢μ+cN/2⁢ⅇj⁢⁢π⁢⁢μ).(2.11)

Flowchart500inFIG. 5summarizes the interpolation between two sample points at an offset μ using a trigonometric polynomial, where the two data samples that are to be interpolated are part of a set of N-data samples (see FIG.2). The flowchart is described as follows.

In step502, a set of N-data samples are received having the two data samples that are to be interpolated.

In step504, coefficients of a trigonometric polynomial are determined based on the set of N data samples, according to equation (2.9). In doing so, the N data samples are multiplied by a complex scaling factor WNknto generate a kthcoefficient for the trigonometric polynomial, wherein WN=e−j2π/N, and wherein n represents an nthdata sample of said N data samples.

In step506, the trigonometric polynomial is evaluated at the offset μ based on equation (2.10) or (2.11).

In step508, the real part of the evaluated trigonometric polynomial is determined and represents the desired interpolation value 202 in FIG.2.

There are three issues that are to be considered in evaluating an interpolation scheme: 1) accuracy of interpolation, 2) complexity of implementation and 3) latency. In following sections, the trigonometric interpolation method is compared to the traditional polynomial method, particularly the Lagrange interpolator, in all these categories.

2.3 Performance Comparisons

Let us first derive the impulse responses of the interpolators. With N samples, N an even integer, the Lagrange formula (2.1) isy⁡(μ)=∑n=-N/2+1N/2⁢y⁡(n)⁢Cn⁡(μ).(2.12)
In addition, the interpolation is performed in the center interval. Thus 0≦μ≦1. Let us define a new function f(t) such that
f(μ−n)=Cn(μ), 0≦μ≦1, −N/2+1≦n<N/2.  (2.13)
Using the example ofFIG. 3, defining t=μ−n, we havef⁡(t)=(C2⁡(t+2)-2≤t<-1C1⁡(t+1)-1≤t<0C0⁡(t)0≤t<1C-1⁡(t-1)1≤t<20otherwise.(2.14)
Thus, the Lagrange formula becomesy⁡(t)=∑n=-N/2+1N/2⁢y⁡(n)⁢Cn⁡(t)=∑n=-∞∞⁢y⁡(n)⁢f⁡(t-n).(2.15)
Therefore, the approach to reconstruct the continuous signal using the Lagrange polynomial is in fact equivalent to applying an interpolation filter f(t) to the available samples, with f(t) being a piecewise polynomial. The interpolator's impulse response f(t) obtained from (2.14) is shown in FIG.6A.

Taking the Fourier transform of f(t), we obtain its frequency response. This allows us to evaluate the interpolation accuracy by examining the frequency response of the interpolation filter. The frequency response702of the Lagrange cubic interpolator (N=4) is shown in FIG.7A. The horizontal axis is the normalized frequency f/Fs, with Fs=1/Ts. An ideal frequency response should have value one in the passband (0≦f/Fs<0.5) and be zero in the stopband (f/F3≧0.5).

For the interpolator using a trigonometric polynomial, we can express y(t) in terms of y(n) by substituting (2.9) into (2.10):y⁡(μ)=⁢1N⁢∑k=-N/2+1N/2-1⁢ck⁢WN-k⁢⁢μ+1N⁢cN/2⁢cos⁢⁢π⁢⁢μ=⁢1N⁢∑k=-N/2+1N/2-1⁢(∑n=-N/2+1N/2⁢y⁡(n)⁢WNkn)⁢WN-k⁢⁢μ+⁢1N⁢(∑n=-N/2+1N/2⁢y⁡(n)⁢WN(N/2)⁢n)⁢cos⁢⁢π⁢⁢μ=⁢1N⁢∑n=-N/2+1N/2⁢y⁡(n)⁢(∑k=-N/2+1N/2-1⁢WNk⁡(n-μ)+(-1)n⁢cos⁢⁢π⁢⁢μ)=⁢1N⁢∑n=-N/2+1N/2⁢y⁡(n)⁢(1+2⁢∑k=1N/2-1⁢cos⁢⁢2⁢⁢π⁢⁢kN⁢(μ-n)+cos⁢⁢π⁡(μ-n)).(2.16)
Definingf⁡(t)=(1+2⁢∑k=1N/2-1⁢cos⁢⁢2⁢⁢π⁢⁢kN⁢t+cos⁢⁢π⁢⁢tt≤N/20t>N/2(2.17)
we havey⁡(μ)=1N⁢∑n=-∞∞⁢y⁡(n)⁢f⁡(μ-n)=1N⁢y⊗fμ.(2.18)
The impulse response f(t) in (2.17) is shown in FIG.6B. The corresponding frequency response704of the trigonometric interpolator (for N=4) is shown inFIG. 7Ain thin lines.

By comparing the frequency responses of the two interpolators, we can see that the trigonometric interpolation response704has a sharper roll-off in the transition band and more rapid attenuation in the stopband than the Lagrange response702. These traits are enhanced as N increases, as demonstrated in FIG.7B. For N=32, the trigonometric response708has a sharper rolloff than the Lagrange response706as shown in FIG.7B.

Next we verify these observations by interpolating the samples of a practical time-domain signal. As an example, we interpolate a baseband signal with raised cosine spectrum and roll-off factor α =1.0, sampled at two samples per symbol period, as shown in FIG.8A.

The interpolation accuracy here is measured as the normalized mean-squared difference between the signal interpolated with an ideal interpolator and the signal interpolated with the practical interpolator. The normalized mean-squared error (NMSE), discussed above, is calculated for both the Lagrange interpolator and the trigonometric interpolator for a range of typical values of N. The results are plotted in FIG.8B.

Our test results demonstrate that the performance is improved with the trigonometric method. Using the same number of samples to interpolate, the proposed method produces a smaller NMSE, and the performance gain becomes greater as the number of samples increases.

2.4 Efficient Implementation Structures

Recalling from Section 2.2, the trigonometric interpolation algorithm includes substantially two steps:

Step1. Given a number of data samples N, calculate the Fourier coefficients ck, k=0, . . . , N/2 using (2.9). In a preferred embodiment, an even number of N data samples are used. In other embodiments, an odd number of data samples are used.

Step2. Compute the synchronized sample y(μ) for any given μ according to (2.11).

The first step involves multiplying the data samples by complex scaling factors WNkn. Since these factors lie on the unit circle, the computation in Step1can be simplified. Let us examine the case when N=4:

Example 2.1: For N=4, the Fourier coefficients are obtained as:
c1=y(−1)+y(0)+y(1)+y(2)
c1=[y(0)−y(2)]+j[−y(1)+y(−1)]
c2=y(0)−y(1)+y(2)−y(−1).  (2.19)
As seen in (2.19), there is no nontrivial scaling multiplier required for N=4.

Example 2.2: We now compute the coefficients ck in (2.9) for N=8. Using the trigonometric identities, we can obtain the following simple form for ck, k=0, . . . , 4:c0=⁢y⁡(-3)+y⁡(-2)+y⁡(-1)+y⁡(0)+⁢y⁢(1)+y⁡(2)+y⁡(3)+y⁡(4)c1=⁢{y⁡(0)-y⁡(4)+⁢[-y⁡(-3)+y⁡(1)_+y⁢(-1)-y⁡(3)_]⁢cos⁡(π/4)}+⁢j⁢{y⁡(-2)-y⁡(2)+⁢[y⁢(-3)-y⁡(1)_+y⁢(-1)-y⁡(3)_]⁢cos⁡(π/4)}c2=⁢{-y⁡(-2)+y⁡(0)-y⁡(2)+y⁡(4)}+⁢j⁢{-y⁡(-3)+y⁡(-1)-y⁡(1)+y⁡(3)}c3=⁢{y⁡(0)-y⁡(4)+⁢[y⁡(-3)-y⁡(1)_+(-y⁡(-1)+y⁡(3))_]⁢cos⁡(π/4)}+⁢j⁢{-y⁡(-2)+y⁡(2)+⁢[y⁡(-3)-y⁡(1)_+(-y⁡(-1)-y⁡(3))_]⁢cos⁡(π/4)}c4=⁢-y⁡(-3)+y⁡(-2)-y⁡(-1)+y⁡(0)-y⁡(1)+y⁡(2)-⁢y⁡(3)+y⁡(4).(2.20)

The only non-trivial scaling multiplications are those multiplications by cos (π/4). It appears that four such multiplications are needed to compute all the complex coefficients ck, k=0, . . . , 4. However, if we examine the data being multiplied by cos (π/4) (those terms embraced by the [ ] brackets), we observe that they are either the sums or differences of the [y(−3)−y(1)] and [y(−1)−y(3)) values. Therefore, we can compute [y(−3)−y(1)] cos(π/4) and [y(−1)−y(3)]cos(π/4), then use these two products to generate the cncoefficients. Thus, only two scaling multiplications are needed in computing all the coefficients.

Having observed the simplicity in the first step, let us focus on the second step. The second step may look complicated because of the complex multiplications ckWN−kμand cN/2e1πμ. However, since |WN−kμ|=|e1πμ|=1, these products are just rotations of points ckand CN/2in the complex plane. Furthermore, this is the same type of operation performed in the phase rotation for carrier recovery by the phase rotator124that is shown in FIG.1D. This suggests that we can reduce the total complexity of the synchronization circuitry by sharing some resources needed by the digital resampler122and the carrier phase rotator124. In one embodiment, a lookup table is utilized to determine the angle rotation associated with the angle2⁢πN⁢k⁢⁢μ
for rotation of the ckcoefficients. In another embodiment, an angle rotator processor is utilized. Both embodiments are discussed further below, and the angle rotator processor is discussed in detail in section 5.

FIG. 10illustrates a trigonometric interpolator1000that is one circuit configuration that implements the trigonometric interpolator equations (2.9)-(2.11), where the number of data samples is N=4. The interpolator1000is not meant to be limiting, as those skilled in the arts may recognize other circuit configurations that implement the equations (2.9)-(2.11). These other circuit configurations are within the scope and spirit of the present invention.

The trigonometric interpolator1000receives input data samples having two data samples that are to be interpolated at an offset μ (see FIG.2). The resulting interpolated value y(μ) represents the interpolated point202in FIG.2. The interpolator1000includes a delay module1004, an adder/subtractor module1006, and an angle rotator module1008, and an adder1012.

The delay module1004includes one or more delay elements1012. The delay elements1012can be configured using known components.

The adder/subtractor module1006includes multiple adders (or subtractors)1014, where subtraction is indicated by a (−) sign.

The angle rotator module includes two angle rotators1010. The angle rotators1010can be configured using an angle rotator processor or a table lookup (e.g. read only memory) as discussed below.

The operation of the trigonometric interpolator1000is discussed further in reference to the flowchart1700inFIG. 17, which is discussed below.

In step1702, the interpolator1000receives a set of N-input data samples. The N-data samples include the two data samples that are to be interpolated at the offset μ relative to one of the data samples, as shown in the FIG.2. InFIG. 2, the interpolation is to be performed between y(0) and y(1) at the offset μ to determine the interpolation value 202.

In step1704, the delay module1004delays the input data samples.

In step1706, the adder/subtractor module1006generates one or more trigonometric coefficients according to the equation (2.9). InFIG. 10, the coefficients are represented by C0, C1, and C2for N=4, where the coefficient C1is a complex coefficient.

In step1708, the angle rotators1010rotate appropriate real and complex coefficients in a complex plane according the offset a, resulting in rotated complex coefficients. More specifically, the angle rotator1010arotates the real coefficient C and the angle rotator1010brotates the complex coefficient C1in the complex plane.

In embodiments, as discussed herein, the angle rotators1010are table look-ups. In which case, a rotation factor is retrieved from the table lookup based on the offset μ, where the rotation factor includes the evaluated cosine and sine functions of2⁢πN⁢k⁢⁢μ
that are shown in the equations (2.21) below. The rotation factor is then multiplied by the corresponding real or complex coefficient, to generate the respective rotated complex coefficient. An interpolator1800having a table lookup ROM1802and a complex multiplier1804are shown inFIG. 18for illustration.

In step1710, the adder1012adds together C0, a real part of the rotated coefficient C1, and a real part of the rotated coefficient C2. The adder1012also scales the output of the adder as necessary according to equation 2.10. The resulting value is the desired interpolation value at the offset a, as represented by point202in FIG.2.

The trigonometric interpolator is not limited to the 4thdegree embodiment that is shown in FIG.10. The trigonometric interpolator can be configured as an Nthdegree interpolator based on N_data points, as represented by the equations (2.9)-(2.11). These other Nthdegree interpolators are within the scope and spirit of the present invention. For example and without limitation,FIG. 1Iillustrates an interpolator1100having N=8. The trigonometric interpolator1100includes: a delay module1102, an adder/subtractor module1104(having two scaling multipliers having coefficients cos (π/4)), an angle rotator module1106, and an adder1108(having a ⅛ scale factor that is not shown). The operation of the interpolator1100will be understood by those skilled in the arts based on the discussion herein.

2.4.1 Using a Lookup Table

For carrier recovery, the phase correction is generally accomplished by looking up the sine and cosine values corresponding to the phase, then by multiplying these values with the complex data. This requires the same operations as the rotation of ckby an angle2⁢πN⁢k⁢⁢μ,
that is:Re⁡(ck⁢WN-k⁢⁢μ)=Re⁡(ck)⁢cos⁢⁢2⁢⁢πN⁢k⁢⁢μ-Im⁡(ck)⁢sin⁢⁢2⁢⁢πN⁢k⁢⁢μ⁢⁢Im⁡(ck⁢WN-k⁢⁢μ)=Re⁡(ck)⁢sin⁢⁢2⁢⁢πN⁢k⁢⁢μ+Im⁡(ck)⁢cos⁢⁢2⁢⁢πN⁢k⁢⁢μ.(2.21)
The sine and cosine table can be used for both the resampler, as in (2.21), and the phase rotator for carrier recovery. In embodiments, a read only memory (ROM) is utilized as the lookup table. However, other embodiments could be utilized in including other types of memories. An interpolator1800utilizing a table lookup ROM1802and complex multiplier1804are shown inFIG. 18for illustration. The ROM table access time is insignificant as compared to the computation time in other parts of the interpolator. Therefore, this method results in low hardware complexity and low computational delay. This implementation will be referred to as the table-lookup method.

2.4.2 Using an Angle Rotation Processor

When a very low complexity implementation is desired at the expense of a slight increase in computational delay, we propose to use an efficient structure for angle rotation, which is described in Section 5. Based on this structure, each angle rotator has a hardware complexity slightly greater than that of two multipliers. In addition, a very small ROM is needed. We will subsequently refer to this particular implementation of our algorithm as the angle-rotation method.

Thus, there are at least two choices for implementing (2.21) as well as the phase rotator for carrier recovery. The trade-off is between complexity and speed. In a base-station where computation power is more affordable, the table lookup method might be used. In a hand-set, where cost is a major concern, an angle rotation processor might be used for both the resampler and the phase rotator, multiplexing the operations.

Now let us compare the complexity of the trigonometric resampler with that of the Lagrange method. Table 2.1 summarizes the comparisons for several typical N values. The numbers are based on the table-lookup method. It indicates that, for the same filter length N the trigonometric interpolation method needs less hardware.

2.5 Delays in the Computation

The critical path of the Farrow structure400(FIG. 4) is now compared to that of the trigonometric interpolator. The Farrow structure implements the Lagrange interpolator as discussed above. The Farrow structure400is shown inFIG. 9(or FIG.4), with the critical path902indicated. The critical path902for this structure includes one scaling multiplier904and N−1 data multipliers906.

In contrast, the critical path for the trigonometric interpolator1000is path1002and it contains just one angle rotation1010, or only one multiplier if the table-lookup method is employed to replace the angle rotator1010. Since the angle rotations for various angles can be carried out in parallel, the critical path does not lengthen as N increases.

Table 2.1 compares the computational delays for the trigonometric interpolator with that of the Lagrange interpolator for various values of N. The delay data for the trigonometric interpolator1000are based on the table-lookup method. As shown inFIG. 10, the trigonometric interpolator (for N=4) has only one multiplier in the critical path, whereas the Lagrange interpolator has 4 multipliers in the critical path. Therefore, the trigonometric interpolator has less latency than the Lagrange interpolator, which is important for voice communications.

2.6 Simplifications of the Preliminary Structures

As mentioned in Section 2.4, to produce each y(μ) we first calculate the Fourier coefficients ck using existing samples, according to (2.9). We then compute Re(ckWN−kμ) to be used in (2.11). This is accomplished either by retrieving WN−kμfrom a lookup table, followed by two real multiplications, or by an angle-rotation processor.

2.6.1 The Simplified Structure for N=4

Let us examine the trigonometric interpolator1000having N=4. To compute Re(ckWN−kμ) and Re(c2W4−2μ) the system requires either two angle rotators1004or two accesses to a lookup table.

If the input samples would happen to be such that c2=0then one fewer angle rotator, or one fewer ROM access, would be required. Of course, the original data samples y(−1), y(0), y(1), and y(2) are not likely to have the special property that c2=(0)−y(1)+y(2)−y(−1)=0. However, if the data samples are changed, then the modified samples can be determined to satisfy c2=0. If the modified samples for interpolation, then the c2angle rotator1010acan be eliminated. However, the interpolation result will not then correspond to the original data samples.

It seems that the data samples can be changed to attain savings in hardware, as long as the interpolation result is fixed so that it corresponds to the original data samples. Of course, it must also cost less in hardware to “fix” the interpolation result than is saved in hardware by using the modified samples.

If the samples y(k) are modified to form {tilde over (y)}(k) according to:
{tilde over (y)}(−1)=y(−1)−K
{tilde over (y)}(0)=y(0)
{tilde over (y)}(1)=y(1)+K
{tilde over (y)}(2)=y(2)+2K(2.22)
then the K value can be adjusted to force the {tilde over (y)}(k) samples to satisfy c2=0, where K is the slope of a straight line1202in FIG.12.

To find K, the c2value that corresponds to the modified samples is determined according to:c2=y~⁡(0)-y~⁡(1)+y~⁡(2)-y~⁡(-1)=y⁡(0)-(y⁡(1)+K)+(y⁡(2)+2⁢K)-(y⁡(-1)-K)=2⁢K-(y⁡(1)+y⁡(-1)-y⁡(0)-y⁡(2)).(2.23)
To force c2=0, requires:K=12⁢(y⁡(1)+y⁡(-1)-y⁡(0)-y⁡(2))(2.24)
Therefore, the c2angle-rotator can be eliminated, and c0and c1are determined accordingly as,
c0=2(y(1)+y(−1))
c1=[2y(0)−y(1)−y(−1)]+j[−2y(1)+y(0)+y(2)]  (2.25)
Then, the interpolated sample isy~⁡(μ)=14⁢c0+12⁢Re⁡(c1⁢ⅇj⁢π2⁢μ).(2.26)

However, {tilde over (y)}(μ) should be adjusted so that it corresponds to the original samples. FromFIG. 12, the values expressing the difference between the original and the modified samples lie on the straight line1202. FromFIG. 13, it follows that the offset due to the modification of samples is Kμ. Therefore, the {tilde over (y)}(μ) value can be compensated by:
y(μ)={tilde over (y)}(μ)−Kμ.(2.27)

Using equations (2.25), (2.26) and (2.27) leads to an interpolator1400as shown in FIG.14. This simplified interpolator structure is not limited to N=4 configurations. In fact, this simplification technique can be applied to an interpolator with an arbitrary N value. To eliminate the angle-rotation needed by Re(cN/2ejπμ) in (2.11), the samples are modified according to
{tilde over (y)}(n)=y(n)+nK, n=integer.  (2.28)
Using (2.9), and then applying (2.28), results incN/2=1N⁢(∑n=-N/2+1N/2⁢(-1)n⁢y~⁡(n))=1N⁢(∑n=-N/2+1N/2⁢(-1)n⁢y⁡(n))+12⁢K.(2.29)
If we chooseK=-2N⁢∑n=-N/2+1N/2⁢(-1)n⁢y⁡(n)(2.30)
we can force CN/2=0.

Referring toFIG. 14, the interpolator1400includes the delay module1004, an adder/subtractor module1402, the angle rotator1010b, a multiplier1404, and an adder1406. The interpolator1400has a simplified structure when compared the interpolator1000(in FIG.10), as the interpolator1400replaces the angle rotator1010bwith a multiplier1404. As discussed above, this can be done because the coefficient CN/2=0 (C20 for N=4) by modification of the data samples, and therefore there is no need to have an angle rotator for CN/2. The operation of the interpolator1400is further discussed in reference to the flowchart1900that follows.

In step1902, the interpolator1400receives a set of N input data samples. The N data samples include two of the data samples that are to be interpolated at the offset μ, as shown in the FIG.2.

In step1904, the delay module1004delays the input data samples.

In step1906, the adder/subtractor module1402modifies one or more of the data samples so that a coefficient cN/2is 0. In embodiments the data samples are modified according to y(n)mod=y(n)+n·K, wherein K is determined by the equation (2.30) above so that cN/2is 0, and wherein y(n) represents the nthdata sample of the N data sample set.

In step1908, the adder/subtractor module1402generates one or more trigonometric coefficients according to modifications to the equation ([2.8]2.9). In the N=4 case, equations (2.25) are implemented by the module1402. InFIG. 14, for N=4, the coefficients are represented by C0and C1, where the coefficient C1is a complex coefficient. By comparing withFIG. 10, it is noted that the C2coefficient is 0. Additionally, the adder/subtractor module1402outputs the K value for further processing. Notice also that inFIG. 14, the output scaling factor has been changed from ¼ to ½. This reflects several other straightforward simplifications that have been made to module1402and in the angle rotator1101b. In embodiments, the steps1906and1908are to be performed simultaneously by the adder/subtractor module1402, as will be understood by those skilled in the relevant arts.

In step1910, the angle rotator1010brotates the complex coefficient C1in a complex plane according the offset μ, resulting in a rotated complex coefficient. In embodiments, as discussed herein, the angle rotator1010bis table look-up. In which case, a complex rotation factor is retrieved from the table lookup based on the offset A, and the resulting rotation factor is then multiplied by the corresponding complex coefficient, to generate the respective rotated complex coefficient. The rotation factor includes the evaluation of the cosine and sine factors that are shown in equations (2.21). Note that since C2=0, the angle rotator1010ais replaced with the multiplier1404.

In the step1912, the multiplier1404multiplies the K factor by the offset μ, to produce a Kμ factor.

In step1914, the adder1406adds together C0and Kμ and a real part of the rotated complex coefficient C1, and scales the sum by the trivial factor ½, to produce the desired interpolation value. The addition of the Kμ factor compensates the desired interpolation value for the modification that was done to the data samples in order to force C to zero in step1906.

The simplified trigonometric interpolator is not limited to the four-sample embodiment that is shown in FIG.14. The simplified trigonometric interpolator can be configured as an N-sample interpolator based on N_data points, as represented by the equations (2.28)-(2.30). These other N-sample interpolators are within the scope and spirit of the present invention. For example and without limitation, an interpolator with N=8 is discussed below.

2.6.2 The Simplified Structure for N=8

The resulting modified structure for N=8 is shown inFIG. 15as interpolator1500. Similar to the interpolator1400, the interpolator1500includes a delay module1504, an adder/subtractor module1506, an angle rotator module1508, a multiplier1510, and an output scaling adder1512. As in the interpolator1400, the multiplier1510substantially replaces an angle rotator module. As in the interpolator1100for N=8 (FIG.11), only two non-trivial scaling multiplications are needed for the modified structure1500.

2.3 Performance Comparisons with Other Structures

How does the simplified interpolator1400(FIG. 14) perform as compared to the interpolator1000(FIG.10)?FIGS. 16A-Cshow the frequency responses, in solid lines, of the Lagrange cubic interpolator400(FIG.4), the interpolator1000(FIG. 10) and the simplified interpolator1400(FIG.14), respectively. For an input signal whose spectrum is a raised cosine with α=0.4, as shown in dashed lines, the amount of interpolation error corresponds to the gray areas. Clearly, the interpolator1400produces less error than the Lagrange cubic interpolator400and the interpolator1000. (FIG. 16Dwill be discussed in Section 4.)

Next, let us verify this performance improvement by interpolating two signals: Signal1, which is the same as signal802inFIG. 8A(M=1.0), and Signal2, which is signal1602inFIG. 16(α=0.4). Then, the NMSE values are compared. We use three interpolators: 1) Lagrange cubic interpolator400; 2) the trigonometric interpolator1000; and 3) and the trigonometric interpolator1400. Also compared is the number of multipliers.

The results in Table 2.2 show that the modified structure for N=4 not only requires less hardware, it also obtains the highest accuracy among the three methods for these practical signals used in our simulation.

TABLE 2.2Comparison of interpolators for N = 4.LagrangeStructure inStructure inN = 4cubicFIG. 10FIG. 14NMSE for Signal 1 in dB−25.80−28.45−29.41NMSE for Signal 2 in dB−31.08−31.21−33.51Nontrivial scaling200multipliersData multipliers3**Multipliers in critical path422*The trigonometric interpolator 1000 employs two one-output angle-rotators, each having the hardware equivalent of slightly more than two multipliers. The trigonometric interpolator 1400 employs one such angle-rotator and one multiplier yielding an equivalent of slightly more than three multipliers.

An important application of the interpolation method and apparatus described in this patent is the following. It is often necessary to increase the Li sampling rate of a given signal by a fixed integer factor. For example, a signal received at a rate of 1000 samples per second might need to be converted to one at a rate of 4000 samples per second, which represents an increase of the sampling rate by the integer factor four. There are methods in common practice for doing such a conversion. One method is a very popular two-step process wherein the first step creates a signal at the desired higher sampling rate but one where simple zero-valued samples (three of them in the example situation just mentioned) are inserted after each input data value. The second step in the interpolation scheme is to pass this “up-sampled” or “data-rate expanded” signal through an appropriately designed lowpass digital filter which, in effect, smoothes the signal by “filling in” data values at the previously inserted zero-valued samples. In the process of doing this filtering operation it might or might not be important that the original data samples remain intact, and when this is important there exist certain special lowpass filters that will not alter those samples.

We can easily adapt the trigonometric interpolator described herein to efficiently create such a sampling rate conversion system, but one that does not require such filtering operations. If we denote the integer factor by which we desire to increase the data rate as L (in the above example, L=4) we proceed as follows. We build the system7800shown below in FIG.78. System7800includes a Delay Module7802and Add/Subtract Module7804(that are similar to those in FIG.10), and such that it can accommodate incoming data at a rate r. We now build L copies of the Angle-Rotation Module7806(similar to that in FIG.10), with each one being fed by the same outputs of the Add/Subtract Module. Within each of these L Angle-Rotation Modules7806we fix the μ value; that is, each one has a different one of the values: 1/L, 2/L, . . . , (1−L)/L. With such fixed μ values, each Angle-Rotation Module7806can be constructed as a set of fixed multipliers (a very special case of the table-lookup method), although any of the Angle-Rotation Module implementations previously discussed can be employed.

As shown inFIG. 78, the L−1 outputs, i.e., the interpolated samples that are offset by the values 1/L, 2/L, . . . , (L−1)/L from the first of the two data points (indicated as μ=0 and μ=1 in the Delay Module ofFIG. 78) are routed to a multiplexer7808, along with the input data point from which all interpolated samples are offset. The multiplexer7808simply selects these samples, in sequence, and provides them to the output at the expanded data rate L×r.

A major advantage of the system7800is that all of the system's components are operated at the (slow) input data rate except the output multiplexer7808. If desired, it would also be possible to employ fewer Angle-Rotation Modules7806, but operating them at a higher data rate, and using several μ values, sequentially, for each. This would result in a system that employed less hardware but one that traded off the hardware savings for a higher data rate operation of such modules.

In this Section we have described an interpolation method that we have devised that uses trigonometric series for interpolation. Comparing the interpolations using the trigonometric polynomial and the Lagrange polynomial of the same degree, the trigonometric-based method achieves higher interpolation accuracy while simultaneously reducing the computation time and the amount of required hardware. Moreover, the trigonometric-based method performs operations that are similar to those of a phase rotator for carrier phase adjustment. This allows a reduction in the overall synchronization circuit complexity by sharing resources.

This trigonometric interpolator yields less computational delay, as compared to algebraic interpolators. To achieve the same throughput rate, this translates into more savings in hardware using the proposed structure, because the data registers that are required by algebraic interpolators to pipeline the computation for a faster rate would not be needed by our structure.

We have also introduced two implementations of the trigonometric interpolation method: one using a lookup table, and one using an angle rotation processor (to be discussed in Section 5).

After introducing a first interpolation method, we have shown that we can trade one angle rotator for a multiplier by conceptually modifying the input samples, then by “correcting” the interpolated value obtained from the “modified” samples. Through this modification, we have obtained a simpler implementation structure while simultaneously improving the performance when interpolating most practical signals. This performance improvement has been demonstrated by comparing the frequency responses of the interpolators and the mean-squared interpolation errors using these interpolators. Our discussion of the optimal digital resampler in Section 4 will be based on this simplified interpolator.

3. Interpolation Filters with Arbitrary Frequency Response

In Section 2, an interpolation method using a trigonometric polynomial was introduced, along with an example of such an interpolation structure of length N=4. In addition to being a very simple structure, our analyses and simulations also demonstrated that the trigonometric interpolator outperformed the interpolator of the same length using a Lagrange polynomial. In this Section, a more systematic approach will be taken to analyze this method from the digital signal processing point of view. A formula giving its impulse response allows us to analyze the frequency response of the interpolation filter. We then show how to modify the algorithm to achieve arbitrary frequency response. The discussions in this Section will provide the framework for the optimal interpolator of Section 4.

3.1 Formulating the Trigonometric Interpolator as an Interpolation Filter

We have shown that, given N equally-spaced samples y(n), a continuous-time signal can be reconstructed asy⁡(t)=∑n=-N/2+1N/2⁢y⁡(n)⁢f⁡(t-n)(3.1)
where f(t) is the impulse response of a continuous-time interpolation filter. As in Section 2, it is assumed that the sampling period is Ts=1. This assumption makes the notation simpler and the results can easily be extended for an arbitrary Ts. In other words, the invention is not limited to a sampling period of T,=1, as other sampling periods could be utilized. In Section 2 we have shown that f(t) can be expressed as:f⁡(t)=(1+2⁢∑k=1N/2-1⁢cos⁢2⁢⁢πN⁢k⁢⁢t+cos⁢⁢π⁢⁢tt≤N/20t>N/2.(3.2)

FIG. 20illustrates f for the trigonometric interpolation filters for N=8 and N=16. By computing the Fourier transform off, we obtain the frequency response of the interpolation filter. The frequency responses for the N=8 and N=16 cases are plotted in FIG.21. Since f(t) is real and symmetric around t=0, its frequency response has zero phase. InFIG. 21, the oscillatory behavior near the band edge is quite obvious. In addition, by comparingFIGS. 21aandb, we can see that as the filter length is increased from N−8 to N=16 the amount of ripple does not decrease. Well known as the Gibbs phenomenon, the magnitude of the ripples does not diminish as the duration of the impulse response is increased.

It is apparent that the amount of oscillation cannot be reduced using the method discussed thus far while only increasing the filter length N. Moreover, it seems that an arbitrary frequency response cannot be achieved using this method. To address these problems, let us examine how the frequency response of this method is determined.

3.2 Analysis of the Frequency Response

Let us examine how the frequency response off for the trigonometric interpolator is obtained using the example inFIG. 22with N=8. According to (3.2), the interpolation filter's impulse response f(t) on the interval-N2≤t≤N2
is a weighted sum of cosine functions. We can view the finite-length filter f in (3.2) as having been obtained by applying a window2206according to the following:w⁡(t)={1t≤N/20t>N/2(3.3)
to an infinitely-long, periodic function2204with period N:fc⁡(t)=⁢1+2⁢∑k=1N/2-1⁢cos⁢2⁢⁢πN⁢k⁢⁢t+cos⁢⁢π⁢⁢t⁢-∞<t<∞=⁢∑k=-N/2+1N/2-1⁢WNk⁢⁢t+12⁢WN-(N/2)⁢t+12⁢WN(N/2)⁢t(3.4)
such that
f(t)=fc(t)w(t), −∞<t<∞.  (3.5)
Thus F the frequency response off can be obtained by convolving Fcand W the Fourier transforms of fcand w, respectively.

The Fourier transform of the periodic function fc(t), −∞<t<∞, isFc⁡(Ω)=∑k=-N/2+1N/2-1⁢δ⁡(Ω-2⁢⁢πN⁢k)+12⁢δ⁡(Ω-π)+12⁢δ⁡(Ω+π)(3.6)
which consists of a sequence of impulses2208. We will subsequently refer to the weights of these impulses as frequency samples. Denoting the weight ofδ⁡(Ω-2⁢⁢πN⁢k)⁢⁢by⁢⁢F^⁡(k),we⁢⁢haveFc⁡(Ω)=∑k=-MM⁢F^⁡(k)⁢δ⁡(Ω-2⁢⁢πN⁢k)(3.7)
whereM≥N2
is an integer. In the case of (3.6),M=N2.
For our particular interpolation filer, according to (3.6), all in-band frequency samples {circumflex over (F)}(k)=1 for |k|<N/2. For |k|>N/2, the out-of-band samples F(k)=0. The two samples in the transition band are {circumflex over (F)}(N/2)={circumflex over (F)}(−N/2)=½. The transition bandwidth is determined by the distance between the last in-band, and the first out-of-band frequency samples.

Since w is a rectangular function, W must be a sinc function2210. Convolving Fcand the sinc function W simply interpolates the frequency samples {circumflex over (F)}(k) to obtain F(Ω), −∞<Ω<∞, shown as response2212. (Here we have plotted the symmetric F only on the positive half of the Ω axis.) We thus haveF^⁡(k)=F⁡(Ω)Ω=2⁢⁢π⁢⁢k/N,-N2≤k≤N2.(3.8)
From response2212, the continuous-frequency response F(Ω) is uniquely determined by an infinite number of equally spaced frequency samples {circumflex over (F)}(k). If we modify the frequency samples2214near the passband edge to let the transition between the passband and stopband be more gradual, as depicted inFIG. 23, then the ripple is decreased.FIG. 23demonstrates gradually reduced samples2302, and the reduction of ripples in the overall response2304, as compared to the response2212in FIG.22. The cost of this improvement is an increased transition bandwidth in the response2304, as compared to the response2212.

If a narrower transition band is desired, we can increase the duration of the filter f(t). This can be seen by comparingFIG. 24, where N=16, with response2304inFIG. 23, in which N=8.

3.3 Implementing the Modified Algorithm

By comparing (3.6) and (3.4) we can see that the frequency sample values, i.e., the weights of the impulses in (3.6), are determined by the weights in the sum in (3.4).

We can modify our original interpolation filter in (3.2) for |t|≦N/2 asfm⁡(t)=F^⁡(0)+2⁢∑k=1M⁢⁢F^⁡(k)⁢cos⁢2⁢πN⁢kt.(3.9)
By expressing (3.9) using the WNnotation, for |t|<N/2 we havefm⁡(t)=F^⁡(0)⁢WN0⁢t+∑k=1M⁢⁢F^⁡(k)⁢(WNkt+WN-kt)=∑k=-MM⁢⁢F^⁡(k)⁢WN-kt.(3.10)
and fm(t)=0for |t|>N/2. Substituting this result into (3.1), and re-ordering terms, we havey⁡(t)=1N⁢∑k=-MM⁢⁢c^k⁢WN-kt⁢⁢where(3.11)c^k=F^⁡(k)⁢(∑n=-N/2+1N/2⁢⁢y⁡(n)⁢WNkn).(3.12)
By comparing (3.12) to (2.9), we can see that, fork=-N2+1,…⁢,N2,⁢c^k=F^⁡(k)⁢ck.(3.13)

Thus, a modified algorithm can be implemented as the following steps:

Step2′: Multiply the coefficients Ck by scale factors {circumflex over (F)}(k) using (3.13).

Step3′: Given a fractional delay p, compute the synchronized samples using (3.11), which, due to Ĉk*=Ĉ−k, can be simplified as:y⁡(μ)=1N⁢Re⁡(c^0+2⁢∑k=1M⁢⁢c^k⁢WN-k⁢⁢μ).(3.14)
It seems that, in Step3′, we need coefficients c(k) (hence c, not only for k≦N/2 but also for k>N/2 while only Ck values for k≦N/2 are computed in Step1′. However, Ckvalues for k>N/2 can be obtained using
ck=ck−mN(3.15)
where m is an integer such that 0≦k−mN≦N/2. We have (3.15) because ckis periodic in k with period N, because ckis obtained from the Fourier transform of the discrete-time signal y(n), −N/2+1≦n≦N/2.

At this point, we have shown that the continuous-time frequency response of the interpolation filter having impulse response f(t) can be improved by modifying the weights {circumflex over (F)}(k) in (3.10). Now a question arises: the modification of the weights would alter the shape of the impulse response of the f(t) filter. How do we guarantee that the resulting filter does not change the original samples?

3.4 Conditions for Zero ISI

In order for f(t) not to alter the original samples when used for interpolation as in (3.1), it must have zero-crossings at integer multiples of the sampling period:f⁡(n)={1n=00n≠0,n⁢⁢an⁢⁢integer.(3.16)
The well-known Nyquist condition for zero ISI (Proakis, J. G.,Digital Communication, McGraw-Hill, New York, N.Y. (1993)) states that the necessary and sufficient condition for (3.16) is∑n=-∞∞⁢⁢F⁡(Ω-2⁢π⁢⁢n)=1⁢-∞<Ω<∞.(3.17)
Since the filter's impulse response f(t) has a finite duration, i.e. f(t)=0 for |t|>N/2, (3.16) holds if and only if the frequency samples F(k) satisfy∑n=-∞∞⁢F^⁡(k-Nn)=1,⁢k=integer.(3.18)
The proof is given in Appendix A.

In summary, we can still guarantee that the modified interpolation filter f does not alter the original samples as long as the modified weights t(k) (frequency samples) satisfy (3.18). Using this constraint, we can design the weights F(k) to meet given frequency response requirements.

Using the approach discussed, one can approximate an arbitrary frequency response by choosing appropriate weights {circumflex over (F)}(k). For example, a desired frequency response Fd(Ω) for an interpolation filter should be unity in the passband and be zero in the stopband, asFd⁡(Ω)={1-π2≤Ω≤π20Ω>π.(3.19)
The interpolation error using our interpolation filter is defined as
e(Ω)=Wt(Ω)|Fd(Ω)−F(Ω)  (3.20)
where Wt(Ω) is a weighting function.

From Section 3.2 we have F(Ω)=Fc(Ω){circle around (×)} sinc(Ω). Thus, we can express F(Ω) in terms of {circumflex over (F)}(k), using (3.7), asF⁡(Ω)=⁢(∑k=-MM⁢F^⁡(k)⁢δ⁡(Ω-2⁢⁢πN))⊗sin⁢⁢c⁡(Ω)=⁢∑k=-MM⁢F^⁡(k)⁢(Ω-2⁢⁢πN⁢k).(3.21)

An optimal interpolation filter can be designed by choosing {circumflex over (F)}(k) to minimize the peak interpolation error, asL∞=,maxΩ⁢{e⁡(Ω)}(3.22)
or the mean-squared interpolation error
L2=∫-∞∞⁢e2⁡(Ω)⁢⁢ⅆΩ(3.23)
subject to the constraint described by (3.18).

By examiningFIGS. 23 and 24, we can see that, by modifying only two frequency samples, those nearest the band edge, a significant improvement is achieved. In these cases we have {circumflex over (F)}(k)=0 for |k|>N/2+1.

In this Section, an interpolation method was presented that achieves arbitrary frequency response by modifying the trigonometric interpolator discussed in Section 2. Using this approach, the performance of a trigonometric interpolation filter can be further improved.

It is interesting to note that this procedure is equivalent to the well-known filter design method using windows.FIG. 25adepicts the impulse responses of the original filter (3.2) as the dashed line, and the modified filter (3.9) as the solid line. By comparing the two impulse responses, we have found a function illustrated inFIG. 25b. If we multiply the original impulse response by this function, we get the impulse response that we obtained by modifying the frequency samples. Therefore, this function is equivalent to a window. According to this interpretation, our frequency domain design method is equivalent to designing a better window than the rectangular window (3.3) in the time domain.

4. Design of Optimal Resamplers

We have thus far discussed digital resampling using interpolation methods. To accomplish this, we conceptually reconstruct the continuous-time signal by fitting a trigonometric polynomial to the existing samples and then re-sample the reconstructed signal by evaluating this polynomial for a given sampling mismatch (or offset) μ. The reconstruction of the continuous-time bandlimited signal y(t) from existing samples y(m) using interpolation filter f(t), according to (3.1), isy⁡(t)=∑m=-N/2+1N/2⁢y⁡(m)⁢f⁡(t-m).(4.1)

Then y is resampled at t=μ asy⁡(μ)=∑n=-N/2+1N/2⁢y⁡(m)⁢f⁡(μ-m)=y⊗fμ(4.2)
where fμ(m)=f(m−μ)

In the previous sections we approached the problem only from the point of view of reconstructing the continuous-time signal, as in (4.1), since we have only examined the frequency response of the continuous-time filter f(t). However, what we actually are interested in is the new sample y(μ) that is obtained by resampling the continuous-time signal at a new sampling instant, t=μ.

Now, a question arises: Even when the frequency response F(Ω) of the continuous-time filter is optimized as in Section 3.5, do we necessarily obtain the minimum error in producing a new sample y(μ) for each μ value?

According to (4.2), the new sample y(μ) is actually obtained by filtering the original samples y(m) using a discrete-time filter whose impulse response fμ(m) depends on a particular delay11. What is the desired frequency response of the fμ(m) filter?

A digital resampler that is used in timing recovery simply compensates for the timing offset in sampling the received signal. Ideally, the fμ(m) filter should not alter the signal spectrum as it simply delays the existing samples by μ. Obviously, the desired frequency response of the discrete-time fμ(m) filter is
Fd(ω, μ)=ejωμ
where ω is the normalized angular frequency. Let us define the frequency response of f82(m) as Fμ(ω). The error in approximating the ideal frequency response Fd(ω, μ) by Fμ(ω) for a given μ value, is
e(ω)=W1(ω)|Fd(ω, μ)−Fμ(ω)|  (4.4)
where Wt(ω) is a weighting function.

We now examine how the discrete-time fractional-delay frequency response Fμ(ω) is obtained. We denote by F(Ω) the Fourier transform of the continuous-time filter f(t). Hence, the Fourier transform of f(t−μ) must be e−jΩμF(Ω). We know that fμ(n)=f(n−μ) are just samples of f(t−μ), where −∞<t<∞. Therefore, according to the sampling theorem (Proakis, J. G.,Digital Signal Processing, Macmillan, New York, N.Y. (1992)), the Fourier transform of fμ(n) isFμ⁡(ω)=∑k=-∞∞⁢ⅇ-j⁡(Ω-2⁢⁢π⁢⁢k)⁢μ⁢F⁡(Ω-2⁢⁢π⁢⁢k)(4.5)
after we replace Ω on the right-hand-side expression by the normalized angular frequency variable ω(ω=Ω since TS=1). This relationship is shown inFIG. 26, where F(Ω) corresponds to the N=8 interpolator of FIG.11. As discussed in Section 3.2, f(t) is symmetric around t=0. This implies that F(Ω) has zero phase. To make the f(t) filter physically realizable, of course, we must introduce a delay of N/2, where N corresponds to the length of the filter. However, this delay simply “shifts” all input samples by N/2, which is an integer because N is even, and it does not change the characteristic of the input signal. Thus, it does not influence the interpolation accuracy. Therefore, to simplify our notation, we just use F(Ω) as a real function. Hence, the phase of the complex function e−j≠μF(Ω) is −Ωμ if F(Ω)≧0, or −Ωμ+π if F(Ω)<0 —the phase depends on μ.FIG. 26shows that the frequency response of the discrete-time filter Fμ(ω) is obtained by first making an infinite number of copies of e−jΩμF(Ω) by shifting this function uniformly in successive amounts of 2π, then by adding these shifted versions to e−jΩμF(Ω). As a sum of complex functions, the shape of Fμ(ω) depends not only on the shape of the continuous-time frequency response F(Ω) but also on the value μ. The dependence of Fμ(ω) on μ is illustrated inFIG. 27, where Fμ(ω) is obtained from the function F(Ω) inFIG. 26, using μ=0.12 and μ=0.5. The magnitude of the ideal fractional-delay frequency response, defined in (4.3), is shown in both FIG.27A andFIG. 27Bas the dashed lines. It is evident that the frequency response2706is worse for μ=0.5 than the frequency response2704is for μ=0.12, since the response2706deviates more from the ideal frequency response2702than does the frequency response2704. Hence, the interpolation error is larger for μ=0.5 than for μ=0.12. We have observed in our simulations that the largest interpolation error occurs when μ=0.5, i.e., when the interpolated sample is exactly in the middle of the two nearest existing samples. As p approaches 0 or 1 (i.e., as the interpolated sample gets closer to an existing sample), the interpolation error becomes smaller. Moreover, the interpolation errors obtained for μ and 1−μ are the same.

In Section 3, we analyzed the relationship between the weights {circumflex over (F)}(k) in (3.9) and the frequency response of the interpolator. We have shown that we can enhance F(Ω) by adjusting the {circumflex over (F)}(k) values. InFIG. 23(N=8), for example, the {circumflex over (F)}(3) and {circumflex over (F)}(5) values correspond to the magnitude of the pulses2302near the band edge. If we adjust {circumflex over (F)}(3) and {circumflex over (F)}(5) such that the transition between the passband and stopband is more gradual, we can achieve a better frequency response.

To further improve the interpolation performance, we could take g into account, by optimizing Fμ(ω) for each μ value. As in Section 3, we could adjust {circumflex over (F)}(k) near the band edge to change the shape of F(Ω), for each μ value, such that the discrete-time frequency response Fμ(Ω), which is obtained from (4.5), best approximates the desired response of (4.3).

As discussed in Section 3, to guarantee that the original samples are not altered using the modified interpolator, the weights {circumflex over (F)}(k) should satisfy (3.18). When N=8, for example, we modify {circumflex over (F)}(3) and {circumflex over (F)}(5) together in order to satisfy (3.18). Here, however, our goal of optimization is to obtain the best approximation for the sample corresponding to a specific μ. Hence we need not be concerned with the issue of altering the original samples in Section 3, where there is only one set of optimized weights for all μ values.

Let us demonstrate this method using the example of N=8. We chose to modify {circumflex over (F)}(3) and {circumflex over (F)}(4). For each given μ, we search for the {circumflex over (F)}(3) and {circumflex over (F)}(4) values such that (4.4)is We denote such {circumflex over (F)}(3) and {circumflex over (F)}(4) values by {circumflex over (F)}μ(3) and {circumflex over (F)}μ(4) respectively, since they are now also dependent on μ.FIGS. 28A and Bshow the modifications to F(Ω) for μ=0.12 and μ=0.5, respectively. The corresponding optimized Fμ(ω) functions are illustrated inFIGS. 28C and 28D, respectively.

To demonstrate the performance improvement, let us use this example: Given μ=0.5, we optimize Fμ(ω) for the signal whose spectrum is shown in dashed lines2902in FIG.29. Comparing the un-optimized frequency response2904with the optimized frequency response2906, the modification clearly produces a better frequency response. More specifically, the response2906is flat in the frequency band where the power of the signal2902is concentrated, and the deviation from the ideal response mostly falls in the “don't care” band.

Similar to Section 3, to implement this improved method, we first compute the coefficients ck from the existing samples as in (2.9). Then, given the μ value, we multiply, e.g., for N=8, the c3and c4values by {circumflex over (F)}μ(3) and {circumflex over (F)}μ(4), respectively. Finally, we compute the synchronized sample y(μ) using (2.11), where c3and c4are replaced by {circumflex over (F)}μ(3) c3and {circumflex over (F)}μ(4) c4, respectively.

We can apply similar modifications to the interpolator with N=4.FIG. 30Ashow the frequency response of the interpolator1000, for μ=0.5, whileFIG. 30Bdisplays the results of a modified interpolator1000, where parameters {circumflex over (F)}(1) and {circumflex over (F)}(2) are optimized, for μ=0.5, to maximize the interpolation accuracy for the signal whose spectrum is shown in dashed lines. As can be seen the optimized response3006is flatter in the part of the spectrum of the signal3002where most of its energy is concentrated than is the un-optimized response3004.

The flowchart3400inFIG. 34generalizes the optimization of the trigonometric optimization procedure. The flowchart3400is similar to the flowchart1700, but includes the additional steps of3402and3404that are described as follows.

In step3402, a factor {circumflex over (F)}μis determined to adjust the frequency response of the trigonometric interpolator so that it is consistent with the frequency response of N-data samples and the offset μ.

In step3404, one or more of the complex coefficients are multiplied by the {circumflex over (F)}μto modify the frequency response of the interpolator so that it is consistent with the input signal and the offset μ.

The optimization routine can also be used with K-modified data samples that leads to the simplified interpolator structures ofFIGS. 14 and 15. The flowchart3500illustrates the {circumflex over (F)}μfactor modification in the context of the flowchart1900.

As will be shown in the section that follows, the steps3402,3404, and1708can be combined into a single step if a table lookup is used to determine the rotation factor. In other words, the sine and cosine values can be multiplied by the {circumflex over (F)}μfactor before they are stored in the ROM.

In Section 2.6, we have presented an efficient algorithm that eliminates one angle-rotation. For example, for N=4, we can “modify” the input samples according to (2.22). With this modification, we can treat the new samples as if the input signal satisfies c2=0. The remaining non-zero coefficients are c0and c1. In the example for N=4 in the previous Section, two parameters, {circumflex over (F)}μ(1) and {circumflex over (F)}μ(2), are optimized to achieve the best approximation of a desired fractional-delay frequency response described by (4.3). Now, with c2=0, we have only one parameter, {circumflex over (F)}μ(1), to choose.

The impulse response of the simplified interpolation filter is derived in Appendix B. From the mathematical expression of the impulse response (B. 5), we can obtain the corresponding frequency response. The frequency responses of the interpolator1400(FIG. 14) before and after applying the {circumflex over (F)}μ(1) modification are shown inFIG. 31A-B, respectively. We can see an improved frequency response3106over the response3104, as the response3106is flatter in the part of the signal3102where its energy is concentrated. Furthermore, it seems that the frequency response3106, where only c1is modified (c2=0!), is as good as the modified response3006inFIG. 30Bwhere both c1and c2are modified.

It may appear that additional hardware is needed to implement the multiplication by, for example, {circumflex over (F)}μ(1) for the simplified N=4 structure. Let us examine the corresponding computations. As we know, we first compute coefficients c0and c1according to (2.25) (c2=0, of course). We then compute y(μ) usingy⁡(μ)=14⁢(c0)+12⁢Re⁡(F^μ⁡(1)⁢c1⁢ⅇj⁢⁢π2⁢μ)-K⁢⁢μ(4.6)
according to (2.26) and (2.27), where K is defined in (2.24). As discussed in Section 2.4, the computationRe⁡(F^μ⁡(1)⁢c1⁢ⅇj⁢π2⁢μ)=Re⁡(c1)⁢(Re⁡(F^μ⁡(1)⁢ⅇj⁢π2)-Im⁡(c1)⁢Im⁡(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)(4.7)
can be accomplished by retrieving theF^μ⁡(1)⁢ⅇj⁢π2⁢μ
value from a ROM lookup table and then multiplying Re(c1)+j Im(c1) by the retrieved value, since both {circumflex over (F)}μ(1) andⅇj⁢π2⁢μ
can be pre-determined for all μ values.

If the angle-rotation method is used, we can use a lookup table to store the {circumflex over (F)}μ(1) values. In real-time, after computingc1⁢ⅇj⁢π2⁢μ
using an angle-rotation processor, we can multiply the result by {circumflex over (F)}μ(1). In this case, if {circumflex over (F)}μ(1) is allowed to be a complex number in optimizing performance, we then need two real multiplications to obtainRe(F^μ⁡(1)⁢c1⁢ⅇj⁢π2⁢μ)
in (4.6). However, if we restrict {circumflex over (F)}μ(1) to be a real number, we can use just one real multiplication asRe(F^μ⁡(1)⁢c1⁢ⅇj⁢π2⁢μ)=F^μ⁡(1)⁢Re(c1⁢ⅇj⁢π2⁢μ)(4.8)

According to Table 4.1, the NMSE using complex and real {circumflex over (F)}μ(1) values are −37.41 dB and −37.08 dB, respectively. Therefore, the performance degradation caused by restricting {circumflex over (F)}μ(1) to be a real number is insignificant.

When the table-lookup method is employed, the implementation structure for the optimal interpolator is the same as that for the interpolator1400, except for the coefficient c1which is now multiplied byF^μ⁡(1)⁢c1⁢ⅇj⁢π2⁢μ
instead ofⅇj⁢π2⁢μ.

The table should therefore contain theRe(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
andIm(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
values, rather than thesin⁢π2⁢μ
andcos⁢π2⁢μ
values used by the interpolator1400. We now show that the size of the table is the same as the one storing the sine and cosine values.

Let us examine the contents of the lookup table.FIG. 32displays theRe(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)⁢⁢and⁢⁢Im(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
values, used by (4.7), where the real values are represented by curve3202, and the imaginary values are represented by the curve3204. These values are monotonic with respect to μ, just like thesin⁢π2⁢μ
andcos⁢π2⁢μ
values for 0≦μ≦1. Moreover, simulations show that, when optimal values of {circumflex over (F)}μ(1) are reached, the real and imaginary components ofF^μ⁡(1)⁢ⅇj⁢π2⁢μ
display the following complementary relationship:Im(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)=Re(F^1-μ⁡(1)⁢ⅇj⁢π2⁢(1-μ)).(4.9)

Therefore, we need only store one of theRe⁡(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
andIm⁡(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
values. The other can be obtained by looking-up the value corresponding to 1−μ. This is the same technique used in storing and retrieving the sine and cosine values with the purpose of reducing the table size.

Various circuit implementations of optimized interpolators having N=4 are illustrated inFIGS. 36-37. These circuit implementations are presented for example purposes only and are not meant to be limiting, as those skilled in the arts will recognize other circuit implementation based on the discussion given herein, including interpolator configurations having different N values.

FIG. 36illustrates an optimized interpolator3600that is based on the simplified interpolator1400(FIG.14). The interpolator3600includes an {circumflex over (F)}μROM3602and a multiplier3604. The ROM3600stores the appropriate {circumflex over (F)}μvalue indexed by μ. The multiplier3604multiples the complex coefficient C, by the appropriate {circumflex over (F)}μvalue, and therefore optimizes the frequency response of the interpolator3600. As discussed above, the order of the angle rotator1010band the multiplier3604can be interchanged so that the rotated complex coefficient is modified by the {circumflex over (F)}μvalue.

FIG. 37illustrates an optimized interpolator3700that is similar to the optimized interpolator3600, except that the F. ROM3602, the multiplier3604, and the angle rotator1010bare combined into to a single ROM3702, that stores theRe⁡(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
andIm⁡(F^μ⁡(1)⁢ⅇj⁢π2⁢μ)
values. Therefore, coefficient optimization and angle rotation are performed in a simultaneous and efficient manner.

It will be apparent to those skilled in the arts that the combined {circumflex over (F)}μand angle rotator ROM3702can be implemented for interpolator configurations that include more than N=4 elements, based on the discussions herein.

4.4 Simulation Results

We have verified the new design with the following simulation. A baseband signal, shown inFIG. 33, with raised cosine spectrum, two samples per symbol and 40% excess bandwidth was generated. Table 4.1 compares the result for N=4 using four interpolation structures: 1) the Lagrange cubic interpolator, 2) the interpolator1000, 3) the interpolator1400, 4) the optimal resampler using a complex {circumflex over (F)}μ(1) value, and 5) the optimal resampler employing a real {circumflex over (F)}μ(1) value.

Using the optimal structure, the NMSE is reduced by 4 dB over the method without optimization (FIG. 14structure). The performance is improved by more than 6 dB compared to the Lagrange cubic interpolator, while the hardware is reduced. Comparing the optimal structure to theFIG. 14structure, a 4 dB performance gain was obtained without increasing the amount of hardware.

The frequency response of an optimized interpolator1400(FIG. 14) using a lookup table is shown in FIG.16D. Also shown inFIG. 16A-Dare the frequency responses of the Lagrange cubic interpolator400, the interpolator1000, and the interpolator1400without optimization. The signal spectrum ofFIG. 33is shown inFIG. 16Din dashed lines1604. The interpolation error corresponds to the gray area1606. FromFIG. 16D, the performance improvement achieved by the optimal interpolator is evident because the gray area1606dis a lower amplitude than the corresponding gray areas1606a-c. In addition, these improvements are accomplished without increasing the amount of hardware.

For a high-performance interpolator, we now turn to the structure described in Section 2.6.2, for N=8. Applying a similar approach for N=4, as just discussed, to the N=8 interpolator of Section 2.6.2, we can multiply the c3coefficient by {circumflex over (F)}μ(3), whose value optimizes the frequency response Fμ(ω) of a fractional-delay filter with delay μ.

In designing the proposed N=8 interpolator, only one parameter {circumflex over (F)}μ(3) was adjusted to minimize the MSE in (3.22).

Table 4.2 shows the simulation results. These results demonstrate that our method has an NMSE more than 16 dB lower than the Lagrange interpolator, and more than 4 dB lower than the Vesma-Saramäki polynomial interpolator in (Vesma, J., and Saramäki, T., “Interpolation filters with arbitrary frequency response for all-digital receivers,” inProc.1996IEEE Int Symp. Circuits Syst. (May 1996), pp. 568-571).

Instead of optimizing F(Ω), the frequency response of the continuous-time interpolation filter, we could optimize Fμ(ω) of the fractional-delay filter for each μvalue. By doing this, better interpolation performance can be achieved, as demonstrated by the simulations.

As for the implementation complexity, when a table-lookup method is employed, the optimal interpolator does not require additional hardware, just table values that implement the coefficient optimization and angle rotation. When the angle rotation method is used, one additional real multiplier is needed.

For N=4, the optimal interpolator attained a 6 dB lower NMSE than the Lagrange cubic interpolator, while requiring less hardware.

5. A High-Speed Angle Rotation Processor

In previous Sections, an interpolation method and apparatus for timing recovery using a trigonometric polynomial has been discussed. The major computation in this method is the angle rotation, such as angle rotator1010(in FIG.10and FIG.14). As mentioned in Section 2.4, these operations, together with the phase rotator for carrier recovery, can be implemented by table-lookup in a ROM containing pre-computed sine and cosine values, followed by four real multipliers to perform the angle rotation (see FIG.18). Herein, going forward, this approach will be referred to as the single-stage angle rotation. Although fast angle rotation can be achieved with efficient multiplier design techniques, for practical precision requirements, the ROM table can be quite large. For applications where low complexity and low power are the major concern, can we further reduce the amount of hardware for angle rotation with slightly more computational delay?

There are various hardware designs that accomplish angle rotations, notably the CORDIC processors (Ahn, Y., et al., “VLSI design of a CORDIC-based derotator,” inProc.1998IEEE Int. Symp. Circuits Syst., Vol. II (May 1998), pp. 449-452; Wang, S., et al., “Hybrid CORDIC algorithms,”IEEE Trans. Comp.46:1202-1207 (1997)), and, recently, an angle-rotation processor (Madisetti, A., et al., “A 100-MHz, 16-b, direct digital frequency synthesizer with a 100-dBc spurious-free dynamic range,”IEEE J. Solid-State Circuits34:1034-1043 (1999)). These algorithms accomplish the rotation through a sequence of subrotations, with the input to each subrotation stage depending on the output of the previous stage. In these cases, the latency is proportional to the precision of the angle.

We now propose a different approach for angle rotation. Here the rotation is partitioned into just two cascaded rotation stages: a coarse rotation and a fine rotation. The two specific amounts of rotation are obtained directly from the original angle without performing iterations as does CORDIC. The critical path is therefore made significantly shorter than that of the CORDIC-type methods. In addition, only a small lookup table is needed.

In this Section, methods and apparatus for two-stage angle rotation will be described. These method and apparatus are meant for example purposes only, and are not meant to be limiting. Those skilled in the arts will recognize other methods and apparatus for two stage angle rotation based on the discussion given herein. These other methods and apparatus for angle rotation are within the scope and spirit of the present invention.

It will be shown that more precision and less hardware can be obtained using the two stage angle rotator compared to the single-stage angle rotator, with slightly more computational delay. We will then show that, given an overall output precision requirement, various simplifications can be applied to the computations within the two stages to reduce the total hardware.

5.1 The angle rotation problem

If we rotate a point in the X-Y plane having coordinates (X0, Y0) counterclockwise, around the origin, by the angle φ, a new point having coordinates (X, Y) is obtained. It is related to the original point (X0, Y0) as:
X=X0cos φ−Y0sin φ
Y=Y0cos φ+X0sin φ  (5.1)

The operation in (5.1) is found in many communication applications, notably in digital mixers which translate a baseband signal to some intermediate frequency and vice versa. In addition to accomplishing (5.1) with CORDIC, a very common implementation is to store pre-computed sine/cosine values in a ROM (Tan, L. and Samueli, H., “A 200-MHz quadrature frequency synthesizer/mixer in 0.8-μm CMOS,”IEEE J. Solid-State Circuits30:193-200 (1995)). Then, in real-time, the computation in (5.1) is accomplished with a ROM access for each given φ followed by four real multiplications. This method avoids the excessive latency of the iterations performed by CORDIC and can yield lower latency than the angle-rotation method (Madisetti, A., “VLSI architectures and IC implementations for bandwidth efficient communications,” Ph.D. dissertation, University of California, Los Angeles (1996)). Furthermore, a very fast circuit can be built, based on efficient multiplier design techniques. However, since the size of the ROM grows exponentially with the precision of φ, a rather large ROM is required to achieve accurate results.

ROM compression can be achieved by exploiting the quarter-wave symmetry of the sine/cosine functions and such trigonometric identities as sin θ=cos(π/2−θ). The angle φ in the full range [0,2π] can be mapped into an angle θε[0,π/4]. This is accomplished by conditionally interchanging the input values and X0and Y0, and conditionally interchanging and negating the output X and Y values (Madisetti, A., “VLSI architectures and IC implementations for bandwidth efficient communications,” Ph.D. dissertation, University of California, Los Angeles (1996)). Thus, we will focus only on θε[0,π/4] and replace φ by θ in(5.1). Defining θ=(π/4){overscore (θ)}, we must have {overscore (θ)}ε[0,1].

Obviously, the sine/cosine ROM samples must be quantized because of the limited storage space for sine/cosine samples. This produces an error in the ROM output when compared to the true (unquantized) sine/cosine value, which will subsequently be referred to as the ROM quantization error. Next we examine how this error affects the output. Let cos θ and sin θ be quantized to N bits, to become [cos θ] and [sin θ], respectively. We have
cos θ=[cos θ]+Δcos θ
sin θ=[sin θ]+Δsin θ(5.2)
where Δcos θand Δmin θare the ROM quantization errors, which satisfy |Δcos θ|<2−Nand |Δsin θ|<2−N. The error in X due to the ROM quantization is the difference between X calculated using infinite-precision sine/cosine values and the quantized values, that isΔX=(X0⁢cos⁢⁢θ-Y0⁢sin⁢⁢θ)-(X0⁡[cos⁢⁢θ]-Y0⁡[sin⁢⁢θ])=X0⁢Δcos⁢⁢θ-Y0⁢Δsin⁢⁢θ.(5.4)
Its upper bound is
|Δx|<(|X0|+|Y0|)2−N.  (5.4)

5.1.2 Rotation by a Small Angle

If the rotation angle happens to be so small that
|θ|<2−N/3(55)
then its sine/cosine values can be approximated as
sin θ˜θ  (5.6)
cos θ˜1−(θ2/2).  (5.7)
For such θ no table is needed. Next, we show how accurate (5.6) and (5.7) are by estimating their approximation errors.

The Taylor expansion of sin θ near θ=0 yields⁢sin⁢⁢θ=θ-sin′⁢ξ6⁢θ3(5.8)
where ξ=hθ, 0≦h≦1. Thus, since
|sin′ξ|=|cos ξ|≦1  (5.9)
and in view of (5.5), an error bound on (5.6) is
|Δsin θ|=|sin θ−θ|≦|θ3/6|<2−N/6.  (5.10)
Similarly, the Taylor expansion of cos θ yieldscos⁢⁢θ=1-12⁢θ2+cos⁢⁢ξ24⁢θ4.(5.11)
Thus, an error bound on (5.7) is
|Δcos θ|=|cos θ−(1−θ2/2)|≦|θ4/24|  (5.12)
which is negligible in comparison to the bound on |Δsin θ|.

5.1.3 Partitioning into Coarse and Fine Rotations

Now the rotation (5.1) is decomposed into two stages: a coarse rotation (5.16) by θMfollowed by a fine rotation (5.15) by θL. With this partitioning (5.5) and (5.6) can be applied to the fine stage:
X=X1(1−θL2/2)−Y1θL
Y=Y1(1−θL2/2)+X1θL(5.17)

A benefit of this partitioning is that the functions cos θMand sin θMin (5.16) depend only on the N/3 most significant bits of the angle {overscore (θ)}, where θ=(π/4){overscore (θ)}. They can be stored in a small lookup table. This results in a significant ROM size reduction. However, the approximation (5.6) introduces additional error. We now seek to achieve an overall precision comparable to that in the implementation having one stage and a large ROM table.

Defining the approximation errors Δsin θL=sin θL−θLand Δcos θL=cos θL−(1−θL2/2), and neglecting terms that are products of error terms or products of an error term and sin θL, which is always small, we calculate the total error in X as the difference between X calculated using (5.15) and (5.16) and X calculated using quantized sin θMand cos θM, and in (5.16) and using (5.17) instead of (5.15). We obtain:ΔX=⁢X0⁡(Δcos⁢⁢θM⁢cos⁢⁢θL+Δcos⁢⁢θL⁢cos⁢⁢θM-Δsin⁢⁢θL⁢sin⁢⁢θM)-⁢Y0⁡(Δsin⁢⁢θM⁢cos⁢⁢θL+Δcos⁢⁢θL⁢sin⁢⁢θM-Δsin⁢⁢θL⁢cos⁢⁢θM).(5.18)

Comparing this error estimate with (5.3) and (5.4) it is evident that, so long as the errors due to Δcos θLand Δsin θLare sufficiently small, the error ΔXin (5.18) can be made comparable to that of (5.4) by reducing the Δcos θMand Δsin θMvalues, i.e., by increasing the number of bits in the sine/cosine samples stored in the ROM. For example, if we add one more bit to the sine/cosine samples, then |Δcos θM|<2−N−1and |Δsin θM|<2−N−1.

Therefore, from (5.18), we have

which is smaller than (5.4). A similar relationship can be found for ΔY. This demonstrates that, if we add one more bit of precision to the ROM for the coarse stage, we can achieve the same precision as that in the one-stage case, but with a significantly smaller ROM.

A straightforward implementation of this method is illustrated by the angle rotator3800in FIG.38. The angle rotator3800includes a ROM3802, butterfly circuits3806and3810, and fine adjustment circuit3804.

The ROM3802stores the cos θMand sin θMvalues, where θMis the most significant part of the input angle θ. In embodiments the input angle θ is normalized and represented by a binary number, so that θMis the most significant word of the binary number, and θLis the least significant word of the binary number.

The first butterfly circuit3806multiplies the input complex number3812by the (cos θM)+ and the (sin θM)+ to perform a coarse rotation, where the ( )+denotes that the appropriate ROM quantization errors have been added to the cos θMand sin θMby the adders3814.

The fine adjustment circuit3804generates a fine adjust value (1−½θL2)O where θLis the least significant word of the input angle θ.

The second butterfly circuit3810multiples the output of circuit3806by θL+and the fine adjustment value from circuit3804, to perform a fine rotation that results in the rotated complex number3814. The + on the θL+denotes that an error value Δsin θLhas been added to improve the accuracy of the fine rotation.

The three error sources Δcos θM, Δsin θMand Δsin θLare shown. The much smaller error source Δcos θLhas been neglected. The thick line depicts the path along which the ROM quantization error Δcos θMpropagates to X. The error Δcos θMis multiplied by X0and then by cos θLas it propagates along this path to become Δcos θMX0cos θLwhen it reaches the output. This matches the error term in (5.18) obtained from our calculation. In subsequent discussions we will use this graphical approach to find the error at the output due to various error sources.

The ROM table3802in the rotator3800contains many fewer sine/cosine samples in comparison to the number of samples needed to implement (5.1) using a conventional (single stage) table-lookup approach. Although the approximation (5.6) introduces additional error, so long as that error is smaller than the conventional ROM quantization error, we can increase the precision of the samples in our small ROM table such that, overall, precision is not sacrificed. In principle, we can reduce the hardware complexity significantly in one block of our structure, with the corresponding accuracy loss compensated by higher precision from another block, and at the cost of a slight increase in the complexity of that block. As a result, the complexity of the overall structure is reduced without loss of accuracy. We will now exploit this idea again to further reduce the computational complexity.

5.2 Simplification of the Coarse Stage

The coarse stage, according to (5.16), involves multiplications of input data X0and Y0by the cos θMand sin θMvalues. Writing sin θMas the binary number
sin θM=0.b1. . . bN/3bN/3+1(5.20)
where bnε{0,1}, we now round sin θMupward, to obtain an (N/3+1)−bit value [sin θM], as
[sin θM]=0.b1. . . bN/3bN/3+1+2−(N/3+13+1),  (5.21)
where N represents the number of bits in the real part and the imaginary part of the input complex number. In other words, the real part has N bits, and the imaginary part has N bits.
Letting θ1be the angle for which
sin θ1=[sin θM]  (5.22)
we must have θ1≧θM. Next we can compute the corresponding cos θ1value. Using sin θ1=[sin θM] and cos θ1values, we actually rotate the point having coordinate (X0, Y0) by θ1instead of θM, as
X1=X0cos θ1−Y0sin θ1
Y1=Y0cos θ1+X0sin θ1.  (5.23)
Since θ1=arc sin([sin θM]) and, of course, θM=arc sin(sin θM), applying the mean value theorem we haveθ1-θM[sin⁢⁢θM]-sin⁢⁢θM-a⁢⁢sin′⁢ξ=11-ξ2(5.24)
where ξ satisfies sin θM≦ξ≦[sin θM]. Since sin θM≦1/(√{square root over (2)}), according to (5.21) we must haveξ≤[sin⁢⁢θM]≤sin⁢⁢θM+2-(N/3+1)≤12+2-(N/3+1).(5.25)
For most applications, N≧9. Thus, according to (5.25), we have is ξ≦0.7696. Applying this value to (5.24),θ1-θM≤11-0.76962×([sin⁢⁢θM]-sin⁢⁢θM)≤1.566×2-(n/3+1).(5.26)
Because 0≦θM≦π/4, we have, for N≧12, that
0≦θ1≦0.0978+π/4=0.8832.  (5.27)

The resulting fine-stage angle is θ−θ1, instead of θL=θ−θM. Thus, as in (Madisetti, A, “VLSI architectures and IC implementations for bandwidth efficient communications,” Ph.D. dissertation, University of California, Los Angeles (996)), a modified fine-stage angle compensates for a simplified coarse-stage angle. Since sin θ1=[sin θM], by rotating by θ1, the (N/3+1)-bit number sin θ1decreases the number of partial products needed in computing X0sin θ1and Y0sin θ1to just over a third of those needed for X0sin θMand Y0sin θM. This simplifies the computation in (5.23). However, if we can also reduce the multiplier size in computing X0cos O, and Y0cos θ1, we can further simplify (5.23). Certainly, truncating the cos θ1value would reduce the number of partial products in computing X0cos θ1and Y0cos θ1. Let us truncate cos θ1to 2N/3 bits to obtain [cos θ]. Then,
0≦Δcos θ1=cos θ1−[cos θ1]<2−2N/3(5.28)
We now have
X1=X0[cos θ1]−Y0sin θ1
Y1=Y0[cos θ1]+X0sin θ1.  (5.29)
Apparently, by truncating cos θ1, smaller multipliers are needed. But the amount of rotation is no longer θ1. We now examine the effect on θ1of using the truncated value [cos θ1] instead of cos θ1asθm=a⁢⁢tan⁢sin⁢⁢θ1[cos⁢⁢θ1].(5.30)
First, we determine how θmis different from θ1due to the truncation of cos θ1. Letting cos θ1and [cos θ1] denote specific values of a variable z we consider the functionΘ⁡(z)=a⁢⁢tan⁢sin⁢⁢θ1z.(5.31)
Hence, θ1and θmare the Θ(z) values corresponding to Z1=cos θ1and Z2=[cos θ1], i.e., θ1=Θ(Z1) and θm=Θ(Z2) According to the mean value theorem, we haveΘ⁡(z1)-Θ⁡(z2)z1-z2=Θ′⁡(ξ)⁢⁢or(5.32)θ1-θmcos⁢⁢θ1-[cos⁢⁢θ1]=a⁢⁢tan′⁢sin⁢⁢θ1ξ(5.33)
where [cos θ1]≦ξ≦cos θ1. The negation of the derivative at anAξ
satisfies-a⁢⁢tan′⁢sin⁢⁢θ1ξ=-sin⁢⁢θ1ξ21+(sin⁢⁢θ1ξ)2=sin⁢⁢θ1(sin⁢⁢θ1)2+ξ2≤sin⁢⁢θ1(sin⁢⁢θ1)2+[cos⁢⁢θ1]2.(5.34)

Combining (5.35) and (5.36), one can verify, for N≧9, that (5.34) satisfies-a⁢⁢tan′⁢sin⁢⁢θ1ξ<0.7976.(5.37)
Thus, according to (5.28) and (5.33), we have
θm−θ1<(cos θ1−[cos θ1])×0.7976≦0.7976×2−2N/3(5.38)
Combining (5.26) and (5.38), and for N≧9, we have
0≦θm−θM≦1.566×2−(N/3+1)+0.7976×2−2N/3<0.8827×2−N/3(5.39)
This is the amount of coarse rotation error, due to coarse-stage simplifications, that a modified fine stage must compensate. Let us examine the bound on the fine-stage angle.

Now, the fine rotation angle is θ1=θ−θminstead of θL. If θ1satisfies
|θ1|<2−N/3(5.40)
then we have |sin θ1−θ1|<2−N/6. That is, the approximations sin θ1˜θ1and cos θ1=1−θ12/2 can be applied as discussed in Section 5.1. Let us now examine the bound on θ1. By definition,0≤θL=π4⁢θ_L<0.7854×2-N/3.(5.41)
Therefore, subtracting (5.39) from (5.41) yields
−0.8827×2−N/3<θL−(θm−θM)<0.7854×2−N/3(5.42)
which implies (5.40) because
θ1=θ−θm=θM+θL−θm=θL−(θm−θM)  (5.43)
Hence, no lookup table is needed for the fine stage.

Next, we examine the magnitude of the complex input sample after rotation. One can verify from (5.29) that
X12+Y12=(X02+Y02)([cos θ1]2+(sin θ1)2(5.44)
Since [cos θ1] is obtained by truncating cos θ1, we must have 0 [cos θ1]<cos θ1, thus
[cos θ1]2+(sin θ1)2≦(cos θ1)2+(sin θ1)2=1  (5.45)
Therefore,
X12+Y12≦X02+Y02(5.46)
To maintain the magnitude, the result X1and Y1must then be multiplied by 1/√{square root over ([cos θ1]2+(sin θ1)2.)} We define a new variable δ[cos θ1]such that1[cos⁢⁢θ1]2+(sin⁢⁢θ1)2=1+δ[cos⁢⁢θ1].(5.47)

Since √{square root over ((cos θ1)2+(sin θ1)2)}{square root over ((cos θ1)2+(sin θ1)2)}=1, and [cos θ1] is very close to cos θ1because of (5.28), we have that √{square root over ([cos θ1]2+(cos θ1)2)} is very close to 1. Thus, the δ[cos θ1]value must be very small. We now examine the bound on δ[cos θ1]. We can write 1√{square root over ([cos θ1]2+(sin θ1)2)} as1[cos⁢⁢θ1]2+(sin⁢⁢θ1)2=(cos⁢⁢θ1)2+(sin⁢⁢θ1)2[cos⁢⁢θ1]2+(sin⁢⁢θ1)2.(5.48)

Because (5.28) and (5.35) imply that Δcos θ1<<[COS θ1], we have Δcos θ12<<[cos θ1]Δcos θ1, hence we can omit Δcos θ12in (5.49). Definingδ=[cos⁢⁢θ1]⁢Δcos⁢⁢θ1[cos⁢⁢θ1]2+(sin⁢⁢θ1)2(5.50)
then (5.49) becomes √{square root over (1+2δ)}. From (5.28) and (5.35) we must have δ>0. Applying the mean-value theorem to √{square root over (1+2δ)}, we have1+2⁢⁢δ-1+0δ-0=11+2⁢⁢ζ≤1(5.51)
where 0≦ζ<δ. Hence,
√{square root over (1+2δ)}≦1+δ  (5.52)
According to (5.35),[cos⁢⁢θ1][cos⁢⁢θ1]2+(sin⁢⁢θ1)2<1(5.53)
and therefore, from (5.28) and (5.50), we have 0≦δ<2-2N/3.

According to (5.40) and (5.54), instead of storing the sin θMand cos θM values in ROM, we may store sin θ1, which has N/3+1 bits for each sample, and [cos θ1], which has 2N/3 bits. Given {overscore (θ)}M, the sin θ1and [cos θ1] values are retrieved from the ROM to be used in performing the coarse rotation. Since the actual angle θmdiffers from the θM=(π/4){overscore (θ)}M, we must also store the θM−θmvalues, so that the fine stage can compensate for the difference. The approximations (5.6) and (5.7) still apply to θbin view of (5.40). In addition, the change of magnitude in the rotation using the sin θ1and [cos θ1] values, as seen in (5.45), must also be compensated. Therefore we store the δ[cos θ1]values in order to scale the coarse-stage output by 1+δ[cos θ1].

We can now implement the coarse rotation stage as in (5.29). Later we will show that the scale factor 1+δ[cos θ1]can be combined with the scaling that will be done for the second stage (i.e., the fine stage) at its output.

To compute θ1we must first convert the normalized {overscore (θ)}Lvalue to the radian value θL, which involves a multiplication by π/4. Since π/4=2−1+2−2+2−5+2−8+2−13+ . . . , if we multiply 0≦{overscore (θ)}L<2−N/3by (2−1+2−2+2−5+2−8), this product and (π/4) {overscore (θ)}Ldiffer by no more than 2−12×2−N/3=2−(N/3+12), which is sufficiently small for a 12-bit system. (And two more bits would suffice for building a 16-bit system.)

5.3 Reduction of Multiplier Size in the Fine Stage

In the fine rotation stage, the computations involved in generating X2areX2=⁢X1⁡(1-θl2/2)-Y1⁢θl=⁢X1-(θl2/2)⁢X1-Y1⁢θl.(5.55)
Since |θ1|<2−N/3it follows that θ1can be expressed as
θ1=S. S . . . S θN/3+1. . . θ2N/3θ2N/3+1(5.56)
where s is the sign bit. The N/3 MSBs do not influence the result. This property helps to reduce the size of the multipliers that implement (5.55). Even more savings in hardware can be achieved by further reducing multiplier size, with just a small loss of accuracy.

Let [Y1] represent the 2N/3 MSBs of Y1as in
Y1=S.y1. . . Y2N/3Y2N/3+1. . . =[Y1]+Δr1.  (5.57)
Then we must have |Δr1|<2−2N/3. The error contributed to the product Y1θ1by using [Y1] instead of Y, is
|Y1θ1−[Y1]θ1|=|Δr1θ1|<2−N.  (5.58)
Therefore, for N-bit precision, the multiplication Y1θ1can be accomplished with a (2N/3)×(2N/3) multiplier.

This method can be applied to the computation of θ12/2. Defining [θ1] as the 2N/3 MSBs of θ1, and letting Δθ1denote the remaining LSBs, we have
[θ1]=s.s . . . sθN/3+1. . . θ2N/3(5.59)
and
Δθ1=θ2N/3+1(5.60)
The error in calculating θ12/2 using [θ1] instead of θ1is
|([θ1]+Δθ1)2−[θ1]2|/2˜|[θ1]Δθ1|<2−N.  (5.61)
Thus θ12can be implemented with an (N/3)×(N/3) multiplier, since the N/3 MSBs of [θ1] are just sign-bits.

As mentioned in Section 5.2, the scale factor 1+δ[cos θ1]can be applied at the output of the fine stage. A straightforward implementation would use the full wordlength of 1+δ[cos θ1]in the product X=X2(1+δ[cos θ1]), which would require a multiplier of size N×N. But this multiplier's size can be reduced as follows: According to (5.54), 0≦δ[cos θ1]<2−2N/3. Moving the factor 1+δ[cos θ1]into the fine stage, we have
X2=X1(1−θ12/2)(1+δ[cos θ1])−Y1θ1(1+δ[cos θ1])  (5.62)
=X1+X1(δ[cos θ1]−θ12/2)−Y1θ1.  (5.63)
The only significant error in approximating (5.62) by (5.63) is the absence of the θ1δ[cos θ1]term in the factor multiplying Y1. But this is tolerable since, according to (5.54) and (5.40),
|θ1δ[cos θ1]|<2−N.  (5.64)
In view of (5.40) we have 0≦θ12<2−2N/3which, combined with (5.54), yields
|δ[cos θ1]−θ12/2|<2−2N/3.  (5.65)
Thus, if we truncate δcos θm−θ12/2 to N bits, only the least significant N/3 bits in the truncated result will be non-sign bits. Therefore, in our computation of X1(δcos θm−θ12/2) in (5.63), if we truncate X1to N/3 bits, we can use an (N/3)×(N/3) multiplier, with the product's error bound being
|δ[cos θ1]−θ12/2|2−N/3<2−N.  (5.66)

By merging the scale factor of the coarse stage into the fine stage, we thus replace multiplications by the scale factor by additions. The final architecture is shown inFIG. 39, where the size of the multipliers is shown in FIG.40.

FIG. 39illustrates an angle rotator3900according to embodiments of the invention. The angle rotator3900includes a ROM3902, a fine adjustment circuit3904, a first butterfly circuit3908, and a second butterfly circuit3910. The angle rotator3900rotates an input complex signal3906according to angle θ to produce a rotated complex output signal3912. The angle θ can be broken down into a most significant portion (or word) θMand least significant portion (word) θL. Note that normalized angle values are shown inFIG. 39, as represented by the {overscore (θ)} nomenclature. However normalized angle values are not required, as will be understood by those skilled in the arts.

The ROM3902stores the following for each corresponding {overscore (θ)}: sin θ1, [cos θ1], δ[cos θ1], and θM−θm, where all of these values have been exactly defined in preceding sections. To summarize, the sin θ1and [cos θ1] values are MSBs of sin θMand cos θ1, respectively. The δ[cos θ1]error value represents the difference between the cos θMand the [cos θ1] value. (The exact definition for δ[cos θ1]is given in (5.47)) Likewise, the (θM−θm) error value represents the difference between sin θMand the sin θ1value. (The exact definition for θmis given in equation (5.30))

The butterfly circuit3908includes multiple multipliers and adders as shown. The implementation of these multipliers and adders is well known to those skilled in the arts. In embodiments, the sizes of the multipliers and adders are as shown in FIG.40. Note that savings are obtained on the size of the multipliers because of bit truncated approximations that are described above. This produces a faster and more efficient angle rotator compared to other angle rotator schemes.

The operation of the angle rotator3900is further described in reference to the flowchart4100. As with all flowcharts herein, the order of the steps is not limiting, as one or more steps can be performed simultaneously (or in a different order) as will be understood by those skilled in the arts.

In step4102, the input complex signal is received.

In step4104, the sin θ1, cos [θ1], δ[cos θ1and θM−θmvalues are retrieved from the ROM3902, based on the rotation angle θ.

In step4106, the butterfly circuit3908multiplies the input complex signal by the sin θ1and [cos θ1] values to perform a coarse rotation of the input complex signal, resulting in an intermediate complex signal at the output of the butterfly circuit3908.

In step4108, the adder3914adds the θLvalue to the error value θM−θmto produce a θ1angle.

In step4110, a fine adjustment circuit3904generates a fine adjust value(δ[cos⁢⁢θ1]-12⁢θl2)
based on the θ1angle and δ[cos θ1]. In step4112, the butterfly circuit3910multiplies the intermediate complex signal by the θ1angle, and the fine adjustment value(δ[cos⁢⁢θ1]-12⁢θl2)
to perform a fine rotation of the intermediate complex number, resulting in the output complex signal3912.

In embodiments, the ROM3902storage space is 2′ words, where N is the bit size of the real or imaginary input complex number3906. Therefore, the overall size of the ROM3902can be quite small compared with other techniques. This occurs because of the two-stage coarse/fine rotation configuration of the angle rotator3900, and saving of storing sin θ1, [cos θ1] instead of sin θ and cos θ. Also, there is another advantage to having a small ROM: in certain technologies it is awkward to implement a ROM. Thus, if only a small ROM is needed, it is possible to implement the ROM's input/output relationship by combinatorial logic circuits instead of employing a ROM. Such circuits will not consume an unreasonable amount of chip area if they need only be equivalent to a small ROM.

5.5 Computational Accuracy and Wordlength

In this section we study the effect of quantization errors on the final output's computational accuracy and the most efficient way to quantize the data for a given accuracy.

In our algorithm, the errors can be classified into three categories. The first category is the quantization of the values in the ROM table. The second category is the error due to the truncation of data before multiplications, to reduce multiplier size. The third type of error is that resulting from approximating sin θ1by θ1. Quantization errors are marked inFIG. 40with an ξ marker as shown. The total error can be obtained by combining the errors propagated from each source. To calculate the propagated error at the output with a given error at the source, we can first identify all paths by which the error reaches the output and then use the approach discussed in Section 5.1.3. Let us first examine all the error sources and determine their effects on X, which is the real component of the output complex signal3912. Table 5.1 displays this information. (Similar results apply to Y.)

The values stored in the ROM are sin θ1, [cos θ1], θM−θmand δ[cos θ1], where sin θ1and [cos θ1] are MSBs of sin θMand cos θ1, respectively. A loss of precision due to ROM quantization error depends only on the number of bits used in representing θM−θmand δ[cos θ1].

The total error in X can be obtained by combining all the terms in the third column of Table 5.1:Y1⁡(ɛ1+ɛ7)+Y1⁢θl⁡(θl26-δ[cos⁢⁢θ1])+X1⁡(ɛ2+ɛ6)-X1⁢θl⁢ɛ5-θ1⁢ɛ3+(δcos⁢⁢θm-θl22)⁢ɛ4+ɛ8(5.67)

Since ξ6in Table 5.1 is a truncation error, we have ξ6≧0. If we quantize δ[cos θ1]by rounding it upward before storing it in ROM, then ξ2≦0. This way such errors tend to cancel each other. Cancelling errors are grouped together in (5.67) since the magnitude of their combined error is no greater than the larger of the two. This yields seven terms in (5.67), each contributing a maximum possible error of 2−N. If the multiplier sizes are as indicated inFIG. 40, the total error in X is bounded by 7×2−N.

From the above analysis it can be seen that the computation errors resulting from hardware reduction have similar magnitudes and no particular source dominates. This seems to provide the best trade-off between the hardware complexity and the accuracy of the entire system.

According to (5.67), the total output error can be reduced by increasing the internal data wordlength and the wordlength of each sample in the ROM. For each bit increase, we get one more bit of precision at the output. Therefore, we can design the processor to have the minimum hardware for a given precision requirement. Next, we give a simulation example to illustrate this method.

Example: A cosine waveform with an error less than 2−12is specified. According to (5.67), we chose N=15, as indicated in FIG.40. We obtained the maximum error to be approximately 5×10−5, which is considerably smaller than 2−12.

InFIG. 40, the ROM is shown as having2N3
words to achieve no more than a total error of 7×2−Nin the X output. If N is not a multiple of 3, we can choose the smallest N′>N that is a multiple of 3. Having2N′3
words in ROM, of course, suffices to achieve the required precision. As discussed before, the total output error is a combination of errors from various sources, such as from quantizing the data before multiplications and from approximating sin θ1by θ1, etc. However, our error bound estimation is rather conservative. Hence, the ROM size can be perturbed to determine the minimum size to satisfy a specific precision requirement. Our experience in designing the angle-rotation processor has shown that, even by roundingN3
down to determine the ROM size, the total error is still less than 7×2−N.

5.6 Comparison with the Single-Stage Mixer

As mentioned earlier, the main advantage of the two-stage angle rotator is that it requires only a small ROM3902. For the single stage angle rotation, the ROM size grows exponentially with the precision of the angle. Thus, our two-stage method is well-suited for applications where more than 14 bits in the input angle are required. In this case, the sine lookup table for the single-stage angle-rotator, even with compression, is too large for high-speed operations (Vankka, J., “Methods of mapping from phase to sine amplitude in direct digital synthesis,”IEEE Trans. Ultrasonics, Ferroelectronics and Freq. Control44:526-534 (1997)). However, the following comparison of our method to a well-known single-stage method with 14-bit input angle shows that even in this case our method has advantages, and this is true even when the single-stage method is optimized for that particular precision requirement.

To compare, we use the quadrature direct digital frequency synthesizer/mixer (QDDFSM) with 14-bit input angle and 12-bit input data that is reported in (Tan, L. and Samueli, H., “A200-MHz quadrature frequency synthesizer/mixer in 0.8-μm CMOS,”IEEE J Solid-State Circuits30:193-200 (1995)). It achieves 84.3 dB spurious free dynamic range (SFDR). According to this method, the sine and cosine values are generated using a DDFS, which employes lookup tables for these values. To reduce the ROM size, ROM compression techniques are used. The DDFS is followed by four 12×12 real multiplications.

For our structure, we chose the internal wordlengths and multiplier sizes as indicated in FIG.42. The phase-accumulator that generates {overscore (θ)} as well as the circuit that maps an angle in the range [0, 2π] into [0, π/4], are described in (Madisetti, A., “VLSI architectures and IC implementation for bandwidth efficient communications,” Ph.D. dissertation, University of California, Los Angeles (1996)). These structures are also employed here in our test. Truncating the 32-bit phase word to 14 bits, this structure has achieved a SFDR of 90.36 dB, as shown in FIG.43. This is 6 dB better than the single stage method.

The integrated circuit that implements this structure is currently being built. A preliminary estimation of its hardware complexity yields a similar transistor count as that of (Tan, L. and Samueli, H.,IEEE J Solid-State Circuits30:193-200 (1995)). Thus, using approximately the same number of transistors, our structure achieves a 6 dB performance gain. Our structure requires a much smaller ROM (17×25=425 bits) in comparison to the single-stage method, which needs a 3072-bit ROM when the ROM compression technique is employed. Since the ROM access is hard to pipeline, it is usually the bottleneck in the data path, thereby limiting the achievable data rate. Hence, one pronounced benefit of having a much smaller ROM would be the much faster ROM access. Also, there is another advantage to having a small ROM: in certain technologies it is awkward to implement a ROM. Thus, if only a small ROM is needed, it is possible to implement the ROM's input/output relationship by combinatorial logic circuits instead of employing a ROM. Such circuits will not consume an unreasonable amount of chip area if they need only be equivalent to a small ROM.

5.7 A Modified Structure When Only One Output is Needed

In some applications, such as the implementation of the trigonometric interpolator discussed in the previous sections, only one output, say X is needed. In such cases, obviously, we can eliminate certain computations used to generate Y. However, using the angle rotator3900, only those generating Y in the fine stage are subject to deletion, while the coarse stage must remain the same, since we need both X1and Y1to generate the X output. Let us seek to further simplify the coarse stage by attempting to eliminate one multiplication by cos θM.

5.7.1 Modifications to the Coarse Stage

If we factor out the cos θMterm of the coarse stage in (5.16), we can then apply the factor cos θMto the output of the second stage in (5.17), because the two operations (scaling and rotation) are permutable, to obtain
X1=X0−Y0tan θM

In this case, we have only two multipliers in the coarse stage (5.68), and the multiplications by the scale factor θMare applied to the output of the fine stage (5.69). Unlike the situation in (5.16) and (5.17), if only one output from the angle rotator, say X, is needed, we can also eliminate one more multiplier—the one that multiplies the coarse stage output with the cos θMfactor. As in Section 5.2, we now seek to simplify the coarse stage in (5.68).

Let tan θmbe tan θMrounded upward at the (N/3)-rd bit. In other words, writing θMas the binary number
tan θM=0.b1. . . bN/3bN/3+1(5.70)
where bnε{0, 1}, tan θmis obtained from tan θMaccording to
tan θm=0.b1. . . bN/3+2−N/3.  (5.71)
Obviously,
0≦tan θm−tan θM≦2−N/3.  (5.72)
The N/3-bit number tan θmdecreases the number of partial products needed in computing X0tan θmand Y0tan θmto at most a third of those needed for X0tan θMand Y0tan θM.

The resulting fine-stage angle is θ1=θ−θm. Thus, as in Section 5.2, a modified fine-stage angle compensates for a simplified coarse-stage angle. If θ1satisfies (5.40), we then have |sin θ1−θ1|<2−N/6. That is, the approximations sin θ1=θ1and cos θ1=1θ12/2 can be applied. The proof that (5.40) holds is as follows:
Proof: According to the mean value theoremtan⁢⁢θm-tan⁢⁢θMθm-θM=tan′⁢ξ(5.73)
where ξ=θM+(θm−θM)h, 0≦h≦1. The derivative tan′ξ satisfies
tan′ξ=1+(tan ξ)2≧1, for every ξ.  (5.74)
Re-arranging (5.73), and using (5.74), we haveθm-θM=tan⁢⁢θm-tan⁢⁢θMtan′⁢ξ≤tan⁢⁢θm-tan⁢⁢θM.(5.75)
Hence, according to (5.72),
0≦θm−θM≦2−N/3.  (5.76)
By definition,
0≦θL<2−N/3.  (5.77)
Therefore, subtracting (5.76) from (5.77) yields
−2−N/3<θL−(θm−θM)<2−N/3.  (5.78)
which is exactly (5.40) because
θ1=θ−θm=θM+θL−θm=θL−(θm−θM).  (5.79)
This concludes our proof.
This indicates that, instead of storing the tan θMvalues in the ROM, we may store tan θm, which has N/3 bits for each sample, and we may store θm−θM. This results in a reduction of the multiplier size in the coarse stage. The difference between θmand θMcan be compensated in the following fine rotation stage. Furthermore, the approximations (5.6) and (5.7) still apply to θ1, in view of (5.40).

We can now implement the coarse rotation stage as follows:
X1=X0−Y0tan θm
Y1=Y0+X0tan θm.  (5.80)
Accordingly, the scale-factor at the output of the fine stage is cos θminstead of cos θM. Since θ1satisfies (5.40), the fine stage simplification is similar to the method described in Section 5.3. Next we examine how the multiplications of the fine-stage output by cos θmcan be simplified.

A straightforward implementation would use the full wordlength of cos θmin the product X=X2cos θm, which would require a multiplier of size N×N. But this multiplier's size can be reduced as follows: By defining [cos θm] as the 2N/3+1 MSBs of cos θmthe scale factor can be written ascos⁢⁢θm=[cos⁢⁢θm]+Δcos⁢⁢θm=[cos⁢⁢θm]⁢(1+Δcos⁢⁢θm[cos⁢⁢θm]).(5.81)

The only significant error approximating (5.83) by (5.84) is the absence of the θ1≡cosmterm in the factor multiplying Y1. But this is tolerable since, according to (5.40) and (5.82),
|θ1δcos θm|<2−N(5.85)
In view of (5.40) we have 0≦θ12<2−2N/3which, combined with (5.82), yields
|δcos θm−θ12/2|<2−N/3(5.86)
Thus, if we truncate δcos θm−θ12/2 to N bits, only the least significant N/3 bits in the truncated result will be non-sign bits. Therefore, in our computation of X1(δcos θm−θ12/2) in (5.84), if we truncate X1to N/3 bits, we can use an (N/3)×(N/3) multiplier, with the product's error bound being
|δcos θm−θ12/2|2−N/3<2−N(5.87)

The factorization of cos θmin (5.81) allows a reduction of the multiplier to approximately ⅔ its original size. In this case, the values of [cos θm] and δcos θmare stored in the ROM instead of cos θm.

The modified structure for one output is illustrated as angle rotator4400in FIG.44. The angle rotator4400includes a ROM4402, a fine adjustment circuit4404, a first butterfly circuit4408, and a second butterfly circuit4410. The angle rotator4400rotates an input complex signal4406according to angle θ to produce a rotated complex output signal4412. As with the rotator3900, the angle θ can be broken down into a most significant portion (or word) θMand least significant portion (word) θL. Note that normalized angle values are shown inFIGS. 39,40,42, and44, as represented by the {overscore (θ)} nomenclature. However normalized angle values are not required, as will be understood by those skilled in the arts.

The ROM4402stores the following for each corresponding normalized θ: tan θm[cos θm], δcos θm, and θM−θm, where all of these values have been exactly defined in preceding sections.

In the butterfly circuit4410, the arithmetic units that are encircled by the line4418can be eliminated when only the X output is needed in the output signal4412. This may be desirable for applications where only one output from the angle rotator4400is needed, such as when implementing a trigonometric interpolator, such as interpolator1000inFIG. 10or interpolator1400in FIG.14.

The operation of the angle rotator4400is further described in reference to the flowchart4500in FIG.45. As with all flowcharts herein, the order of the steps is not limiting, as one or more steps can be performed simultaneously (or in a different order) as will be understood by those skilled in the arts.

In step4502, the input complex signal4406is received.

In step4504, the tan θm, [cos θm], δcos θm, and θM−θmvalues are retrieved from the ROM4402, based on the rotation angle θ (or the normalized value {overscore (θ)}).

In step4506, the butterfly circuit4408multiplies the input complex signal4406by tan θmto perform a coarse rotation of the input complex number, resulting in an intermediate complex signal at the output of the butterfly circuit4408.

In step4508, the adder4414adds the θLvalue to the error value θM−θmto produce a θ1angle.

In step4510, a fine adjustment circuit4404generates a fine adjust value(δ[cos⁢⁢θ1]-12⁢θl2)
based on the θ1angle and δcos θm.

In step4512, the butterfly circuit4410multiplies the intermediate complex signal by the θ1angle, and the fine adjustment value(δ[cos⁢⁢θ1]-12⁢θl2)
to perform a fine rotation of the intermediate complex signal, resulting in the output complex signal.

In step4514, the X value for the output complex signal is scaled by the [cos θm] value, resulting in the output complex number4412. As discussed above, the elements inside the outline4418can be eliminated if only the X value of signal4412is desired. Alternatively, similar elements could be eliminated from the butterfly circuit4410if only the Y value of signal4412was desired.

5.8 Application of Angle Rot on Processors

This subsection describes exemplary applications for angle rotator processors. These applications are provided for example purposes only and are not meant to be limiting, as those skilled in the arts will recognize other applications based on the discussions given herein. These other applications are within the scope and spirit of the present invention.

One application for the angle rotation processor is the Quadrature Direct Digital Frequency Synthesizer/Mixer (QDDFSM), including a few special cases that are candidates for the angle rotator algorithm. One is the case when only one of the outputs (X or Y) is desired, as shown by angle rotator4400(FIG.44). As shown inFIG. 44, this is accomplished by simply deleting the hardware required for the computation of the unused output. Yet another special case of QDDFSM is the Direct Digital Frequency Synthesizer (DDFS). In DDFS configuration we simply fix the input vector (X0, Y0) to be (1,0). This enables the complete elimination of the coarse stage by taking advantage of the fact that 1×A=A and 0×A=0. In the following section we will concentrate our discussion on the QDDFSM, since it is the general case, while keeping in mind the special cases and the associated hardware reductions mentioned above.

5.8.1 Using the Angle Rotation Processor in a Quadrature Direct Digital Frequency Synthesizer/Mixer

The frequency synthesis and mixing operation can be described with the following pair of equations, which relate an input with x-y coordinates (X0, Y0) and a frequency control word (fcw) for the synthesizer, to an output with new x-y coordinates (X, Y). The following pair of equations establishes the relationship between (X0, Y0), fcw, and (X, Y).
X=[X0×cos(fcw×n)]−[Y0×sin(fcw×n)]
Y=[Y0×cos(fcw×n)]+[X0×sin(fcw×n)]  (5.88)
where n is the time index

Per (5.88), since the sine and cosine functions are periodic with period27(i.e., fcw×n=<fcw×n>2π=φ, where < >is a modulo operator) an overflowing adder is used as a phase accumulator to compute φ from the input few, as shown by the the phase accumulator4600in FIG.46.

Now, for any given time instance n, we have a corresponding angle φ from the phase accumulator, hence the original pair of equations (5.88) for QDDFSM can be rewritten in terms of the angle φ as follows.
X=[X0×cos φ]−[Y0×sin φ]
Y=[Y0×cos φ]+[X0×sin φ  (5.89)

Note that the expressions (5.89) are exactly those of an angle rotator expressed by equations (5.1). By applying a phase accumulator fed by an fcw, we have converted the QDDFSM into an angle rotation application. The only conflict between the above expressions and the angle rotation processor is that the angle rotation processor takes an angle θ in the range [0, π/4], while the angle φ in the above expressions is in the interval [0,2π).

5. $1.1 A General Angle Rotator for Arbitrary Input Angles

Let us consider the changes necessary to make the angle rotation processor use an input angle φ that may lie outside the [0, π/4) range. Fortunately, a simple interchange operation at the input of the coarse stage, and an interchange/negate operation at the output of the fine stage is all we need in order to map ¢ into an angle θ in the range [0, π/4] and use it as the input to the angle rotator. Even though the input angle θ is in the range [0, π/4], the rotation by 0 along with the interchange and interchange/negate operations make the overall rotation of the input (X0, Y0) equivalent to a rotation by the original angle θ in the full range [0,2π). The latter is possible because of the convenient symmetry properties of sine and cosine functions over the range [0, 2π].

For example, sin φ=−sin (φ−π) and cos φ−−cos(φ−π), while sin φ=cos(φ−π/2) and cos φ=−sin(φ−π/2), and finally, for 0<φ<π/4, if we write π/4+φ for φ then sin(π/4+φ)=cos(π/4−φ) and cos(π/4+φ)=sin(π/4−φ). Using the first pair of trigonometric identities, we can map φ into the range [0, π]) by simply performing a negate operation at the output of the angle rotator. Using the second pair of identities along with the first pair enables one to map φ into the range [0, π/2) by performing negate and interchange operations at the output of the angle rotator. Finally, using all three pairs of identities, the angle φ can be mapped into the range [0, σ/4) by performing an interchange operation at the input of the angle rotator, along with interchange and negate operations at the output of the angle rotator. Note that all of these interchange and negate operations are conditioned only on the issue of which octant φ is in. This means that if φ is a normalized angle, then the interchange and negate decisions depend only on the top three MSB bits of φ. The following tables show the interchange and negate operations required for all eight octants (specified by the three MSB bits of φ). It is evident, as well, that other interchange and negate criteria for the input and output would also be suitable.

This table indicates when an interchange operation is required at the input and when an interchange operation is required at the output of the angle rotator.

The following table indicates when a negation operation is required at the output of the angle rotator.

Negation ofNegation ofOctant of φ (3 MSBs of φ)output Xoutput Y1-st octant (000)2-nd octant (001)Negate output X3-rd octant (010)Negate output Y4-th octant (011)Negate output XNegate output Y5-th octant (100)Negate output XNegate output Y6-th octant (101)Negate output Y7-th octant (110)Negate output X8-th octant (111)
Note that the flag for input interchange is simply the 3rd MSB bit of φ, while the flag for output interchange is just the 2nd MSB bit of φ. Finally, to produce the remapped angle θ in the range [0, π/4) for the angle rotation processor, we simply take the remaining bits of +after stripping the top two MSBs and performing a conditional subtract operation to produce e. More specifically, if the MSB bit (after stripping the two MSB bits) is low, i.e., the angle is in an even octant (numbering them 0, . . . , 7), we pass the angle unchanged, otherwise we perform a “two s-complement type” inversion of the angle. Note here that after such remapping operation, the MSB bit of θ is set to one only in the case when θ =π/4. This fact is useful in determining the required amount of lookup table in the angle rotation processor. In other words, even though the MSB bit of θ is an address to the lookup table, since we know that when it is ‘1’ the remaining bits have to all be ‘0’ we only need to allocate a single address for that case (as opposed to increasing the size of the lookup table by an entire factor of two).

5.8.1.2 Adapting the General Angle Rotator to Make a QDDFSM

The structure of the QDDFSM using an angle rotation processor3900is depicted in the FIG.4700. It simply requires the employment of a phase accumulator4702and a conditional subtract4704to provide an input angle from the input frequency control word fcw. We refer to the system ofFIG. 47with the phase accumulator excluded as a General Angle Rotator. It has the capability to receive an angle in the interval [0, 2π) and to perform an angle rotation of the input data (X0, Y0) by that angle. We show a general angle rotator inFIG. 48, but one in which further structural simplification has been made. The method of performing these simplifications will be discussed next.

5.8.2 How to Use the Conditionally Negating Multipliers in the General Angle Rotator

For a moment assume we have a powerful technique for making conditionally negating multipliers. What we mean by that is a multiplier which takes a negate flag to produce an output depending on that negate flag as follows: The output is simply the product of the input signals if the flag is low (0) and the output is the negative of the product of the input signals if the flag is high (1).

Each one of the two outputs in the coarse and fine stages is computed with two multipliers and one adder as shown in FIG.47. These multipliers and the adder are implemented in a single Carry-Save Adder (CSA) tree, with the partial products being generated from Booth decode modules corresponding to the two multipliers. This technique of employing a single tree eliminates the need for intermediate carry propagation from each multiplier and makes the propagation delay of each rotation stage very short. Note that the single CSA tree implementation is possible since the multipliers are operating in parallel. Furthermore, because the structure that is needed to compute one output of a rotation stage is identical to the structure required by the other output (with the exception of the minus sign), a single CSA tree can easily be interleaved between the two outputs for a significant amount of hardware savings. The minus sign at the output of the multiplier can be implemented very efficiently by the technique described in the following sections (using the conditionally negating multiplier). The negation or non-negation of the multiplier output can be controlled with a flag that changes between the two cycles of the interleave operation.

The angle at the output of the conditional subtract module4704inFIG. 47is in the range [0, π/4]. As already discussed, the outputs for the angles outside this range are constructed by mapping the angle into the range [0, π/4] while conditionally interchanging the inputs (inputs to the coarse stage) and conditionally interchanging and negating the outputs (outputs of the fine stage) of the angle rotator. A negation at the output of the fine stage simply means changing the output signs of the multipliers and negating the input of the adder coming from the input of the fine rotation stage. Changing the output signs of the multipliers is once again accomplished by using conditionally negating multipliers. The negation of the input to the fine rotation stage can easily be implemented with XOR gates and a conditional high or low bit insertion into the CSA tree at the position corresponding to the LSB location of the input. Since this conditional high or low bit is inserted in the CSA tree, there is no additional carry propagation introduced for the negation of the input. Note that the latter technique eliminates any circuitry required to implement the conditional negation of the outputs, and hence eliminates any carry propagations associated with two's complement numbers.

Furthermore, the conditional interchange of the outputs can be implemented by conditionally interchanging the inputs of the fine rotation stage and appropriately controlling the signs of the multiplier outputs in the fine stage. The conditional interchange of the fine stage inputs can be propagated to the inputs of the coarse stage with the same line of reasoning. Remember that the inputs to the coarse stage were conditionally interchanged according to the three MSBs of the input angle anyway. In conclusion, the conditional interchange and negation operations of the outputs can be implemented by modifying only the condition of the interchange at the inputs of the coarse stage and appropriately controlling the multiplier output signs by using conditionally negating multipliers (which we had to do for interleaving anyway). This eliminates the conditional negate and interchange block at the output of the fine stage entirely (i.e., it eliminates muxes and two's complement negators), and also eliminates the need for storing and pipelining control signals (i.e., it eliminates registers) to perform the conditional interchange and negation operations at the output. The resulting General Angle Rotator4800is now depicted in the following FIG.48.

There are many algorithms for digital multiplication. One of the most popular is the Booth multiplier. The essence of the Booth multiplier is in the decoding scheme performed on the multiplicand to reduce the number of partial products which, when added together, produce the desired product. For an N×M Booth multiplier, where N is the wordlength of the multiplier, and M is the wordlength of the multiplicand, there will be ceiling(N/2) Booth decoders. Each Booth decoder will take three bits from the multiplier (with one bit overlapping the decoders on both sides) and will manipulate the multiplicand according to the Booth decoding table5000shown in FIG.50. Some relevant details for a 10×M Booth multiplier are depicted inFIG. 49, especially how the multiplier bits feed into the Booth decoders to produce the five partial products which, when added, compute the result (the product of the multiplier and the multiplicand).

5.8.2.2 How to Make a Negating Booth Multiplier

Suppose we wish to make a multiplier that produces the negative of the product. More specifically, suppose we wish to multiply two signals N and M and get —C=−(N×M). The latter can be accomplished in a number of different ways. The most obvious is perhaps to use a regular multiplier to produce the product C=(N×M) and then negate C to achieve —C=−(N×M). In case of two's complement representation, this approach requires an additional carry propagation chain through the negator, which is costly in terms of speed and additional hardware associated with a negating circuit. Another approach, described below, is more favorable in a few key aspects.

The product C is essentially the result of adding a number of partial products, which are generated by the Booth decode blocks as described in the previous section. Therefore, we can write the following sum expression for C:C=∑i=1n⁢⁢pi(5.90)
where p, are then (in the 10×M example above n=5) partial products generated from the n Booth decoders. Note that, in order to negate C, we can negate all of the partial products and proceed with the summation of the negated partial products to produce —C. The expression for —C is then the following:-C=∑i=1n⁢⁢-pi,(5.91)
where −p1are the negated n partial products generated from the n Booth decoders. Let us investigate how the Booth decoders need to change to produce the desired negated partial products. All we need to do is to change the decoding table5000from that ofFIG. 50, to the decoding table5100in FIG.51. Note that the difference between the tables is only in the partial product columns and, more specifically, the partial product column5102of table5100is the negative of the partial product column5002of table5000. This means that by simply modifying the Booth decode table to the negating Booth decode table shown inFIG. 51, the result will be the negative of the product, as desired, with absolutely no additional hardware and absolutely no speed penalty. An example for a 10×M negating Booth multiplier5200is shown in FIG.52.

5.8.2.3 How to make a Conditionally Negating Booth Multiplier

A particularly interesting case arises when one wishes the multiplier product to be negated sometimes, and normal (non-negated) the other times. One can extend the idea presented in the previous section to craft the following powerful technique. Let us investigate the original Booth decode table5000depicted in FIG.50and the negating Booth decode table5100ofFIG. 51a bit more closely. Note the horizontal line of symmetry that runs through the midline of both decoding tables. This line of symmetry suggests that we can create the negating Booth decode table5100from the original Booth decode table5000by simply inverting the three bits (b2b1b0). For example, if the three bits (b2b1b0) are (010), then, according to the original Booth decode table, the corresponding partial product is A, where A is the multiplicand. If we invert the three bits (b2b1b0) as suggested above, we will have (1 0 1) and the corresponding partial product will be—A, exactly what is needed for a negated partial product.

Given a signal F which specifies when the output of the multiplier should be negated and when not (F=0 implies regular multiplication, F═I implies negating multiplication), F can simply be X OR d with the three bits (b2b1b0) at the input of the regular Booth decoders to make a new conditionally negating Booth decoder, hence a conditionally negating multiplier. The details of a conditionally negating Booth decoder5300are captured in FIG.53. Note that with a minimal amount of hardware (N XOR gates for an N×M multiplier, which is insignificant compared to the hardware cost of the entire multiplier), we have the means to control the sign of the multiplier product. Also note that the overall latency of the multiplier is increased insignificantly since the latency through a single XOR gate is much smaller than the latency through the entire multiplier. Furthermore, the latency of a single XOR gate is much smaller than the latency associated with a carry propagation chain that would be necessary if one built such a circuit with a two's complement negator. A 10×M conditionally negating multiplier5400is shown in FIG.54.

5.8.3 Using the Angle Rotation Processor in a Quadrature Direct Digital Frequency Synthesizer

As mentioned above, the angle rotator is useful in implementing various forms of direct digital synthesizers. In this case, all starting points for the angle rotations are X0=1, Y0=0 (with, of course, the various usual interchange/negation requirements).FIG. 55shows a quadrature direct digital synthesizer (QDDS)5500, a system having two outputs, one being samples of a cosine waveform and the other being samples of a sine waveform. An exact 90 degree phase offset between the two waveforms is obtained by the QDDS, and numerous applications for such a device are well known. No X0and Y0input samples are shown in theFIG. 55system. These fixed values have been “built in” and used to greatly simplify the coarse rotation stage.

Notice that the angle rotator5502is preceded by a system5504that generates a data stream of input rotation angles, a so-called overflowing phase accumulator5506, and its input is a single fixed data word that precisely controls the frequency of the output sample waveforms. The three MSBs of each phase accumulator output word, of course, assess the approximate size of the angle that is being used as a rotation angle (i.e., these three bits show how many octants the rotation angle spans), and they are stripped off to control the interchange/negation operations that are appropriate for obtaining the desired output samples. Also, the third MSB is used, as described previously, to determine whether or not to perform a “two's complement type” inversion of the LSBs. One other operation is required by the “Conditional Subtract” module5508shown inFIG. 55; in addition to stripping off the three MSBs, it appends one MSB having the value zero except in the case where a rotation angle of exactly π/4 is required. In that case, the appended MSB is one and all other ROM-address bits are zero.

A special case of the QDDS system, one having only a single output data stream, which could be either of the two, but which we call the “cosine-only” case, is also useful for various well-known applications. FIG.56andFIG. 57show two specializations of the angle rotator circuits previously discussed to implement the cosine-only DDS. The system5600inFIG. 56results from specializing the angle-rotation system3900in FIG.39. The system5700inFIG. 57is a specialization of the angle rotator4400in FIG.44.

Based on the design method discussed, for a given accuracy requirement, an architecture with the least amount of hardware is produced by balancing the precision of intermediate computations and the complexity of each arithmetic block, while keeping the output error within the specified bound. Furthermore, our architecture consolidates all operations into a small number of reduced-size multipliers. This permits us to take advantage of many efficient techniques that have been developed for multiplier implementation, such as Booth encoding, thereby yielding a smaller and faster circuit than those previously proposed.

Simulations and preliminary complexity estimation show that, even comparing to the method of (Tan, L. and Samueli, H.,IEEE J Solid-State Circuits30:193-200 (1995)) that is optimized for a 14-bit input angle, our method achieved 6 dB more SFDR while using approximately the same number of transistors as those needed by (Tan, L. and Samueli, H.,IEEE J Solid-State Circuits30:193-200 (1995)). In addition, since our structure employs only a small ROM, it overcomes the problem of slow access time that occurs when large ROMs are used, thereby facilitating a higher data rate. Using the two-stage method, when a higher precision is needed, it is very straightforward to satisfy such a requirement, since more accurate results can be attained simply by increasing the wordlength and the multiplier size. For the single-stage method, however, when high precision is desired, the required lookup table is likely to be too large to be practical, particularly for high-speed operation.

6. Symbol Synchronization for Bursty Transmissions

We have thus far discussed methods that provide efficient implementations of the resampler for symbol synchronization in a digital receiver using trigonometric interpolation as well as the phase rotator for carrier recovery. To produce the correct samples, a timing recovery circuit must supply the resampler with symbol timing information, as shown in FIG.1D. We will now consider how this can be accomplished. 6.1 Initial Parameter Estimations for Burst Modems

There are many methods to derive timing information from the received signal. According to their topologies, synchronization circuits can be divided into two categories: there are feedback and feedforward schemes. Feedback structures usually have very good tracking performance, and they work quite well in continuous mode transmissions. For packet data systems used by third-generation mobile communications, where the transmission is bursty, it is essential to acquire initial synchronization parameters rapidly from the observation of a short signal-segment.

A typical packet format is shown in FIG.58. It includes a short preamble5802followed by user data5804. The preamble5802is a set of known modulation symbols added to the user data packet at the transmitter with the intention of assisting the receiver in acquisition.

There are many approaches to burst demodulation, depending on the specific system requirements. In one approach (S. Gardner, “Burst modem design techniques, part 1,” Electron. Eng.71:85-92 (September 1999); Gardner, S., “Burst modem design techniques, part 2,”Electron Eng.71:75-83 (December 1999)), the receiver first detects the presence of the preamble, using a correlator, whose output should produce a large magnitude when the preamble is present. It then estimates the symbol timing. If the sampling frequency error is small, the total change of the timing phase from the start of the short preamble to the end is negligible. Next, it estimates the initial carrier frequency and phase. The above steps assume that the impairment caused by the channel is small enough that the modem can successfully track the timing carrier phase prior to equalization. Otherwise, equalizer training prior to the timing and carrier recovery is needed.

With a typical preamble of 8 to 32 symbols, depending on the required system performance, for QPSK modulation, rapid acquisition is desired. Feedforward timing estimation is known to have rapid acquisition, since it produces a one-shot estimate instead of tracking the initial timing through a feedback loop.

A well-known method, digital square timing recovery (Oerder M., and Meyr, H.,IEEE Trans. Comm.36:605-612 (1988)), has shown rapid acquisition, but it requires oversampling of the signal at, typically, four times the symbol rate, which imposes a demand for higher processing speed on the subsequent digital operations. Moreover, it does not work well for signals employing small excess bandwidth. However, pulses with small excess bandwidth are of interest for bandwidth-efficient modulation.

For applications where low power and low complexity are the major requirements, such as in personal communications, it is desirable to sample the signal at the lowest possible rate and to have the synchronizer be as simple as possible. In this section, a synchronizer is proposed that needs just two samples per symbol period. In addition, it has been shown to work well for small excess bandwidth, which is important for spectral efficiency. Using this method, the estimations of the symbol timing and the carrier phase can be carried out independently of each other. Hence, they can be carried out in parallel. Using the proposed structure, the timing and carrier-phase estimators can be implemented efficiently by means of direct computation (instead of a search, as is employed, for example, by (Sabel, L., and Cowley, W., “A recursive algorithm for the estimation of symbol timing in PSK burst modems,” inProc. Globecom1992, vol. 1(1992), pp. 360-364) using an efficient rectangular-to-polar converter (to be discussed in Section 7). This yields a very small computation load. Thus, this structure is well suited for low-power, low-complexity and high-data-rate applications, such as those in multimedia mobile communications.

6.2 Background Information

The system model 5900 used in developing the symbol timing and carrier phase recovery algorithm described in this section is shown in FIG.59.

Here h(t) is a real-valued, unit-energy square-root Nyquist pulse and w(t) is complex white Gaussian noise with independent real and imaginary components, each having power spectral density N0/2.

As mentioned in Section 6.1, a typical data packet for a burst modem consists of a short preamble5802followed by user data5804. According to the approach of (Gardner, S.,Electron. Eng.71:85-92 (September 1999)), the matched filter output is sampled every T1=T/2 seconds, i.e., at twice the symbol rate. The receiver then detects the presence of the preamble in the received signal by correlating the conjugate of the known preamble sequence am, whose length is L, with the sampled data x(nT) asrxx⁡(n)=∑m=0L-1⁢⁢am*⁢x⁡(nTs+2⁢mTs).(6.1)

The correlator output rxx(n) should produce a large magnitude |rxx(n)| when the preamble is encountered. It then estimates the initial synchronization parameters, namely the symbol timing and the carrier phase, assuming the transmitter/receiver frequency mismatches are insignificant.

The complex envelope x(t) of the received signal, after the matched filter, isx⁡(t)=ⅇjθ⁢∑k=-∞∞⁢⁢ak⁢g⁡(t-kT-τ)+v⁡(t)(6.2)
where {αk} is a sequence of independent equally-probable symbols with E[|αk|2]=1. We also have that v(t)=w(t){circle around (×)}h(−t) and that g(t)=h(t){circle around (×)}h(−t) is a Nyquist pulse. The time delay τ and the carrier phase θ are both unknown.

To estimate the data sequence αkwe want sample values of x(t) at t=mT+τ, with m an integer, whereas only the samples x(nT1) are available after sampling x(t) by a fixed clock.

Now let us examine how the correlator output relates to symbol timing and carrier phase. Inserting (6.2) into (6.1) yieldsrxx⁡(n)=∑m=0L-1⁢⁢∑k=-∞∞⁢⁢am*⁢ak⁢g⁡(nTs+2⁢mTs-kT-τ)⁢ⅇjθ+∑m=0L-1⁢⁢am*⁢v⁡(nTs+2⁢mTs).(6.3)
Since the data are independent, and they are independent of the noise, we haveE⁡[am*⁢ak]={1k=m0k≠m(6.4)
According to (6.4) and (6.5), and because T=2T, the expectation of rxx(n) with respect to the data and the noise is (for simplicity, we omit the constant real scale factor L)
E[rxx(n)]=ejθg(nTs−τ).  (6.6)

Thus, the mean value of the complex preamble correlator output actually equals the sample of the delayed signaling pulse g(t), with delay being τ, rotated by the angle θ. This is shown inFIG. 60for θ=0, where g(t) is a raised cosine Nyquist pulse with α=0.35. The total timing delay τ can be expressed as
τ=n0Ts+μ  (6.7)
where the integer n0represents the portion of τ that corresponds to an integer multiple of the sampling interval Tsand 0≦μ≦Tsis the sampling time mismatch.

Most practical signaling pulses g(t) are symmetrical and their peak value occurs at g(0). If θ is known, using these properties, we can estimate the sampling time mismatch μ from the correlator output rxx(n). In the next section we will discuss such an algorithm. We will derive this algorithm by first assuming that θ=0. Then we will discuss how the method can be carried out independently of the carrier phase. Simultaneously, we also derive a phase estimation algorithm that is independent of the symbol timing.

From (6.6), with θ=0, we have
E[rxx(n)]=g(nTs−τ).  (6.8)
According to (6.7) and (6.8), if the transmission delay τ is exactly an integer multiple of Tswe must have μ=0, and thus rxx(n0) must correspond to the peak g(0). Otherwise, we have μ≠0, with rxx(n0) and rxx(n0+1) being the two correlator output values nearest the peak value g(0), as shown in FIG.60. That is, rxx(n0) and rxx(n0+1) must be the two largest correlator outputs. Therefore, once the largest correlator output is located, we can obtain n0, the integer part of τ.

We now turn to finding11. Without loss of generality, let us assume Ts=1. Replacing n by n0+n we have, according to (6.8) and (6.7),
E[rxx(n0+n)]=g((n0+n)−τ)=g(n−μ).  (6.9)
For simplicity in our discussion on finding the fractional delay μ, we henceforth drop the index n0, which corresponds to an integer multiple of sample delays, from our notation. Next we define R(ejω) as the Fourier transform of rxx(n):R⁡(ⅇjω)=∑n=-∞∞⁢⁢rxx⁡(n)⁢ⅇjω⁢⁢n.(6.10)
The expectation of R(ejω) can be expressed as
E[R(ejω)]=E[FT(rxx(n))]=FT(E[rxx(n)]).  (6.11)
According to (6.9), and (C.4) in Appendix C, we have
E[R(ejω)]=FT(g(n−μ))=ejωμG(ejω)  (6.12)
where G(ejω) is the Fourier transform of g(n). Since g(n) is symmetrical, G(ejω) must have zero phase. Thus, according to (6.12),
arg(E[R(ejω)])=arg(ejωμG(ejω))=ωμ.  (6.13)
Evaluating (6.13) at X=π/2, we can obtain an estimate of μ asμ=2π⁢arg⁡(R⁡(ⅇjπ/2)).(6.14)
Therefore, the unknown sampling mismatch p can be obtained by taking the Fourier transform of rxx(n) and evaluating the phase corresponding to X=π/2.

To make the implementation of (6.14) realistic, we should truncate the sequence rxx(n) before taking its Fourier transform. For example, using only the four samples rxx(−1), rxx(0), rxx(1), and rxx(2), we have
RT(ejπ/2)=[rxx(0)−rxx(2)]+j[rxx(−1)−rxx(1)].  (6.15)
Using the correlator output, the μ value can be obtained by first computing RT(ejπ/2) according to (6.15), and then from the following:μ=2π⁢arg⁡(RT⁡(ⅇjπ/2)).(6.16)
For low precision requirements, this operation can be accomplished using a small CORDIC processor (Chen, A., et al., “Modified CORDIC demodulator implementation for digital IF-sampled receiver,” inProc. Globecom1995, vol.2(November 1995), pp. 1450-1454) or a ROM lookup table (Boutin, N.,IEEE Trans. Consumer Electron. CE-38:5-9 (1992)). With high accuracy requirements, however, the CORDIC processor will have long delays, while the table-lookup method will certainly require a very large ROM. In this case, we propose to use the rectangular-to-polar converter which will be discussed in Section 7. This rectangular-to-polar converter requires two small ROMs and it consolidates the operations into small array-multipliers, which can yield a smaller and faster circuit using well-known efficient multiplier implementation techniques.

A synchronizer6100for implementing the synchronization scheme described above is illustrated in FIG.61. The synchronizer6100includes a correlator6102, a Fourier Transform module6104, and a rectangular-to-polar converter6106. The Fourier transform module6104includes various delays and adders that are known to those skilled in the arts. The rectangular-to-polar converter is described further in Section 7.

The synchronizer6100receives data samples associated with sampling one or more received symbols and determines an offset πμ/2, where μ represents a synchronization offset of the data samples relative to the incoming symbols. The operation of synchronizer6100is described in reference to the flowchart6200, as follows.

In step6202, a set of complex data samples is received.

In step6204, the correlator6102correlates the complex data samples with a complex conjugate of a preamble data set (am*), resulting in correlated complex data samples.

In step6206, the Fourier transform module6104determines the Fourier transform of the correlated data samples signal, according to equations (6.10)-(6.13) and related equations;

In step6208, the Fourier transform module6104evaluates the Fourier transform of the correlated data samples at n/2, generating a complex signal representing a complex number;

In step6210, the rectangular-to-polar converter6106determines an angle in a complex plane associated with the complex number of step6210, where the angle represents synchronization between the data samples and the incoming symbols.

In step6212, the angle from step6210is scaled by 2/π to determine the synchronization offset.

6.4 Bias in Symbol Timing Estimation due to Truncating the Sequence

By truncating the sequence rxx(n) before taking the Fourier transform, we have produced a very simple structure to compute μ. However, since RT(ejω) differs from R(ejω) we must determine how this difference would affect the estimated μ value. The truncated sequence rT(n) is related to the original sequence rxx(n) as
rT(n)=rxx(n)w(n)  (6.17)
where w(n) is a rectangular function whose Fourier transform W(ejω) is a sinc function. Thus,
RT(ejω)=R(ejω){circle around (×)}W(ejω).  (6.18
Taking the expectation of (6.18) we haveE⁡[RT⁡(ⅇjω)]=E⁡[R⁡(ⅇjω)⊗W⁡(ⅇjω)]=E⁡[R⁡(ⅇjω)]⊗W⁡(ⅇjω).(6.19)
Obviously, the μ value obtained using RT(ejω) in (6.16) would be different from that obtained using R(ejω). This will introduce a non-zero timing-jitter mean (bias) to the μ value obtained using RT(ejω) instead of R(ejω). But the phase difference of the expected values of RT(ejπ/2) and R(ejπ/2) can be computed for a given g(t).
The procedure is as follows:1. Given the pulse waveform g(t), obtain, for each value μ, the samples g(n−μ), n=−1, . . . , 2.2. Compute RT(ejπ/2) using these samples g(n−τ) according to (6.15).3. Find the value {circumflex over (μ)} according to (6.16). The difference between the desired value μ and the value {circumflex over (μ)} computed using finite samples g(n−μ), n=−1, . . . , 2, is the bias.
This bias is illustrated inFIG. 63, where g(t) is a raised cosine Nyquist pulse with rolloff factor α =0.1.

FromFIG. 63, the bias is a function of μ and it can be precalculated and stored in a ROM in the receiver. Although the size of the ROM depends on the precision of μ, for typical precision requirements on μ the ROM can be quite small. Let us illustrate this point using an example: If an 8-bit accuracy is desired for the bias, the bias value corresponding to the three most significant bits (MSBs) in μ is indistinguishable from that corresponding to the full-precision μ value. Hence, we can use only the 3 MSBs in μ to determine the bias, thereby needing only 8 words in the bias lookup table.

Thus, for each of our symbol timing detector output samples, we can obtain the corresponding bias value from the ROM table, then subtract this bias from the original timing estimate to obtain an unbiased estimate.

We have thus far restricted our discussion to the timing recovery algorithm for θ=0. We now consider how this algorithm can be made to accommodate an arbitrary carrier phase0.

According to (6.6), with the Ts=1 normalization, the complex correlator output rxx(n) is dependent on θ. Although the expectation of its magnitude
E[|rxx(n)|]=|g(n−τ)|  (6.20)
does not depend on θ, it is non-trivial to compute the magnitude of rxx(n) from its real and imaginary components. Expressing rxx(n) in terms of its real and imaginary components, according to (6.6), we have
E[rxx(n)]=g(n−μ) cos θ+jg(n−μ) sin θ.  (6.21)
Thus,
E[Re[rxx(n)]]=g(n−μ) cos θ  (6.22)
E[Im[rxx(n)]]=g(n−μ) sin θ.  (6.23)

Since the carrier phase0does not depend on μ we can treat it as a constant scale factor in Re[rxx(n)] and Im[rxx(n)] when we are only concerned with extracting the timing information.

Clearly, therefore, instead of using the magnitude of the complex rxx(n) value, we can use one of its real and imaginary parts, which are available at the output of the preamble correlator.

We, of course, must decide which of Re[rxx(n)] and Im[rxx(n)] to use. If the unknown phase θ is such that cos θ˜0 it is certainly desirable to use Im[rxx(n)] instead of Re[rxx(n)], and vise versa. But we don't know the θ value thus far. How do we decide which one to use?

From (6.22) and (6.23) we can see that the relative magnitudes of cos θ and sin θ can be obtained from the real and imaginary components of rxx(n). For example, if |Re[rxx(n)]|>|Im[rxx(n)]| we certainly have that |cos θ|>|sin θ|, thus we should use the real part of the correlator output to find μ. Henceforth we denote the appropriate (real or imaginary) part of rxx(n) by {circumflex over (r)}xx(n).

6.6 Carrier Phase Computation

Next, let us examine the problem of extracting the carrier phase. From (6.6) we can see that the phase of the complex number E[rxx(n)] does not depend on μ. Moreover, the carrier phase can simply be obtained by extracting the phase of rxx(n). In order to achieve the best precision, it is desirable to choose the rxx(n) value with the largest magnitude for carrier phase estimation. For example, if rxx(n0) is the correlator output with largest squared-magnitude, we choose rxx(n0) to compute
θ=arg(rxx(n0)).  (6.24)
One advantage of this approach is that the symbol timing and carrier phase estimations are independent of each other. They can thus be carried out in parallel.

As for symbol timing estimation in (6.16), the computation in (6.24) can be accomplished efficiently using the rectangular-to-polar converter to be discussed in Section 7.

A synchronizer6400for determining timing and phase offsets is shown in FIG.64. Similar to synchronizer6100, the synchronizer6400receives data samples associated with sampling one or more received symbols and determines a timing offset πμ/2, where μ represents a synchronization offset between the data samples and the incoming symbols. Additionally, the synchronizer6400determines a carrier phase offset represented by θ. The synchronizer6400includes the correlator6102, sample selectors6404and6406, the Fourier transform module6104, and two rectangular-to-polar converters6106. The operation of synchronizer6400is described in reference to the flowchart6500inFIGS. 65A-B, as follows. The order of the steps in flowchart6500is not limiting, as one or more steps can be performed simultaneously or in a different order, as will be understood by those skilled in the relevant arts.

In step6502, a set of complex data samples is received.

In step6504, the correlator6102correlates the complex data samples with a complex conjugate of a preamble data set (am*), resulting in correlated complex data samples. Each correlated complex data sample includes a real sample and an imaginary sample.

In step6506, the sample set selector6404selects either the set of real correlated samples or the set of imaginary correlated samples. In embodiments, the set with the larger magnitude is selected.

In step6508, the Fourier transform module6104determines the Fourier transform of the selected real or imaginary data samples, according to equations (6.10)-(6.13) and related equations;

In step6510, the Fourier transform module6104evaluates the Fourier transform at π/2, generating a complex signal representing a complex number;

In step6512, the rectangular-to-polar converter6106adetermines an angle in a complex plane associated with the complex number of step6510, where the angle represents synchronization between the data samples and the incoming symbols.

In step6514, the angle from step6512is scaled by 2/π to determine the synchronization offset.

In step6516, the selector6406selects the largest correlator complex output. This selection can be based on an examination of one of the parts (real, imaginary) of the data sequence.

In step6518, the rectangular-to-polar converter6106bdetermines an angle in a complex plane associated with complex output of step6516, where the angle represents the carrier phase offset θ.

6.7 Simulation Result

We have used the above procedures to estimate the timing delay and the carrier phase of binary PAM symbols. The pulse shape was raised cosine with rolloff factor α=0.4. The block size was L=32 preamble symbols. To demonstrate its performance for signals with small excess bandwidth, we also tested this method with α=0.1. For a carrier phase offset θ=45°, we ran the simulation repeatedly using the synchronizer6400, each time using a μ value randomly chosen between 0 and 1.

In addition to synchronizer6400, we have also used the following two well-known methods to estimate the sampling mismatch:1) the DFT-based square-timing recovery (Oerder M., and Meyr, H., IEEE Trans. Comm. 36:605-612 (1988)),2) the method of (Gardner, S., Electron. Eng. 71:75-83 (December 1999)) that maps rxx(n0+1)/rxx(n0)—the ratio of the two correlation values nearest the peak (see FIG.60)—to the sampling mismatch value μ.

The variances of the timing jitter using these estimation methods for α=0.4 and α=0.1 are plotted in FIG.66andFIG. 67, respectively. The corresponding Cramer-Rao bounds (CRB)—the theoretical lower bounds of estimation errors (Meyr, H., et al.,Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, Wiley, New York, N.Y. (1998)>are also shown. We can see that, in both cases, the timing-jitter variance using the proposed synchronizer is quite close to the theoretical bound. It clearly outperforms the other two methods, even for signals employing small excess bandwidth, as seen in FIG.67.

The variance of the phase estimation error is depicted in FIG.68. It shows that, using the proposed method, the phase estimation error agrees quite well with the theoretical bound.

A synchronizer for initial symbol timing and carrier phase estimation using preambles has been presented. This synchronizer requires just two samples per symbol. Since the two estimations are independent of each other, they can be carried out simultaneously. These characteristics would ease the demand for computational speed for high-data-rate applications. Moreover, this synchronizer has demonstrated very good timing estimation performance even for signals with small excess bandwidth, which is essential for bandwidth efficient communications. The parameter estimations can be implemented very efficiently using the synchronizer6400. Due to its simplicity, this method is attractive for applications where low power and low complexity are desired, such as in a hand-held transceiver.

7. A High-Speed Processor for Rectangular-to-Polar Conversion

As discussed previously, the rapid acquisition characteristic of feedforward symbol synchronizers is essential to symbol synchronization for burst modems. Many feedforward structures require the evaluation of the phase of a complex number. That is, an efficient implementation of the phase extraction process is crucial. In order to handle a wide range of communications problems (Section 8), a general rectangular-to-polar conversion problem is considered.

There are several well-known implementations for a rectangular to polar coordinate conversion, i.e. obtaining the magnitude and phase of a complex number. One method uses a ROM lookup table with both the real and imaginary components as input. This is practical only for low bit-accuracy requirements, as the ROM size grows exponentially with an increasing number of input bits. To reduce the ROM size, we can first divide the imaginary by the real component, then use the quotient to index the lookup table. But the hardware for a full-speed divider is very complicated and power consuming. An iterative divider implemented using shifting and subtraction requires less hardware, but it is usually quite slow. Recently, CORDIC has been applied in this coordinate conversion (Chen, A., and Yang, S., “Reduced complexity CORDIC demodulator implementation for D-AMPS and digital IF-sampled receiver,” inProc. Globecom1998, vol. 3 (1998), pp. 1491-1496). However, due to the sequential nature of CORDIC, it is difficult to pipeline, thus limiting the throughput rate.

In burst-mode communication systems, rapid carrier and clock synchronization is crucial (Andronico, M., et al., “A new algorithm for fast synchronization in a burst mode PSK demodulator,” inProc.1995IEEE Int. Conf: Comm., vol. 3 (June 1995), pp. 1641-1646). Therefore, a fast rectangular-to-polar conversion is desired. In this section, we present an apparatus and method that implements the angle computation for rectangular-to-polar conversion with low latency and low hardware cost. This processor and the polar-to-rectangular processor presented in Section 5 (See rotator3900in FIG.39), together, can perform the M-ary PSK modulation devised in (Critchlow, D., “The design and simulation of a modulatable direct digital synthesizer with non-iterative coordinate transformation and noise shaping filter,” M. S. thesis, University of California, San Diego (1989)).

7.1 Partitioning the Angle

FIG. 69displays a point in the Cartesian X-Y plane having coordinates (X0, Y0), wherein X0and Y0represent the real and imaginary parts of an input complex signal. The angle φ can be computed as
φ=tan−1(Y0/X0)  (7.1)
In deriving the core of our algorithm, we assume the dividend and divisor satisfy
X0≧Y0≧0.  (7.2)
We will discuss how to extend the result to arbitrary values in Section 7.4. To achieve the highest precision for given hardware, the inputs X0and Y0should be scaled such that
1≦X0<2(7.3)

A straightforward method for fast implementation of(7.1) can be devised as follows:1) Obtain the reciprocal of X0from a lookup table.2) Compute Y0×(1/X0) with a fast multiplier.3) Use this product to index an arctangent table for φ.
However, the size of the two tables grows exponentially with increased precision requirements on φ, and rather large tables would be required to achieve accurate results. Therefore, for high-precision applications, such an implementation seems impractical.

If we approximate 1/X0by the reciprocal of the most significant bits (MSBs) of X0, denoted by [X0], then the required reciprocal table is much smaller. We can then multiply the table output by Y0to yield Y0/[X0], which is an approximation of Y0/X0. This quotient can then be used to index an arctangent table. Similar to the reciprocal table, a much smaller arctangent table is needed if we use only the MSBs of Y0/[X0], denoted by [Y0/[X0]], to address the table, which returns φ1=tan−1([Y0/[X0]]). Obviously, this result is just an approximation to φ. We will subsequently refer to the computation of φ1as the coarse computation stage.

Let φ2be the difference between φ and φ1. Using the trigonometric identity
tan φ2=tan (φ−φ1)=(tan φ−tan φ1)/(1+tan φ×tan φ1)  (7.4)
and the definitions tan φ=Y0X0and tan φ1=[Y0/[X0]], we havetan⁢⁢ϕ2=Y0/X0-[Y0/[X0]]1+(Y0/X0)×[Y0/[X0]]=Y0-X0×[Y0/[X0]]X0+Y0×[Y0/[X0]].(7.5)

Using this relationship, φ2, can be determined from [Y0/[X0]], the coarse computation results. Therefore, the desired result φ can be obtained by adding the fine correction angle φ2to the coarse approximation φ1. This procedure of finding φ2will subsequently be referred to as the fine computation stage.

By partitioning the computation of (7.1) into two stages, the table size in the coarse stage can be reduced significantly at the expense of additional computations, which are handled by the fine stage. Let us now examine the complexity of the fine stage. To find φ2, we can first compute
X1=X0+Y0×[Y0/[X0]]
Y1=Y0−X0×[Y0/[X0]]  (7.6)
and then find φ2as
φ2=tan−1(Y1/Xl)  (7.7)

The computation in (7.6) involves only adders and multipliers, while (7.7) requires lookup tables. Moreover, it seems we can't use the same coarse-stage tables because they have low resolution and thus can't satisfy the high precision requirements for the fine angle φ2Now let us analyze φ2to see if there is any property that can help in this situation.

If φ1is a good approximation of φ, then φ2=(φ−φ1, is close to zero. In view of (7.7), Y1/X1, should be very small too. This property helps us in two respects: 1) The difference between Y1/X1and Y1/[X1] is much smaller than that between 1/X1and 1/[X1]. This suggests that if we use the same low resolution reciprocal table as in the coarse stage, the error contributed to the final result will be very small. We will demonstrate this in the next section. 2) If Y1/X1is sufficiently small to satisfy
|Y1/X1|=|tan φ2|<2−N/3(7.8)
where N denotes the desired number of bits in φ, then
φ2=tan−1(Y1/X1)˜Y1/X1(7.9)
and we can compute φ2without using an arctangent table. This is explained as follows:

From the Taylor expansion of tan−1(Y1/X1) near Y1/X1=0, we obtain
tan−1(Y1/X1)=Y1/X1−(Y1/X1)3/3+0((Y1/X1)5)  (7.10)
Since 0((Y1/X1)5) is negligible in comparison to (Y1/X1)3/3, it can be omitted. Therefore, if Y1/X1is used to approximate tan−1(Y1/X1), an error
Δtan=tan−1(Y1/X1)−Y1/X1=−(Y1/X1)3/3  (7.11)
will occur. However, according to (7.8), Δtanis bounded by
|Δtan|<2−N/3  (7.12)
which is very small. This indicates that the approximation (7.9) is quite accurate if (7.8) is satisfied.

From the above analysis, no additional tables are needed for the fine stage if φ1is sufficiently close to φ. On the other hand, the better that φ1approximates φ, the larger the tables required for its computation become. As mentioned previously, table size grows exponentially as the precision increases. A good trade-off is obtained when the result φ1of the coarse stage is just close enough to φ that (7.8) is satisfied, thereby eliminating the additional tables in the fine stage. A detailed description of a rectangular-to-polar converter that implements the algorithm follows.

FIG. 71illustrates a rectangular-to-polar converter7100that implements the coarse and fine rotation described in section 7 herein, including equation (7.1)-(7.53). The converter7100receives a complex input signal7102(that represents a complex number having X0and Y0components) and determines the angle φ, which represents the position of the complex signal7102in the complex plane. In doing so, the converter7100determines a coarse angle computation that is represented by the angle φ1, and performs a fine angle computation represented by the angle φ2. Once φ1is determined, the input complex number7102is conceptually rotated back toward the X-axis to an intermediate complex signal7115as represented inFIG. 72, and φ2is determined from intermediate complex signal7115. The angles φ1and φ2are added together to determine p.

The converter7100includes: an input mux7104, reciprocal ROM7106, output demux7108, an arctan ROM7110, a multiplier7112, a butterfly circuit7114, a scaling shifter7116, a fine angle computation stage7124, and an adder7126. The fine angle computation includes a multiplier set7118, a one's complementer7120, and a multiplier7122.

The ROM7106stores reciprocal values of [X0], wherein [X0] is defined as the most significant bits (MSB) of X0of the input signal7102. The reciprocal of [X0] is represented as Z0, for ease of reference. As will be shown, the ROM7106is re-used to determine the reciprocal of [X1], where X1is the real part of the intermediate complex number7115shown in FIG.71and FIG.72. The reciprocal of [X1] is represented as Z1, for ease of reference. In embodiments, the ROM7106has 2N/3+1storage spaces, where N is the number of bits that represents X0(and Y0) of the input signal7102.

The input mux7104chooses between [X0] and [X0] as an input to the reciprocal ROM7106, according to the control7128. The output demux7108couples an output of the ROM7106to Z0or Z1according to the control7128. The control7128assures that Z0receives the stored reciprocal value for [X0], and that Z1receives the stored reciprocal value for [X1].

The arctan ROM7110stores the coarse approximation angle (p, based on a [Y0Z0] input. Therefore, a coarse stage can be described as including the ROM ROM7110, the ROM7106, and the multiplier7112, as they are used in the coarse angle computation.

The operation of the converter7100is described further with reference to the flowchart7300, as follows. The order of the steps in the flowchart7300is not limiting as one or more of the steps can be performed simultaneously, or in different order.

In step7302, the input complex signal7102having a X0component and a Y0component is received. In embodiments, the X0and Y0components are N-bit binary numbers.

In step7304, the control7128causes Z to be retrieved from the ROM7106, where Z0represents 1/[X0], and wherein [X0] is the MSBs of X0

In step7306, the multiplier7112multiplies Y0of the input complex number7102by Z0, resulting in a [Z0Y0] component. The [Z0Y,] component is an approximation of Y0/X0.

In step7308, the coarse angle φ1is retrieved from the ROM7110based on [Z0Y0], and is sent to the adder7126. Note that the coarse stage can be described as including the ROM7110, the ROM7106, and the multiplier7112, as they are used in the coarse angle computation.

In step7310, the butterfly circuit7114multiplies the input complex signal7102by [Z0Y0]. This causes the input complex signal7102to be rotated in the complex plane toward the real axis to produce the intermediate complex signal7115(representing an intermediate complex number), having a real X, component and an imaginary Y, component.

In step7312, the scaler7116scales the X1component of the intermediate complex signal so that it is compatible with the reciprocal values stored in the ROM7106. The scaler also scales the Y1component by the same amount.

In step7314, the control7128causes Z1to be retrieved from the ROM7106based on [X1], where Z1represents 1/[X1], and wherein [X1] is the MSBs of X1. Note, that the ROM7106is efficiently used twice to calculate two different reciprocals Z0and Z1, thereby reducing overall memory size.

In step7316, the fine angle computation stage7124determines the fine angle φ2based on Z1and the scaled intermediate complex number7115. In doing so the Newton-Raphson method is emulated in hardware to estimate φ2, which is the arctan of Z1Y1. More specifically, multiplier set7118multiples X1Y1by Z1. The ones' (approximating two's) complement7120is then determined for X1Z1. After which, the multiplier7127multiplies (2−X1Z1) by Y1Z1, to determine tan φ2. Since φ2is a small angle, the value tan φ2is used as an approximation of φ2.

In step7316, the φ1and φ2are added together to get φ.

A more detailed description of the algorithm follows.

7.2 The Two-Stage Algorithm

In this section we first analyze how the coarse approximation error φ2=φ−φ1depends upon the precision of the tables7106and7110, in order to determine the amount of hardware that must be allocated to the coarse stage. Next we explore ways to simplify the computations in the fine stage.

7.2.1 Simplification in the Coarse Computation Stage

The main concern in the coarse stage design is how the lookup table values are generated to produce as precise results as possible for a given table size. As mentioned previously, there are two lookup tables:

The input to this table, 1≦X0<2, can be expressed as
X0=1.x1x2. . . xm. . . xN(7.13)
where only bits x1through xmare used to index the table. To generate the table value, if we merely truncate X0as
[X0]=1.x1x2. . . xm(7.14)
then the quantization error ΔX0=X0−[X0] is bounded by
0<ΔX0<2−m.  (7.15)
Thus, the difference between the table value and 1/X0,
1/X0−1/[X0]=([X0]−X0)/([X0]X0)˜−ΔX0/X02(7.16)
is bounded by
−2−m<1/X0−1/[X0]≦0.  (7.17)
But if we generate the table value corresponding to
[X0]=1.x1x2. . . xm1  (7.18)
with a bit “1” appended as the LSB, then the quantization error in (7.15) is centered around zero:
−2−m−1<ΔX0<2−m−1(7.19)
hence, the error in the reciprocal is also centered around zero:
−2−m−1<1/X0−1/[X0]≦2−m−1.  (7.20)
Comparing (7.20) to (7.17), the maximum absolute error is reduced. This is the technique introduced in (Fowler, D. L., and Smith, J. E., “An accurate high speed implementation of division by reciprocal approximation,” inProc.9th Symp. on Computer Arithmetic(1989), pp. 60-67).
Since the output of the table will be multiplied by Y0, the fewer the bits in the table value, the smaller the required multiplier hardware. Let the table value Z0be generated by rounding 1/[X0] to n bits:
ZO=0.1z2z3. . . zn.  (7.21)
The quantization error ΔZ0=1/[X0]−Z0is then bounded by
−2−n−1<ΔZ0<2−n−1.  (7.22)

Once we have obtained Z0from the reciprocal table, we can get an approximation to the quotient Y0/X0by computing Y0Z0. This result is then used to address the arctangent table for φ1.

In order to use a very small table, Y0Z0is rounded to k bits to the right of the radix point to become [Y0Z0], with the rounding error bounded by
−2−k−1<ΔY0Z0=Y0Z0−[Y0Z0]<2−k−1.  (7.23)
Then, [Y0Z0] is used to index the arctangent table, which returns the coarse angle φ1=tan−1([Y0Z0]).

Now we must determine the minimum m, n and k values such that (7.8) is satisfied. First, let us examine X1and Y1which are computed using [Y0Z0] as
X1=X0+Y0[Y0Z0]
Y1=Y0−X0[Y0Z0].  (7.24)

Dividing (7.25) by (7.24), and then dividing both the numerator and denominator by X0, we have⁢Y1/X1=(Y0/X0-[Y0⁢Z0])/(1+(Y0/X0)⁡[Y0⁢Z0])≤Y0/X0-[Y0⁢Z0].(7.26)
The inequality is true because X0≧Y0≧0 and [Y0Z0]≧0. Taking into account all the quantization errors in (7.20), (7.22) and (7.23), we can express Y0/X0in terms of [Y0Z0] asY0⁡(1/X0)≈⁢Y0⁡(1/[X0]-Δx0/X02)=⁢Y0⁡((Z0+Δz0)-Δx0/X02)=⁢Y0⁢Z0+Y0⁢Δz0-Y0⁡(Δx0/X02)=⁢[Y0⁢Z0]+ΔY0⁢Z0+Y0⁢Δz0-Y0⁡(Δx0/X02).(7.27)
Substituting this result into (7.26), we have
|Y1/X1|≦|ΔY0Z0+Y0ΔZ0−Y0(ΔX0/X02)|.  (7.28)
Since Y0(ΔX0/X02)=Y0/X0)(ΔX0/X0), from (7.2) and (7.19),
−2−m−1<Y0(ΔX0/X02)<2−m−1.  (7.29)
Also, according to (7.2) and (7.22), we have
−2−n<Y0ΔZ0<2−n.  (7.30)
Applying (7.23), (7.29) and (7.30) to (7.28), we obtain |Y1X1|<2−m−1+2−n+2−k−1. If we choose m≧N/3+1, n≧N/3+2and k≧N/3+1, then
|Y1/X1|<0.75×2−N/3.  (7.31)
Therefore, since the inputs X1and Y1to the fine stage satisfy (7.8), no additional tables are needed for the fine stage. Henceforth we choose m=N/3+1, n=N/3+2 and k=N/3+1.

7.2.2 Hardware Reduction in the Fine Computation Stage7124

Since (7.8) is satisfied, we can obtain the fine angle Φ2by computing the quotient Y1/X1. From (7.24), we have X0≦X1≦X0+Y0, hence 1<X1<4. In order to use the same reciprocal table as in the coarse stage, X1should be scaled such that
1≦X1<2.  (7.32)
This can be satisfied by shifting X1to the right if X1≧2. Of course Y1should also be shifted accordingly so that Y1/X1remains unchanged.

As in the coarse stage, the reciprocal table accepts N/3+1 MSBs of X1and returns Z1. We define the reciprocal error δ1=1/X1−Z1. Since the same reciprocal table is used as in the coarse stage, δ1and δ0must have the same bound. Since
δ1=δ0=1/X0−Z0=1/X0−1/[X0]+ΔZ0(7.33)
we can use (7.20) and (7.22) to obtain
−0.75×2−N/3−1<δ1<0.75×2−N/3−1.  (7.34)
The bound on Y1can be found using (7.31) and (7.32):
|Y1|<0.75×2−N/3+1.  (7.35)
Now we can obtain the final error bound in approximating X1/Y1by Y1Z1, according to (7.34) and (7.35), as
|Y1/X1−Y1X1|=Y1δ1|<(0.75)2×2−2N/3.  (7.36)

Clearly, this approximation error is too large. To reduce the maximum error below 2−N, the bound on |δ1| should be approximately 2−2N/3, which would require the reciprocal table to accept 2N/3 bits as input. That is, the table needed for such a high-resolution input would be significantly larger than the one already employed by the coarse stage.

To overcome this difficulty, we can apply the Newton-Raphson iteration method (Koren, I.,Computer Arithmetic Algorithms, Prentice Hall Englewood Cliffs, N.J. (1993)) to reduce the initial approximation error81. We DOW briefly explain how this method works. First, let us define the following function:
f(Z)=1/Z−X1(7.37)
Obviously, we can obtain Z=1/X1by solving f(Z)=0. In other words, we can find 1/X1by searching for the Z value such that f(Z) intersects the Z-axis, as shown in FIG.70A.

Shifting the Z-axis down by X1, we obtain a new functionf2⁡(Z)=1Z,
shown in FIG.70B. At Z1=1/[X1], Z1being the initial guess, the slope of f1(Z)=1/Z isf1′⁡(Z1)=-1Z12.(7.38)
The tangent, shown as the dashed line7102, intersects the f1(Z)=X1line at a new point Z2. FromFIG. 70B, Z2is much closer to the desired value 1/X1than the initial guess Z1. Let us now find Z2. According toFIG. 70B, we must haveX1-1/Z1Z2-Z1=-1Z12.(7.39)
Expressing Z2in terms of Z1and X1we have
Z2=Z1(2−X1Z1).  (7.40)
Thus, we can obtain Z2, a more accurate approximation of 1/X1, from Z1. One may wonder how accurate Z2is in approximating1/X1. Let us now examine the approximation error bound.

According to (7.32), (7.34) and (7.35), after one Newton-Raphson iteration, the error in Y1Z1is reduced to
|Y1/X1−Y1Z2|=Y1X1δ12|<(0.75)3×2−N.  (7.42)
Thus, a rather accurate result is obtained with just one iteration.
Finally, the fine angle can be computed by multiplying Z2by Y1:
φ2˜Y1Z2=Y1Z1(2−X1Z1)  (7.43)

Although there are three multipliers involved in (7.43), the size of these multipliers can be reduced with just a slight accuracy loss by truncating the data before feeding them to the multipliers. The computational procedure of (7.43) is as follows:

1) The inputs to the fine stage, X1and Y1are truncated to 2N/3+2 and N+3 bits to the right of their radix points, respectively. Since the N/3−1 MSBs in Y1are just sign bits, as indicated by (7.35), they do not influence the complexity of the multiplier that produces Y1Z1. The corresponding quantization errors are bounded by
0≦ΔX1<2−2N/3−1(7.44)
0≦ΔY1<2−N−3(7.45)

3) To form 2−X1Z1instead of generating the two's complement of X1Z1, we can use the ones complement with only an insignificant error. Since this error is much smaller, in comparison to the truncation error in the next step, we can neglect it.

4) The product Y1Z1is truncated to N+3 bits. We would also truncate the ones complement of X1Z1. But since the inverted LSBs of X1Z1will be discarded, we can truncate X1Z1to 2N/3+2 bits and then take its ones vcomplement. The corresponding quantization errors, as discussed above, are:
0≦ΔX1Z1<2−2N/3−2(7.46)
0≦Δt1Z1<2−N−3(7.47)

After including all the error sources due to simplification, we now analyze the effects of these errors on the final result φ2. Taking the errors into account, we can rewrite (7.43) as:
φ2˜((Y1−Δy1)(1/X1−δ1)+Δy1Z1)(2−(X1−ΔX1)(1/X1−δ1)+ΔX1Z1)  (7.48)
Expanding this product and neglecting terms whose magnitudes are insignificant, we have
φ2˜Y1/X1−Y1X1δ12+(Y1X12)ΔX1−(1/X1)ΔY1+(Y1/X1)(ΔX1Z1.  (7.49)

As mentioned in Section 7.1, Y1/X1is an approximation of tan−1(Y1/X1). Its approximation error, defined in (7.8), is bounded by
|Δtan|=|(Y1/X1)3/3|<(0.75)3×2−N/3.  (7.50)
Replacing Y1/X1by tan−1(Y1/X1)+(Y1/X1)3/3 in (7.49), we have
φ2˜tan−1(Y1/X1)+(Y1/X1)3/3−Y1X1δ12+(Y1/X12ΔX1−(1/X1)ΔY1+(Y1/X1)ΔX1Z1+ΔY1Z1.  (7.51)
The total error, ε=φ2−tan−1(Y1/X1), is
ε=(Y1/X1){(Y1/X1)2/3−(X1δ1)2+ΔX1/X1+ΔX1Z1}−(1/X1)ΔY1+ΔY1Z1.  (7.52)
All terms in the subtotal (Y1/X1)2/3−(X11δ1)2+ΔX1/X1+ΔX1Z1are non-negative. Thus, the lower bound of this subtotal is the minimum value of −(X1δ1)2, which is −0.752×2−2N/3=−0.56×2−2N/3, according to (7.34). Correspondingly, its upper bound is the sum of the maximum values of the other three terms: (0.752/3+2−2+2−2)×2−2N/3=0.68×2−2N/3.

Finally, we can obtain the total error bound as:
|ε|<0.75×2−N/3×0.68×2−2N/3+2−N−3+2−N−3=0.76×2−N.  (7.53)

Once the angle of the vector (X0,Y0) is known, its magnitude can be obtained by multiplying X0by 1/cos φ, whose values can be precalculated and stored in a ROM thereby requiring only a single multiplication. However, if we use all the available bits to index the ROM table, it is likely that a very large ROM will be needed.

As we know from the preceding discussion, the coarse angle φ1is an approximation of φ. Similarly 1/cos φ1approximates 1/cos φ. Therefore, we can expand the coarse-stage ROM7110to include also the 1/cos φ1values. That is, for each input [Y0Z0], the coarse-stage ROM would output both φ1=tan−1([Y0Z0]) and 1/cos φ1. Since X0and Y0satisfy (7.2) and (7.3), the 1/cos φ value is within the interval [1, √{square root over (2)}].

For many applications, the magnitude value is used only to adjust the scaling of some signal level, and high precision is not necessary. For applications where a higher precision is desired, we propose the following approach:

First, instead of using the above-mentioned table of 1/cos φ1values, we pre-calculate and store in ROM the 1/cos φMvalues, where φMcontains only the m MSBs of φ. Obviously a small table, one of comparable size to the 1/cos φ1table, is needed. Then, we can look up the table entries for the two nearest values to φ, namely φ′Mand φM1=φM+2−m. Then a better approximation of 1/cos φcan be obtained by interpolating between the table values 1/cos φMand 1/cos φ′M1as1/cos⁢⁢ϕ≈1⁢cos⁢⁢ϕM+1/cos⁢⁢ϕM′-1/cos⁢⁢ϕMϕM′-ϕM×(ϕ-ϕM).(7.54)
Let φL=φ−φM. Obviously, φLsimply contains the LSBs of φ. We can now rewrite (7.54) as
1/cos φ˜1/cos φM+(1/cos φ′M−1/cos φM)×φL×2m(7.55)
which involves only a multiplication and a shift operation, in addition to two adders.

7.4 Converting Arbitrary Inputs

In previous sections we have restricted the input values to lie within the bounds of (7.2) and (7.3). However, if the coordinates of (X0, Y0) do not satisfy that condition, we must map the given point to one whose coordinates do. Of course, the resulting angle must be modified accordingly. To do that, we replace X0and Y0by their absolute values. This maps (X0,Y0) into the first quadrant. Next, the larger of |X0| and |Y0| is used as the denominator in (7.1) and the other as the numerator. This places the corresponding angle in the interval [0, π/4]. We can now use the procedure discussed previously to obtain φ. Once we get φ, we can find the angle φ that corresponds to the original coordinates from φ. First, if originally |X0|<|Y0| we should map φ to [π/4, π/2] using φ′=π/2−φ. Otherwise φ′=φ. We then map this result to the original quadrant according to Table 7.1.

Next, let us examine the affect of the above-mentioned mapping on the magnitude calculation. Since the negation and exchange of the original X0and Y0values do not change the magnitude, whose value is (X02+Y02)1/2, the result obtained using the X0and Y0values after the mapping needs no correction. However, if the input values were scaled to satisfy (7.3), we then need to scale the computed magnitude to the original scale of X0and Y0.

7.5 Test Result

We have verified our error bound estimation by a bit-level simulation of the rectangular-to-polar converter7100. To test the core algorithm described in Section 7.2, we generated the pair of inputs X0and Y0randomly within the range described by (7.2) and (7.3). This test was run repeatedly over many different values of X0and Y0, and the maximum error value was recorded. Choosing N=9for this simulation, the error bound estimate according to (7.53) is 0.0015. Our test results yielded the error bounds [0.00014, 0.000511, well within the calculated bound.

An efficient rectangular-to-polar converter is described. The angle computation of a complex number is partitioned into coarse and fine computational stages. Very small arctangent and reciprocal tables are used to obtain a coarse angle. These tables should provide just enough precision such that the remaining fine angle is small enough to approximately equal its tangent value. Therefore the fine angle can be obtained without a look-up table. The computations are consolidated into a few small multipliers, given a precision requirement. While a low-precision magnitude can be obtained quite simply, a high-precision result can be achieved by combining the angle computation with the angle rotation processor3900of Section 5.

The applications of this converter include the implementation of the converter6106in the symbol synchronizer6100and the synchronizer6400. However, the converter is not limited to symbol synchronization. It also provides efficient implementation of computational tasks for many communication systems, such as constant-amplitude FSK and PSK modems (Chen, A, and Yang, S., “Reduced complexity CORDIC demodulator implementation for D-AMPS and digital IF-sampled receiver,” inProc. Globecom1998, vol. 3 (1998), pp. 1491-1496; Boutin, N.,IEEE Trans. Consumer Electron38:5-9 (1992)), DMT modems (Arivoli, T., et al., “A single chip DMT modem for high-speed WLANs,” inProc.1998Custom Integrated Circuits Conf. (May 1998), pp. 9-11), as well as carrier synchronization (Andronico, M., et al., “A new algorithm for fast synchronization in a burst mode PSK demodulator,” inProc.1995IEEE Int Conf Comm., vol. 3 (June 1995), pp. 1641-1646; Fitz, M.P., and Lindsey, W. C.,IEEE Trans. Comm.40:1644-1653 (1992) where the computation of phase and magnitude from the rectangular coordinates is essential.

8. Exemplary Computer System

Embodiments of invention may be implemented using hardware, software or a combination thereof and may be implemented in a computer system or other processing system. In fact, in one embodiment, the invention is directed toward a software and/or hardware embodiment in a computer system. An example computer system7702is shown in FIG.77. The computer system7702includes one or more processors, such as processor7704. The processor7704is connected to a communication bus7706. The invention can be implemented in various software embodiments that can operate in this example computer system. After reading this description, it will become apparent to a person skilled in the relevant art how to implement the invention using other computer systems and/or computer architectures.

Computer system7702also includes a main memory7708, preferably a random access memory (RAM), and can also include a secondary memory or secondary storage7710. The secondary memory7710can include, for example, a hard disk drive7712and a removable storage drive7714, representing a floppy disk drive, a magnetic tape drive, an optical disk drive, etc. The removable storage drive7714reads from and/or writes to a removable storage unit7716in a well known manner. Removable storage unit7716, represents a floppy disk, magnetic tape, optical disk, etc. which is read by and written to by removable storage drive7714. As will be appreciated, the removable storage unit7716includes a computer usable storage medium having stored therein computer software and/or data.

In alternative embodiments, secondary memory7710may include other similar means for allowing computer software and data to be loaded into computer system7702. Such means can include, for example, a removable storage unit7720and an storage interface7718. Examples of such can include a program cartridge and cartridge interface (such as that found in video game devices), a removable memory chip (such as an EPROM, or PROM) and associated socket, and other removable storage units7720and interfaces7718which allow software and data to be transferred from the removable storage unit7720to the computer system7702.

Computer system7702can also include a communications interface7722. Communications interface7722allows software and data to be transferred between computer system7702and external devices7726. Examples of communications interface7722can include a modem, a network interface (such as an Ethernet card), a communications port, a PCMCIA slot and card, etc. Software and data transferred via communications interface7722are in the form of signals, which can be electronic, electromagnetic, optical or other signals capable of being received by the communications interface7722. These signals are provided to the communications interface7722via a channel7724. This channel7724can be implemented using wire or cable, fiber optics, a phone line, a cellular phone link, an RF link and other communications channels.

Computer system7702may also include well known peripherals7703including a display monitor, a keyboard, a printers and facsimile, and a pointing device such a computer mouse, track ball, etc.

In this document, the terms “computer program medium” and “computer usable medium” are used to generally refer to media such as the removable storage devices7716and7718, a hard disk installed in hard disk drive7712, semiconductor memory devices including RAM and ROM, and associated signals. These computer program products are means for providing software (including computer programs that embody the invention) and/or data to computer system7702.

Computer programs (also called computer control logic or computer program logic) are generally stored in main memory7708and/or secondary memory7710and executed therefrom. Computer programs can also be received via communications interface7722. Such computer programs, when executed, enable the computer system7702to perform the features of the present invention as discussed herein. In particular, the computer programs, when executed, enable the processor7704to perform the features of the present invention. Accordingly, such computer programs represent controllers of the computer system7702.

In an embodiment where the invention is implemented using software, the software may be stored in a computer program product and loaded into computer system7702using removable storage drive7714, hard drive7712or communications interface7722. The control logic (software), when executed by the processor7704, causes the processor7704to perform the functions of the invention as described herein.

In another embodiment, the invention is implemented primarily in hardware using, for example, hardware components such as application specific integrated circuits (ASICs), stand alone processors, and/or digital signal processors (DSPs). Implementation of the hardware state machine so as to perform the functions described herein will be apparent to persons skilled in the relevant art(s). In embodiments, the invention can exist as software operating on these hardware platforms.

The following Appendices are included.

9.1 Appendix A: Proof of the Zero ISI Condition.

In Section 3.4 we have examined the condition for a real-valued function f(t) to have zero crossings at integer multiples of the sampling period, i.e., for satisfying (3.16). We have stated that f(t) satisfies (3.16) if and only if {circumflex over (F)}(k), defined as samples of F(Ω) in (3.8) (F is the frequency response of f), satisfy (3.18). Here, −∞<k<∞ is an integer. We now provide the proof.
Proof: First let us define a periodic, of period N, extension of f(t) asfc⁡(t)=∑n=-∞∞⁢f⁡(t-Nn).(A⁢.1)
Its Fourier transform consists of a sequence of impulsesFc⁡(Ω)=∑k=-∞∞⁢F^⁡(k)⁢δ⁡(Ω-2⁢⁢πN⁢k).(A⁢.2)
Next, consider an impulse chain.c⁡(t)=∑n=-∞∞⁢δ⁡(t-n).(A⁢.3)
whose Fourier transform isC⁡(Ω)=∑k=-∞∞⁢δ⁡(Ω-2⁢⁢π⁢⁢k).(A⁢.4)

The convolution Fc{circle around (×)}C can be expressed asFc⁡(Ω)⊗C⁡(Ω)=⁢Fc⁡(Ω)⊗∑m=-∞∞⁢δ⁡(Ω-2⁢⁢π⁢⁢m)=⁢∑m=-∞∞⁢Fc⁡(Ω-2⁢⁢π⁢⁢m)(A⁢.5)

Therefore, we have the following relationships(3.16)⇔⁢fc⁡(t)={1t=Nm,m⁢⁢an⁢⁢integer0all⁢⁢other⁢⁢integers.⇔⁢fc⁡(t)=∑m=-∞∞⁢δ⁡(t-Nm)⇔⁢Fc⁡(Ω)⊗C⁡(Ω)=∑k=-∞∞⁢δ⁡(Ω-2⁢πN⁢k)⁢(A⁢.6)⇔⁢∑k=-∞∞⁢(∑m=-∞∞⁢F^⁡(k-Nm))⁢δ⁡(Ω-2⁢πN⁢k)=∑k=-∞∞⁢δ⁡(Ω-2⁢πN⁢k)⇔⁢∑m=-∞∞⁢F^⁡(k-Nm)=1⇔⁢(3.18)

This concludes the proof.

9.2 Appendix B: Impulse Response of the Simplified Interpolators

In Section 2, after introducing a preliminary interpolation method, we have shown that we can trade one angle rotator for a multiplier by conceptually modifying the input samples, then by “correcting” the interpolated value obtained from the “modified” samples. A simpler implementation structure as well as better performance in interpolating most practical signals have been facilitated. We now derive the impulse response of this interpolation filter. As discussed in Section 2, the interpolated sample is computed asy⁡(μ)=1N⁢∑k=-N/2+1N/2-1⁢ck⁢WN-k⁢⁢μ-K⁢⁢μ(B⁢.1)
where K is defined in (2.30), andck=∑m=-N/2+1N/2⁢y~⁡(m)⁢W[m]⁢N_km=∑m=-N/2+1N/2⁢(y⁡(m)+mK)⁢WNkm(B⁢.2)
where k=0, . . . , N/2−1.
Substituting (2.30) into (B.2), we haveck=⁢∑m=-N/2+1N/2⁢y⁡(m)⁢WNkm-∑m=-N/2+1N/2⁢(m⁢2N⁢∑n=-N/2+1N/2⁢(-1)n⁢y⁡(n))⁢WNkm=⁢∑m=-N/2+1N/2⁢y⁡(m)⁢WNkm-∑n=-N/2+1N/2⁢(∑m=-N/2+1N/2⁢m⁢2N⁢WNkm)⁢(-1)n⁢y⁡(n)=⁢∑m=-N/2+1N/2⁢y⁡(m)⁢WNkm-∑m=-N/2+1N/2⁢(∑n=-N/2+1N/2⁢n⁢2N⁢WNkm)⁢(-1)m⁢y⁡(m)(B⁢.3)
Replacing K and ckin (B.1) by (2.30) and (B.3), respectively, we havey⁡(μ)=⁢1N⁢∑k=-N/2+1N/2-1⁢∑m=-N/2+1N/2⁢(WNkm-⁢∑n=-N/2+1N/2⁢n⁢⁢2N⁢WNkm⁡(-1)m)⁢y⁡(m)⁢WN-k⁢⁢μ+⁢2N⁢∑m=-N/2+1N/2⁢(-1)m⁢y⁡(m)⁢⁢μ=⁢∑m=-N/2+1N/2⁢y⁡(m)⁢(1N⁢∑k=-N/2+1N/2⁢WNk⁡(m-μ)-⁢2N⁢(-1)m⁢(∑n=-N/2+1N/2-1⁢nN⁢∑k=-N/2+1N/2-1⁢WNk⁡(n-μ)-μ))=⁢∑m=-N/2+1N/2⁢y⁡(m)⁢f⁡(μ-m).(B⁢.4)
where f(t), the impulse response of the simplified interpolation filter discussed in Section 2.6.1, is now defined asf⁡(t)=⁢1N⁢∑k=-N/2+1N/2-1⁢WN-kt-⁢2N⁢(-1)m⁢(∑n=-N/2+1N/2-1⁢nN⁢∑k=-N/2+1N/2-1⁢WNk⁡(n-t-m)-(t+m))(B⁢.5)
for −m≦t≦1−m, m=−N/2+1, . . . , N/2. Otherwise, f(t)=0.

The frequency response, of course, can be obtained by taking the Fourier transform of f(t). To modify the frequency response of f(t), we can multiply the ckcoefficient in (B.3) by a value denoted by {circumflex over (F)}μ(k). In designing an optimal interpolation filter as discussed in Section 4, we search for the {circumflex over (F)}μ(k) value that minimizes (4.4), i.e., Fμ(ω) most accurately approximates the desired response (4.3).

Since, in Section 6, the g(nTs) are samples of the continuous time pulse g(t), assuming, without loss of generality, that Ts=1, it is well-known (Freeman, H.,Discrete-Time Systems, Wiley, New York, N.Y. (1965)) that the Fourier transforms are related asG⁡(ⅇj⁢⁢ω)=∑k=-∞∞⁢G^⁡(ω+2⁢⁢π⁢⁢k)(C⁢.1)
where G(ejω) and Ĝ(ω) are the Fourier transforms of g(n) and g(t), respectively. Since g(t) is bandlimited, i.e., |Ĝ(ω)|=0 for |ω|>π, we have
G(ejω)=Ĝ(ω), −π≦ω≦π.  (C.2)
Using the Fourier transform's time-shifting property, the Fourier transform of g(t−μ) is
ej ωμĜ(ω).  (C.3)
Since the g(n-μ) are samples of g(t-μ), for the sane reason as the above, their Fourier transforms are the same in the interval −π≦ω≦π, as in (C.2). Thus, according to (C.2) and (C.3) we have
FT(g(n−μ))=ejωμG(ejω), −π≦ω≦π.  (C.4)

When we first discussed the interpolation problem in Section 2, we focused on interpolating between the two samples in the middle of the set of samples used to generate a synchronized sample. What is the impact on interpolation performance when we interpolate in an interval not at the center of the samples being used?FIG. 74shows such an example for N=4, where the interpolation is performed between y(0) and y(1) using y(−2),y(−1),y(0) and y(1) (as opposed to using y(−1),y(0),y(1) and y(2), as seen in FIG.2.

Using the procedure described in Section 2, given N samples y(n), n=−N+2, . . . , 1, we first compute the Fourier coefficients asck=∑n=-N+21⁢y⁡(i)⁢WNkn,k=-N2+1,…⁢,N2.(D⁢.1)

Comparing (D.1) to (2.9), their only difference is the range of the summation. As in Section 2, for a given offset 0 μl<1, the synchronized sample y(g) can be computed as:y⁡(μ)=1N⁢Re⁡(c0+2⁢∑k=1N/2-1⁢ck⁢WN-k⁢⁢μ+cN/2⁢ⅇj⁢⁢π⁢⁢μ).(D⁢.2)

We can express y(μ) in terms of y(n) by substituting (D.2) into (D. 1), asy⁡(μ)=⁢1N⁢∑n=-N+21⁢y⁡(n)⁢(1+2⁢∑k=1N/2-1⁢cos⁢⁢2⁢π⁢⁢kN⁢(μ-n)+cos⁢⁢π⁡(μ-n))=⁢1N⁢∑n=-N+21⁢y⁡(n)⁢f⁡(μ-n)⁢⁢where(D⁢.3)f⁡(t)=(1+2⁢∑k=1N/2-1⁢cos⁢2⁢⁢π⁢⁢kN⁢t+cos⁢⁢π⁢⁢t-1≤t≤-N+2⁢0⁢⁢otherwise(D⁢.4)
is the impulse response of the corresponding interpolation filter. For N=4, f(t) is plotted in FIG.75A. Taking the Fourier transform of f(t), we obtain the corresponding frequency response, which is shown in FIG.76A.

Comparing theFIG. 76Afrequency response toFIG. 7A, both for N=4, we can see that the interpolation performance degraded significantly, as shown by the ripples in the passband and large sidelobes in the stopband inFIG. 76A, when the interpolation is not done in the center interval.

However, using the optimization method discussed in Section 4, we can “reshape” the impulse response At) such that the corresponding frequency response is a better approximation to the ideal interpolation frequency response.

The impulse response of an optimized interpolation filter for a non-center interval is illustrated in FIG.75B. The corresponding frequency response is shown inFIG. 76B, which is clearly better thanFIG. 76A, since it has less ripple in the passband and more attenuation in the stopband.

Using samples y(−N+2), . . . , y(−1), and y(0), to interpolate obviously reduces the latency in generating synchronized samples, as compared to using samples y(−N/2+1), . . . , y(0), . . . , y(N/2), since the interpolator does not have to wait until samples y(1), . . . , y(N/2), become available. In applications where low latency takes a higher priority than interpolation accuracy, this approach will be useful.

9.5 Appendix E

Example implementations of the methods, systems and components of the invention have been described herein. As noted elsewhere, these example implementations have been described for illustrative purposes only, and are not limiting. Other implementation embodiments are possible and covered by the invention, such as but not limited to software and software/hardware implementations of the systems and components of the invention. Such implementation embodiments will be apparent to persons skilled in the relevant art(s) based on the teachings contained herein.