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A frequency-designed fractional delay FIR structure, which is suitable for software radio applications, is presented. The design method is based on frequency optimization of a combination of modified Farrow and mutirate structures. As a result the optimization frequency range is made only in half of desired total bandwidth. According to the obtained results the proposed fractional delay structure allows online desired fractional delay update, with a high fractional delay value resolution.

Software-defined radio represents a major change in the design paradigm for radios, [

A software-defined radio platform is designed to make mobile systems more flexible with respect to the bandwidth requirements of different mobile standards. This flexibility is achieved by performing channel selection in the digital domain through sample rate conversion (SRC) with programmable digital filters. Fractional Delay (FD) filters are key components used to perform nonrational SRC [

Additionally, Software radio systems employ direct conversion receivers with asynchronous sampling such that the actual sampling instants are not synchronized with the incoming symbols. In order to evaluate the received symbols a digital symbol synchronization is implemented through FD filter structures [

One of the key requirements for FD in software radio applications is to have the flexibility to change among different communication protocols and to be able to perform fractional delay value update on line, known as variable fractional delay (VFD) filters [

There are several FD design methods [

Farrow structure.

There are two main FD design approaches based on the polynomial approach. The first one is completely time domain design based on either Lagrange interpolation [

The second FD design approach is made in frequency domain using optimization techniques for coefficients computing. The main advantage of this design approach is an improved control on frequency specifications. This is because three design parameters are available: polynomial order

The use of frequency domain design methods for FD filters with a wide bandwidth requires that an optimization method be applied over a large frequency range. On the other hand large filter length

This paper describes the use of a multirate structure in a frequency design approach in order to reduce the optimization workload in coefficients computing for FD filters with a wide bandwidth, high fractional delay resolution, and online fractional delay value update capability. In this way a flexible frequency design method with a reduced optimization workload as well as a resulting structure with a reduced number of arithmetic operations per output sample is obtained.

The used frequency design method is the modified Farrow structure [

Section

The frequency design method in [

FIR filters

In the modified Farrow structure the FIR filters

Each filter

In the same way it is possible to approximate the input signal through Taylor series in a modified Farrow structure. The

The

The arithmetic complexity of the resulting implemented structure is an important factor to be considered. The comparative parameters are the following:

number of multipliers per output sample (MPS),

number of additions per output sample (APS).

In the modified Farrow structure the MPS_{1} and APS_{1} are given as

The multirate structure in [

The multirate structure shown in Figure

Multirate structure for FD filter.

Resulting structure for FD filter.

The proposed method for FD filter design with a wide bandwidth and high fractional delay resolution is based on a frequency domain optimization approach, described in the second section, applied to the FD multirate structure, described in last the section.

As mentioned before the maximum frequency of the FD filter in the multirate structure is half of the desired bandwidth. In this way the frequency optimization is made only on the half of the required passband

In Figure

Initial structure of the proposed method.

Resulting proposed structure.

The

In order to reduce the total number of arithmetic operations per output sample the filter

The design method was implemented in MATLAB. An illustrative design example is presented with an FD filter bandwidth of 0.9

The FD filter design using WLS method [

For the proposed design method an interpolator filter

In the proposed method the frequency optimization is applied up to

Differentiators ideal responses (dot line) and approximations (solid lines). (a) Proposed method

The all band magnitude responses and group delays for fractional delay values range from 0.0080 to 0.0090 using the direct frequency FD design method and the proposed method results are shown in Figures

All band magnitude responses for (a) proposed method and (b) direct all band optimization.

All band fractional delay responses for (a) proposed method and (b) direct all band optimization.

According to obtained results the proposed method has smaller number of operations per output sample. In order to compare the achieved proposed method approximation with the one obtained with existing methods, the frequency domain error

The obtained maximum absolute error and the root mean square error are presented in Table

Comparison of approximation errors for several methods.

Methods | ||
---|---|---|

WLS design [ | ||

Improved WLS [ | ||

Discretization-free [ | ||

Variable Factional Delay [ | ||

Direct Taylor approximation [ | ||

Proposed method |

A frequency optimization design approach for wide bandwidth and high fractional value resolution FD filters has been proposed. These specifications coupled with the capability of updating the fractional delay value in real-time make the resulting structure suitable to perform important physical layer functions for software-defined radio applications, such as digital symbol synchronization and sample rate conversion.

The obtained results show that the design method notably reduces the coefficients computing workload. The resulting structure allows fractional delay values of 1/10000 of sample and a bandwidth of