Vankka J. - Digital Synthesizers and Transmitters for Software Radio (2000)(en)

resulting in a limited range of filter characteristics.

Filter sharpening can be used to improve the response of a CIC filter. This technique applies the same filter several times to an input to improve both passband and stopband characteristics. The interested reader is referred to [Kwe97] for more details.

11.11 Pipelining/Interleaving

The pipelining/interleaving (P/I) technique can be used to reduce the amount of hardware when the same filtering operation is performed for several independent data streams [Par89], [Jia97]. The idea is to use the same hardware for all K data streams, as shown in Figure 11-11. This may be achieved with some additional logic. The data streams are interleaved in a serial form in time. The filtering operations are made with a K times higher clock frequency; after filtering, the data stream is de-interleaved back into K parallel data streams. This serial-in-time computation requires K register elements instead of one register in the filter structure. In addition to that, extra logic is needed in the interleave/de-interleave interfaces. The expense of this technique is the increased power consumption. This increase accrues because the number of delay elements remains same, but the clock frequency is doubled. The increased power consumption due to the doubled clock frequency in the other elements such as in the taps and adders is compensated by the halved number of elements.

REFERENCES

[Che82] P. R. Chevillat, and G. Ungerboeck, "Optimum FIR Transmitter and Receiver Filters for Data Transmission over Band-Limited Channels," IEEE Trans. Commun., Vol. 30, No. 8, pp. 1909-1915, Aug. 1982.

[Che99] C. Chen, and A. Willson, "A Trellis Search Algorithm for the Design of FIR Filters with Signed-Powers-of-Two Coefficients," IEEE Transactions on Circuits and Systems-II, Vol. 46, No. 1, pp. 29-39, Jan. 1999.

[Con99] G. A. Constantinides, P. Y. K. Cheung, and W. Luk, "Truncation noise in Fixed-Point SFGs [digital filters]," Electron. Lett., Vol. 35, No. 23,

Figure 11-11 Digital filtering of K independent signal sequences using a single filter.

pp. 2012 -2014, Nov. 1999.

[Dua00] J. N. Duan, L. Ko, and B. Daneshrad, "A Highly Versatile Beamforming ASIC for Application in Broad-Band Fixed Wireless Access Systems," IEEE J. of Solid State Circuits, Vol. 35, No. 3, pp. 391-400, Mar. 2000.

[Fli94] N. J. Fliege, "Multirate Digital Signal Processing," John Wiley & Sons, 1994.

[Gra01] E. Grayver, and B. Daneshrad, "VLSI implementation of a 100- W Multirate FSK Receiver," IEEE J. of Solid State Circuits, Vol. 36, No. 11, pp. 1821-1828, Nov. 2001.

[Har96] R. Hartley, "Subexpression Sharing in Filters Using Canonic Signed Digit Multipliers," IEEE Transactions on Circuits and Systems-II, Vol. 43, No. 10, pp. 677-688, Oct. 1996.

[Hat01] M. Hatamian, "Efficient FIR Filter for High-Speed Communication," U. S. Patent 6,272,173, Aug. 7, 2001.

[Hau93] J. C. Hausman, R. R. Harnden, E. G. Cohen, and H. G. Mills, "Programmable Canonic Signed Digit Filter Chip," U. S. Patent 5,262,974, Nov. 16, 1993.

[Haw96] R. Hawley, B. Wong, T. Lin, J. Laskowski, and H. Samueli, "Design Techniques for Silicon Compiler Implementations of High-Speed FIR Digital Filters," IEEE Journal of Solid-State Circuits, Vol. 31, No. 5, pp.

656-667, May 1996.

[Hog81] E. B. Hogenauer, "An Economial Class of Digital Filters for Decimation and Interpolation," IEEE Trans. Acoust., Speech, Signal Process., Vol. ASSP-29, No. 2, pp. 155-162, Apr. 1981.

[Hwa75] S. Hwang, "On Monotonicity of Lp and lp Norms," IEEE Transactions on Acoustics, Speech, and Signal Processing, Dec. 1975, pp. 593-594. [Jia97] Z. Jiang, and A. Willson, "Efficient Digital Filtering Architectures Using Pipelining/Interleaving," IEEE Transactions on Circuits and SystemsII, Vol. 44, No. 2, pp. 110-119, Feb. 1997.

[Kho01] K. Y. Khoo, Z. Yu, and A. Willson, "Design of Optimal Hybrid Form FIR Filter," IEEE International Symposium on Circuits and Systems 2001, Vol. 2, pp. 621-624.

[Kho96] K. Y. Khoo, A. Kwentus, and A. N. Willson Jr., "A Programmable FIR Digital Filter using CSD Coefficients," IEEE J. of Solid State Circuits, Vol. 31, No. 6, pp. 869-874, June 1996.

[Kwe97] A. Y. Kwentus, Z. Jiang, and Alan N. Wilson, Jr. , "Application of Filter Sharpening to Cascaded Integrator-Comb Decimation Filters," IEEE Transactions on Signal Processing, Vol. 45, No. 2, pp. 457-467, 1997. [Lee96] H. R. Lee, C. W. Jen, and C. M. Liu, "A New Hardware-Efficient Architecture for Programmable FIR Filters," IEEE Transactions on Circuits and Systems-II, Vol. 43, No. 9, pp. 637-644, Sept. 1996.

[Zha01] T. Zhangwen, Z. Z. Zhanpeng, Z. Jie, and M. Hao "A High-Speed, Programmable, CSD Coefficient FIR Filter," in Proc. 13th Annual IEEE International ASIC/SOC Conference, Sept. 2001, pp. 397-400.

[Zha88] Q. Zhao, and Y. Tadokoro, "A Simple Design of FIR Filters with Powers-of-Two Coefficients," IEEE Transactions on Circuits and Systems, Vol. 35, No. 5, pp. 566-570, May 1988.

the multistage techniques are most efficient for these kind of solutions [Mit98]. In [Gra01], a similar approach in which the decimation ratio M can be a mixed integer/fractional number is used. In this system, the nearest interpolated value is selected as an output value and the exact value at the time instant is not calculated, which introduces some timing jitter. The methods described in [Che93] and [Che98] allow the decimation and interpolation factors to be prime numbers while preserving the multistage efficiency. Programmable cascaded integrator-comb filters are used for interpolation with a ratio of variable rational numbers in [Hen99] and [Ana01].

12.1 Interpolation for Timing Adjustment

When a sampling rate is to be converted, new samples using the known input samples need to be calculated. An interpolator calculates one interpolant y(kTout) at a time using N input samples. Let's first consider a continuoustime signal that is sampled with a sampling frequency Fs = 1/Ts. The output interpolants of an ideal interpolator are the same as if the original continu- ous-time signal had been sampled at the sampling frequency Fout = 1/Tout [Ves96a].

The time of the interpolant is governed by the fractional interval k in Figure 12-1 and defined by

x(nTs)

= y(nTout)

x(t)

k+ Ts

k Ts

Figure 12-1 Interpolation in time-domain.

where x[(n i)Ts] and ha[( i+ k)Ts] for i = -N/2+1,...,N/2 are the N input samples, i.e. basepoints, and the N samples of the continuous impulse response of the interpolation filter, respectively.

12.2 Interpolation Filter with Polynomial-Based Impulse Response

As can be seen from Figure 12-1, in case of the arbitrary re-sampling rate, the fractional interval k has to be controlled on-line during the computation. Equation (12.2) states that the continuous-time impulse response ha(t) is a function of k. This means that the filter coefficients of the interpolation filter must change for every new value of k. This can be achieved by calculating the coefficients for all possible values of k and storing these into a lookup table. A new set of coefficients is then read into the filter structure every time k changes. Of course, this leads to a need for a huge coefficient memory, particularly when using long filters and a high resolution of the fractional interval. To avoid the very inefficient implementation using the coefficient memory, the filter coefficients can be calculated on-line using a polynomial-based interpolation.

The idea of the polynomial-based interpolator is that the continuous-time impulse response ha(t) of the interpolation filter is expressed in each interval of length Ts by means of polynomial [Ves99]

for i = -N/2+1, -N/2+2,..., N/2 and for k [0,1). cm(i) denotes the polynomial coefficients for ha[(i + k)Ts] and M is the degree of polynomial. The coefficients cm(i) are constants and independent of k. This leads to the most attractive feature of the polynomial-based interpolation filters; they can be implemented very efficiently using the so-called Farrow structure [Far98] described in Section 12.3.

Table 12-1 Filter coefficients for a cubic Lagrange interpolator.

interpolator, when using it as a sampling rate converter, is the phase linearity. Even though the underlying continuous time impulse response of the Lagrange interpolator is symmetric and thus linear phase, the phase response of the interpolator is not linear for all k. Fortunately, it is phase linear on average. The average phase linearity can also be seen from the phase responses of the cubic Lagrange interpolation filter for k =0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1 depicted in Figure 12-4.

12.3 Farrow Structure

The Farrow structure is a very efficient implementation structure for a poly- nomial-based interpolator. It suits well in real-time systems because it is fast and it has fixed coefficient values. A Farrow structure can be found for any polynomial-based filter [Ves99].

Substituting (12.5) into (12.2), the following is obtained

Figure 12-2 Impulse responses of linear and cubic Lagrange interpolators.

−10

Figure 12-3 Magnitude responses of linear and cubic Lagrange interpolators.

i N / 2+1

(12.7) and (12.8) indicate that the polynomial-based interpolation filter

Figure 12-4 Phase response of cubic Lagrange interpolator for different k values.