POLYNOMIAL-BASED INTERPOLATION FILTERS FOR DSP APPLICATIONS

Tampere University of Technology 51

Table 1. Requirements for the structures under consideration

Multiplications/s No. of No. of
multipliers I&Ds

TF 4Fin+70Fout 74 5

PTF 12Fin+49Fout 61 13

TF+FIR 3Fin+59Fout 38 4

TF+FIR1+FIR2 2Fin+55Fout 37 3

TF is the transposed Farrow structure, PTF is the prolonged transposed

Farrow structure, TF+FIR is the transposed Farrow structure in cascade with

a one-stage FIR decimator, and TF+FIR1+FIR2 is the transposed Farrow

structure in cascade with a two stage FIR decimator.

Tampere University of Technology 52

Direct design

Magnitude in dB 0 TF
PTF

−20

−40

−60

−80

−100
012345678

Frequency relative to F

out

Fig. 23. Magnitude responses for the transposed Farrow structure
(TF) and the prolonged transposed Farrow structure (PTF).

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Design with fixed two stage decimator after the
transposed Farrow structure

Magnitude in dB 0 FIR 1
FIR 2
Farrow

−20 Overall

−40

−60

−80

−100

−120
012345678

Frequency in Fout
Fig. 24. The magnitude responses for both FIR filters in the two-

stage decimator, the Farrow structure, and the overall system.

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Response in rational sample rate conversion

• When Fout / Fin = K / L > 1 is rational, µl becomes
periodic, with the periodicity of K. The set of µ values

contains K members and can be written as

{µ(0)+ k/K | k=0…K-1, 0≤ µ(0)<1/K} (49)

• Rational SRC can be modelled as a cascade of an
upsampler, a discrete-time LTI filter, and a
downsampler. This filter is a sampled version of the
filter ha(t) of the analog model.

• Takes into account the sampler of the analog model!

• The formula horrors of page 42 are mostly avoided.
=> Simpler formulas, simpler programs.

• BUT: A filter designed using the discrete-time model
may not be good for other conversion factors!

x(n) ↑K FIR filter z(j) y(l)
Fin KFin Hd(z) KFin
↓L Fout=KFin/L

Fig. 25. Model for rational sample rate conversion.

Tampere University of Technology 55

Emulating Continuous-Time Signal Processing

• The Farrow structure can be easily generalized to
process digitally the reconstructed signal ya(t).

• These applications include, among others,
interpolating the derivative or integral of ya(t).

• Derivative is widely utilized, for example, in finding
the location of a local maximum or minimum of a
signal.

• Integral can be used to calculate the energy of a signal
over a given time interval.

• We concentrate on determining the derivative of ya(t).

Farrow Structure for Determining the Derivative of
ya(t)

• In the intervals nTin≤t<(n+1)Tin for n=0, 1, 2,···
ya(t) can be expressed as

M (50)

∑ [ ]ya (t) t=(n+µ)Tin = p(n, µ ) = vm (n) 2µ − 1 m
m=0

where the vm(n)’s are the output samples of the FIR
branch filters in the Modified Farrow structure.

• The derivative of ya(t) in the intervals is thus given by

Tampere University of Technology 56

M

= vm (n)2m 2µ −1

m=0
d ya (t)

∑ [ ]d t
t =(n + µ )Tin = d p(n, µ) m −1. (51)
dµ

• The derivative of ya(t) at t = (n+µ)Tin can be

determined by multiplying the vm(n)’s by

2m(2µ−1)m−1, instead of (2µ −1) m in the Modified

Farrow structure.

• If the frequency response of the interpolation filter is

Ha(ω), then the corresponding response for the
differentiator is jωHa(ω).

• The location of an extremum can be found by setting

(51) to zero and solving µ.

|Ha(ω)| |jωHa(ω)|

1 1

ωω

Fig. 26. Ideal frequency responses for interpolator and
differentiator.

Tampere University of Technology 57

Example on the Derivative Approximation

• Design parameters for the differentiator are: L=5,

N=8, ωp∈[0, 0.35fs],ωs=0.65fs, Wp=0.035,and Ws=1.

Impulse response for dh(t)/dt

1.5

1

0.5

Amplitude 0

−0.5

−1

−1.5
−4 −3 −2 −1 0 1 2 3 4
Time / T

Fig. 27. Impulse response ha(t) for the differentiator.

2.5

(a)

2

Amplitude 1.5

1

0.5

0 3
0 0.5 1 1.5 2 2.5

Frequency f / Fs

Fig. 28. Amplitude response.

Tampere University of Technology 58

Symbol Synchronization in Digital Receivers

r(t) r(n) Matched filter x(n) Interpolator y(l) â(l)
hR(n)
ADC h(k,µl) Decision

∼ Fin=1/Tin nl µl

Timing
estimation

Fig. 29. Digital receiver with non-synchronized sampling.

• The sampling of the received signal is performed by a
fixed sampling clock, and thus, sampling is not
synchronized to the received symbols.
⇒ Timing adjustment must be done by digital
methods after sampling.

• Can be done by using an interpolation filter.

• Advantages of nonsynchronized sampling:
−Separates the analog and digital parts
(no feedback across the A/D interface).
−Easy to change the sampling rate.
−Sampling rate does not have to be a multiple of
the symbol rate (only high enough to avoid
aliasing).
−No need for a complex PLL circuit.

Tampere University of Technology 59

DESIGN EXAMPLE

• The input signal x(n) has a raised cosine pulse shape

with the excess bandwidth of α=0.5.

• The sample rate Fin is twice the symbol rate.

⇒ The highest baseband frequency component of

x(n) is (1+α)Fin /4=0.375Fin.

• The interpolator is designed in the least-mean-square
sense with the following specifications: N=8, M=5,
the passband edge fp=0.375Fin, and the stopband edge
fs =0.625Fin.

• The following pages show the characteristics for the
resulting interpolator as well as for the linear and third
order Lagrange interpolator.

Tampere University of Technology 60

L2 design

Magnitude in dB0

−10

−20

−30

−40

−50

−60

−70

−80
0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency in F

in

Fig. 30. The magnitude response for the L2 interpolator (solid line)
and the amplitude spectrum of the reconstructed signal ya(t) (dark

area).

Tampere University of Technology 61

Linear interpolation filter

Magnitude in dB0

−10

−20

−30

−40

−50

−60

−70

−80
0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency in Fin
Fig. 31. The magnitude response for the linear interpolation filter
(solid line) and the amplitude spectrum of the reconstructed signal

ya(t) (dark area).

Tampere University of Technology 62

Cubic Lagrange interpolation filter

Magnitude in dB0

−10

−20

−30

−40

−50

−60

−70

−80
0 0.5 1 1.5 2 2.5 3 3.5 4

Frequency in F

in

Fig. 32. The magnitude response for the cubic Lagrange
interpolation filter (solid line) and the amplitude spectrum of the

reconstructed signal ya(t) (dark area).

Tampere University of Technology 63

Some remarks

• Cubic Lagrange or even linear interpolation may be
sufficient for the cases where the oversampling factor
is high, e.g. >>2.

• However, if there are signal components close to
Fin/2, i.e., the oversampling factor is small (<2), the
linear and cubic Lagrange interpolations are not good
enough.

• The complexity of the Farrow structure can be
reduced by increasing the sampling rate (by 2, 4, 8,…)
before the Farrow structure.

• It is also possible to optimize this fixed interpolation
and the Farrow structure together.

Tampere University of Technology 64

References

[1] T. I. Laakso, V. Välimäki, M. Karjalainen, and U. K. Laine, “Splitting the unit delay,” IEEE Signal
Processing Magazine, vol. 13, pp. 30-60, Jan. 1996.

[2] C. W. Farrow, “A continuously variable digital delay element,” in Proc. IEEE Int. Symp. Circuits
& Syst., Espoo, Finland, June 1988, pp. 2641-2645.

[3] F. M. Gardner, “Interpolation in digital modems - Part I: Fundamentals,” IEEE Trans. Commun.,
vol. 41, pp. 501-507, Mar. 1993.

[4] L. Erup, F. M. Gardner, and R. A. Harris, “Interpolation in digital modems - Part II: Implementa-
tion and performance,” IEEE Trans. Commun., vol. 41, pp. 998-1008, June 1993.

[5] D. Kincaid and W. Cheney, Numerical Analysis. Pacific Grove, 1991.
[6] J. Vesma and T. Saramäki, “Interpolation filters with arbitrary frequency response for all-digital

receivers,” in Proc. IEEE Int. Symp. Circuits & Syst., Atlanta, GA, May 1996, pp. 568-571.
[7] J. Vesma, M. Renfors, and J. Rinne, “Comparison of efficient interpolation techniques for symbol

timing recovery,” in Proc. IEEE Globecom 96, London, UK, Nov. 1996, pp. 953-957.
[8] J. Vesma and T. Saramäki, “Optimization and efficient implementation of FIR filters with

adjustable fractional delay,” in Proc. IEEE Int. Symp. Circuits & Syst., Hong Kong, June 1997, pp.
2256-2259.
[9] H. Ridha, J. Vesma, T. Saramäki, and M. Renfors, “Derivative approximations for sampled signals
based on polynomial interpolation,” in Proc. 13th Int. Conf. on Digital Signal Processing,
Santorini, Greece, July 1997, pp. 939-942.
[10] H. Ridha, J. Vesma, M. Renfors, and T. Saramäki, “Discrete-time simulation of continuous-time
systems using generalized interpolation techniques,” in Proc. 1997 Summer Computer Simulation
Conference, Arlington, Virginia, USA, July 1997, pp. 914-919.
[11] V. Tuukkanen, J. Vesma, and M. Renfors, “Combined interpolation and maximum likelihood
symbol timing recovery in digital receivers,” to be presented in 1997 IEEE Int. Conference on
Universal Personal Communications, San Diego, CA, USA, Oct. 1997.
[12] T. Saramäki and M. Ritoniemi, "An efficient approach for conversion between arbitrary sampling
frequencies," in Proc. IEEE Int. Symp. Circuits & Syst., Atlanta, GA, May 1996, pp. 285-288.
[13] J. Vesma, R. Hamila, T. Saramäki, and M. Renfors, “Design of polynomial interpolation filters
based on Taylor series,” in Proc. IX European Signal Processing Conf., Rhodes, Greece, Sep.
1998, pp. 283-286.
[14] J. Vesma, R. Hamila, M. Renfors, and T. Saramäki, “Continuous-time signal processing based on
polynomial approximation,” in Proc. IEEE Int. Symp. on Circuits and Systems, Monterey, CA,
USA, May 1998, vol. 5, pp. 61-65.
[15] D. Fu and A. N. Willson, Jr., “Interpolation in timing recovery using a trigonometric polynomial
and its implementation,” in IEEE Globecom 1998 Communications Mini Conference Record,
Sydney, Australia, Nov. 1998, pp. 173−178.

[16] f. harris, “Performance and design considerations of Farrow filter used for arbitrary resampling,” in
Proc. 13th Int. Conf. on Digital Signal Processing, Santorini, Greece, July 1997, pp. 595−599.

[17] G. Oetken, “A new approach for the design of digital interpolation filters,” IEEE Trans. Acoust.,
Speech, Signal Process., vol. ASSP−27, pp. 637−643, Dec. 1979.

[18] T. A. Ramstad, “Digital methods for conversion between arbitrary sampling frequencies,” IEEE
Trans. Acoust. Speech, Signal Processing, vol. ASSP−32, pp. 577−591, June 1984.

[19] T. A. Ramstad, “Fractional rate decimator and interpolator design,” in Proc. IX European Signal
Processing Conf., Rhodes, Greece, Sep. 1998, pp. 1949−1952.

Tampere University of Technology 65

[20] R. W. Schafer and L. R. Rabiner, “A digital signal processing approach to interpolation,” Proc.
IEEE, vol. 61, pp. 692−702, June 1973.

[21] M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image
representation and interpolation,” Trans. Pat. Anal., Mach. Int., vol. 13, pp. 277−285, Mar. 1991.

[22] M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and
asymptotic equivalence with Shannon’s sampling theorem,” IEEE Trans. Information Theory, vol.

38, pp. 95−103, Jan. 1992.
[23] J. Vesma, Timing Adjustment in Digital Receivers Using Interpolation. M.Sc. Thesis, Tampere,

Finland: Tampere University of Tech., Department of Information Technology, Nov. 1995.
[24] V. Välimäki, Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters. Doctoral

thesis, Espoo, Finland: Helsinki University of Technology, Dec. 1995.

[25] S. R. Dooley and A. K. Nandi, “On explicit time delay estimation using the Farrow structure,”
Signal Processing, vol. 72, pp. 53−57, Jan. 1999.

[26] J. Vesma, “A frequency-domain approach to polynomial-based interpolation and the Farrow
structure,” to appear IEEE Trans. on Circuits and Systems II, March 2000.

[27] J. Vesma, Optimization and Applications of Polynomial-Based Interpolation Filters. Dr. Tech.

Thesis, Tampere, Finland: Tampere University of Tech., Department of Information Technology,

May 1999

[28] D. Babic, J. Vesma, T. Saramäki, M. Renfors, “Implementation of the transposed Farrow
structure,” in Proc. 2002 IEEE Int. Symp. Circuits and Systems, Scotsdale, Arizona, USA, 2002,

vol. 4, pp. 4−8.
[29] D. Babic, T. Saramäki and M. Renfors, “Conversion between arbitrary sampling frequencies using

polynomial-based interpolation filters,” in Proc. Int. Workshop on Spectral Methods and Multirate Signal Processing,

SMMSP’02, Toulouse, France, September 2002, pp. 57−64.

TLT-5806
Receiver Architectures
and Signal Processing

Addition to the lecture:
Polynomial-Based Interpolation

Filters for DSP Applications

NEWTON INTERPOLATION FOR
FRACTIONAL-DELAY FILTERING

Vesa Lehtinen
Department of Communications Engineering

Tampere University of Technology
P.O.Box 553, 33101 Tampere, Finland

vesa.lehtinen@tut.fi

TLT-5806 Receiver Architectures and Signal Processing: Newton Interpolation for Fractional-Delay Filtering

The Newton structure

x[n] –D 1 – z–1 -–----D---2---+-----1- –-----D------+--N---N------–-----1-
1 – z–1 1 – z–1

y[n] Building block

Fig. 1: The Newton structure for fractional-delay filte ing.

• Efficient implementation o Newton’s backward difference for-
mula for interpolation
– Anticausal version would implement Newton’s forward
difference formula
– Discrete-time counterpart for Taylor series
– Sometimes incorrectly referred to as the Taylor structure

• Equivalent to Lagrange & Pascal interpolation
• Proposed by various authors at different times
• D : total delay in sample intervals

– Keep close to N / 2 for best response.
– Cf. Farrow: D not in [0,1)
N : interpolator order, i.e., number of stages
• Easy to see: y[n]=x[n–D] for integer D.
• Demonstrated by newtondemo.m

Vesa Lehtinen 2/3 29.10.2008

TLT-5806 Receiver Architectures and Signal Processing: Newton Interpolation for Fractional-Delay Filtering

Advantages of the Newton structure:

• Simple: multiplier-free subfilter , few multipliers
• Wide delay range: far beyond one sample interval

– Order-delay-response trade-off possible
• Modular structure: good for VLSI, interpolator order can be

easily changed
• O(N) complexity! (vs. Farrow’s O(N2))

=> high orders feasible

Disadvantages:

• Poor response (Lagrange)
– Requires an oversampled signal

• High order (due to poor response)
=> Longer delay in causal realisations

References

[1] L. Elden, L. Wittmeyer-Koch, and H.B. Nielsen, Introduction to Numerical Com-
putation. Studentlitteratur, Lund, 2004, pp. 107–113.

[2] http://mathworld.wolfram.com/UmbralCalculus.html
[3] S. Tassart and Ph. Depalle, “Fractional delays using Lagrange interpolators,“ in

Proc. Nordic Acoustic Meeting, Helsinki, Finland, 12–14 June, 1996.
[4] C. Candan, “An efficient filt ing structure for Lagrange interpolation,” in IEEE

Signal Processing Letters, Vol. 14, No. 1, Jan 2007, pp. 17–19.
[5] T.J. Goodman, M.F. Aburdene, “Interpolation Using the Discrete Pascal Trans-

form,” Proc. 40th Annual Conf. Information Sciences and Systems, 22–24
March 2006, pp. 1079–1083.
[6] Vesa Lehtinen, Markku Renfors, "Structures for Interpolation, Decimation, and
Nonuniform Sampling Based on Newton's Interpolation Formula," in Proc.
Sampling Theory and Applications (SAMPTA), Marseille, France, 18-22 May
2009. http://hal.archives-ouvertes.fr/hal-00451769/

Vesa Lehtinen 3/3 29.10.2008